3.101 \(\int \frac{x}{(a^2+2 a b x^3+b^2 x^6)^{3/2}} \, dx\)

Optimal. Leaf size=277 \[ \frac{x^2}{6 a \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{2 x^2}{9 a^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{2 \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{7/3} b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{2 \left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{7/3} b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

[Out]

(2*x^2)/(9*a^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + x^2/(6*a*(a + b*x^3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - (2*(
a + b*x^3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(7/3)*b^(2/3)*Sqrt[a^2 + 2*a*b*x^3
+ b^2*x^6]) - (2*(a + b*x^3)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(7/3)*b^(2/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) +
((a + b*x^3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(27*a^(7/3)*b^(2/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^
6])

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Rubi [A]  time = 0.14049, antiderivative size = 277, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1355, 290, 292, 31, 634, 617, 204, 628} \[ \frac{x^2}{6 a \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{2 x^2}{9 a^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{2 \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{7/3} b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{2 \left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{7/3} b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^2)/(9*a^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + x^2/(6*a*(a + b*x^3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - (2*(
a + b*x^3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(7/3)*b^(2/3)*Sqrt[a^2 + 2*a*b*x^3
+ b^2*x^6]) - (2*(a + b*x^3)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(7/3)*b^(2/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) +
((a + b*x^3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(27*a^(7/3)*b^(2/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^
6])

Rule 1355

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^
(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^n)^(2*p), x], x] /; Fr
eeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rule 290

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(
a*c*n*(p + 1)), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[
{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 292

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> -Dist[(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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 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 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 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]

Rubi steps

\begin{align*} \int \frac{x}{\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx &=\frac{\left (b^2 \left (a b+b^2 x^3\right )\right ) \int \frac{x}{\left (a b+b^2 x^3\right )^3} \, dx}{\sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{x^2}{6 a \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (2 b \left (a b+b^2 x^3\right )\right ) \int \frac{x}{\left (a b+b^2 x^3\right )^2} \, dx}{3 a \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{2 x^2}{9 a^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{x^2}{6 a \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (2 \left (a b+b^2 x^3\right )\right ) \int \frac{x}{a b+b^2 x^3} \, dx}{9 a^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{2 x^2}{9 a^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{x^2}{6 a \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (2 \left (a b+b^2 x^3\right )\right ) \int \frac{1}{\sqrt [3]{a} \sqrt [3]{b}+b^{2/3} x} \, dx}{27 a^{7/3} b \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (2 \left (a b+b^2 x^3\right )\right ) \int \frac{\sqrt [3]{a} \sqrt [3]{b}+b^{2/3} x}{a^{2/3} b^{2/3}-\sqrt [3]{a} b x+b^{4/3} x^2} \, dx}{27 a^{7/3} b \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{2 x^2}{9 a^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{x^2}{6 a \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{2 \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (a b+b^2 x^3\right ) \int \frac{-\sqrt [3]{a} b+2 b^{4/3} x}{a^{2/3} b^{2/3}-\sqrt [3]{a} b x+b^{4/3} x^2} \, dx}{27 a^{7/3} b^{5/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (a b+b^2 x^3\right ) \int \frac{1}{a^{2/3} b^{2/3}-\sqrt [3]{a} b x+b^{4/3} x^2} \, dx}{9 a^2 b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{2 x^2}{9 a^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{x^2}{6 a \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{2 \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{7/3} b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (2 \left (a b+b^2 x^3\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{9 a^{7/3} b^{5/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{2 x^2}{9 a^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{x^2}{6 a \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{2 \left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{7/3} b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{2 \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{7/3} b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ \end{align*}

Mathematica [A]  time = 0.0711672, size = 237, normalized size = 0.86 \[ \frac{21 a^{4/3} b^{2/3} x^2+2 b^2 x^6 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+4 a b x^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 a^2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+12 \sqrt [3]{a} b^{5/3} x^5-4 \left (a+b x^3\right )^2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-4 \sqrt{3} \left (a+b x^3\right )^2 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{54 a^{7/3} b^{2/3} \left (a+b x^3\right ) \sqrt{\left (a+b x^3\right )^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(21*a^(4/3)*b^(2/3)*x^2 + 12*a^(1/3)*b^(5/3)*x^5 - 4*Sqrt[3]*(a + b*x^3)^2*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/
Sqrt[3]] - 4*(a + b*x^3)^2*Log[a^(1/3) + b^(1/3)*x] + 2*a^2*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + 4
*a*b*x^3*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + 2*b^2*x^6*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*
x^2])/(54*a^(7/3)*b^(2/3)*(a + b*x^3)*Sqrt[(a + b*x^3)^2])

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Maple [A]  time = 0.009, size = 301, normalized size = 1.1 \begin{align*}{\frac{b{x}^{3}+a}{54\,b{a}^{2}} \left ( -4\,\sqrt{3}\arctan \left ( 1/3\,{\sqrt{3} \left ( -2\,x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \right ){x}^{6}{b}^{2}-4\,\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){x}^{6}{b}^{2}+2\,\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{2/3} \right ){x}^{6}{b}^{2}+12\,\sqrt [3]{{\frac{a}{b}}}{x}^{5}{b}^{2}-8\,\sqrt{3}\arctan \left ( 1/3\,{\sqrt{3} \left ( -2\,x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \right ){x}^{3}ab-8\,\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){x}^{3}ab+4\,\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{2/3} \right ){x}^{3}ab+21\,\sqrt [3]{{\frac{a}{b}}}{x}^{2}ab-4\,\sqrt{3}\arctan \left ( 1/3\,{\sqrt{3} \left ( -2\,x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \right ){a}^{2}-4\,\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){a}^{2}+2\,\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{2/3} \right ){a}^{2} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/54*(-4*3^(1/2)*arctan(1/3*3^(1/2)*(-2*x+(a/b)^(1/3))/(a/b)^(1/3))*x^6*b^2-4*ln(x+(a/b)^(1/3))*x^6*b^2+2*ln(x
^2-(a/b)^(1/3)*x+(a/b)^(2/3))*x^6*b^2+12*(a/b)^(1/3)*x^5*b^2-8*3^(1/2)*arctan(1/3*3^(1/2)*(-2*x+(a/b)^(1/3))/(
a/b)^(1/3))*x^3*a*b-8*ln(x+(a/b)^(1/3))*x^3*a*b+4*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))*x^3*a*b+21*(a/b)^(1/3)*x^2
*a*b-4*3^(1/2)*arctan(1/3*3^(1/2)*(-2*x+(a/b)^(1/3))/(a/b)^(1/3))*a^2-4*ln(x+(a/b)^(1/3))*a^2+2*ln(x^2-(a/b)^(
1/3)*x+(a/b)^(2/3))*a^2)*(b*x^3+a)/(a/b)^(1/3)/b/a^2/((b*x^3+a)^2)^(3/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.90036, size = 1164, normalized size = 4.2 \begin{align*} \left [\frac{12 \, a b^{3} x^{5} + 21 \, a^{2} b^{2} x^{2} + 6 \, \sqrt{\frac{1}{3}}{\left (a b^{3} x^{6} + 2 \, a^{2} b^{2} x^{3} + a^{3} b\right )} \sqrt{\frac{\left (-a b^{2}\right )^{\frac{1}{3}}}{a}} \log \left (\frac{2 \, b^{2} x^{3} - a b + 3 \, \sqrt{\frac{1}{3}}{\left (a b x + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} x^{2} + \left (-a b^{2}\right )^{\frac{1}{3}} a\right )} \sqrt{\frac{\left (-a b^{2}\right )^{\frac{1}{3}}}{a}} - 3 \, \left (-a b^{2}\right )^{\frac{2}{3}} x}{b x^{3} + a}\right ) + 2 \,{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (-a b^{2}\right )^{\frac{2}{3}} \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac{1}{3}} b x + \left (-a b^{2}\right )^{\frac{2}{3}}\right ) - 4 \,{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (-a b^{2}\right )^{\frac{2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac{1}{3}}\right )}{54 \,{\left (a^{3} b^{4} x^{6} + 2 \, a^{4} b^{3} x^{3} + a^{5} b^{2}\right )}}, \frac{12 \, a b^{3} x^{5} + 21 \, a^{2} b^{2} x^{2} + 12 \, \sqrt{\frac{1}{3}}{\left (a b^{3} x^{6} + 2 \, a^{2} b^{2} x^{3} + a^{3} b\right )} \sqrt{-\frac{\left (-a b^{2}\right )^{\frac{1}{3}}}{a}} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (2 \, b x + \left (-a b^{2}\right )^{\frac{1}{3}}\right )} \sqrt{-\frac{\left (-a b^{2}\right )^{\frac{1}{3}}}{a}}}{b}\right ) + 2 \,{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (-a b^{2}\right )^{\frac{2}{3}} \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac{1}{3}} b x + \left (-a b^{2}\right )^{\frac{2}{3}}\right ) - 4 \,{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (-a b^{2}\right )^{\frac{2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac{1}{3}}\right )}{54 \,{\left (a^{3} b^{4} x^{6} + 2 \, a^{4} b^{3} x^{3} + a^{5} b^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/54*(12*a*b^3*x^5 + 21*a^2*b^2*x^2 + 6*sqrt(1/3)*(a*b^3*x^6 + 2*a^2*b^2*x^3 + a^3*b)*sqrt((-a*b^2)^(1/3)/a)*
log((2*b^2*x^3 - a*b + 3*sqrt(1/3)*(a*b*x + 2*(-a*b^2)^(2/3)*x^2 + (-a*b^2)^(1/3)*a)*sqrt((-a*b^2)^(1/3)/a) -
3*(-a*b^2)^(2/3)*x)/(b*x^3 + a)) + 2*(b^2*x^6 + 2*a*b*x^3 + a^2)*(-a*b^2)^(2/3)*log(b^2*x^2 + (-a*b^2)^(1/3)*b
*x + (-a*b^2)^(2/3)) - 4*(b^2*x^6 + 2*a*b*x^3 + a^2)*(-a*b^2)^(2/3)*log(b*x - (-a*b^2)^(1/3)))/(a^3*b^4*x^6 +
2*a^4*b^3*x^3 + a^5*b^2), 1/54*(12*a*b^3*x^5 + 21*a^2*b^2*x^2 + 12*sqrt(1/3)*(a*b^3*x^6 + 2*a^2*b^2*x^3 + a^3*
b)*sqrt(-(-a*b^2)^(1/3)/a)*arctan(sqrt(1/3)*(2*b*x + (-a*b^2)^(1/3))*sqrt(-(-a*b^2)^(1/3)/a)/b) + 2*(b^2*x^6 +
2*a*b*x^3 + a^2)*(-a*b^2)^(2/3)*log(b^2*x^2 + (-a*b^2)^(1/3)*b*x + (-a*b^2)^(2/3)) - 4*(b^2*x^6 + 2*a*b*x^3 +
a^2)*(-a*b^2)^(2/3)*log(b*x - (-a*b^2)^(1/3)))/(a^3*b^4*x^6 + 2*a^4*b^3*x^3 + a^5*b^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b**2*x**6+2*a*b*x**3+a**2)**(3/2),x)

[Out]

Integral(x/((a + b*x**3)**2)**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x