3.30.26 \(\int \frac {-x^2+\sqrt {1+2 x^2}+(1+2 x^2)^{5/2}}{x^2-x (1+2 x^2)^{3/2}} \, dx\)

Optimal. Leaf size=337 \[ \frac {1}{2} \text {RootSum}\left [\text {$\#$1}^6+2 \sqrt {2} \text {$\#$1}^4+3 \text {$\#$1}^4-2 \sqrt {2} \text {$\#$1}^2+3 \text {$\#$1}^2+1\& ,\frac {2 \text {$\#$1}^5 \log \left (-\text {$\#$1}+\sqrt {2 x^2+1}-\sqrt {2} x\right )+\text {$\#$1}^4 \log \left (-\text {$\#$1}+\sqrt {2 x^2+1}-\sqrt {2} x\right )+4 \sqrt {2} \text {$\#$1}^3 \log \left (-\text {$\#$1}+\sqrt {2 x^2+1}-\sqrt {2} x\right )+2 \text {$\#$1}^3 \log \left (-\text {$\#$1}+\sqrt {2 x^2+1}-\sqrt {2} x\right )+4 \text {$\#$1} \log \left (-\text {$\#$1}+\sqrt {2 x^2+1}-\sqrt {2} x\right )-\log \left (-\text {$\#$1}+\sqrt {2 x^2+1}-\sqrt {2} x\right )}{3 \text {$\#$1}^5+4 \sqrt {2} \text {$\#$1}^3+6 \text {$\#$1}^3-2 \sqrt {2} \text {$\#$1}+3 \text {$\#$1}}\& \right ]-x^2+\frac {1}{2} \left (2+\sqrt {2}\right ) \log \left (\sqrt {2 x^2+1}-\sqrt {2} x\right )-2 \log \left (\sqrt {2} x \sqrt {2 x^2+1}-2 x^2\right ) \]

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Rubi [C]  time = 2.52, antiderivative size = 776, normalized size of antiderivative = 2.30, number of steps used = 112, number of rules used = 37, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.755, Rules used = {6742, 402, 215, 377, 206, 14, 444, 43, 1692, 208, 205, 1247, 634, 617, 204, 628, 1251, 773, 12, 416, 528, 523, 266, 701, 800, 203, 50, 63, 1169, 618, 699, 1127, 1161, 1164, 824, 707, 1094} \begin {gather*} -x^2+\frac {1}{5} \log \left (x^2+1\right )+\frac {1}{40} \sqrt {1+5 \sqrt {2}} \log \left (4 x^2-2 \sqrt {2 \left (\sqrt {2}-1\right )} x+\sqrt {2}\right )-\frac {1}{40} \sqrt {1+5 \sqrt {2}} \log \left (4 x^2+2 \sqrt {2 \left (\sqrt {2}-1\right )} x+\sqrt {2}\right )+\frac {\log \left (2 \left (2 x^2+1\right )-2 \sqrt {1+\sqrt {2}} \sqrt {2 x^2+1}+\sqrt {2}\right )}{20 \sqrt {2 \left (1+\sqrt {2}\right )}}+\frac {3 \log \left (2 \left (2 x^2+1\right )-2 \sqrt {1+\sqrt {2}} \sqrt {2 x^2+1}+\sqrt {2}\right )}{40 \sqrt {1+\sqrt {2}}}-\frac {\log \left (\sqrt {2} \left (2 x^2+1\right )+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {2 x^2+1}+1\right )}{20 \sqrt {2 \left (1+\sqrt {2}\right )}}-\frac {3 \log \left (\sqrt {2} \left (2 x^2+1\right )+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {2 x^2+1}+1\right )}{40 \sqrt {1+\sqrt {2}}}-\left (\frac {2}{5}-\frac {3 i}{10}\right ) \tan ^{-1}\left (\frac {(1+i) x}{\sqrt {2 x^2+1}}\right )+\frac {1}{5} \tan ^{-1}\left (\sqrt {2 x^2+1}\right )-\frac {2}{5} \tan ^{-1}\left (4 x^2+1\right )+\frac {1}{10} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {1+\sqrt {2}}-2 \sqrt {2 x^2+1}}{\sqrt {\sqrt {2}-1}}\right )-\frac {3 \tan ^{-1}\left (\frac {\sqrt {1+\sqrt {2}}-2 \sqrt {2 x^2+1}}{\sqrt {\sqrt {2}-1}}\right )}{20 \sqrt {\sqrt {2}-1}}-\frac {1}{10} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tan ^{-1}\left (\frac {2 \sqrt {2 x^2+1}+\sqrt {1+\sqrt {2}}}{\sqrt {\sqrt {2}-1}}\right )+\frac {3 \tan ^{-1}\left (\frac {2 \sqrt {2 x^2+1}+\sqrt {1+\sqrt {2}}}{\sqrt {\sqrt {2}-1}}\right )}{20 \sqrt {\sqrt {2}-1}}+\frac {2}{5} \tanh ^{-1}\left (\frac {x}{\sqrt {2 x^2+1}}\right )-\left (\frac {3}{10}-\frac {2 i}{5}\right ) \tanh ^{-1}\left (\frac {(1+i) x}{\sqrt {2 x^2+1}}\right )+\frac {3}{20} \log \left (8 x^4+4 x^2+1\right )-2 \log (x)-\frac {1}{5} \tan ^{-1}(x)-\frac {1}{20} \sqrt {5 \sqrt {2}-1} \tan ^{-1}\left (-2 \sqrt {2 \left (\sqrt {2}-1\right )} x-\sqrt {2}+1\right )+\frac {1}{20} \sqrt {5 \sqrt {2}-1} \tan ^{-1}\left (2 \sqrt {2 \left (\sqrt {2}-1\right )} x-\sqrt {2}+1\right )-\frac {\sinh ^{-1}\left (\sqrt {2} x\right )}{\sqrt {2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[(-x^2 + Sqrt[1 + 2*x^2] + (1 + 2*x^2)^(5/2))/(x^2 - x*(1 + 2*x^2)^(3/2)),x]

[Out]

-x^2 - ArcSinh[Sqrt[2]*x]/Sqrt[2] - ArcTan[x]/5 - (Sqrt[-1 + 5*Sqrt[2]]*ArcTan[1 - Sqrt[2] - 2*Sqrt[2*(-1 + Sq
rt[2])]*x])/20 + (Sqrt[-1 + 5*Sqrt[2]]*ArcTan[1 - Sqrt[2] + 2*Sqrt[2*(-1 + Sqrt[2])]*x])/20 - (2/5 - (3*I)/10)
*ArcTan[((1 + I)*x)/Sqrt[1 + 2*x^2]] + ArcTan[Sqrt[1 + 2*x^2]]/5 - (2*ArcTan[1 + 4*x^2])/5 - (3*ArcTan[(Sqrt[1
+ Sqrt[2]] - 2*Sqrt[1 + 2*x^2])/Sqrt[-1 + Sqrt[2]]])/(20*Sqrt[-1 + Sqrt[2]]) + (Sqrt[(1 + Sqrt[2])/2]*ArcTan[
(Sqrt[1 + Sqrt[2]] - 2*Sqrt[1 + 2*x^2])/Sqrt[-1 + Sqrt[2]]])/10 + (3*ArcTan[(Sqrt[1 + Sqrt[2]] + 2*Sqrt[1 + 2*
x^2])/Sqrt[-1 + Sqrt[2]]])/(20*Sqrt[-1 + Sqrt[2]]) - (Sqrt[(1 + Sqrt[2])/2]*ArcTan[(Sqrt[1 + Sqrt[2]] + 2*Sqrt
[1 + 2*x^2])/Sqrt[-1 + Sqrt[2]]])/10 + (2*ArcTanh[x/Sqrt[1 + 2*x^2]])/5 - (3/10 - (2*I)/5)*ArcTanh[((1 + I)*x)
/Sqrt[1 + 2*x^2]] - 2*Log[x] + Log[1 + x^2]/5 + (Sqrt[1 + 5*Sqrt[2]]*Log[Sqrt[2] - 2*Sqrt[2*(-1 + Sqrt[2])]*x
+ 4*x^2])/40 - (Sqrt[1 + 5*Sqrt[2]]*Log[Sqrt[2] + 2*Sqrt[2*(-1 + Sqrt[2])]*x + 4*x^2])/40 + (3*Log[1 + 4*x^2 +
8*x^4])/20 + (3*Log[Sqrt[2] - 2*Sqrt[1 + Sqrt[2]]*Sqrt[1 + 2*x^2] + 2*(1 + 2*x^2)])/(40*Sqrt[1 + Sqrt[2]]) +
Log[Sqrt[2] - 2*Sqrt[1 + Sqrt[2]]*Sqrt[1 + 2*x^2] + 2*(1 + 2*x^2)]/(20*Sqrt[2*(1 + Sqrt[2])]) - (3*Log[1 + Sqr
t[2*(1 + Sqrt[2])]*Sqrt[1 + 2*x^2] + Sqrt[2]*(1 + 2*x^2)])/(40*Sqrt[1 + Sqrt[2]]) - Log[1 + Sqrt[2*(1 + Sqrt[2
])]*Sqrt[1 + 2*x^2] + Sqrt[2]*(1 + 2*x^2)]/(20*Sqrt[2*(1 + Sqrt[2])])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 402

Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Dist[b/d, Int[(a + b*x^2)^(p - 1), x], x]
- Dist[(b*c - a*d)/d, Int[(a + b*x^2)^(p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d,
0] && GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4])

Rule 416

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1)*(c
+ d*x^n)^(q - 1))/(b*(n*(p + q) + 1)), x] + Dist[1/(b*(n*(p + q) + 1)), Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)
*Simp[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; F
reeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntB
inomialQ[a, b, c, d, n, p, q, x]

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 523

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]

Rule 528

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[
(f*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(b*(n*(p + q + 1) + 1)), x] + Dist[1/(b*(n*(p + q + 1) + 1)), Int[(a +
b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f + b*e*n*(p + q + 1)) + (d*(b*e - a*f) + f*n*q*(b*c - a*d) + b*
d*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && GtQ[q, 0] && NeQ[n*(p + q + 1) + 1
, 0]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 699

Int[Sqrt[(d_.) + (e_.)*(x_)]/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[2*e, Subst[Int[x^2/(c*d^2
- b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^
2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]

Rule 701

Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[(d + e*x)
^m, a + b*x + c*x^2, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2,
0] && NeQ[2*c*d - b*e, 0] && IGtQ[m, 1] && (NeQ[d, 0] || GtQ[m, 2])

Rule 707

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2*e, Subst[Int[1/(c*d^
2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[
b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]

Rule 773

Int[(((d_.) + (e_.)*(x_))*((f_) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*g*x)/
c, x] + Dist[1/c, Int[(c*d*f - a*e*g + (c*e*f + c*d*g - b*e*g)*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c,
d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 824

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(g
*(d + e*x)^m)/(c*m), x] + Dist[1/c, Int[((d + e*x)^(m - 1)*Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x])
/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*
e^2, 0] && FractionQ[m] && GtQ[m, 0]

Rule 1094

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}
, Dist[1/(2*c*q*r), Int[(r - x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(r + x)/(q + r*x + x^2), x], x
]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]

Rule 1127

Int[(x_)^2/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, Dist[1/2, Int[(q + x^2)/(
a + b*x^2 + c*x^4), x], x] - Dist[1/2, Int[(q - x^2)/(a + b*x^2 + c*x^4), x], x]] /; FreeQ[{a, b, c}, x] && Lt
Q[b^2 - 4*a*c, 0] && PosQ[a*c]

Rule 1161

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e - b/c, 2]},
Dist[e/(2*c), Int[1/Simp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /
; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && (GtQ[(2*d)/e - b/c, 0] || ( !Lt
Q[(2*d)/e - b/c, 0] && EqQ[d - e*Rt[a/c, 2], 0]))

Rule 1164

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e - b/c, 2]},
Dist[e/(2*c*q), Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x
- x^2, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] &&  !GtQ[b^2
- 4*a*c, 0]

Rule 1169

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r =
Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(
d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]

Rule 1247

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rule 1251

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rule 1692

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandInteg
rand[Px*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, q}, x] && PolyQ[Px, x^2] && NeQ[b
^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {-x^2+\sqrt {1+2 x^2}+\left (1+2 x^2\right )^{5/2}}{x^2-x \left (1+2 x^2\right )^{3/2}} \, dx &=\int \left (-\frac {\sqrt {1+2 x^2}}{x \left (-x+\sqrt {1+2 x^2}+2 x^2 \sqrt {1+2 x^2}\right )}-\frac {\left (1+2 x^2\right )^{5/2}}{x \left (-x+\sqrt {1+2 x^2}+2 x^2 \sqrt {1+2 x^2}\right )}-\frac {x}{x-\left (1+2 x^2\right )^{3/2}}\right ) \, dx\\ &=-\int \frac {\sqrt {1+2 x^2}}{x \left (-x+\sqrt {1+2 x^2}+2 x^2 \sqrt {1+2 x^2}\right )} \, dx-\int \frac {\left (1+2 x^2\right )^{5/2}}{x \left (-x+\sqrt {1+2 x^2}+2 x^2 \sqrt {1+2 x^2}\right )} \, dx-\int \frac {x}{x-\left (1+2 x^2\right )^{3/2}} \, dx\\ &=-\int \left (\frac {1}{5 \left (1+x^2\right )}+\frac {x \sqrt {1+2 x^2}}{5 \left (1+x^2\right )}+\frac {-1-8 x^2}{5 \left (1+4 x^2+8 x^4\right )}-\frac {6 x \sqrt {1+2 x^2}}{5 \left (1+4 x^2+8 x^4\right )}-\frac {8 x^3 \sqrt {1+2 x^2}}{5 \left (1+4 x^2+8 x^4\right )}\right ) \, dx-\int \left (\frac {\sqrt {1+2 x^2}}{5 \left (1+x^2\right )}+\frac {1+2 x^2}{x}+\frac {x \left (1+2 x^2\right )}{5 \left (1+x^2\right )}-\frac {4 \left (-1+2 x^2\right ) \sqrt {1+2 x^2}}{5 \left (1+4 x^2+8 x^4\right )}-\frac {16 x \left (1+2 x^2\right )}{5 \left (1+4 x^2+8 x^4\right )}-\frac {48 x^3 \left (1+2 x^2\right )}{5 \left (1+4 x^2+8 x^4\right )}\right ) \, dx-\int \left (\frac {\left (1+2 x^2\right )^{5/2}}{5 \left (1+x^2\right )}+\frac {\left (1+2 x^2\right )^3}{x}+\frac {x \left (1+2 x^2\right )^3}{5 \left (1+x^2\right )}-\frac {4 \left (-1+2 x^2\right ) \left (1+2 x^2\right )^{5/2}}{5 \left (1+4 x^2+8 x^4\right )}-\frac {16 x \left (1+2 x^2\right )^3}{5 \left (1+4 x^2+8 x^4\right )}-\frac {48 x^3 \left (1+2 x^2\right )^3}{5 \left (1+4 x^2+8 x^4\right )}\right ) \, dx\\ &=-\left (\frac {1}{5} \int \frac {1}{1+x^2} \, dx\right )-\frac {1}{5} \int \frac {\sqrt {1+2 x^2}}{1+x^2} \, dx-\frac {1}{5} \int \frac {x \sqrt {1+2 x^2}}{1+x^2} \, dx-\frac {1}{5} \int \frac {x \left (1+2 x^2\right )}{1+x^2} \, dx-\frac {1}{5} \int \frac {\left (1+2 x^2\right )^{5/2}}{1+x^2} \, dx-\frac {1}{5} \int \frac {x \left (1+2 x^2\right )^3}{1+x^2} \, dx-\frac {1}{5} \int \frac {-1-8 x^2}{1+4 x^2+8 x^4} \, dx+\frac {4}{5} \int \frac {\left (-1+2 x^2\right ) \sqrt {1+2 x^2}}{1+4 x^2+8 x^4} \, dx+\frac {4}{5} \int \frac {\left (-1+2 x^2\right ) \left (1+2 x^2\right )^{5/2}}{1+4 x^2+8 x^4} \, dx+\frac {6}{5} \int \frac {x \sqrt {1+2 x^2}}{1+4 x^2+8 x^4} \, dx+\frac {8}{5} \int \frac {x^3 \sqrt {1+2 x^2}}{1+4 x^2+8 x^4} \, dx+\frac {16}{5} \int \frac {x \left (1+2 x^2\right )}{1+4 x^2+8 x^4} \, dx+\frac {16}{5} \int \frac {x \left (1+2 x^2\right )^3}{1+4 x^2+8 x^4} \, dx+\frac {48}{5} \int \frac {x^3 \left (1+2 x^2\right )}{1+4 x^2+8 x^4} \, dx+\frac {48}{5} \int \frac {x^3 \left (1+2 x^2\right )^3}{1+4 x^2+8 x^4} \, dx-\int \frac {1+2 x^2}{x} \, dx-\int \frac {\left (1+2 x^2\right )^3}{x} \, dx\\ &=-\frac {1}{10} x \left (1+2 x^2\right )^{3/2}-\frac {1}{5} \tan ^{-1}(x)-\frac {1}{20} \int \frac {\left (2-2 x^2\right ) \sqrt {1+2 x^2}}{1+x^2} \, dx-\frac {1}{10} \operatorname {Subst}\left (\int \frac {\sqrt {1+2 x}}{1+x} \, dx,x,x^2\right )-\frac {1}{10} \operatorname {Subst}\left (\int \frac {1+2 x}{1+x} \, dx,x,x^2\right )-\frac {1}{10} \operatorname {Subst}\left (\int \frac {(1+2 x)^3}{1+x} \, dx,x,x^2\right )+\frac {1}{5} \int \frac {1}{\left (1+x^2\right ) \sqrt {1+2 x^2}} \, dx-\frac {2}{5} \int \frac {1}{\sqrt {1+2 x^2}} \, dx-\frac {1}{2} \operatorname {Subst}\left (\int \frac {(1+2 x)^3}{x} \, dx,x,x^2\right )+\frac {3}{5} \operatorname {Subst}\left (\int \frac {\sqrt {1+2 x}}{1+4 x+8 x^2} \, dx,x,x^2\right )+\frac {4}{5} \int \left (\frac {(2+6 i) \sqrt {1+2 x^2}}{(4-4 i)+16 x^2}+\frac {(2-6 i) \sqrt {1+2 x^2}}{(4+4 i)+16 x^2}\right ) \, dx+\frac {4}{5} \int \left (\frac {(2+6 i) \left (1+2 x^2\right )^{5/2}}{(4-4 i)+16 x^2}+\frac {(2-6 i) \left (1+2 x^2\right )^{5/2}}{(4+4 i)+16 x^2}\right ) \, dx+\frac {4}{5} \operatorname {Subst}\left (\int \frac {x \sqrt {1+2 x}}{1+4 x+8 x^2} \, dx,x,x^2\right )+\frac {8}{5} \operatorname {Subst}\left (\int \frac {1+2 x}{1+4 x+8 x^2} \, dx,x,x^2\right )+\frac {8}{5} \operatorname {Subst}\left (\int \frac {(1+2 x)^3}{1+4 x+8 x^2} \, dx,x,x^2\right )+\frac {24}{5} \operatorname {Subst}\left (\int \frac {x (1+2 x)}{1+4 x+8 x^2} \, dx,x,x^2\right )+\frac {24}{5} \operatorname {Subst}\left (\int \frac {x (1+2 x)^3}{1+4 x+8 x^2} \, dx,x,x^2\right )-\frac {\int \frac {-\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )}-\left (-1+2 \sqrt {2}\right ) x}{\frac {1}{2 \sqrt {2}}-\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} x+x^2} \, dx}{20 \sqrt {-1+\sqrt {2}}}-\frac {\int \frac {-\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )}+\left (-1+2 \sqrt {2}\right ) x}{\frac {1}{2 \sqrt {2}}+\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} x+x^2} \, dx}{20 \sqrt {-1+\sqrt {2}}}-\int \left (\frac {1}{x}+2 x\right ) \, dx\\ &=\frac {x^2}{5}+\frac {1}{20} x \sqrt {1+2 x^2}-\frac {1}{10} x \left (1+2 x^2\right )^{3/2}-\frac {1}{5} \sqrt {2} \sinh ^{-1}\left (\sqrt {2} x\right )-\frac {1}{5} \tan ^{-1}(x)-\log (x)-\frac {1}{40} \int \frac {6+14 x^2}{\left (1+x^2\right ) \sqrt {1+2 x^2}} \, dx-\frac {1}{10} \operatorname {Subst}\left (\int \left (2+\frac {1}{-1-x}\right ) \, dx,x,x^2\right )+\frac {1}{10} \operatorname {Subst}\left (\int \frac {1}{(1+x) \sqrt {1+2 x}} \, dx,x,x^2\right )+\frac {1}{10} \operatorname {Subst}\left (\int -\frac {2}{\sqrt {1+2 x} \left (1+4 x+8 x^2\right )} \, dx,x,x^2\right )-\frac {1}{10} \operatorname {Subst}\left (\int \left (2+\frac {1}{-1-x}+4 x+8 x^2\right ) \, dx,x,x^2\right )+\frac {1}{5} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {1+2 x^2}}\right )+\frac {1}{5} \operatorname {Subst}\left (\int \frac {4+16 x}{1+4 x+8 x^2} \, dx,x,x^2\right )-\frac {1}{2} \operatorname {Subst}\left (\int \left (6+\frac {1}{x}+12 x+8 x^2\right ) \, dx,x,x^2\right )+\frac {3}{5} \operatorname {Subst}\left (\int -\frac {2}{1+4 x+8 x^2} \, dx,x,x^2\right )+\frac {4}{5} \operatorname {Subst}\left (\int \frac {1}{1+4 x+8 x^2} \, dx,x,x^2\right )+\frac {8}{5} \operatorname {Subst}\left (\int \left (1+x+\frac {x}{1+4 x+8 x^2}\right ) \, dx,x,x^2\right )+\left (\frac {8}{5}-\frac {24 i}{5}\right ) \int \frac {\sqrt {1+2 x^2}}{(4+4 i)+16 x^2} \, dx+\left (\frac {8}{5}-\frac {24 i}{5}\right ) \int \frac {\left (1+2 x^2\right )^{5/2}}{(4+4 i)+16 x^2} \, dx+\left (\frac {8}{5}+\frac {24 i}{5}\right ) \int \frac {\sqrt {1+2 x^2}}{(4-4 i)+16 x^2} \, dx+\left (\frac {8}{5}+\frac {24 i}{5}\right ) \int \frac {\left (1+2 x^2\right )^{5/2}}{(4-4 i)+16 x^2} \, dx+\frac {12}{5} \operatorname {Subst}\left (\int \frac {x^2}{4-8 x^2+8 x^4} \, dx,x,\sqrt {1+2 x^2}\right )+\frac {24}{5} \operatorname {Subst}\left (\int \left (\frac {1}{8}+x+x^2-\frac {1+4 x}{8 \left (1+4 x+8 x^2\right )}\right ) \, dx,x,x^2\right )+\frac {1}{80} \left (4+\sqrt {2}\right ) \int \frac {1}{\frac {1}{2 \sqrt {2}}-\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} x+x^2} \, dx+\frac {1}{80} \left (4+\sqrt {2}\right ) \int \frac {1}{\frac {1}{2 \sqrt {2}}+\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} x+x^2} \, dx+\frac {1}{40} \sqrt {1+5 \sqrt {2}} \int \frac {-\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )}+2 x}{\frac {1}{2 \sqrt {2}}-\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} x+x^2} \, dx-\frac {1}{40} \sqrt {1+5 \sqrt {2}} \int \frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )}+2 x}{\frac {1}{2 \sqrt {2}}+\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} x+x^2} \, dx\\ &=-x^2+\frac {1}{20} x \sqrt {1+2 x^2}-\frac {1}{5} \sqrt {2} \sinh ^{-1}\left (\sqrt {2} x\right )-\frac {1}{5} \tan ^{-1}(x)+\frac {1}{5} \tanh ^{-1}\left (\frac {x}{\sqrt {1+2 x^2}}\right )-2 \log (x)+\frac {1}{5} \log \left (1+x^2\right )+\frac {1}{40} \sqrt {1+5 \sqrt {2}} \log \left (\sqrt {2}-2 \sqrt {2 \left (-1+\sqrt {2}\right )} x+4 x^2\right )-\frac {1}{40} \sqrt {1+5 \sqrt {2}} \log \left (\sqrt {2}+2 \sqrt {2 \left (-1+\sqrt {2}\right )} x+4 x^2\right )+\frac {1}{5} \log \left (1+4 x^2+8 x^4\right )+\left (-\frac {8}{5}-\frac {16 i}{5}\right ) \int \frac {1}{\sqrt {1+2 x^2} \left ((4+4 i)+16 x^2\right )} \, dx+\left (-\frac {8}{5}+\frac {16 i}{5}\right ) \int \frac {1}{\sqrt {1+2 x^2} \left ((4-4 i)+16 x^2\right )} \, dx+\left (\frac {1}{40}-\frac {3 i}{40}\right ) \int \frac {\sqrt {1+2 x^2} \left ((56-8 i)+(160-64 i) x^2\right )}{(4+4 i)+16 x^2} \, dx+\left (\frac {1}{40}+\frac {3 i}{40}\right ) \int \frac {\sqrt {1+2 x^2} \left ((56+8 i)+(160+64 i) x^2\right )}{(4-4 i)+16 x^2} \, dx+\frac {1}{10} \operatorname {Subst}\left (\int \frac {1}{\frac {1}{2}+\frac {x^2}{2}} \, dx,x,\sqrt {1+2 x^2}\right )+\frac {1}{5} \int \frac {1}{\left (1+x^2\right ) \sqrt {1+2 x^2}} \, dx-\frac {1}{5} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+2 x} \left (1+4 x+8 x^2\right )} \, dx,x,x^2\right )+\left (\frac {1}{5}-\frac {3 i}{5}\right ) \int \frac {1}{\sqrt {1+2 x^2}} \, dx+\left (\frac {1}{5}+\frac {3 i}{5}\right ) \int \frac {1}{\sqrt {1+2 x^2}} \, dx-\frac {7}{20} \int \frac {1}{\sqrt {1+2 x^2}} \, dx-\frac {2}{5} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+4 x^2\right )-\frac {3}{5} \operatorname {Subst}\left (\int \frac {1+4 x}{1+4 x+8 x^2} \, dx,x,x^2\right )-\frac {6}{5} \operatorname {Subst}\left (\int \frac {1}{1+4 x+8 x^2} \, dx,x,x^2\right )-\frac {6}{5} \operatorname {Subst}\left (\int \frac {\frac {1}{\sqrt {2}}-x^2}{4-8 x^2+8 x^4} \, dx,x,\sqrt {1+2 x^2}\right )+\frac {6}{5} \operatorname {Subst}\left (\int \frac {\frac {1}{\sqrt {2}}+x^2}{4-8 x^2+8 x^4} \, dx,x,\sqrt {1+2 x^2}\right )+\frac {8}{5} \operatorname {Subst}\left (\int \frac {x}{1+4 x+8 x^2} \, dx,x,x^2\right )-\frac {1}{40} \left (4+\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{2} \left (-1-\sqrt {2}\right )-x^2} \, dx,x,-\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )}+2 x\right )-\frac {1}{40} \left (4+\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{2} \left (-1-\sqrt {2}\right )-x^2} \, dx,x,\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )}+2 x\right )\\ &=-x^2+\frac {\sinh ^{-1}\left (\sqrt {2} x\right )}{20 \sqrt {2}}-\frac {1}{5} \sqrt {2} \sinh ^{-1}\left (\sqrt {2} x\right )-\frac {1}{5} \tan ^{-1}(x)-\frac {1}{20} \sqrt {-1+5 \sqrt {2}} \tan ^{-1}\left (1-\sqrt {2}-2 \sqrt {2 \left (-1+\sqrt {2}\right )} x\right )+\frac {1}{20} \sqrt {-1+5 \sqrt {2}} \tan ^{-1}\left (1-\sqrt {2}+2 \sqrt {2 \left (-1+\sqrt {2}\right )} x\right )+\frac {1}{5} \tan ^{-1}\left (\sqrt {1+2 x^2}\right )+\frac {2}{5} \tan ^{-1}\left (1+4 x^2\right )+\frac {1}{5} \tanh ^{-1}\left (\frac {x}{\sqrt {1+2 x^2}}\right )-2 \log (x)+\frac {1}{5} \log \left (1+x^2\right )+\frac {1}{40} \sqrt {1+5 \sqrt {2}} \log \left (\sqrt {2}-2 \sqrt {2 \left (-1+\sqrt {2}\right )} x+4 x^2\right )-\frac {1}{40} \sqrt {1+5 \sqrt {2}} \log \left (\sqrt {2}+2 \sqrt {2 \left (-1+\sqrt {2}\right )} x+4 x^2\right )+\frac {1}{20} \log \left (1+4 x^2+8 x^4\right )+\left (-\frac {8}{5}-\frac {16 i}{5}\right ) \operatorname {Subst}\left (\int \frac {1}{(4+4 i)+(8-8 i) x^2} \, dx,x,\frac {x}{\sqrt {1+2 x^2}}\right )+\left (-\frac {8}{5}+\frac {16 i}{5}\right ) \operatorname {Subst}\left (\int \frac {1}{(4-4 i)+(8+8 i) x^2} \, dx,x,\frac {x}{\sqrt {1+2 x^2}}\right )+\left (\frac {1}{1280}-\frac {3 i}{1280}\right ) \int \frac {(896-640 i)+(2560-3072 i) x^2}{\sqrt {1+2 x^2} \left ((4+4 i)+16 x^2\right )} \, dx+\left (\frac {1}{1280}+\frac {3 i}{1280}\right ) \int \frac {(896+640 i)+(2560+3072 i) x^2}{\sqrt {1+2 x^2} \left ((4-4 i)+16 x^2\right )} \, dx+\frac {3}{40} \operatorname {Subst}\left (\int \frac {1}{\frac {1}{\sqrt {2}}-\sqrt {1+\sqrt {2}} x+x^2} \, dx,x,\sqrt {1+2 x^2}\right )+\frac {3}{40} \operatorname {Subst}\left (\int \frac {1}{\frac {1}{\sqrt {2}}+\sqrt {1+\sqrt {2}} x+x^2} \, dx,x,\sqrt {1+2 x^2}\right )+\frac {1}{10} \operatorname {Subst}\left (\int \frac {4+16 x}{1+4 x+8 x^2} \, dx,x,x^2\right )+\frac {1}{5} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {1+2 x^2}}\right )-\frac {2}{5} \operatorname {Subst}\left (\int \frac {1}{1+4 x+8 x^2} \, dx,x,x^2\right )+\frac {3}{5} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+4 x^2\right )-\frac {4}{5} \operatorname {Subst}\left (\int \frac {1}{4-8 x^2+8 x^4} \, dx,x,\sqrt {1+2 x^2}\right )+\frac {3 \operatorname {Subst}\left (\int \frac {\sqrt {1+\sqrt {2}}+2 x}{-\frac {1}{\sqrt {2}}-\sqrt {1+\sqrt {2}} x-x^2} \, dx,x,\sqrt {1+2 x^2}\right )}{40 \sqrt {1+\sqrt {2}}}+\frac {3 \operatorname {Subst}\left (\int \frac {\sqrt {1+\sqrt {2}}-2 x}{-\frac {1}{\sqrt {2}}+\sqrt {1+\sqrt {2}} x-x^2} \, dx,x,\sqrt {1+2 x^2}\right )}{40 \sqrt {1+\sqrt {2}}}\\ &=-x^2+\frac {\sinh ^{-1}\left (\sqrt {2} x\right )}{20 \sqrt {2}}-\frac {1}{5} \sqrt {2} \sinh ^{-1}\left (\sqrt {2} x\right )-\frac {1}{5} \tan ^{-1}(x)-\frac {1}{20} \sqrt {-1+5 \sqrt {2}} \tan ^{-1}\left (1-\sqrt {2}-2 \sqrt {2 \left (-1+\sqrt {2}\right )} x\right )+\frac {1}{20} \sqrt {-1+5 \sqrt {2}} \tan ^{-1}\left (1-\sqrt {2}+2 \sqrt {2 \left (-1+\sqrt {2}\right )} x\right )-\left (\frac {1}{5}-\frac {2 i}{5}\right ) \tan ^{-1}\left (\frac {(1+i) x}{\sqrt {1+2 x^2}}\right )+\frac {1}{5} \tan ^{-1}\left (\sqrt {1+2 x^2}\right )-\frac {1}{5} \tan ^{-1}\left (1+4 x^2\right )+\frac {2}{5} \tanh ^{-1}\left (\frac {x}{\sqrt {1+2 x^2}}\right )-\left (\frac {2}{5}-\frac {i}{5}\right ) \tanh ^{-1}\left (\frac {(1+i) x}{\sqrt {1+2 x^2}}\right )-2 \log (x)+\frac {1}{5} \log \left (1+x^2\right )+\frac {1}{40} \sqrt {1+5 \sqrt {2}} \log \left (\sqrt {2}-2 \sqrt {2 \left (-1+\sqrt {2}\right )} x+4 x^2\right )-\frac {1}{40} \sqrt {1+5 \sqrt {2}} \log \left (\sqrt {2}+2 \sqrt {2 \left (-1+\sqrt {2}\right )} x+4 x^2\right )+\frac {3}{20} \log \left (1+4 x^2+8 x^4\right )+\frac {3 \log \left (\sqrt {2}-2 \sqrt {1+\sqrt {2}} \sqrt {1+2 x^2}+2 \left (1+2 x^2\right )\right )}{40 \sqrt {1+\sqrt {2}}}-\frac {3 \log \left (1+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+2 x^2}+\sqrt {2} \left (1+2 x^2\right )\right )}{40 \sqrt {1+\sqrt {2}}}+\left (-\frac {8}{5}-\frac {4 i}{5}\right ) \int \frac {1}{\sqrt {1+2 x^2} \left ((4-4 i)+16 x^2\right )} \, dx+\left (-\frac {8}{5}+\frac {4 i}{5}\right ) \int \frac {1}{\sqrt {1+2 x^2} \left ((4+4 i)+16 x^2\right )} \, dx+\left (-\frac {13}{40}-\frac {21 i}{40}\right ) \int \frac {1}{\sqrt {1+2 x^2}} \, dx+\left (-\frac {13}{40}+\frac {21 i}{40}\right ) \int \frac {1}{\sqrt {1+2 x^2}} \, dx-\frac {3}{20} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2}-x^2} \, dx,x,-\sqrt {1+\sqrt {2}}+2 \sqrt {1+2 x^2}\right )-\frac {3}{20} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2}-x^2} \, dx,x,\sqrt {1+\sqrt {2}}+2 \sqrt {1+2 x^2}\right )+\frac {1}{5} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+4 x^2\right )-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {1+\sqrt {2}}-x}{\frac {1}{\sqrt {2}}-\sqrt {1+\sqrt {2}} x+x^2} \, dx,x,\sqrt {1+2 x^2}\right )}{10 \sqrt {2 \left (1+\sqrt {2}\right )}}-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {1+\sqrt {2}}+x}{\frac {1}{\sqrt {2}}+\sqrt {1+\sqrt {2}} x+x^2} \, dx,x,\sqrt {1+2 x^2}\right )}{10 \sqrt {2 \left (1+\sqrt {2}\right )}}\\ &=-x^2-\frac {3 \sinh ^{-1}\left (\sqrt {2} x\right )}{5 \sqrt {2}}-\frac {1}{5} \sqrt {2} \sinh ^{-1}\left (\sqrt {2} x\right )-\frac {1}{5} \tan ^{-1}(x)-\frac {1}{20} \sqrt {-1+5 \sqrt {2}} \tan ^{-1}\left (1-\sqrt {2}-2 \sqrt {2 \left (-1+\sqrt {2}\right )} x\right )+\frac {1}{20} \sqrt {-1+5 \sqrt {2}} \tan ^{-1}\left (1-\sqrt {2}+2 \sqrt {2 \left (-1+\sqrt {2}\right )} x\right )-\left (\frac {1}{5}-\frac {2 i}{5}\right ) \tan ^{-1}\left (\frac {(1+i) x}{\sqrt {1+2 x^2}}\right )+\frac {1}{5} \tan ^{-1}\left (\sqrt {1+2 x^2}\right )-\frac {2}{5} \tan ^{-1}\left (1+4 x^2\right )-\frac {3 \tan ^{-1}\left (\frac {\sqrt {1+\sqrt {2}}-2 \sqrt {1+2 x^2}}{\sqrt {-1+\sqrt {2}}}\right )}{20 \sqrt {-1+\sqrt {2}}}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {1+\sqrt {2}}+2 \sqrt {1+2 x^2}}{\sqrt {-1+\sqrt {2}}}\right )}{20 \sqrt {-1+\sqrt {2}}}+\frac {2}{5} \tanh ^{-1}\left (\frac {x}{\sqrt {1+2 x^2}}\right )-\left (\frac {2}{5}-\frac {i}{5}\right ) \tanh ^{-1}\left (\frac {(1+i) x}{\sqrt {1+2 x^2}}\right )-2 \log (x)+\frac {1}{5} \log \left (1+x^2\right )+\frac {1}{40} \sqrt {1+5 \sqrt {2}} \log \left (\sqrt {2}-2 \sqrt {2 \left (-1+\sqrt {2}\right )} x+4 x^2\right )-\frac {1}{40} \sqrt {1+5 \sqrt {2}} \log \left (\sqrt {2}+2 \sqrt {2 \left (-1+\sqrt {2}\right )} x+4 x^2\right )+\frac {3}{20} \log \left (1+4 x^2+8 x^4\right )+\frac {3 \log \left (\sqrt {2}-2 \sqrt {1+\sqrt {2}} \sqrt {1+2 x^2}+2 \left (1+2 x^2\right )\right )}{40 \sqrt {1+\sqrt {2}}}-\frac {3 \log \left (1+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+2 x^2}+\sqrt {2} \left (1+2 x^2\right )\right )}{40 \sqrt {1+\sqrt {2}}}+\left (-\frac {8}{5}-\frac {4 i}{5}\right ) \operatorname {Subst}\left (\int \frac {1}{(4-4 i)+(8+8 i) x^2} \, dx,x,\frac {x}{\sqrt {1+2 x^2}}\right )+\left (-\frac {8}{5}+\frac {4 i}{5}\right ) \operatorname {Subst}\left (\int \frac {1}{(4+4 i)+(8-8 i) x^2} \, dx,x,\frac {x}{\sqrt {1+2 x^2}}\right )-\frac {\operatorname {Subst}\left (\int \frac {1}{\frac {1}{\sqrt {2}}-\sqrt {1+\sqrt {2}} x+x^2} \, dx,x,\sqrt {1+2 x^2}\right )}{20 \sqrt {2}}-\frac {\operatorname {Subst}\left (\int \frac {1}{\frac {1}{\sqrt {2}}+\sqrt {1+\sqrt {2}} x+x^2} \, dx,x,\sqrt {1+2 x^2}\right )}{20 \sqrt {2}}+\frac {\operatorname {Subst}\left (\int \frac {-\sqrt {1+\sqrt {2}}+2 x}{\frac {1}{\sqrt {2}}-\sqrt {1+\sqrt {2}} x+x^2} \, dx,x,\sqrt {1+2 x^2}\right )}{20 \sqrt {2 \left (1+\sqrt {2}\right )}}-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {1+\sqrt {2}}+2 x}{\frac {1}{\sqrt {2}}+\sqrt {1+\sqrt {2}} x+x^2} \, dx,x,\sqrt {1+2 x^2}\right )}{20 \sqrt {2 \left (1+\sqrt {2}\right )}}\\ &=-x^2-\frac {3 \sinh ^{-1}\left (\sqrt {2} x\right )}{5 \sqrt {2}}-\frac {1}{5} \sqrt {2} \sinh ^{-1}\left (\sqrt {2} x\right )-\frac {1}{5} \tan ^{-1}(x)-\frac {1}{20} \sqrt {-1+5 \sqrt {2}} \tan ^{-1}\left (1-\sqrt {2}-2 \sqrt {2 \left (-1+\sqrt {2}\right )} x\right )+\frac {1}{20} \sqrt {-1+5 \sqrt {2}} \tan ^{-1}\left (1-\sqrt {2}+2 \sqrt {2 \left (-1+\sqrt {2}\right )} x\right )-\left (\frac {2}{5}-\frac {3 i}{10}\right ) \tan ^{-1}\left (\frac {(1+i) x}{\sqrt {1+2 x^2}}\right )+\frac {1}{5} \tan ^{-1}\left (\sqrt {1+2 x^2}\right )-\frac {2}{5} \tan ^{-1}\left (1+4 x^2\right )-\frac {3 \tan ^{-1}\left (\frac {\sqrt {1+\sqrt {2}}-2 \sqrt {1+2 x^2}}{\sqrt {-1+\sqrt {2}}}\right )}{20 \sqrt {-1+\sqrt {2}}}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {1+\sqrt {2}}+2 \sqrt {1+2 x^2}}{\sqrt {-1+\sqrt {2}}}\right )}{20 \sqrt {-1+\sqrt {2}}}+\frac {2}{5} \tanh ^{-1}\left (\frac {x}{\sqrt {1+2 x^2}}\right )-\left (\frac {3}{10}-\frac {2 i}{5}\right ) \tanh ^{-1}\left (\frac {(1+i) x}{\sqrt {1+2 x^2}}\right )-2 \log (x)+\frac {1}{5} \log \left (1+x^2\right )+\frac {1}{40} \sqrt {1+5 \sqrt {2}} \log \left (\sqrt {2}-2 \sqrt {2 \left (-1+\sqrt {2}\right )} x+4 x^2\right )-\frac {1}{40} \sqrt {1+5 \sqrt {2}} \log \left (\sqrt {2}+2 \sqrt {2 \left (-1+\sqrt {2}\right )} x+4 x^2\right )+\frac {3}{20} \log \left (1+4 x^2+8 x^4\right )+\frac {3 \log \left (\sqrt {2}-2 \sqrt {1+\sqrt {2}} \sqrt {1+2 x^2}+2 \left (1+2 x^2\right )\right )}{40 \sqrt {1+\sqrt {2}}}+\frac {\log \left (\sqrt {2}-2 \sqrt {1+\sqrt {2}} \sqrt {1+2 x^2}+2 \left (1+2 x^2\right )\right )}{20 \sqrt {2 \left (1+\sqrt {2}\right )}}-\frac {3 \log \left (1+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+2 x^2}+\sqrt {2} \left (1+2 x^2\right )\right )}{40 \sqrt {1+\sqrt {2}}}-\frac {\log \left (1+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+2 x^2}+\sqrt {2} \left (1+2 x^2\right )\right )}{20 \sqrt {2 \left (1+\sqrt {2}\right )}}+\frac {\operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2}-x^2} \, dx,x,-\sqrt {1+\sqrt {2}}+2 \sqrt {1+2 x^2}\right )}{10 \sqrt {2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2}-x^2} \, dx,x,\sqrt {1+\sqrt {2}}+2 \sqrt {1+2 x^2}\right )}{10 \sqrt {2}}\\ &=-x^2-\frac {3 \sinh ^{-1}\left (\sqrt {2} x\right )}{5 \sqrt {2}}-\frac {1}{5} \sqrt {2} \sinh ^{-1}\left (\sqrt {2} x\right )-\frac {1}{5} \tan ^{-1}(x)-\frac {1}{20} \sqrt {-1+5 \sqrt {2}} \tan ^{-1}\left (1-\sqrt {2}-2 \sqrt {2 \left (-1+\sqrt {2}\right )} x\right )+\frac {1}{20} \sqrt {-1+5 \sqrt {2}} \tan ^{-1}\left (1-\sqrt {2}+2 \sqrt {2 \left (-1+\sqrt {2}\right )} x\right )-\left (\frac {2}{5}-\frac {3 i}{10}\right ) \tan ^{-1}\left (\frac {(1+i) x}{\sqrt {1+2 x^2}}\right )+\frac {1}{5} \tan ^{-1}\left (\sqrt {1+2 x^2}\right )-\frac {2}{5} \tan ^{-1}\left (1+4 x^2\right )-\frac {3 \tan ^{-1}\left (\frac {\sqrt {1+\sqrt {2}}-2 \sqrt {1+2 x^2}}{\sqrt {-1+\sqrt {2}}}\right )}{20 \sqrt {-1+\sqrt {2}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {1+\sqrt {2}}-2 \sqrt {1+2 x^2}}{\sqrt {-1+\sqrt {2}}}\right )}{10 \sqrt {2 \left (-1+\sqrt {2}\right )}}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {1+\sqrt {2}}+2 \sqrt {1+2 x^2}}{\sqrt {-1+\sqrt {2}}}\right )}{20 \sqrt {-1+\sqrt {2}}}-\frac {\tan ^{-1}\left (\frac {\sqrt {1+\sqrt {2}}+2 \sqrt {1+2 x^2}}{\sqrt {-1+\sqrt {2}}}\right )}{10 \sqrt {2 \left (-1+\sqrt {2}\right )}}+\frac {2}{5} \tanh ^{-1}\left (\frac {x}{\sqrt {1+2 x^2}}\right )-\left (\frac {3}{10}-\frac {2 i}{5}\right ) \tanh ^{-1}\left (\frac {(1+i) x}{\sqrt {1+2 x^2}}\right )-2 \log (x)+\frac {1}{5} \log \left (1+x^2\right )+\frac {1}{40} \sqrt {1+5 \sqrt {2}} \log \left (\sqrt {2}-2 \sqrt {2 \left (-1+\sqrt {2}\right )} x+4 x^2\right )-\frac {1}{40} \sqrt {1+5 \sqrt {2}} \log \left (\sqrt {2}+2 \sqrt {2 \left (-1+\sqrt {2}\right )} x+4 x^2\right )+\frac {3}{20} \log \left (1+4 x^2+8 x^4\right )+\frac {3 \log \left (\sqrt {2}-2 \sqrt {1+\sqrt {2}} \sqrt {1+2 x^2}+2 \left (1+2 x^2\right )\right )}{40 \sqrt {1+\sqrt {2}}}+\frac {\log \left (\sqrt {2}-2 \sqrt {1+\sqrt {2}} \sqrt {1+2 x^2}+2 \left (1+2 x^2\right )\right )}{20 \sqrt {2 \left (1+\sqrt {2}\right )}}-\frac {3 \log \left (1+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+2 x^2}+\sqrt {2} \left (1+2 x^2\right )\right )}{40 \sqrt {1+\sqrt {2}}}-\frac {\log \left (1+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+2 x^2}+\sqrt {2} \left (1+2 x^2\right )\right )}{20 \sqrt {2 \left (1+\sqrt {2}\right )}}\\ \end {align*}

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Mathematica [C]  time = 7.14, size = 3855, normalized size = 11.44 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(-x^2 + Sqrt[1 + 2*x^2] + (1 + 2*x^2)^(5/2))/(x^2 - x*(1 + 2*x^2)^(3/2)),x]

[Out]

(x^3*(-x + (1 + 2*x^2)^(3/2)))/(x^2 - x*(1 + 2*x^2)^(3/2)) + (x*(-x + (1 + 2*x^2)^(3/2))*ArcSinh[Sqrt[2]*x])/(
Sqrt[2]*(x^2 - x*(1 + 2*x^2)^(3/2))) + (x*(-x + (1 + 2*x^2)^(3/2))*ArcTan[x])/(5*(x^2 - x*(1 + 2*x^2)^(3/2)))
- ((1/5 + I/10)*x*(-x + (1 + 2*x^2)^(3/2))*ArcTan[(2*x)/Sqrt[1 - I]])/(Sqrt[1 - I]*(x^2 - x*(1 + 2*x^2)^(3/2))
) - ((1/5 - I/10)*x*(-x + (1 + 2*x^2)^(3/2))*ArcTan[(2*x)/Sqrt[1 + I]])/(Sqrt[1 + I]*(x^2 - x*(1 + 2*x^2)^(3/2
))) + ((1/5 - (3*I)/20)*x*(-x + (1 + 2*x^2)^(3/2))*ArcTan[(2*x^2)/(1 + 2*x^2)])/(x^2 - x*(1 + 2*x^2)^(3/2)) -
((1/5 + (3*I)/20)*x*(-x + (1 + 2*x^2)^(3/2))*ArcTan[(1 + 2*x^2)/(2*x^2)])/(x^2 - x*(1 + 2*x^2)^(3/2)) + ((1/40
+ (3*I)/40)*((4 + 2*I) + Sqrt[-2 - 2*I])*x*(-x + (1 + 2*x^2)^(3/2))*ArcTan[((-7 - 10*I) - 2*(-1 - I)^(3/2)*Sq
rt[2] - (36 + 20*I)*x + (75 + 19*I)*Sqrt[-2 - 2*I]*x + (330 + 266*I)*x^2 + (116 + 64*I)*Sqrt[-2 - 2*I]*x^2 + (
176 + 320*I)*x^3 - (300 + 104*I)*Sqrt[-2 - 2*I]*x^3 - (192 + 160*I)*x^4 + 88*(-1 - I)^(3/2)*Sqrt[2]*x^4)/((49
+ 40*I) - 21*(-1 - I)^(3/2)*Sqrt[2] + (92 + 260*I)*x - (170 + 68*I)*Sqrt[-2 - 2*I]*x - (410 + 402*I)*x^2 - (23
2 + 268*I)*Sqrt[-2 - 2*I]*x^2 - (192 + 560*I)*x^3 + (280 + 88*I)*Sqrt[-2 - 2*I]*x^3 + (144 + 80*I)*x^4 - 96*(-
1 - I)^(3/2)*Sqrt[2]*x^4)])/(Sqrt[-1 - I]*Sqrt[1 + I]*(x^2 - x*(1 + 2*x^2)^(3/2))) + ((1/40 + (3*I)/40)*((-4 -
2*I) + Sqrt[-2 - 2*I])*x*(-x + (1 + 2*x^2)^(3/2))*ArcTan[((7 + 10*I) - 2*(-1 - I)^(3/2)*Sqrt[2] + (36 + 20*I)
*x + (75 + 19*I)*Sqrt[-2 - 2*I]*x - (330 + 266*I)*x^2 + (116 + 64*I)*Sqrt[-2 - 2*I]*x^2 - (176 + 320*I)*x^3 -
(300 + 104*I)*Sqrt[-2 - 2*I]*x^3 + (192 + 160*I)*x^4 + 88*(-1 - I)^(3/2)*Sqrt[2]*x^4)/((-49 - 40*I) - 21*(-1 -
I)^(3/2)*Sqrt[2] - (92 + 260*I)*x - (170 + 68*I)*Sqrt[-2 - 2*I]*x + (410 + 402*I)*x^2 - (232 + 268*I)*Sqrt[-2
- 2*I]*x^2 + (192 + 560*I)*x^3 + (280 + 88*I)*Sqrt[-2 - 2*I]*x^3 - (144 + 80*I)*x^4 - 96*(-1 - I)^(3/2)*Sqrt[
2]*x^4)])/(Sqrt[-1 - I]*Sqrt[1 + I]*(x^2 - x*(1 + 2*x^2)^(3/2))) - ((1/10 + I/5)*x*(-x + (1 + 2*x^2)^(3/2))*Ar
cTan[(-4 - 4*x^2 - 5*Sqrt[1 + 2*x^2])/(-3 + 2*x^2)])/(x^2 - x*(1 + 2*x^2)^(3/2)) - ((1/10 - I/5)*x*(-x + (1 +
2*x^2)^(3/2))*ArcTan[(4 + 4*x^2 - 5*Sqrt[1 + 2*x^2])/(-3 + 2*x^2)])/(x^2 - x*(1 + 2*x^2)^(3/2)) - ((1/20 - I/4
0)*((-2 - 4*I) + Sqrt[2 - 2*I])*x*(-x + (1 + 2*x^2)^(3/2))*ArcTan[((10 + 7*I) - 22/Sqrt[1/2 - I/2] + (52 + 36*
I)*x - (47 + 21*I)*Sqrt[2 - 2*I]*x + (214 + 166*I)*x^2 - (60 + 96*I)*Sqrt[2 - 2*I]*x^2 + (32 + 176*I)*x^3 - (5
2 + 136*I)*Sqrt[2 - 2*I]*x^3 + (160 + 192*I)*x^4 - (32*x^4)/Sqrt[1/2 - I/2] - (110 - 10*I)*x*Sqrt[1 + 2*x^2] +
20*Sqrt[2 - 2*I]*x*Sqrt[1 + 2*x^2] - 80*x^2*Sqrt[1 + 2*x^2] + (120 + 100*I)*Sqrt[2 - 2*I]*x^2*Sqrt[1 + 2*x^2]
- (240 + 200*I)*x^3*Sqrt[1 + 2*x^2] + (80*Sqrt[2]*x^3*Sqrt[1 + 2*x^2])/Sqrt[1 - I])/((-10 - I) + 6/Sqrt[1/2 -
I/2] - (56 - 32*I)*x + (26 + 8*I)*Sqrt[2 - 2*I]*x - (22 + 58*I)*x^2 + (80 + 68*I)*Sqrt[2 - 2*I]*x^2 - (176 +
48*I)*x^3 + (36 + 68*I)*Sqrt[2 - 2*I]*x^3 + (120 - 296*I)*x^4 + (48 + 128*I)*Sqrt[2 - 2*I]*x^4)])/(x^2 - x*(1
+ 2*x^2)^(3/2)) - ((1/40 + I/20)*((2 + 4*I) + Sqrt[2 - 2*I])*x*(-x + (1 + 2*x^2)^(3/2))*ArcTanh[((-7 + 10*I) -
11*(1 - I)^(3/2)*Sqrt[2] - (36 - 52*I)*x - (21 - 47*I)*Sqrt[2 - 2*I]*x - (166 - 214*I)*x^2 - (96 - 60*I)*Sqrt
[2 - 2*I]*x^2 - (176 - 32*I)*x^3 - (136 - 52*I)*Sqrt[2 - 2*I]*x^3 - (192 - 160*I)*x^4 - 16*(1 - I)^(3/2)*Sqrt[
2]*x^4 + (10 + 110*I)*x*Sqrt[1 + 2*x^2] + (20*I)*Sqrt[2 - 2*I]*x*Sqrt[1 + 2*x^2] + (80*I)*x^2*Sqrt[1 + 2*x^2]
- (100 - 120*I)*Sqrt[2 - 2*I]*x^2*Sqrt[1 + 2*x^2] - (200 - 240*I)*x^3*Sqrt[1 + 2*x^2] - 40*(1 - I)^(3/2)*Sqrt[
2]*x^3*Sqrt[1 + 2*x^2])/((10 + I) + 6/Sqrt[1/2 - I/2] + (56 - 32*I)*x + (26 + 8*I)*Sqrt[2 - 2*I]*x + (22 + 58*
I)*x^2 + (80 + 68*I)*Sqrt[2 - 2*I]*x^2 + (176 + 48*I)*x^3 + (36 + 68*I)*Sqrt[2 - 2*I]*x^3 - (120 - 296*I)*x^4
+ (48 + 128*I)*Sqrt[2 - 2*I]*x^4)])/(x^2 - x*(1 + 2*x^2)^(3/2)) - ((3/80 - I/80)*((-4 - 2*I) + Sqrt[-2 - 2*I])
*x*(-x + (1 + 2*x^2)^(3/2))*Log[(Sqrt[-2 - 2*I] - (2 + 2*I)*x)^2*(Sqrt[-2 - 2*I] - (2 - 2*I)*x)^2])/(Sqrt[-1 -
I]*Sqrt[1 + I]*(x^2 - x*(1 + 2*x^2)^(3/2))) - ((1/80 + I/40)*((-2 - 4*I) + Sqrt[2 - 2*I])*x*(-x + (1 + 2*x^2)
^(3/2))*Log[(Sqrt[2 - 2*I] - (2 + 2*I)*x)^2*(Sqrt[2 - 2*I] - (2 - 2*I)*x)^2])/(x^2 - x*(1 + 2*x^2)^(3/2)) + (2
*x*(-x + (1 + 2*x^2)^(3/2))*Log[x])/(x^2 - x*(1 + 2*x^2)^(3/2)) - ((I/10)*x*(-x + (1 + 2*x^2)^(3/2))*Log[(-I +
x)*(I + x)])/(x^2 - x*(1 + 2*x^2)^(3/2)) - ((3/80 - I/80)*((4 + 2*I) + Sqrt[-2 - 2*I])*x*(-x + (1 + 2*x^2)^(3
/2))*Log[(Sqrt[-2 - 2*I] + (2 - 2*I)*x)^2*(Sqrt[-2 - 2*I] + (2 + 2*I)*x)^2])/(Sqrt[-1 - I]*Sqrt[1 + I]*(x^2 -
x*(1 + 2*x^2)^(3/2))) - ((1/80 + I/40)*((2 + 4*I) + Sqrt[2 - 2*I])*x*(-x + (1 + 2*x^2)^(3/2))*Log[(Sqrt[2 - 2*
I] + (2 - 2*I)*x)^2*(Sqrt[2 - 2*I] + (2 + 2*I)*x)^2])/(x^2 - x*(1 + 2*x^2)^(3/2)) - (x*(-x + (1 + 2*x^2)^(3/2)
)*Log[1 + x^2])/(5*(x^2 - x*(1 + 2*x^2)^(3/2))) - (3*x*(-x + (1 + 2*x^2)^(3/2))*Log[1 + 4*x^2 + 8*x^4])/(20*(x
^2 - x*(1 + 2*x^2)^(3/2))) + ((1/10 + I/20)*x*(-x + (1 + 2*x^2)^(3/2))*Log[1 + 3*x^2 - 2*x*Sqrt[1 + 2*x^2]])/(
x^2 - x*(1 + 2*x^2)^(3/2)) - ((1/10 - I/20)*x*(-x + (1 + 2*x^2)^(3/2))*Log[1 + 3*x^2 + 2*x*Sqrt[1 + 2*x^2]])/(
x^2 - x*(1 + 2*x^2)^(3/2)) + ((1/80 + I/40)*((-2 - 4*I) + Sqrt[2 - 2*I])*x*(-x + (1 + 2*x^2)^(3/2))*Log[(7 + 1
6*I) - (1 + 3*I)*Sqrt[2 - 2*I] + (4 + 12*I)*x + (9 - 23*I)*Sqrt[2 - 2*I]*x - (16 - 68*I)*x^2 + (4 - 12*I)*Sqrt
[2 - 2*I]*x^2 + (16*I)*x^3 + (20 - 24*I)*Sqrt[2 - 2*I]*x^3 - (40 - 48*I)*x^4 + 8*(1 - I)^(3/2)*Sqrt[2]*x^4 - (
4 + 4*I)*Sqrt[1 + 2*x^2] + (1 + 11*I)*Sqrt[2 - 2*I]*Sqrt[1 + 2*x^2] - (4 + 44*I)*x*Sqrt[1 + 2*x^2] + (8*I)*Sqr
t[2 - 2*I]*x*Sqrt[1 + 2*x^2] - (16*I)*x^2*Sqrt[1 + 2*x^2] - (20 - 24*I)*Sqrt[2 - 2*I]*x^2*Sqrt[1 + 2*x^2]])/(x
^2 - x*(1 + 2*x^2)^(3/2)) + ((1/80 + I/40)*((2 + 4*I) + Sqrt[2 - 2*I])*x*(-x + (1 + 2*x^2)^(3/2))*Log[(7 + 16*
I) + (1 + 3*I)*Sqrt[2 - 2*I] + (4 + 12*I)*x - (9 - 23*I)*Sqrt[2 - 2*I]*x - (16 - 68*I)*x^2 - (4 - 12*I)*Sqrt[2
- 2*I]*x^2 + (16*I)*x^3 - (20 - 24*I)*Sqrt[2 - 2*I]*x^3 - (40 - 48*I)*x^4 - 8*(1 - I)^(3/2)*Sqrt[2]*x^4 + (4
+ 4*I)*Sqrt[1 + 2*x^2] + (1 + 11*I)*Sqrt[2 - 2*I]*Sqrt[1 + 2*x^2] + (4 + 44*I)*x*Sqrt[1 + 2*x^2] + (8*I)*Sqrt[
2 - 2*I]*x*Sqrt[1 + 2*x^2] + (16*I)*x^2*Sqrt[1 + 2*x^2] - (20 - 24*I)*Sqrt[2 - 2*I]*x^2*Sqrt[1 + 2*x^2]])/(x^2
- x*(1 + 2*x^2)^(3/2)) + ((3/80 - I/80)*((-4 - 2*I) + Sqrt[-2 - 2*I])*x*(-x + (1 + 2*x^2)^(3/2))*Log[(14 + 3*
I) - (6 + 2*I)*Sqrt[-2 - 2*I] + (40 + 8*I)*x + (25 - 21*I)*Sqrt[-2 - 2*I]*x - (64 - 80*I)*x^2 - (64*x^2)/Sqrt[
-1/2 - I/2] - (32 - 64*I)*x^3 + (24 + 52*I)*Sqrt[-2 - 2*I]*x^3 - (40 + 32*I)*x^4 - 16*(-1 - I)^(3/2)*Sqrt[2]*x
^4 - 8*Sqrt[-1 - I]*Sqrt[1 + I]*Sqrt[1 + 2*x^2] + (9 - I)*Sqrt[2 + 2*I]*Sqrt[1 + 2*x^2] + (24 - 30*I)*Sqrt[-1
- I]*Sqrt[1 + I]*x*Sqrt[1 + 2*x^2] + 24*Sqrt[2 + 2*I]*x*Sqrt[1 + 2*x^2] - (64*Sqrt[1 + I]*x^2*Sqrt[1 + 2*x^2])
/Sqrt[-1 - I] - (32 - 40*I)*Sqrt[2 + 2*I]*x^2*Sqrt[1 + 2*x^2] + (4 + 36*I)*Sqrt[-1 - I]*Sqrt[1 + I]*x^3*Sqrt[1
+ 2*x^2] - (32*Sqrt[2]*x^3*Sqrt[1 + 2*x^2])/Sqrt[1 + I]])/(Sqrt[-1 - I]*Sqrt[1 + I]*(x^2 - x*(1 + 2*x^2)^(3/2
))) + ((3/80 - I/80)*((4 + 2*I) + Sqrt[-2 - 2*I])*x*(-x + (1 + 2*x^2)^(3/2))*Log[(-14 - 3*I) - (6 + 2*I)*Sqrt[
-2 - 2*I] - (40 + 8*I)*x + (25 - 21*I)*Sqrt[-2 - 2*I]*x + (64 - 80*I)*x^2 - (64*x^2)/Sqrt[-1/2 - I/2] + (32 -
64*I)*x^3 + (24 + 52*I)*Sqrt[-2 - 2*I]*x^3 + (40 + 32*I)*x^4 - 16*(-1 - I)^(3/2)*Sqrt[2]*x^4 - 8*Sqrt[-1 - I]*
Sqrt[1 + I]*Sqrt[1 + 2*x^2] - (9 - I)*Sqrt[2 + 2*I]*Sqrt[1 + 2*x^2] + (24 - 30*I)*Sqrt[-1 - I]*Sqrt[1 + I]*x*S
qrt[1 + 2*x^2] - 24*Sqrt[2 + 2*I]*x*Sqrt[1 + 2*x^2] - (64*Sqrt[1 + I]*x^2*Sqrt[1 + 2*x^2])/Sqrt[-1 - I] + (32
- 40*I)*Sqrt[2 + 2*I]*x^2*Sqrt[1 + 2*x^2] + (4 + 36*I)*Sqrt[-1 - I]*Sqrt[1 + I]*x^3*Sqrt[1 + 2*x^2] + (32*Sqrt
[2]*x^3*Sqrt[1 + 2*x^2])/Sqrt[1 + I]])/(Sqrt[-1 - I]*Sqrt[1 + I]*(x^2 - x*(1 + 2*x^2)^(3/2)))

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IntegrateAlgebraic [A]  time = 1.06, size = 337, normalized size = 1.00 \begin {gather*} -x^2+\frac {1}{2} \left (2+\sqrt {2}\right ) \log \left (-\sqrt {2} x+\sqrt {1+2 x^2}\right )-2 \log \left (-2 x^2+\sqrt {2} x \sqrt {1+2 x^2}\right )+\frac {1}{2} \text {RootSum}\left [1+3 \text {$\#$1}^2-2 \sqrt {2} \text {$\#$1}^2+3 \text {$\#$1}^4+2 \sqrt {2} \text {$\#$1}^4+\text {$\#$1}^6\&,\frac {-\log \left (-\sqrt {2} x+\sqrt {1+2 x^2}-\text {$\#$1}\right )+4 \log \left (-\sqrt {2} x+\sqrt {1+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+2 \log \left (-\sqrt {2} x+\sqrt {1+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^3+4 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {1+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^3+\log \left (-\sqrt {2} x+\sqrt {1+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^4+2 \log \left (-\sqrt {2} x+\sqrt {1+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^5}{3 \text {$\#$1}-2 \sqrt {2} \text {$\#$1}+6 \text {$\#$1}^3+4 \sqrt {2} \text {$\#$1}^3+3 \text {$\#$1}^5}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-x^2 + Sqrt[1 + 2*x^2] + (1 + 2*x^2)^(5/2))/(x^2 - x*(1 + 2*x^2)^(3/2)),x]

[Out]

-x^2 + ((2 + Sqrt[2])*Log[-(Sqrt[2]*x) + Sqrt[1 + 2*x^2]])/2 - 2*Log[-2*x^2 + Sqrt[2]*x*Sqrt[1 + 2*x^2]] + Roo
tSum[1 + 3*#1^2 - 2*Sqrt[2]*#1^2 + 3*#1^4 + 2*Sqrt[2]*#1^4 + #1^6 & , (-Log[-(Sqrt[2]*x) + Sqrt[1 + 2*x^2] - #
1] + 4*Log[-(Sqrt[2]*x) + Sqrt[1 + 2*x^2] - #1]*#1 + 2*Log[-(Sqrt[2]*x) + Sqrt[1 + 2*x^2] - #1]*#1^3 + 4*Sqrt[
2]*Log[-(Sqrt[2]*x) + Sqrt[1 + 2*x^2] - #1]*#1^3 + Log[-(Sqrt[2]*x) + Sqrt[1 + 2*x^2] - #1]*#1^4 + 2*Log[-(Sqr
t[2]*x) + Sqrt[1 + 2*x^2] - #1]*#1^5)/(3*#1 - 2*Sqrt[2]*#1 + 6*#1^3 + 4*Sqrt[2]*#1^3 + 3*#1^5) & ]/2

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fricas [B]  time = 1.67, size = 2052, normalized size = 6.09

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+(2*x^2+1)^(1/2)+(2*x^2+1)^(5/2))/(x^2-x*(2*x^2+1)^(3/2)),x, algorithm="fricas")

[Out]

-x^2 - 1/20*(10*sqrt(7/200*I + 1/200) + 4*I - 3)*log(17/25*(10*sqrt(7/200*I + 1/200) + 4*I - 3)^3 - 11/25*(10*
sqrt(7/200*I + 1/200) + 4*I - 3)^2 + 167*x - 54*sqrt(7/200*I + 1/200) - 108/5*I - 254/5) - 1/20*(10*sqrt(-7/20
0*I + 1/200) - 4*I - 3)*log(-17/25*(10*sqrt(7/200*I + 1/200) + 4*I - 3)^3 - 2/25*(85*sqrt(7/200*I + 1/200) + 3
4*I + 82)*(10*sqrt(-7/200*I + 1/200) - 4*I - 3)^2 - 204/25*(10*sqrt(7/200*I + 1/200) + 4*I - 3)^2 - 1/25*(17*(
10*sqrt(7/200*I + 1/200) + 4*I - 3)^2 + 2040*sqrt(7/200*I + 1/200) + 816*I + 968)*(10*sqrt(-7/200*I + 1/200) -
4*I - 3) + 167*x - 578*sqrt(7/200*I + 1/200) - 1156/5*I - 658/5) + 1/20*(5*sqrt(7/200*I + 1/200) + 5*sqrt(-7/
200*I + 1/200) - sqrt(-3/4*(10*sqrt(7/200*I + 1/200) + 4*I - 3)^2 - 1/2*(10*sqrt(7/200*I + 1/200) + 4*I + 9)*(
10*sqrt(-7/200*I + 1/200) - 4*I - 3) - 3/4*(10*sqrt(-7/200*I + 1/200) - 4*I - 3)^2 - 60*sqrt(7/200*I + 1/200)
- 24*I - 31) + 3)*log(1/25*(85*sqrt(7/200*I + 1/200) + 34*I + 82)*(10*sqrt(-7/200*I + 1/200) - 4*I - 3)^2 + 43
/10*(10*sqrt(7/200*I + 1/200) + 4*I - 3)^2 + 1/50*(17*(10*sqrt(7/200*I + 1/200) + 4*I - 3)^2 + 2040*sqrt(7/200
*I + 1/200) + 816*I + 968)*(10*sqrt(-7/200*I + 1/200) - 4*I - 3) + 1/25*sqrt(-3/4*(10*sqrt(7/200*I + 1/200) +
4*I - 3)^2 - 1/2*(10*sqrt(7/200*I + 1/200) + 4*I + 9)*(10*sqrt(-7/200*I + 1/200) - 4*I - 3) - 3/4*(10*sqrt(-7/
200*I + 1/200) - 4*I - 3)^2 - 60*sqrt(7/200*I + 1/200) - 24*I - 31)*(2*(85*sqrt(7/200*I + 1/200) + 34*I + 82)*
(10*sqrt(-7/200*I + 1/200) - 4*I - 3) + 2150*sqrt(7/200*I + 1/200) + 860*I + 355) + 167*x + 316*sqrt(7/200*I +
1/200) + 632/5*I + 456/5) + 1/20*(5*sqrt(7/200*I + 1/200) + 5*sqrt(-7/200*I + 1/200) + sqrt(-3/4*(10*sqrt(7/2
00*I + 1/200) + 4*I - 3)^2 - 1/2*(10*sqrt(7/200*I + 1/200) + 4*I + 9)*(10*sqrt(-7/200*I + 1/200) - 4*I - 3) -
3/4*(10*sqrt(-7/200*I + 1/200) - 4*I - 3)^2 - 60*sqrt(7/200*I + 1/200) - 24*I - 31) + 3)*log(1/25*(85*sqrt(7/2
00*I + 1/200) + 34*I + 82)*(10*sqrt(-7/200*I + 1/200) - 4*I - 3)^2 + 43/10*(10*sqrt(7/200*I + 1/200) + 4*I - 3
)^2 + 1/50*(17*(10*sqrt(7/200*I + 1/200) + 4*I - 3)^2 + 2040*sqrt(7/200*I + 1/200) + 816*I + 968)*(10*sqrt(-7/
200*I + 1/200) - 4*I - 3) - 1/25*sqrt(-3/4*(10*sqrt(7/200*I + 1/200) + 4*I - 3)^2 - 1/2*(10*sqrt(7/200*I + 1/2
00) + 4*I + 9)*(10*sqrt(-7/200*I + 1/200) - 4*I - 3) - 3/4*(10*sqrt(-7/200*I + 1/200) - 4*I - 3)^2 - 60*sqrt(7
/200*I + 1/200) - 24*I - 31)*(2*(85*sqrt(7/200*I + 1/200) + 34*I + 82)*(10*sqrt(-7/200*I + 1/200) - 4*I - 3) +
2150*sqrt(7/200*I + 1/200) + 860*I + 355) + 167*x + 316*sqrt(7/200*I + 1/200) + 632/5*I + 456/5) - 1/20*(10*s
qrt(-7/200*I + 1/200) - 4*I - 3)*log(-1/25*(76*x*(10*sqrt(7/200*I + 1/200) + 4*I - 3)^3 + 912*x*(10*sqrt(7/200
*I + 1/200) + 4*I - 3)^2 + 2*(38*x*(10*sqrt(7/200*I + 1/200) + 4*I - 3) + 235*x)*(10*sqrt(-7/200*I + 1/200) -
4*I - 3)^2 + 6460*x*(10*sqrt(7/200*I + 1/200) + 4*I - 3) + 2*(38*x*(10*sqrt(7/200*I + 1/200) + 4*I - 3)^2 + 45
6*x*(10*sqrt(7/200*I + 1/200) + 4*I - 3) + 2795*x)*(10*sqrt(-7/200*I + 1/200) - 4*I - 3) + 30650*x - 4175*sqrt
(2*x^2 + 1) + 4175)/x) - 1/20*(10*sqrt(7/200*I + 1/200) + 4*I - 3)*log(1/25*(76*x*(10*sqrt(7/200*I + 1/200) +
4*I - 3)^3 + 442*x*(10*sqrt(7/200*I + 1/200) + 4*I - 3)^2 + 870*x*(10*sqrt(7/200*I + 1/200) + 4*I - 3) - 4050*
x + 4175*sqrt(2*x^2 + 1) - 4175)/x) - 1/20*(10*sqrt(7/200*I + 1/200) - 4*I + 3)*log(1/5*(28*x*(10*sqrt(-7/200*
I + 1/200) + 4*I + 3)^3 + 2*(14*x*(10*sqrt(-7/200*I + 1/200) + 4*I + 3) - 69*x)*(10*sqrt(7/200*I + 1/200) - 4*
I + 3)^2 - 336*x*(10*sqrt(-7/200*I + 1/200) + 4*I + 3)^2 + 2*(14*x*(10*sqrt(-7/200*I + 1/200) + 4*I + 3)^2 - 1
68*x*(10*sqrt(-7/200*I + 1/200) + 4*I + 3) + 643*x)*(10*sqrt(7/200*I + 1/200) - 4*I + 3) + 2380*x*(10*sqrt(-7/
200*I + 1/200) + 4*I + 3) - 6370*x + 835*sqrt(2*x^2 + 1) - 835)/x) - 1/20*(10*sqrt(-7/200*I + 1/200) + 4*I + 3
)*log(-1/5*(28*x*(10*sqrt(-7/200*I + 1/200) + 4*I + 3)^3 - 198*x*(10*sqrt(-7/200*I + 1/200) + 4*I + 3)^2 + 109
4*x*(10*sqrt(-7/200*I + 1/200) + 4*I + 3) - 3430*x - 835*sqrt(2*x^2 + 1) + 835)/x) + 1/20*(5*sqrt(7/200*I + 1/
200) + 5*sqrt(-7/200*I + 1/200) - sqrt(-3/4*(10*sqrt(7/200*I + 1/200) - 4*I + 3)^2 - 3/4*(10*sqrt(-7/200*I + 1
/200) + 4*I + 3)^2 - 1/2*(10*sqrt(7/200*I + 1/200) - 4*I + 3)*(10*sqrt(-7/200*I + 1/200) + 4*I - 9) + 60*sqrt(
-7/200*I + 1/200) + 24*I - 31) - 3)*log(-1/5*((14*x*(10*sqrt(-7/200*I + 1/200) + 4*I + 3) - 69*x)*(10*sqrt(7/2
00*I + 1/200) - 4*I + 3)^2 - 69*x*(10*sqrt(-7/200*I + 1/200) + 4*I + 3)^2 + (14*x*(10*sqrt(-7/200*I + 1/200) +
4*I + 3)^2 - 168*x*(10*sqrt(-7/200*I + 1/200) + 4*I + 3) + 643*x)*(10*sqrt(7/200*I + 1/200) - 4*I + 3) + 643*
x*(10*sqrt(-7/200*I + 1/200) + 4*I + 3) + 2*((14*x*(10*sqrt(-7/200*I + 1/200) + 4*I + 3) - 69*x)*(10*sqrt(7/20
0*I + 1/200) - 4*I + 3) - 69*x*(10*sqrt(-7/200*I + 1/200) + 4*I + 3) + 185*x)*sqrt(-3/4*(10*sqrt(7/200*I + 1/2
00) - 4*I + 3)^2 - 3/4*(10*sqrt(-7/200*I + 1/200) + 4*I + 3)^2 - 1/2*(10*sqrt(7/200*I + 1/200) - 4*I + 3)*(10*
sqrt(-7/200*I + 1/200) + 4*I - 9) + 60*sqrt(-7/200*I + 1/200) + 24*I - 31) - 3140*x - 835*sqrt(2*x^2 + 1) + 83
5)/x) + 1/20*(5*sqrt(7/200*I + 1/200) + 5*sqrt(-7/200*I + 1/200) + sqrt(-3/4*(10*sqrt(7/200*I + 1/200) - 4*I +
3)^2 - 3/4*(10*sqrt(-7/200*I + 1/200) + 4*I + 3)^2 - 1/2*(10*sqrt(7/200*I + 1/200) - 4*I + 3)*(10*sqrt(-7/200
*I + 1/200) + 4*I - 9) + 60*sqrt(-7/200*I + 1/200) + 24*I - 31) - 3)*log(-1/5*((14*x*(10*sqrt(-7/200*I + 1/200
) + 4*I + 3) - 69*x)*(10*sqrt(7/200*I + 1/200) - 4*I + 3)^2 - 69*x*(10*sqrt(-7/200*I + 1/200) + 4*I + 3)^2 + (
14*x*(10*sqrt(-7/200*I + 1/200) + 4*I + 3)^2 - 168*x*(10*sqrt(-7/200*I + 1/200) + 4*I + 3) + 643*x)*(10*sqrt(7
/200*I + 1/200) - 4*I + 3) + 643*x*(10*sqrt(-7/200*I + 1/200) + 4*I + 3) - 2*((14*x*(10*sqrt(-7/200*I + 1/200)
+ 4*I + 3) - 69*x)*(10*sqrt(7/200*I + 1/200) - 4*I + 3) - 69*x*(10*sqrt(-7/200*I + 1/200) + 4*I + 3) + 185*x)
*sqrt(-3/4*(10*sqrt(7/200*I + 1/200) - 4*I + 3)^2 - 3/4*(10*sqrt(-7/200*I + 1/200) + 4*I + 3)^2 - 1/2*(10*sqrt
(7/200*I + 1/200) - 4*I + 3)*(10*sqrt(-7/200*I + 1/200) + 4*I - 9) + 60*sqrt(-7/200*I + 1/200) + 24*I - 31) -
3140*x - 835*sqrt(2*x^2 + 1) + 835)/x) + 1/20*(5*sqrt(7/200*I + 1/200) + 5*sqrt(-7/200*I + 1/200) - sqrt(-3/4*
(10*sqrt(7/200*I + 1/200) + 4*I - 3)^2 - 1/2*(10*sqrt(7/200*I + 1/200) + 4*I + 9)*(10*sqrt(-7/200*I + 1/200) -
4*I - 3) - 3/4*(10*sqrt(-7/200*I + 1/200) - 4*I - 3)^2 - 60*sqrt(7/200*I + 1/200) - 24*I - 31) + 3)*log(1/25*
(235*x*(10*sqrt(7/200*I + 1/200) + 4*I - 3)^2 + (38*x*(10*sqrt(7/200*I + 1/200) + 4*I - 3) + 235*x)*(10*sqrt(-
7/200*I + 1/200) - 4*I - 3)^2 + 2795*x*(10*sqrt(7/200*I + 1/200) + 4*I - 3) + (38*x*(10*sqrt(7/200*I + 1/200)
+ 4*I - 3)^2 + 456*x*(10*sqrt(7/200*I + 1/200) + 4*I - 3) + 2795*x)*(10*sqrt(-7/200*I + 1/200) - 4*I - 3) + 2*
(235*x*(10*sqrt(7/200*I + 1/200) + 4*I - 3) + (38*x*(10*sqrt(7/200*I + 1/200) + 4*I - 3) + 235*x)*(10*sqrt(-7/
200*I + 1/200) - 4*I - 3) + 25*x)*sqrt(-3/4*(10*sqrt(7/200*I + 1/200) + 4*I - 3)^2 - 1/2*(10*sqrt(7/200*I + 1/
200) + 4*I + 9)*(10*sqrt(-7/200*I + 1/200) - 4*I - 3) - 3/4*(10*sqrt(-7/200*I + 1/200) - 4*I - 3)^2 - 60*sqrt(
7/200*I + 1/200) - 24*I - 31) + 9000*x + 4175*sqrt(2*x^2 + 1) - 4175)/x) + 1/20*(5*sqrt(7/200*I + 1/200) + 5*s
qrt(-7/200*I + 1/200) + sqrt(-3/4*(10*sqrt(7/200*I + 1/200) + 4*I - 3)^2 - 1/2*(10*sqrt(7/200*I + 1/200) + 4*I
+ 9)*(10*sqrt(-7/200*I + 1/200) - 4*I - 3) - 3/4*(10*sqrt(-7/200*I + 1/200) - 4*I - 3)^2 - 60*sqrt(7/200*I +
1/200) - 24*I - 31) + 3)*log(1/25*(235*x*(10*sqrt(7/200*I + 1/200) + 4*I - 3)^2 + (38*x*(10*sqrt(7/200*I + 1/2
00) + 4*I - 3) + 235*x)*(10*sqrt(-7/200*I + 1/200) - 4*I - 3)^2 + 2795*x*(10*sqrt(7/200*I + 1/200) + 4*I - 3)
+ (38*x*(10*sqrt(7/200*I + 1/200) + 4*I - 3)^2 + 456*x*(10*sqrt(7/200*I + 1/200) + 4*I - 3) + 2795*x)*(10*sqrt
(-7/200*I + 1/200) - 4*I - 3) - 2*(235*x*(10*sqrt(7/200*I + 1/200) + 4*I - 3) + (38*x*(10*sqrt(7/200*I + 1/200
) + 4*I - 3) + 235*x)*(10*sqrt(-7/200*I + 1/200) - 4*I - 3) + 25*x)*sqrt(-3/4*(10*sqrt(7/200*I + 1/200) + 4*I
- 3)^2 - 1/2*(10*sqrt(7/200*I + 1/200) + 4*I + 9)*(10*sqrt(-7/200*I + 1/200) - 4*I - 3) - 3/4*(10*sqrt(-7/200*
I + 1/200) - 4*I - 3)^2 - 60*sqrt(7/200*I + 1/200) - 24*I - 31) + 9000*x + 4175*sqrt(2*x^2 + 1) - 4175)/x) + 1
/2*sqrt(2)*log(sqrt(2)*x - sqrt(2*x^2 + 1)) - 1/5*arctan(x) - 1/5*arctan((x + sqrt(2*x^2 + 1) - 1)/x) + 1/5*ar
ctan(-(x - sqrt(2*x^2 + 1) + 1)/x) + 1/5*log(x^2 + 1) - 2*log(x) - 1/5*log((2*x^2 - sqrt(2*x^2 + 1)*(x + 1) +
x + 1)/x^2) + 1/5*log((2*x^2 + sqrt(2*x^2 + 1)*(x - 1) - x + 1)/x^2)

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giac [B]  time = 1.02, size = 929, normalized size = 2.76

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+(2*x^2+1)^(1/2)+(2*x^2+1)^(5/2))/(x^2-x*(2*x^2+1)^(3/2)),x, algorithm="giac")

[Out]

-x^2 + 1/20*(sqrt(5*sqrt(2) - 1) + 8)*arctan(-1/2*sqrt(2)*(2*sqrt(2)*x + sqrt(2)*(2*sqrt(2) + 3)^(1/4) - 2*sqr
t(2*x^2 + 1))/(2*sqrt(2) + 3)^(1/4)) + 1/20*(sqrt(5*sqrt(2) - 1) - 8)*arctan(-1/2*sqrt(2)*(2*sqrt(2)*x - sqrt(
2)*(2*sqrt(2) + 3)^(1/4) - 2*sqrt(2*x^2 + 1))/(2*sqrt(2) + 3)^(1/4)) + 1/20*(sqrt(5*sqrt(2) - 1) + 8)*arctan(8
*(1/8)^(3/4)*(2*x + (1/8)^(1/4)*sqrt(-sqrt(2) + 2))/sqrt(sqrt(2) + 2)) + 1/20*(sqrt(5*sqrt(2) - 1) - 8)*arctan
(8*(1/8)^(3/4)*(2*x - (1/8)^(1/4)*sqrt(-sqrt(2) + 2))/sqrt(sqrt(2) + 2)) - 1/20*(sqrt(5*sqrt(2) - 1) - 8)*arct
an(-1/2*sqrt(2)*(2*sqrt(2)*x + sqrt(2)*(-2*sqrt(2) + 3)^(1/4) - 2*sqrt(2*x^2 + 1))/(-2*sqrt(2) + 3)^(1/4)) - 1
/20*(sqrt(5*sqrt(2) - 1) + 8)*arctan(-1/2*sqrt(2)*(2*sqrt(2)*x - sqrt(2)*(-2*sqrt(2) + 3)^(1/4) - 2*sqrt(2*x^2
+ 1))/(-2*sqrt(2) + 3)^(1/4)) + 1/40*sqrt(5*sqrt(2) + 1)*log((sqrt(2)*x - sqrt(2*x^2 + 1))^2 + sqrt(2)*(sqrt(
2)*x - sqrt(2*x^2 + 1))*(2*sqrt(2) + 3)^(1/4) + sqrt(2*sqrt(2) + 3)) - 1/40*sqrt(5*sqrt(2) + 1)*log((sqrt(2)*x
- sqrt(2*x^2 + 1))^2 - sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 + 1))*(2*sqrt(2) + 3)^(1/4) + sqrt(2*sqrt(2) + 3)) + 1
/40*sqrt(5*sqrt(2) + 1)*log((sqrt(2)*x - sqrt(2*x^2 + 1))^2 + sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 + 1))*(-2*sqrt(2
) + 3)^(1/4) + sqrt(-2*sqrt(2) + 3)) - 1/40*sqrt(5*sqrt(2) + 1)*log((sqrt(2)*x - sqrt(2*x^2 + 1))^2 - sqrt(2)*
(sqrt(2)*x - sqrt(2*x^2 + 1))*(-2*sqrt(2) + 3)^(1/4) + sqrt(-2*sqrt(2) + 3)) - 1/40*sqrt(5*sqrt(2) + 1)*log(x^
2 + (1/8)^(1/4)*x*sqrt(-sqrt(2) + 2) + 1/2*sqrt(1/2)) + 1/40*sqrt(5*sqrt(2) + 1)*log(x^2 - (1/8)^(1/4)*x*sqrt(
-sqrt(2) + 2) + 1/2*sqrt(1/2)) - (50305164660422142002238655969020*sqrt(2) - 71142246120180725728612927680401)
*log(-sqrt(2)*x + sqrt(2*x^2 + 1))/(71142246120180725728612927680401*sqrt(2) - 1006103293208442840044773119380
40) - 1/5*arctan(x) + 1/5*arctan(-(sqrt(2)*x - sqrt(2*x^2 + 1))/(sqrt(2) + 1)) - 1/5*arctan(-(sqrt(2)*x - sqrt
(2*x^2 + 1))/(sqrt(2) - 1)) - 3/20*log((sqrt(2)*x - sqrt(2*x^2 + 1))^4 + 2*sqrt(2) + 3) + 3/20*log((sqrt(2)*x
- sqrt(2*x^2 + 1))^4 - 2*sqrt(2) + 3) + 3/20*log(8*x^4 + 4*x^2 + 1) + 1/5*log((sqrt(2)*x - sqrt(2*x^2 + 1))^2
+ 2*sqrt(2) + 3) - 1/5*log((sqrt(2)*x - sqrt(2*x^2 + 1))^2 - 2*sqrt(2) + 3) + 1/5*log(x^2 + 1) - 2*log(abs(x))

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maple [B]  time = 0.26, size = 886, normalized size = 2.63 \[\frac {x \left (2 x^{2}+1\right )^{\frac {3}{2}}}{40}-\frac {\sqrt {2 x^{2}+1}\, x}{20}-\frac {\sqrt {2}\, x^{2}}{20}-\frac {\sqrt {2}}{320}-\frac {\sqrt {2}\, x^{4}}{10}-\frac {\sqrt {2}\, x}{10}-\frac {\sqrt {2 x^{2}+1}}{10}-\frac {\arctan \relax (x )}{5}+\frac {\ln \left (x^{2}+1\right )}{5}-2 \ln \relax (x )-x^{2}+\frac {4 \arctan \left (\frac {2 \left (-\sqrt {2}\, x +\sqrt {2 x^{2}+1}\right )^{2}}{-2+2 \sqrt {2}}\right ) \sqrt {2}}{5 \left (-2+2 \sqrt {2}\right )}-\frac {3 \ln \left (\left (-\sqrt {2}\, x +\sqrt {2 x^{2}+1}\right )^{4}+2 \sqrt {2}+3\right )}{20}+\frac {4 \arctan \left (\frac {2 \left (-\sqrt {2}\, x +\sqrt {2 x^{2}+1}\right )^{2}}{2+2 \sqrt {2}}\right ) \sqrt {2}}{5 \left (2+2 \sqrt {2}\right )}+\frac {4 \arctan \left (\frac {2 \left (-\sqrt {2}\, x +\sqrt {2 x^{2}+1}\right )^{2}}{2+2 \sqrt {2}}\right )}{5 \left (2+2 \sqrt {2}\right )}-\frac {4 \arctan \left (\frac {2 \left (-\sqrt {2}\, x +\sqrt {2 x^{2}+1}\right )^{2}}{-2+2 \sqrt {2}}\right )}{5 \left (-2+2 \sqrt {2}\right )}+\frac {3 \ln \left (\left (-\sqrt {2}\, x +\sqrt {2 x^{2}+1}\right )^{4}+3-2 \sqrt {2}\right )}{20}+\frac {\sqrt {2}\, \ln \left (-\sqrt {2}\, x +\sqrt {2 x^{2}+1}\right )}{8}+\frac {3 \ln \left (8 x^{4}+4 x^{2}+1\right )}{20}-\frac {2 \arctan \left (4 x^{2}+1\right )}{5}-\frac {3 \ln \left (\sqrt {2}+2 \sqrt {\sqrt {2}-1}\, \sqrt {2}\, x +4 x^{2}\right ) \sqrt {\sqrt {2}-1}}{40}+\frac {3 \ln \left (-2 \sqrt {\sqrt {2}-1}\, \sqrt {2}\, x +4 x^{2}+\sqrt {2}\right ) \sqrt {\sqrt {2}-1}}{40}-\frac {3 x^{3} \sqrt {2 x^{2}+1}}{20}+\frac {\sqrt {2}}{320 \left (-\sqrt {2}\, x +\sqrt {2 x^{2}+1}\right )^{4}}-\frac {\ln \left (\sqrt {2}+2 \sqrt {\sqrt {2}-1}\, \sqrt {2}\, x +4 x^{2}\right ) \sqrt {2}\, \sqrt {\sqrt {2}-1}}{40}+\frac {\arctan \left (\frac {2 \sqrt {2}\, \sqrt {\sqrt {2}-1}+8 x}{2 \sqrt {2+2 \sqrt {2}}}\right ) \sqrt {2}}{20 \sqrt {2+2 \sqrt {2}}}+\frac {\ln \left (-2 \sqrt {\sqrt {2}-1}\, \sqrt {2}\, x +4 x^{2}+\sqrt {2}\right ) \sqrt {2}\, \sqrt {\sqrt {2}-1}}{40}+\frac {\arctan \left (\frac {-2 \sqrt {2}\, \sqrt {\sqrt {2}-1}+8 x}{2 \sqrt {2+2 \sqrt {2}}}\right ) \sqrt {2}}{20 \sqrt {2+2 \sqrt {2}}}+\frac {\arctan \left (\sqrt {2 x^{2}+1}\right )}{5}+\frac {1}{-10 \sqrt {2}\, x +10 \sqrt {2 x^{2}+1}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (2 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+25\right )}{\sum }\textit {\_R} \ln \left (35 \left (-\sqrt {2}\, x +\sqrt {2 x^{2}+1}\right )^{2}+\left (4 \textit {\_R}^{3}-26 \textit {\_R} \right ) \left (-\sqrt {2}\, x +\sqrt {2 x^{2}+1}\right )+35\right )\right )}{20}+\frac {2 \arctanh \left (\frac {x}{\sqrt {2 x^{2}+1}}\right )}{5}-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (8 \textit {\_Z}^{4}+28 \textit {\_Z}^{2}+25\right )}{\sum }\textit {\_R} \ln \left (5 \left (-\sqrt {2}\, x +\sqrt {2 x^{2}+1}\right )^{2}+\left (-24 \textit {\_R}^{3}-44 \textit {\_R} \right ) \left (-\sqrt {2}\, x +\sqrt {2 x^{2}+1}\right )+5\right )\right )}{10}-\frac {3 \sqrt {2}\, \arcsinh \left (\sqrt {2}\, x \right )}{8}+\frac {\arctan \left (\frac {-2 \sqrt {2}\, \sqrt {\sqrt {2}-1}+8 x}{2 \sqrt {2+2 \sqrt {2}}}\right )}{5 \sqrt {2+2 \sqrt {2}}}+\frac {\arctan \left (\frac {2 \sqrt {2}\, \sqrt {\sqrt {2}-1}+8 x}{2 \sqrt {2+2 \sqrt {2}}}\right )}{5 \sqrt {2+2 \sqrt {2}}}\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^2+(2*x^2+1)^(1/2)+(2*x^2+1)^(5/2))/(x^2-x*(2*x^2+1)^(3/2)),x)

[Out]

1/40*x*(2*x^2+1)^(3/2)-1/20*(2*x^2+1)^(1/2)*x-1/20*2^(1/2)*x^2-1/320*2^(1/2)-1/10*2^(1/2)*x^4-1/10*2^(1/2)*x-1
/10*(2*x^2+1)^(1/2)-1/5*arctan(x)+1/5*ln(x^2+1)-2*ln(x)-x^2+1/20*sum(_R*ln(35*(-2^(1/2)*x+(2*x^2+1)^(1/2))^2+(
4*_R^3-26*_R)*(-2^(1/2)*x+(2*x^2+1)^(1/2))+35),_R=RootOf(2*_Z^4+2*_Z^2+25))+4/5/(-2+2*2^(1/2))*arctan(2*(-2^(1
/2)*x+(2*x^2+1)^(1/2))^2/(-2+2*2^(1/2)))*2^(1/2)-3/20*ln((-2^(1/2)*x+(2*x^2+1)^(1/2))^4+2*2^(1/2)+3)+4/5/(2+2*
2^(1/2))*arctan(2*(-2^(1/2)*x+(2*x^2+1)^(1/2))^2/(2+2*2^(1/2)))*2^(1/2)+4/5/(2+2*2^(1/2))*arctan(2*(-2^(1/2)*x
+(2*x^2+1)^(1/2))^2/(2+2*2^(1/2)))-4/5/(-2+2*2^(1/2))*arctan(2*(-2^(1/2)*x+(2*x^2+1)^(1/2))^2/(-2+2*2^(1/2)))+
2/5*arctanh(x/(2*x^2+1)^(1/2))+3/20*ln((-2^(1/2)*x+(2*x^2+1)^(1/2))^4+3-2*2^(1/2))-3/8*2^(1/2)*arcsinh(2^(1/2)
*x)+1/8*2^(1/2)*ln(-2^(1/2)*x+(2*x^2+1)^(1/2))+3/20*ln(8*x^4+4*x^2+1)-2/5*arctan(4*x^2+1)-3/40*ln(2^(1/2)+2*(2
^(1/2)-1)^(1/2)*2^(1/2)*x+4*x^2)*(2^(1/2)-1)^(1/2)+3/40*ln(-2*(2^(1/2)-1)^(1/2)*2^(1/2)*x+4*x^2+2^(1/2))*(2^(1
/2)-1)^(1/2)-3/20*x^3*(2*x^2+1)^(1/2)+1/320*2^(1/2)/(-2^(1/2)*x+(2*x^2+1)^(1/2))^4-1/40*ln(2^(1/2)+2*(2^(1/2)-
1)^(1/2)*2^(1/2)*x+4*x^2)*2^(1/2)*(2^(1/2)-1)^(1/2)+1/20/(2+2*2^(1/2))^(1/2)*arctan(1/2*(2*2^(1/2)*(2^(1/2)-1)
^(1/2)+8*x)/(2+2*2^(1/2))^(1/2))*2^(1/2)+1/40*ln(-2*(2^(1/2)-1)^(1/2)*2^(1/2)*x+4*x^2+2^(1/2))*2^(1/2)*(2^(1/2
)-1)^(1/2)+1/20/(2+2*2^(1/2))^(1/2)*arctan(1/2*(-2*2^(1/2)*(2^(1/2)-1)^(1/2)+8*x)/(2+2*2^(1/2))^(1/2))*2^(1/2)
+1/10/(-2^(1/2)*x+(2*x^2+1)^(1/2))-1/10*sum(_R*ln(5*(-2^(1/2)*x+(2*x^2+1)^(1/2))^2+(-24*_R^3-44*_R)*(-2^(1/2)*
x+(2*x^2+1)^(1/2))+5),_R=RootOf(8*_Z^4+28*_Z^2+25))+1/5*arctan((2*x^2+1)^(1/2))+1/5/(2+2*2^(1/2))^(1/2)*arctan
(1/2*(-2*2^(1/2)*(2^(1/2)-1)^(1/2)+8*x)/(2+2*2^(1/2))^(1/2))+1/5/(2+2*2^(1/2))^(1/2)*arctan(1/2*(2*2^(1/2)*(2^
(1/2)-1)^(1/2)+8*x)/(2+2*2^(1/2))^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -x^{2} - \int -\frac {256 \, x^{16} - 128 \, x^{15} + 1024 \, x^{14} - 448 \, x^{13} + 1792 \, x^{12} - 640 \, x^{11} + 1824 \, x^{10} - 496 \, x^{9} + 1188 \, x^{8} - 234 \, x^{7} + 508 \, x^{6} - 69 \, x^{5} + 142 \, x^{4} - 12 \, x^{3} + 24 \, x^{2} - x + 2}{384 \, x^{15} + 1344 \, x^{13} + 2176 \, x^{11} + 2000 \, x^{9} + 1086 \, x^{7} + 335 \, x^{5} + 52 \, x^{3} - {\left (256 \, x^{16} + 1024 \, x^{14} + 1984 \, x^{12} + 2272 \, x^{10} + 1636 \, x^{8} + 724 \, x^{6} + 181 \, x^{4} + 22 \, x^{2} + 1\right )} \sqrt {2 \, x^{2} + 1} + 3 \, x}\,{d x} + \frac {2}{3} \, \int \frac {8 \, x^{5} + 14 \, x^{3} - 3 \, x^{2} + 8 \, x}{8 \, x^{6} + 12 \, x^{4} + 9 \, x^{2} + 1}\,{d x} + \frac {1}{6} \, \log \left (2 \, x^{2} + 1\right ) - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+(2*x^2+1)^(1/2)+(2*x^2+1)^(5/2))/(x^2-x*(2*x^2+1)^(3/2)),x, algorithm="maxima")

[Out]

-x^2 - integrate(-(256*x^16 - 128*x^15 + 1024*x^14 - 448*x^13 + 1792*x^12 - 640*x^11 + 1824*x^10 - 496*x^9 + 1
188*x^8 - 234*x^7 + 508*x^6 - 69*x^5 + 142*x^4 - 12*x^3 + 24*x^2 - x + 2)/(384*x^15 + 1344*x^13 + 2176*x^11 +
2000*x^9 + 1086*x^7 + 335*x^5 + 52*x^3 - (256*x^16 + 1024*x^14 + 1984*x^12 + 2272*x^10 + 1636*x^8 + 724*x^6 +
181*x^4 + 22*x^2 + 1)*sqrt(2*x^2 + 1) + 3*x), x) + 2/3*integrate((8*x^5 + 14*x^3 - 3*x^2 + 8*x)/(8*x^6 + 12*x^
4 + 9*x^2 + 1), x) + 1/6*log(2*x^2 + 1) - 2*log(x)

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mupad [B]  time = 2.47, size = 540, normalized size = 1.60

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((2*x^2 + 1)^(1/2) - x^2 + (2*x^2 + 1)^(5/2))/(x*(2*x^2 + 1)^(3/2) - x^2),x)

[Out]

log(x + 1i)*(2/5 - 1i/5) - log(x - (2^(1/2)*(x^2 + 1/2)^(1/2))/2 + 1i/2)*(1/5 - 1i/10) + log(x + (2^(1/2)*(x^2
+ 1/2)^(1/2))/2 - 1i/2)*(1/5 + 1i/10) - 2*log(x) + log(x + (- 1/4 - 1i/4)^(1/2))*((- 1/4 - 1i/4)^(3/2)*(1/5 +
2i/5) + (3/20 - 1i/5)) - (2^(1/2)*asinh(2^(1/2)*x))/2 - log(x - (- 1 + 1i)^(1/2)/2)*((- 1/4 + 1i/4)^(3/2)*(1/
5 - 2i/5) - (3/20 + 1i/5)) + log(x + (- 1 + 1i)^(1/2)/2)*((- 1/4 + 1i/4)^(3/2)*(1/5 - 2i/5) + (3/20 + 1i/5)) -
log(x - (- 1/4 - 1i/4)^(1/2))*((- 1/4 - 1i/4)^(3/2)*(1/5 + 2i/5) - (3/20 - 1i/5)) - x^2 - (2^(1/2)*(log((1/4
- 1i/4)^(1/2)*(x^2 + 1/2)^(1/2) - (- 1/4 - 1i/4)^(1/2)*x + 1/2) - log(x + (- 1/4 - 1i/4)^(1/2)))*((- 1/4 - 1i/
4)^(1/2) + 4*(- 1/4 - 1i/4)^(3/2) + 4*(- 1/4 - 1i/4)^(5/2) + (1/4 - 3i/4)))/(2*(1/4 - 1i/4)^(1/2)*(10*(- 1/4 -
1i/4)^(1/2) + 48*(- 1/4 - 1i/4)^(3/2) + 48*(- 1/4 - 1i/4)^(5/2))) + (2^(1/2)*(log(x - (- 1 + 1i)^(1/2)/2) - l
og((- 1/4 + 1i/4)^(1/2)*x + (1/4 + 1i/4)^(1/2)*(x^2 + 1/2)^(1/2) + 1/2))*((- 1/4 + 1i/4)^(1/2) + 4*(- 1/4 + 1i
/4)^(3/2) + 4*(- 1/4 + 1i/4)^(5/2) - (1/4 + 3i/4)))/(2*(1/4 + 1i/4)^(1/2)*(10*(- 1/4 + 1i/4)^(1/2) + 48*(- 1/4
+ 1i/4)^(3/2) + 48*(- 1/4 + 1i/4)^(5/2))) + (2^(1/2)*(log(x + (- 1 + 1i)^(1/2)/2) - log((1/4 + 1i/4)^(1/2)*(x
^2 + 1/2)^(1/2) - (- 1/4 + 1i/4)^(1/2)*x + 1/2))*((- 1/4 + 1i/4)^(1/2) + 4*(- 1/4 + 1i/4)^(3/2) + 4*(- 1/4 + 1
i/4)^(5/2) + (1/4 + 3i/4)))/(2*(1/4 + 1i/4)^(1/2)*(10*(- 1/4 + 1i/4)^(1/2) + 48*(- 1/4 + 1i/4)^(3/2) + 48*(- 1
/4 + 1i/4)^(5/2))) + (2^(1/2)*(log(x - (- 1/4 - 1i/4)^(1/2)) - log((- 1/4 - 1i/4)^(1/2)*x + (1/4 - 1i/4)^(1/2)
*(x^2 + 1/2)^(1/2) + 1/2))*((- 1/4 - 1i/4)^(1/2) + 4*(- 1/4 - 1i/4)^(3/2) + 4*(- 1/4 - 1i/4)^(5/2) - (1/4 - 3i
/4)))/(2*(1/4 - 1i/4)^(1/2)*(10*(- 1/4 - 1i/4)^(1/2) + 48*(- 1/4 - 1i/4)^(3/2) + 48*(- 1/4 - 1i/4)^(5/2)))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**2+(2*x**2+1)**(1/2)+(2*x**2+1)**(5/2))/(x**2-x*(2*x**2+1)**(3/2)),x)

[Out]

Timed out

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