\(\int \frac {-x^2+\sqrt {1+2 x^2}+(1+2 x^2)^{5/2}}{x^2-x (1+2 x^2)^{3/2}} \, dx\) [2926]

Optimal result

Integrand size = 49, antiderivative size = 337 \[ \int \frac {-x^2+\sqrt {1+2 x^2}+\left (1+2 x^2\right )^{5/2}}{x^2-x \left (1+2 x^2\right )^{3/2}} \, dx=-x^2+\frac {1}{2} \left (2+\sqrt {2}\right ) \log \left (-\sqrt {2} x+\sqrt {1+2 x^2}\right )-2 \log \left (-2 x^2+\sqrt {2} x \sqrt {1+2 x^2}\right )+\frac {1}{2} \text {RootSum}\left [1+3 \text {$\#$1}^2-2 \sqrt {2} \text {$\#$1}^2+3 \text {$\#$1}^4+2 \sqrt {2} \text {$\#$1}^4+\text {$\#$1}^6\&,\frac {-\log \left (-\sqrt {2} x+\sqrt {1+2 x^2}-\text {$\#$1}\right )+4 \log \left (-\sqrt {2} x+\sqrt {1+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+2 \log \left (-\sqrt {2} x+\sqrt {1+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^3+4 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {1+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^3+\log \left (-\sqrt {2} x+\sqrt {1+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^4+2 \log \left (-\sqrt {2} x+\sqrt {1+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^5}{3 \text {$\#$1}-2 \sqrt {2} \text {$\#$1}+6 \text {$\#$1}^3+4 \sqrt {2} \text {$\#$1}^3+3 \text {$\#$1}^5}\&\right ] \]

[Out]

Unintegrable

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.63 (sec) , antiderivative size = 776, normalized size of antiderivative = 2.30, number of steps used = 112, number of rules used = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.755, Rules used = {6874, 399, 221, 385, 212, 14, 455, 45, 1706, 214, 211, 1261, 648, 631, 210, 642, 1265, 787, 12, 427, 542, 537, 272, 715, 814, 209, 52, 65, 1183, 632, 713, 1141, 1175, 1178, 838, 721, 1108} \[ \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=-\frac {\text {arcsinh}\left (\sqrt {2} x\right )}{\sqrt {2}}-\left (\frac {2}{5}-\frac {3 i}{10}\right ) \arctan \left (\frac {(1+i) x}{\sqrt {2 x^2+1}}\right )+\frac {1}{5} \arctan \left (\sqrt {2 x^2+1}\right )-\frac {2}{5} \arctan \left (4 x^2+1\right )+\frac {1}{10} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {1+\sqrt {2}}-2 \sqrt {2 x^2+1}}{\sqrt {\sqrt {2}-1}}\right )-\frac {3 \arctan \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 )} \arctan \left (\frac {2 \sqrt {2 x^2+1}+\sqrt {1+\sqrt {2}}}{\sqrt {\sqrt {2}-1}}\right )+\frac {3 \arctan \left (\frac {2 \sqrt {2 x^2+1}+\sqrt {1+\sqrt {2}}}{\sqrt {\sqrt {2}-1}}\right )}{20 \sqrt {\sqrt {2}-1}}-\frac {\arctan (x)}{5}-\frac {1}{20} \sqrt {5 \sqrt {2}-1} \arctan \left (-2 \sqrt {2 \left (\sqrt {2}-1\right )} x-\sqrt {2}+1\right )+\frac {1}{20} \sqrt {5 \sqrt {2}-1} \arctan \left (2 \sqrt {2 \left (\sqrt {2}-1\right )} x-\sqrt {2}+1\right )+\frac {2}{5} \text {arctanh}\left (\frac {x}{\sqrt {2 x^2+1}}\right )-\left (\frac {3}{10}-\frac {2 i}{5}\right ) \text {arctanh}\left (\frac {(1+i) x}{\sqrt {2 x^2+1}}\right )-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}}}+\frac {3}{20} \log \left (8 x^4+4 x^2+1\right )-2 \log (x) \]

[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 45

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 52

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 65

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 209

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

Rule 210

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

Rule 211

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

Rule 212

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

Rule 214

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

Rule 221

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 272

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 385

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 399

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

Rule 427

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 455

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 537

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 542

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 631

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 632

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 642

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 648

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 713

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 715

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 721

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 787

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 814

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 838

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 1108

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 1141

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 1175

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 1178

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 1183

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 1261

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 1265

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 1706

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 6874

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

Rubi steps \begin{align*} \text {integral}& = \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 {\arctan (x)}{5}-\frac {1}{20} \int \frac {\left (2-2 x^2\right ) \sqrt {1+2 x^2}}{1+x^2} \, dx-\frac {1}{10} \text {Subst}\left (\int \frac {\sqrt {1+2 x}}{1+x} \, dx,x,x^2\right )-\frac {1}{10} \text {Subst}\left (\int \frac {1+2 x}{1+x} \, dx,x,x^2\right )-\frac {1}{10} \text {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} \text {Subst}\left (\int \frac {(1+2 x)^3}{x} \, dx,x,x^2\right )+\frac {3}{5} \text {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} \text {Subst}\left (\int \frac {x \sqrt {1+2 x}}{1+4 x+8 x^2} \, dx,x,x^2\right )+\frac {8}{5} \text {Subst}\left (\int \frac {1+2 x}{1+4 x+8 x^2} \, dx,x,x^2\right )+\frac {8}{5} \text {Subst}\left (\int \frac {(1+2 x)^3}{1+4 x+8 x^2} \, dx,x,x^2\right )+\frac {24}{5} \text {Subst}\left (\int \frac {x (1+2 x)}{1+4 x+8 x^2} \, dx,x,x^2\right )+\frac {24}{5} \text {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} \text {arcsinh}\left (\sqrt {2} x\right )-\frac {\arctan (x)}{5}-\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} \text {Subst}\left (\int \left (2+\frac {1}{-1-x}\right ) \, dx,x,x^2\right )+\frac {1}{10} \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {1+2 x}} \, dx,x,x^2\right )+\frac {1}{10} \text {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} \text {Subst}\left (\int \left (2+\frac {1}{-1-x}+4 x+8 x^2\right ) \, dx,x,x^2\right )+\frac {1}{5} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {1+2 x^2}}\right )+\frac {1}{5} \text {Subst}\left (\int \frac {4+16 x}{1+4 x+8 x^2} \, dx,x,x^2\right )-\frac {1}{2} \text {Subst}\left (\int \left (6+\frac {1}{x}+12 x+8 x^2\right ) \, dx,x,x^2\right )+\frac {3}{5} \text {Subst}\left (\int -\frac {2}{1+4 x+8 x^2} \, dx,x,x^2\right )+\frac {4}{5} \text {Subst}\left (\int \frac {1}{1+4 x+8 x^2} \, dx,x,x^2\right )+\frac {8}{5} \text {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} \text {Subst}\left (\int \frac {x^2}{4-8 x^2+8 x^4} \, dx,x,\sqrt {1+2 x^2}\right )+\frac {24}{5} \text {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} \text {arcsinh}\left (\sqrt {2} x\right )-\frac {\arctan (x)}{5}+\frac {1}{5} \text {arctanh}\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} \text {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} \text {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} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+4 x^2\right )-\frac {3}{5} \text {Subst}\left (\int \frac {1+4 x}{1+4 x+8 x^2} \, dx,x,x^2\right )-\frac {6}{5} \text {Subst}\left (\int \frac {1}{1+4 x+8 x^2} \, dx,x,x^2\right )-\frac {6}{5} \text {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} \text {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} \text {Subst}\left (\int \frac {x}{1+4 x+8 x^2} \, dx,x,x^2\right )-\frac {1}{40} \left (4+\sqrt {2}\right ) \text {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 ) \text {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 {\text {arcsinh}\left (\sqrt {2} x\right )}{20 \sqrt {2}}-\frac {1}{5} \sqrt {2} \text {arcsinh}\left (\sqrt {2} x\right )-\frac {\arctan (x)}{5}-\frac {1}{20} \sqrt {-1+5 \sqrt {2}} \arctan \left (1-\sqrt {2}-2 \sqrt {2 \left (-1+\sqrt {2}\right )} x\right )+\frac {1}{20} \sqrt {-1+5 \sqrt {2}} \arctan \left (1-\sqrt {2}+2 \sqrt {2 \left (-1+\sqrt {2}\right )} x\right )+\frac {1}{5} \arctan \left (\sqrt {1+2 x^2}\right )+\frac {2}{5} \arctan \left (1+4 x^2\right )+\frac {1}{5} \text {arctanh}\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 ) \text {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 ) \text {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} \text {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} \text {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} \text {Subst}\left (\int \frac {4+16 x}{1+4 x+8 x^2} \, dx,x,x^2\right )+\frac {1}{5} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {1+2 x^2}}\right )-\frac {2}{5} \text {Subst}\left (\int \frac {1}{1+4 x+8 x^2} \, dx,x,x^2\right )+\frac {3}{5} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+4 x^2\right )-\frac {4}{5} \text {Subst}\left (\int \frac {1}{4-8 x^2+8 x^4} \, dx,x,\sqrt {1+2 x^2}\right )+\frac {3 \text {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 \text {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 {\text {arcsinh}\left (\sqrt {2} x\right )}{20 \sqrt {2}}-\frac {1}{5} \sqrt {2} \text {arcsinh}\left (\sqrt {2} x\right )-\frac {\arctan (x)}{5}-\frac {1}{20} \sqrt {-1+5 \sqrt {2}} \arctan \left (1-\sqrt {2}-2 \sqrt {2 \left (-1+\sqrt {2}\right )} x\right )+\frac {1}{20} \sqrt {-1+5 \sqrt {2}} \arctan \left (1-\sqrt {2}+2 \sqrt {2 \left (-1+\sqrt {2}\right )} x\right )-\left (\frac {1}{5}-\frac {2 i}{5}\right ) \arctan \left (\frac {(1+i) x}{\sqrt {1+2 x^2}}\right )+\frac {1}{5} \arctan \left (\sqrt {1+2 x^2}\right )-\frac {1}{5} \arctan \left (1+4 x^2\right )+\frac {2}{5} \text {arctanh}\left (\frac {x}{\sqrt {1+2 x^2}}\right )-\left (\frac {2}{5}-\frac {i}{5}\right ) \text {arctanh}\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} \text {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} \text {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} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+4 x^2\right )-\frac {\text {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 {\text {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 \text {arcsinh}\left (\sqrt {2} x\right )}{5 \sqrt {2}}-\frac {1}{5} \sqrt {2} \text {arcsinh}\left (\sqrt {2} x\right )-\frac {\arctan (x)}{5}-\frac {1}{20} \sqrt {-1+5 \sqrt {2}} \arctan \left (1-\sqrt {2}-2 \sqrt {2 \left (-1+\sqrt {2}\right )} x\right )+\frac {1}{20} \sqrt {-1+5 \sqrt {2}} \arctan \left (1-\sqrt {2}+2 \sqrt {2 \left (-1+\sqrt {2}\right )} x\right )-\left (\frac {1}{5}-\frac {2 i}{5}\right ) \arctan \left (\frac {(1+i) x}{\sqrt {1+2 x^2}}\right )+\frac {1}{5} \arctan \left (\sqrt {1+2 x^2}\right )-\frac {2}{5} \arctan \left (1+4 x^2\right )-\frac {3 \arctan \left (\frac {\sqrt {1+\sqrt {2}}-2 \sqrt {1+2 x^2}}{\sqrt {-1+\sqrt {2}}}\right )}{20 \sqrt {-1+\sqrt {2}}}+\frac {3 \arctan \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} \text {arctanh}\left (\frac {x}{\sqrt {1+2 x^2}}\right )-\left (\frac {2}{5}-\frac {i}{5}\right ) \text {arctanh}\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 ) \text {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 ) \text {Subst}\left (\int \frac {1}{(4+4 i)+(8-8 i) x^2} \, dx,x,\frac {x}{\sqrt {1+2 x^2}}\right )-\frac {\text {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 {\text {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 {\text {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 {\text {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 \text {arcsinh}\left (\sqrt {2} x\right )}{5 \sqrt {2}}-\frac {1}{5} \sqrt {2} \text {arcsinh}\left (\sqrt {2} x\right )-\frac {\arctan (x)}{5}-\frac {1}{20} \sqrt {-1+5 \sqrt {2}} \arctan \left (1-\sqrt {2}-2 \sqrt {2 \left (-1+\sqrt {2}\right )} x\right )+\frac {1}{20} \sqrt {-1+5 \sqrt {2}} \arctan \left (1-\sqrt {2}+2 \sqrt {2 \left (-1+\sqrt {2}\right )} x\right )-\left (\frac {2}{5}-\frac {3 i}{10}\right ) \arctan \left (\frac {(1+i) x}{\sqrt {1+2 x^2}}\right )+\frac {1}{5} \arctan \left (\sqrt {1+2 x^2}\right )-\frac {2}{5} \arctan \left (1+4 x^2\right )-\frac {3 \arctan \left (\frac {\sqrt {1+\sqrt {2}}-2 \sqrt {1+2 x^2}}{\sqrt {-1+\sqrt {2}}}\right )}{20 \sqrt {-1+\sqrt {2}}}+\frac {3 \arctan \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} \text {arctanh}\left (\frac {x}{\sqrt {1+2 x^2}}\right )-\left (\frac {3}{10}-\frac {2 i}{5}\right ) \text {arctanh}\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 {\text {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 {\text {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 \text {arcsinh}\left (\sqrt {2} x\right )}{5 \sqrt {2}}-\frac {1}{5} \sqrt {2} \text {arcsinh}\left (\sqrt {2} x\right )-\frac {\arctan (x)}{5}-\frac {1}{20} \sqrt {-1+5 \sqrt {2}} \arctan \left (1-\sqrt {2}-2 \sqrt {2 \left (-1+\sqrt {2}\right )} x\right )+\frac {1}{20} \sqrt {-1+5 \sqrt {2}} \arctan \left (1-\sqrt {2}+2 \sqrt {2 \left (-1+\sqrt {2}\right )} x\right )-\left (\frac {2}{5}-\frac {3 i}{10}\right ) \arctan \left (\frac {(1+i) x}{\sqrt {1+2 x^2}}\right )+\frac {1}{5} \arctan \left (\sqrt {1+2 x^2}\right )-\frac {2}{5} \arctan \left (1+4 x^2\right )-\frac {3 \arctan \left (\frac {\sqrt {1+\sqrt {2}}-2 \sqrt {1+2 x^2}}{\sqrt {-1+\sqrt {2}}}\right )}{20 \sqrt {-1+\sqrt {2}}}+\frac {\arctan \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 \arctan \left (\frac {\sqrt {1+\sqrt {2}}+2 \sqrt {1+2 x^2}}{\sqrt {-1+\sqrt {2}}}\right )}{20 \sqrt {-1+\sqrt {2}}}-\frac {\arctan \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} \text {arctanh}\left (\frac {x}{\sqrt {1+2 x^2}}\right )-\left (\frac {3}{10}-\frac {2 i}{5}\right ) \text {arctanh}\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*}

Mathematica [A] (verified)

Time = 0.79 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.97 \[ \int \frac {-x^2+\sqrt {1+2 x^2}+\left (1+2 x^2\right )^{5/2}}{x^2-x \left (1+2 x^2\right )^{3/2}} \, dx=-x^2+\left (1+\frac {1}{\sqrt {2}}\right ) \log \left (-\sqrt {2} x+\sqrt {1+2 x^2}\right )-2 \log \left (x \left (-2 x+\sqrt {2+4 x^2}\right )\right )+\frac {1}{2} \text {RootSum}\left [1+3 \text {$\#$1}^2-2 \sqrt {2} \text {$\#$1}^2+3 \text {$\#$1}^4+2 \sqrt {2} \text {$\#$1}^4+\text {$\#$1}^6\&,\frac {-\log \left (-\sqrt {2} x+\sqrt {1+2 x^2}-\text {$\#$1}\right )+4 \log \left (-\sqrt {2} x+\sqrt {1+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}+2 \log \left (-\sqrt {2} x+\sqrt {1+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^3+4 \sqrt {2} \log \left (-\sqrt {2} x+\sqrt {1+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^3+\log \left (-\sqrt {2} x+\sqrt {1+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^4+2 \log \left (-\sqrt {2} x+\sqrt {1+2 x^2}-\text {$\#$1}\right ) \text {$\#$1}^5}{3 \text {$\#$1}-2 \sqrt {2} \text {$\#$1}+6 \text {$\#$1}^3+4 \sqrt {2} \text {$\#$1}^3+3 \text {$\#$1}^5}\&\right ] \]

[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^2 + (1 + 1/Sqrt[2])*Log[-(Sqrt[2]*x) + Sqrt[1 + 2*x^2]] - 2*Log[x*(-2*x + Sqrt[2 + 4*x^2])] + RootSum[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[-(S
qrt[2]*x) + Sqrt[1 + 2*x^2] - #1]*#1^3 + Log[-(Sqrt[2]*x) + Sqrt[1 + 2*x^2] - #1]*#1^4 + 2*Log[-(Sqrt[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

Maple [F(-1)]

Timed out.

hanged

[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]

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)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 1.08 (sec) , antiderivative size = 2052, normalized size of antiderivative = 6.09 \[ \int \frac {-x^2+\sqrt {1+2 x^2}+\left (1+2 x^2\right )^{5/2}}{x^2-x \left (1+2 x^2\right )^{3/2}} \, dx=\text {Too large to display} \]

[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)

Sympy [N/A]

Not integrable

Time = 102.66 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.53 \[ \int \frac {-x^2+\sqrt {1+2 x^2}+\left (1+2 x^2\right )^{5/2}}{x^2-x \left (1+2 x^2\right )^{3/2}} \, dx=- \int \left (- \frac {x^{2}}{2 x^{3} \sqrt {2 x^{2} + 1} - x^{2} + x \sqrt {2 x^{2} + 1}}\right )\, dx - \int \frac {2 \sqrt {2 x^{2} + 1}}{2 x^{3} \sqrt {2 x^{2} + 1} - x^{2} + x \sqrt {2 x^{2} + 1}}\, dx - \int \frac {4 x^{2} \sqrt {2 x^{2} + 1}}{2 x^{3} \sqrt {2 x^{2} + 1} - x^{2} + x \sqrt {2 x^{2} + 1}}\, dx - \int \frac {4 x^{4} \sqrt {2 x^{2} + 1}}{2 x^{3} \sqrt {2 x^{2} + 1} - x^{2} + x \sqrt {2 x^{2} + 1}}\, dx \]

[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]

-Integral(-x**2/(2*x**3*sqrt(2*x**2 + 1) - x**2 + x*sqrt(2*x**2 + 1)), x) - Integral(2*sqrt(2*x**2 + 1)/(2*x**
3*sqrt(2*x**2 + 1) - x**2 + x*sqrt(2*x**2 + 1)), x) - Integral(4*x**2*sqrt(2*x**2 + 1)/(2*x**3*sqrt(2*x**2 + 1
) - x**2 + x*sqrt(2*x**2 + 1)), x) - Integral(4*x**4*sqrt(2*x**2 + 1)/(2*x**3*sqrt(2*x**2 + 1) - x**2 + x*sqrt
(2*x**2 + 1)), x)

Maxima [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.72 \[ \int \frac {-x^2+\sqrt {1+2 x^2}+\left (1+2 x^2\right )^{5/2}}{x^2-x \left (1+2 x^2\right )^{3/2}} \, dx=\int { -\frac {{\left (2 \, x^{2} + 1\right )}^{\frac {5}{2}} - x^{2} + \sqrt {2 \, x^{2} + 1}}{{\left (2 \, x^{2} + 1\right )}^{\frac {3}{2}} x - x^{2}} \,d x } \]

[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)

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 1.02 (sec) , antiderivative size = 929, normalized size of antiderivative = 2.76 \[ \int \frac {-x^2+\sqrt {1+2 x^2}+\left (1+2 x^2\right )^{5/2}}{x^2-x \left (1+2 x^2\right )^{3/2}} \, dx=\text {Too large to display} \]

[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))

Mupad [B] (verification not implemented)

Time = 7.76 (sec) , antiderivative size = 540, normalized size of antiderivative = 1.60 \[ \int \frac {-x^2+\sqrt {1+2 x^2}+\left (1+2 x^2\right )^{5/2}}{x^2-x \left (1+2 x^2\right )^{3/2}} \, dx=\text {Too large to display} \]

[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)))