3.28.9 \(\int \frac {\sqrt {b^2+a^2 x^3} (2 b^2+c x^3+a^2 x^6)}{x^7 (b^2+a^2 x^6)} \, dx\) [2709]

Optimal. Leaf size=247 \[ \frac {\sqrt {b^2+a^2 x^3} \left (-2 b^2-a^2 x^3-2 c x^3\right )}{6 b^2 x^6}+\frac {\sqrt {a-i b} \left ((-1)^{3/4} a^2 b-\sqrt [4]{-1} a c\right ) \text {ArcTan}\left (\frac {\sqrt [4]{-1} \sqrt {b^2+a^2 x^3}}{\sqrt {a-i b} \sqrt {b}}\right )}{3 b^{5/2}}-\frac {\sqrt {a+i b} \left (-\sqrt [4]{-1} a^2 b+(-1)^{3/4} a c\right ) \text {ArcTan}\left (\frac {(-1)^{3/4} \sqrt {b^2+a^2 x^3}}{\sqrt {a+i b} \sqrt {b}}\right )}{3 b^{5/2}}+\frac {\left (a^4+4 a^2 b^2-2 a^2 c\right ) \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{6 b^3} \]

[Out]

1/6*(a^2*x^3+b^2)^(1/2)*(-a^2*x^3-2*c*x^3-2*b^2)/b^2/x^6+1/3*(a-I*b)^(1/2)*((-1)^(3/4)*a^2*b-(-1)^(1/4)*a*c)*a
rctan((-1)^(1/4)*(a^2*x^3+b^2)^(1/2)/(a-I*b)^(1/2)/b^(1/2))/b^(5/2)-1/3*(a+I*b)^(1/2)*(-(-1)^(1/4)*a^2*b+(-1)^
(3/4)*a*c)*arctan((-1)^(3/4)*(a^2*x^3+b^2)^(1/2)/(a+I*b)^(1/2)/b^(1/2))/b^(5/2)+1/6*(a^4+4*a^2*b^2-2*a^2*c)*ar
ctanh((a^2*x^3+b^2)^(1/2)/b)/b^3

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(744\) vs. \(2(247)=494\).
time = 2.53, antiderivative size = 744, normalized size of antiderivative = 3.01, number of steps used = 27, number of rules used = 15, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6857, 272, 43, 44, 65, 214, 52, 6847, 839, 841, 1183, 648, 632, 212, 642} \begin {gather*} -\frac {c \sqrt {a^2 x^3+b^2}}{3 b^2 x^3}-\frac {a^2 \sqrt {a^2 x^3+b^2}}{6 b^2 x^3}+\frac {2 a^2 \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{b}\right )}{3 b}-\frac {\sqrt {a^2 x^3+b^2}}{3 x^6}-\frac {a^2 c \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{b}\right )}{3 b^3}+\frac {a^2 \left (\sqrt {a^2+b^2} \left (b^2-c\right )+a^2 b+b^3\right ) \log \left (-\sqrt {2} \sqrt {b} \sqrt {\sqrt {a^2+b^2}+b} \sqrt {a^2 x^3+b^2}+b \left (\sqrt {a^2+b^2}+b\right )+a^2 x^3\right )}{6 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+b}}-\frac {a^2 \left (\sqrt {a^2+b^2} \left (b^2-c\right )+a^2 b+b^3\right ) \log \left (\sqrt {2} \sqrt {b} \sqrt {\sqrt {a^2+b^2}+b} \sqrt {a^2 x^3+b^2}+b \left (\sqrt {a^2+b^2}+b\right )+a^2 x^3\right )}{6 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+b}}-\frac {a^2 \left (-\sqrt {a^2+b^2} \left (b^2-c\right )+a^2 b+b^3\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {\sqrt {a^2+b^2}+b}-\sqrt {2} \sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {b-\sqrt {a^2+b^2}}}\right )}{3 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {b-\sqrt {a^2+b^2}}}+\frac {a^2 \left (-\sqrt {a^2+b^2} \left (b^2-c\right )+a^2 b+b^3\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a^2 x^3+b^2}+\sqrt {b} \sqrt {\sqrt {a^2+b^2}+b}}{\sqrt {b} \sqrt {b-\sqrt {a^2+b^2}}}\right )}{3 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {b-\sqrt {a^2+b^2}}}+\frac {a^4 \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{b}\right )}{6 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sqrt[b^2 + a^2*x^3]*(2*b^2 + c*x^3 + a^2*x^6))/(x^7*(b^2 + a^2*x^6)),x]

[Out]

-1/3*Sqrt[b^2 + a^2*x^3]/x^6 - (a^2*Sqrt[b^2 + a^2*x^3])/(6*b^2*x^3) - (c*Sqrt[b^2 + a^2*x^3])/(3*b^2*x^3) + (
a^4*ArcTanh[Sqrt[b^2 + a^2*x^3]/b])/(6*b^3) + (2*a^2*ArcTanh[Sqrt[b^2 + a^2*x^3]/b])/(3*b) - (a^2*c*ArcTanh[Sq
rt[b^2 + a^2*x^3]/b])/(3*b^3) - (a^2*(a^2*b + b^3 - Sqrt[a^2 + b^2]*(b^2 - c))*ArcTanh[(Sqrt[b]*Sqrt[b + Sqrt[
a^2 + b^2]] - Sqrt[2]*Sqrt[b^2 + a^2*x^3])/(Sqrt[b]*Sqrt[b - Sqrt[a^2 + b^2]])])/(3*Sqrt[2]*b^(5/2)*Sqrt[a^2 +
b^2]*Sqrt[b - Sqrt[a^2 + b^2]]) + (a^2*(a^2*b + b^3 - Sqrt[a^2 + b^2]*(b^2 - c))*ArcTanh[(Sqrt[b]*Sqrt[b + Sq
rt[a^2 + b^2]] + Sqrt[2]*Sqrt[b^2 + a^2*x^3])/(Sqrt[b]*Sqrt[b - Sqrt[a^2 + b^2]])])/(3*Sqrt[2]*b^(5/2)*Sqrt[a^
2 + b^2]*Sqrt[b - Sqrt[a^2 + b^2]]) + (a^2*(a^2*b + b^3 + Sqrt[a^2 + b^2]*(b^2 - c))*Log[b*(b + Sqrt[a^2 + b^2
]) + a^2*x^3 - Sqrt[2]*Sqrt[b]*Sqrt[b + Sqrt[a^2 + b^2]]*Sqrt[b^2 + a^2*x^3]])/(6*Sqrt[2]*b^(5/2)*Sqrt[a^2 + b
^2]*Sqrt[b + Sqrt[a^2 + b^2]]) - (a^2*(a^2*b + b^3 + Sqrt[a^2 + b^2]*(b^2 - c))*Log[b*(b + Sqrt[a^2 + b^2]) +
a^2*x^3 + Sqrt[2]*Sqrt[b]*Sqrt[b + Sqrt[a^2 + b^2]]*Sqrt[b^2 + a^2*x^3]])/(6*Sqrt[2]*b^(5/2)*Sqrt[a^2 + b^2]*S
qrt[b + Sqrt[a^2 + b^2]])

Rule 43

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

Rule 44

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 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 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 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 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 839

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (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 + c*e*f)*x, x]/(a + c*x^2)), x], x] /
; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && FractionQ[m] && GtQ[m, 0]

Rule 841

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2, Subst[Int[(e*f
- d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 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 6847

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\sqrt {b^2+a^2 x^3} \left (2 b^2+c x^3+a^2 x^6\right )}{x^7 \left (b^2+a^2 x^6\right )} \, dx &=\int \left (\frac {2 \sqrt {b^2+a^2 x^3}}{x^7}+\frac {c \sqrt {b^2+a^2 x^3}}{b^2 x^4}-\frac {a^2 \sqrt {b^2+a^2 x^3}}{b^2 x}+\frac {a^2 x^2 \sqrt {b^2+a^2 x^3} \left (-c+a^2 x^3\right )}{b^2 \left (b^2+a^2 x^6\right )}\right ) \, dx\\ &=2 \int \frac {\sqrt {b^2+a^2 x^3}}{x^7} \, dx-\frac {a^2 \int \frac {\sqrt {b^2+a^2 x^3}}{x} \, dx}{b^2}+\frac {a^2 \int \frac {x^2 \sqrt {b^2+a^2 x^3} \left (-c+a^2 x^3\right )}{b^2+a^2 x^6} \, dx}{b^2}+\frac {c \int \frac {\sqrt {b^2+a^2 x^3}}{x^4} \, dx}{b^2}\\ &=\frac {2}{3} \text {Subst}\left (\int \frac {\sqrt {b^2+a^2 x}}{x^3} \, dx,x,x^3\right )-\frac {a^2 \text {Subst}\left (\int \frac {\sqrt {b^2+a^2 x}}{x} \, dx,x,x^3\right )}{3 b^2}+\frac {a^2 \text {Subst}\left (\int \frac {\sqrt {b^2+a^2 x} \left (-c+a^2 x\right )}{b^2+a^2 x^2} \, dx,x,x^3\right )}{3 b^2}+\frac {c \text {Subst}\left (\int \frac {\sqrt {b^2+a^2 x}}{x^2} \, dx,x,x^3\right )}{3 b^2}\\ &=-\frac {\sqrt {b^2+a^2 x^3}}{3 x^6}-\frac {c \sqrt {b^2+a^2 x^3}}{3 b^2 x^3}+\frac {1}{6} a^2 \text {Subst}\left (\int \frac {1}{x^2 \sqrt {b^2+a^2 x}} \, dx,x,x^3\right )-\frac {1}{3} a^2 \text {Subst}\left (\int \frac {1}{x \sqrt {b^2+a^2 x}} \, dx,x,x^3\right )+\frac {\text {Subst}\left (\int \frac {-a^2 b^2 \left (a^2+c\right )+a^4 \left (b^2-c\right ) x}{\sqrt {b^2+a^2 x} \left (b^2+a^2 x^2\right )} \, dx,x,x^3\right )}{3 b^2}+\frac {\left (a^2 c\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {b^2+a^2 x}} \, dx,x,x^3\right )}{6 b^2}\\ &=-\frac {\sqrt {b^2+a^2 x^3}}{3 x^6}-\frac {a^2 \sqrt {b^2+a^2 x^3}}{6 b^2 x^3}-\frac {c \sqrt {b^2+a^2 x^3}}{3 b^2 x^3}-\frac {2}{3} \text {Subst}\left (\int \frac {1}{-\frac {b^2}{a^2}+\frac {x^2}{a^2}} \, dx,x,\sqrt {b^2+a^2 x^3}\right )+\frac {2 \text {Subst}\left (\int \frac {-a^4 b^2 \left (b^2-c\right )-a^4 b^2 \left (a^2+c\right )+a^4 \left (b^2-c\right ) x^2}{a^4 b^2+a^2 b^4-2 a^2 b^2 x^2+a^2 x^4} \, dx,x,\sqrt {b^2+a^2 x^3}\right )}{3 b^2}-\frac {a^4 \text {Subst}\left (\int \frac {1}{x \sqrt {b^2+a^2 x}} \, dx,x,x^3\right )}{12 b^2}+\frac {c \text {Subst}\left (\int \frac {1}{-\frac {b^2}{a^2}+\frac {x^2}{a^2}} \, dx,x,\sqrt {b^2+a^2 x^3}\right )}{3 b^2}\\ &=-\frac {\sqrt {b^2+a^2 x^3}}{3 x^6}-\frac {a^2 \sqrt {b^2+a^2 x^3}}{6 b^2 x^3}-\frac {c \sqrt {b^2+a^2 x^3}}{3 b^2 x^3}+\frac {2 a^2 \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{3 b}-\frac {a^2 c \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{3 b^3}-\frac {a^2 \text {Subst}\left (\int \frac {1}{-\frac {b^2}{a^2}+\frac {x^2}{a^2}} \, dx,x,\sqrt {b^2+a^2 x^3}\right )}{6 b^2}+\frac {\text {Subst}\left (\int \frac {\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}} \left (-a^4 b^2 \left (b^2-c\right )-a^4 b^2 \left (a^2+c\right )\right )-\left (-a^4 b^2 \left (b^2-c\right )-a^4 b \sqrt {a^2+b^2} \left (b^2-c\right )-a^4 b^2 \left (a^2+c\right )\right ) x}{b \sqrt {a^2+b^2}-\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {b^2+a^2 x^3}\right )}{3 \sqrt {2} a^2 b^{7/2} \sqrt {a^2+b^2} \sqrt {b+\sqrt {a^2+b^2}}}+\frac {\text {Subst}\left (\int \frac {\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}} \left (-a^4 b^2 \left (b^2-c\right )-a^4 b^2 \left (a^2+c\right )\right )+\left (-a^4 b^2 \left (b^2-c\right )-a^4 b \sqrt {a^2+b^2} \left (b^2-c\right )-a^4 b^2 \left (a^2+c\right )\right ) x}{b \sqrt {a^2+b^2}+\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {b^2+a^2 x^3}\right )}{3 \sqrt {2} a^2 b^{7/2} \sqrt {a^2+b^2} \sqrt {b+\sqrt {a^2+b^2}}}\\ &=-\frac {\sqrt {b^2+a^2 x^3}}{3 x^6}-\frac {a^2 \sqrt {b^2+a^2 x^3}}{6 b^2 x^3}-\frac {c \sqrt {b^2+a^2 x^3}}{3 b^2 x^3}+\frac {a^4 \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{6 b^3}+\frac {2 a^2 \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{3 b}-\frac {a^2 c \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{3 b^3}-\frac {\left (a^2 \left (a^2 b+b^3-\sqrt {a^2+b^2} \left (b^2-c\right )\right )\right ) \text {Subst}\left (\int \frac {1}{b \sqrt {a^2+b^2}-\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {b^2+a^2 x^3}\right )}{6 b^2 \sqrt {a^2+b^2}}-\frac {\left (a^2 \left (a^2 b+b^3-\sqrt {a^2+b^2} \left (b^2-c\right )\right )\right ) \text {Subst}\left (\int \frac {1}{b \sqrt {a^2+b^2}+\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {b^2+a^2 x^3}\right )}{6 b^2 \sqrt {a^2+b^2}}+\frac {\left (a^2 \left (a^2 b+b^3+\sqrt {a^2+b^2} \left (b^2-c\right )\right )\right ) \text {Subst}\left (\int \frac {-\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}}+2 x}{b \sqrt {a^2+b^2}-\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {b^2+a^2 x^3}\right )}{6 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {b+\sqrt {a^2+b^2}}}-\frac {\left (a^2 \left (a^2 b+b^3+\sqrt {a^2+b^2} \left (b^2-c\right )\right )\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}}+2 x}{b \sqrt {a^2+b^2}+\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {b^2+a^2 x^3}\right )}{6 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {b+\sqrt {a^2+b^2}}}\\ &=-\frac {\sqrt {b^2+a^2 x^3}}{3 x^6}-\frac {a^2 \sqrt {b^2+a^2 x^3}}{6 b^2 x^3}-\frac {c \sqrt {b^2+a^2 x^3}}{3 b^2 x^3}+\frac {a^4 \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{6 b^3}+\frac {2 a^2 \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{3 b}-\frac {a^2 c \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{3 b^3}+\frac {a^2 \left (a^2 b+b^3+\sqrt {a^2+b^2} \left (b^2-c\right )\right ) \log \left (b \left (b+\sqrt {a^2+b^2}\right )+a^2 x^3-\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}} \sqrt {b^2+a^2 x^3}\right )}{6 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {b+\sqrt {a^2+b^2}}}-\frac {a^2 \left (a^2 b+b^3+\sqrt {a^2+b^2} \left (b^2-c\right )\right ) \log \left (b \left (b+\sqrt {a^2+b^2}\right )+a^2 x^3+\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}} \sqrt {b^2+a^2 x^3}\right )}{6 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {b+\sqrt {a^2+b^2}}}+\frac {\left (a^2 \left (a^2 b+b^3-\sqrt {a^2+b^2} \left (b^2-c\right )\right )\right ) \text {Subst}\left (\int \frac {1}{2 b \left (b-\sqrt {a^2+b^2}\right )-x^2} \, dx,x,-\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}}+2 \sqrt {b^2+a^2 x^3}\right )}{3 b^2 \sqrt {a^2+b^2}}+\frac {\left (a^2 \left (a^2 b+b^3-\sqrt {a^2+b^2} \left (b^2-c\right )\right )\right ) \text {Subst}\left (\int \frac {1}{2 b \left (b-\sqrt {a^2+b^2}\right )-x^2} \, dx,x,\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}}+2 \sqrt {b^2+a^2 x^3}\right )}{3 b^2 \sqrt {a^2+b^2}}\\ &=-\frac {\sqrt {b^2+a^2 x^3}}{3 x^6}-\frac {a^2 \sqrt {b^2+a^2 x^3}}{6 b^2 x^3}-\frac {c \sqrt {b^2+a^2 x^3}}{3 b^2 x^3}+\frac {a^4 \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{6 b^3}+\frac {2 a^2 \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{3 b}-\frac {a^2 c \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{3 b^3}-\frac {a^2 \left (a^2 b+b^3-\sqrt {a^2+b^2} \left (b^2-c\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {b+\sqrt {a^2+b^2}}-\sqrt {2} \sqrt {b^2+a^2 x^3}}{\sqrt {b} \sqrt {b-\sqrt {a^2+b^2}}}\right )}{3 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {b-\sqrt {a^2+b^2}}}+\frac {a^2 \left (a^2 b+b^3-\sqrt {a^2+b^2} \left (b^2-c\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {b+\sqrt {a^2+b^2}}+\sqrt {2} \sqrt {b^2+a^2 x^3}}{\sqrt {b} \sqrt {b-\sqrt {a^2+b^2}}}\right )}{3 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {b-\sqrt {a^2+b^2}}}+\frac {a^2 \left (a^2 b+b^3+\sqrt {a^2+b^2} \left (b^2-c\right )\right ) \log \left (b \left (b+\sqrt {a^2+b^2}\right )+a^2 x^3-\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}} \sqrt {b^2+a^2 x^3}\right )}{6 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {b+\sqrt {a^2+b^2}}}-\frac {a^2 \left (a^2 b+b^3+\sqrt {a^2+b^2} \left (b^2-c\right )\right ) \log \left (b \left (b+\sqrt {a^2+b^2}\right )+a^2 x^3+\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}} \sqrt {b^2+a^2 x^3}\right )}{6 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {b+\sqrt {a^2+b^2}}}\\ \end {align*}

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Mathematica [A]
time = 0.80, size = 224, normalized size = 0.91 \begin {gather*} \frac {-\frac {b \sqrt {b^2+a^2 x^3} \left (2 b^2+\left (a^2+2 c\right ) x^3\right )}{x^6}+2 (-1)^{3/4} a \sqrt {a-i b} \sqrt {b} (a b+i c) \text {ArcTan}\left (\frac {\sqrt [4]{-1} \sqrt {b^2+a^2 x^3}}{\sqrt {a-i b} \sqrt {b}}\right )+2 \sqrt [4]{-1} a \sqrt {a+i b} \sqrt {b} (a b-i c) \text {ArcTan}\left (\frac {(-1)^{3/4} \sqrt {b^2+a^2 x^3}}{\sqrt {a+i b} \sqrt {b}}\right )+a^2 \left (a^2+4 b^2-2 c\right ) \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{6 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[b^2 + a^2*x^3]*(2*b^2 + c*x^3 + a^2*x^6))/(x^7*(b^2 + a^2*x^6)),x]

[Out]

(-((b*Sqrt[b^2 + a^2*x^3]*(2*b^2 + (a^2 + 2*c)*x^3))/x^6) + 2*(-1)^(3/4)*a*Sqrt[a - I*b]*Sqrt[b]*(a*b + I*c)*A
rcTan[((-1)^(1/4)*Sqrt[b^2 + a^2*x^3])/(Sqrt[a - I*b]*Sqrt[b])] + 2*(-1)^(1/4)*a*Sqrt[a + I*b]*Sqrt[b]*(a*b -
I*c)*ArcTan[((-1)^(3/4)*Sqrt[b^2 + a^2*x^3])/(Sqrt[a + I*b]*Sqrt[b])] + a^2*(a^2 + 4*b^2 - 2*c)*ArcTanh[Sqrt[b
^2 + a^2*x^3]/b])/(6*b^3)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.33, size = 864, normalized size = 3.50

method	result	size
risch	\(-\frac {\sqrt {a^{2} x^{3}+b^{2}}\, \left (a^{2} x^{3}+2 c \,x^{3}+2 b^{2}\right )}{6 b^{2} x^{6}}-\frac {a^{2} \left (-\frac {2 i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{6} a^{2}+b^{2}\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{3} a^{2} b^{2}+\underline {\hspace {1.25 ex}}\alpha ^{3} a^{2} c +a^{2} b^{2}+b^{2} c \right ) \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i a \left (2 x +\frac {-i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}+\left (-a \,b^{2}\right )^{\frac {1}{3}}}{a}\right )}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {a \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{a}\right )}{-3 \left (-a \,b^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i a \left (2 x +\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}+\left (-a \,b^{2}\right )^{\frac {1}{3}}}{a}\right )}{2 \left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \left (2 a^{2} \left (a^{2} \underline {\hspace {1.25 ex}}\alpha ^{5}-b^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}\right )+i a^{3} \left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{4} \sqrt {3}-i a^{2} \underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {2}{3}}-a^{3} \left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{4}-a^{2} \underline {\hspace {1.25 ex}}\alpha ^{3} \left (-a \,b^{2}\right )^{\frac {2}{3}}-i \left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, b^{2} a +i \left (-a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}\, b^{2}+\left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \,b^{2} a +\left (-a \,b^{2}\right )^{\frac {2}{3}} b^{2}\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 a}\right ) \sqrt {3}\, a}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \frac {2 i \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{5} a^{3}-i \left (-a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{4} a^{2}+i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{3} a^{2} b^{2}-3 \left (-a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{4} a^{2}-2 i \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a \,b^{2}+i \left (-a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha \,b^{2}-3 \underline {\hspace {1.25 ex}}\alpha ^{3} a^{2} b^{2}-i \sqrt {3}\, b^{4}+3 \left (-a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha \,b^{2}+3 b^{4}}{2 b^{2} \left (a^{2}+b^{2}\right )}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{a \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 a}\right )}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha ^{3} \left (a^{2}+b^{2}\right ) \sqrt {a^{2} x^{3}+b^{2}}}\right )}{3 a^{3} b^{2}}-\frac {2 \left (a^{2}+4 b^{2}-2 c \right ) \arctanh \left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {b^{2}}}\right )}{3 \sqrt {b^{2}}}\right )}{4 b^{2}}\)	\(746\)
default	\(\text {Expression too large to display}\)	\(864\)
elliptic	\(\text {Expression too large to display}\)	\(12546\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2*x^3+b^2)^(1/2)*(a^2*x^6+c*x^3+2*b^2)/x^7/(a^2*x^6+b^2),x,method=_RETURNVERBOSE)

[Out]

-1/3*(a^2*x^3+b^2)^(1/2)/x^6-1/6*a^2/b^2*(a^2*x^3+b^2)^(1/2)/x^3+1/6/b^2*a^4*arctanh((a^2*x^3+b^2)^(1/2)/(b^2)
^(1/2))/(b^2)^(1/2)-a^2/b^2*(2/3*(a^2*x^3+b^2)^(1/2)-2/3*b^2*arctanh((a^2*x^3+b^2)^(1/2)/(b^2)^(1/2))/(b^2)^(1
/2))+a^2/b^2*(2/3*(a^2*x^3+b^2)^(1/2)+1/6*I/a^3/b^2*2^(1/2)*sum((-_alpha^3*a^2*b^2+_alpha^3*a^2*c+a^2*b^2+b^2*
c)/_alpha^3/(a^2+b^2)*(-a*b^2)^(1/3)*(1/2*I*a*(2*x+1/a*(-I*3^(1/2)*(-a*b^2)^(1/3)+(-a*b^2)^(1/3)))/(-a*b^2)^(1
/3))^(1/2)*(a*(x-1/a*(-a*b^2)^(1/3))/(-3*(-a*b^2)^(1/3)+I*3^(1/2)*(-a*b^2)^(1/3)))^(1/2)*(-1/2*I*a*(2*x+1/a*(I
*3^(1/2)*(-a*b^2)^(1/3)+(-a*b^2)^(1/3)))/(-a*b^2)^(1/3))^(1/2)/(a^2*x^3+b^2)^(1/2)*(2*a^2*(_alpha^5*a^2-_alpha
^2*b^2)+I*a^3*(-a*b^2)^(1/3)*_alpha^4*3^(1/2)-I*a^2*_alpha^3*3^(1/2)*(-a*b^2)^(2/3)-a^3*(-a*b^2)^(1/3)*_alpha^
4-a^2*_alpha^3*(-a*b^2)^(2/3)-I*(-a*b^2)^(1/3)*_alpha*3^(1/2)*b^2*a+I*(-a*b^2)^(2/3)*3^(1/2)*b^2+(-a*b^2)^(1/3
)*_alpha*b^2*a+(-a*b^2)^(2/3)*b^2)*EllipticPi(1/3*3^(1/2)*(I*(x+1/2/a*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/a*(-a*b^2)^
(1/3))*3^(1/2)*a/(-a*b^2)^(1/3))^(1/2),1/2*(2*I*(-a*b^2)^(1/3)*3^(1/2)*_alpha^5*a^3-I*(-a*b^2)^(2/3)*3^(1/2)*_
alpha^4*a^2+I*3^(1/2)*_alpha^3*a^2*b^2-3*(-a*b^2)^(2/3)*_alpha^4*a^2-2*I*(-a*b^2)^(1/3)*3^(1/2)*_alpha^2*a*b^2
+I*(-a*b^2)^(2/3)*3^(1/2)*_alpha*b^2-3*_alpha^3*a^2*b^2-I*3^(1/2)*b^4+3*(-a*b^2)^(2/3)*_alpha*b^2+3*b^4)/b^2/(
a^2+b^2),(I*3^(1/2)/a*(-a*b^2)^(1/3)/(-3/2/a*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/a*(-a*b^2)^(1/3)))^(1/2)),_alpha=Roo
tOf(_Z^6*a^2+b^2)))+c/b^2*(-1/3*(a^2*x^3+b^2)^(1/2)/x^3-1/3*a^2*arctanh((a^2*x^3+b^2)^(1/2)/(b^2)^(1/2))/(b^2)
^(1/2))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*x^3+b^2)^(1/2)*(a^2*x^6+c*x^3+2*b^2)/x^7/(a^2*x^6+b^2),x, algorithm="maxima")

[Out]

integrate((a^2*x^6 + c*x^3 + 2*b^2)*sqrt(a^2*x^3 + b^2)/((a^2*x^6 + b^2)*x^7), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 10278 vs. \(2 (194) = 388\).
time = 4.83, size = 10278, normalized size = 41.61 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*x^3+b^2)^(1/2)*(a^2*x^6+c*x^3+2*b^2)/x^7/(a^2*x^6+b^2),x, algorithm="fricas")

[Out]

-1/12*(4*sqrt(2)*b^13*x^6*sqrt((a^8*b^4 + a^6*b^6 + (a^4 + a^2*b^2)*c^4 + 2*(a^6*b^2 + a^4*b^4)*c^2 - (a^2*b^8
- 2*a^2*b^6*c - b^6*c^2)*sqrt((a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/b^10))/(
a^8*b^4 + 4*a^6*b^4*c - 4*a^4*b^2*c^3 + a^4*c^4 - 2*(a^6*b^2 - 2*a^4*b^4)*c^2))*((a^10*b^4 + a^8*b^6 + (a^6 +
a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/b^10)^(3/4)*sqrt((a^10*b^4 + 4*a^8*b^4*c - 4*a^6*b^2*c^3 + a^6*c^4 -
2*(a^8*b^2 - 2*a^6*b^4)*c^2)/b^10)*arctan((sqrt(2)*sqrt(a^2*x^3 + b^2)*((a^8*b^22 + 2*a^6*b^22*c + 2*a^4*b^20
*c^3 - a^4*b^18*c^4)*sqrt((a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/b^10)*sqrt((a
^10*b^4 + 4*a^8*b^4*c - 4*a^6*b^2*c^3 + a^6*c^4 - 2*(a^8*b^2 - 2*a^6*b^4)*c^2)/b^10) + (a^12*b^20 - a^10*b^18*
c^2 - 5*a^8*b^16*c^4 - 3*a^6*b^14*c^6 + a^6*b^12*c^7 + (a^8*b^14 + 2*a^6*b^16)*c^5 - (a^10*b^16 - 4*a^8*b^18)*
c^3 - (a^12*b^18 - 2*a^10*b^20)*c)*sqrt((a^10*b^4 + 4*a^8*b^4*c - 4*a^6*b^2*c^3 + a^6*c^4 - 2*(a^8*b^2 - 2*a^6
*b^4)*c^2)/b^10))*sqrt((a^8*b^4 + a^6*b^6 + (a^4 + a^2*b^2)*c^4 + 2*(a^6*b^2 + a^4*b^4)*c^2 - (a^2*b^8 - 2*a^2
*b^6*c - b^6*c^2)*sqrt((a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/b^10))/(a^8*b^4
+ 4*a^6*b^4*c - 4*a^4*b^2*c^3 + a^4*c^4 - 2*(a^6*b^2 - 2*a^4*b^4)*c^2))*((a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)
*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/b^10)^(3/4) + sqrt(2)*(b^18*sqrt((a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 +
2*(a^8*b^2 + a^6*b^4)*c^2)/b^10)*sqrt((a^10*b^4 + 4*a^8*b^4*c - 4*a^6*b^2*c^3 + a^6*c^4 - 2*(a^8*b^2 - 2*a^6*
b^4)*c^2)/b^10) + (a^4*b^16 - a^4*b^14*c + a^2*b^14*c^2 - a^2*b^12*c^3)*sqrt((a^10*b^4 + 4*a^8*b^4*c - 4*a^6*b
^2*c^3 + a^6*c^4 - 2*(a^8*b^2 - 2*a^6*b^4)*c^2)/b^10))*sqrt((a^8*b^4 + a^6*b^6 + (a^4 + a^2*b^2)*c^4 + 2*(a^6*
b^2 + a^4*b^4)*c^2 - (a^2*b^8 - 2*a^2*b^6*c - b^6*c^2)*sqrt((a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8
*b^2 + a^6*b^4)*c^2)/b^10))/(a^8*b^4 + 4*a^6*b^4*c - 4*a^4*b^2*c^3 + a^4*c^4 - 2*(a^6*b^2 - 2*a^4*b^4)*c^2))*s
qrt((a^18*b^10 + a^16*b^12 + (a^10*b^2 + a^8*b^4)*c^8 - 4*(a^10*b^4 + a^8*b^6)*c^7 + 4*(a^10*b^6 + a^8*b^8)*c^
6 - 4*(a^12*b^6 + a^10*b^8)*c^5 - 2*(a^14*b^6 - 3*a^12*b^8 - 4*a^10*b^10)*c^4 + 4*(a^14*b^8 + a^12*b^10)*c^3 +
(a^20*b^8 + a^18*b^10 + (a^12 + a^10*b^2)*c^8 - 4*(a^12*b^2 + a^10*b^4)*c^7 + 4*(a^12*b^4 + a^10*b^6)*c^6 - 4
*(a^14*b^4 + a^12*b^6)*c^5 - 2*(a^16*b^4 - 3*a^14*b^6 - 4*a^12*b^8)*c^4 + 4*(a^16*b^6 + a^14*b^8)*c^3 + 4*(a^1
6*b^8 + a^14*b^10)*c^2 + 4*(a^18*b^8 + a^16*b^10)*c)*x^3 + 4*(a^14*b^10 + a^12*b^12)*c^2 + sqrt(2)*(a^16*b^10
+ a^14*b^12 + (a^10*b^4 + a^8*b^6)*c^6 - 4*(a^10*b^6 + a^8*b^8)*c^5 - (a^12*b^6 - 3*a^10*b^8 - 4*a^8*b^10)*c^4
- (a^14*b^8 - 3*a^12*b^10 - 4*a^10*b^12)*c^2 + 4*(a^14*b^10 + a^12*b^12)*c + (a^10*b^14 + 5*a^6*b^10*c^4 - a^
6*b^8*c^5 + 2*(a^8*b^10 - 4*a^6*b^12)*c^3 - 2*(3*a^8*b^12 - 2*a^6*b^14)*c^2 - (a^10*b^12 - 4*a^8*b^14)*c)*sqrt
((a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/b^10))*sqrt(a^2*x^3 + b^2)*sqrt((a^8*b
^4 + a^6*b^6 + (a^4 + a^2*b^2)*c^4 + 2*(a^6*b^2 + a^4*b^4)*c^2 - (a^2*b^8 - 2*a^2*b^6*c - b^6*c^2)*sqrt((a^10*
b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/b^10))/(a^8*b^4 + 4*a^6*b^4*c - 4*a^4*b^2*c^3
+ a^4*c^4 - 2*(a^6*b^2 - 2*a^4*b^4)*c^2))*((a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*
c^2)/b^10)^(1/4) + 4*(a^16*b^10 + a^14*b^12)*c + (a^14*b^12 + a^12*b^14 + (a^8*b^6 + a^6*b^8)*c^6 - 4*(a^8*b^8
+ a^6*b^10)*c^5 - (a^10*b^8 - 3*a^8*b^10 - 4*a^6*b^12)*c^4 - (a^12*b^10 - 3*a^10*b^12 - 4*a^8*b^14)*c^2 + 4*(
a^12*b^12 + a^10*b^14)*c)*sqrt((a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/b^10))/(
a^2 + b^2))*((a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/b^10)^(3/4) + (a^16*b^18 +
a^14*b^20 - (a^8*b^10 + a^6*b^12)*c^8 + 2*(a^8*b^12 + a^6*b^14)*c^7 - 2*(a^10*b^12 + a^8*b^14)*c^6 + 6*(a^10*
b^14 + a^8*b^16)*c^5 + 6*(a^12*b^16 + a^10*b^18)*c^3 + 2*(a^14*b^16 + a^12*b^18)*c^2 + 2*(a^14*b^18 + a^12*b^2
0)*c)*sqrt((a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/b^10)*sqrt((a^10*b^4 + 4*a^8
*b^4*c - 4*a^6*b^2*c^3 + a^6*c^4 - 2*(a^8*b^2 - 2*a^6*b^4)*c^2)/b^10) + (a^20*b^16 + a^18*b^18 - (a^10*b^6 + a
^8*b^8)*c^10 + 2*(a^10*b^8 + a^8*b^10)*c^9 - 3*(a^12*b^8 + a^10*b^10)*c^8 + 8*(a^12*b^10 + a^10*b^12)*c^7 - 2*
(a^14*b^10 + a^12*b^12)*c^6 + 12*(a^14*b^12 + a^12*b^14)*c^5 + 2*(a^16*b^12 + a^14*b^14)*c^4 + 8*(a^16*b^14 +
a^14*b^16)*c^3 + 3*(a^18*b^14 + a^16*b^16)*c^2 + 2*(a^18*b^16 + a^16*b^18)*c)*sqrt((a^10*b^4 + 4*a^8*b^4*c - 4
*a^6*b^2*c^3 + a^6*c^4 - 2*(a^8*b^2 - 2*a^6*b^4)*c^2)/b^10))/(a^26*b^12 + a^24*b^14 + (a^14 + a^12*b^2)*c^12 -
4*(a^14*b^2 + a^12*b^4)*c^11 + 2*(a^16*b^2 + 3*a^14*b^4 + 2*a^12*b^6)*c^10 - 12*(a^16*b^4 + a^14*b^6)*c^9 - (
a^18*b^4 - 15*a^16*b^6 - 16*a^14*b^8)*c^8 - 8*(a^18*b^6 + a^16*b^8)*c^7 - 4*(a^20*b^6 - 5*a^18*b^8 - 6*a^16*b^
10)*c^6 + 8*(a^20*b^8 + a^18*b^10)*c^5 - (a^22*b^8 - 15*a^20*b^10 - 16*a^18*b^12)*c^4 + 12*(a^22*b^10 + a^20*b
^12)*c^3 + 2*(a^24*b^10 + 3*a^22*b^12 + 2*a^20*...

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Sympy [F(-1)] Timed out
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((a**2*x**3+b**2)**(1/2)*(a**2*x**6+c*x**3+2*b**2)/x**7/(a**2*x**6+b**2),x)

[Out]

Timed out

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*x^3+b^2)^(1/2)*(a^2*x^6+c*x^3+2*b^2)/x^7/(a^2*x^6+b^2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(c
onst gen &

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Mupad [B]
time = 14.50, size = 274, normalized size = 1.11 \begin {gather*} \frac {a^2\,\ln \left (\frac {{\left (b+\sqrt {a^2\,x^3+b^2}\right )}^3\,\left (b-\sqrt {a^2\,x^3+b^2}\right )}{x^6}\right )\,\left (a^2+4\,b^2-2\,c\right )}{12\,b^3}-\frac {\sqrt {a^2\,x^3+b^2}\,\left (a^2+2\,c\right )}{6\,b^2\,x^3}-\frac {\sqrt {a^2\,x^3+b^2}}{3\,x^6}+\frac {a\,\ln \left (\frac {2\,b^2-a\,b\,1{}\mathrm {i}+a^2\,x^3+\sqrt {b}\,\sqrt {a^2\,x^3+b^2}\,\sqrt {-b+a\,1{}\mathrm {i}}\,2{}\mathrm {i}}{a\,x^3+b\,1{}\mathrm {i}}\right )\,\left (-a\,b+c\,1{}\mathrm {i}\right )\,\sqrt {-b+a\,1{}\mathrm {i}}\,1{}\mathrm {i}}{6\,b^{5/2}}+\frac {a\,\ln \left (\frac {a\,b\,1{}\mathrm {i}+2\,b^2+a^2\,x^3-2\,\sqrt {b}\,\sqrt {a^2\,x^3+b^2}\,\sqrt {b+a\,1{}\mathrm {i}}}{-a\,x^3+b\,1{}\mathrm {i}}\right )\,\left (a\,b+c\,1{}\mathrm {i}\right )\,\sqrt {b+a\,1{}\mathrm {i}}}{6\,b^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b^2 + a^2*x^3)^(1/2)*(c*x^3 + 2*b^2 + a^2*x^6))/(x^7*(b^2 + a^2*x^6)),x)

[Out]

(a^2*log(((b + (b^2 + a^2*x^3)^(1/2))^3*(b - (b^2 + a^2*x^3)^(1/2)))/x^6)*(a^2 - 2*c + 4*b^2))/(12*b^3) - ((b^
2 + a^2*x^3)^(1/2)*(2*c + a^2))/(6*b^2*x^3) - (b^2 + a^2*x^3)^(1/2)/(3*x^6) + (a*log((2*b^2 - a*b*1i + a^2*x^3
+ b^(1/2)*(b^2 + a^2*x^3)^(1/2)*(a*1i - b)^(1/2)*2i)/(b*1i + a*x^3))*(c*1i - a*b)*(a*1i - b)^(1/2)*1i)/(6*b^(
5/2)) + (a*log((a*b*1i + 2*b^2 + a^2*x^3 - 2*b^(1/2)*(b^2 + a^2*x^3)^(1/2)*(a*1i + b)^(1/2))/(b*1i - a*x^3))*(
c*1i + a*b)*(a*1i + b)^(1/2))/(6*b^(5/2))

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