3.104 \(\int \frac{1}{x^2 (a^2+2 a b x^3+b^2 x^6)^{3/2}} \, dx\)

Optimal. Leaf size=316 \[ -\frac{14 \left (a+b x^3\right )}{9 a^3 x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{7}{18 a^2 x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{6 a x \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )}+\frac{14 \sqrt [3]{b} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{7 \sqrt [3]{b} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{10/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{14 \sqrt [3]{b} \left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{10/3} \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

[Out]

7/(18*a^2*x*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + 1/(6*a*x*(a + b*x^3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - (14*(a
+ b*x^3))/(9*a^3*x*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (14*b^(1/3)*(a + b*x^3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(
Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(10/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (14*b^(1/3)*(a + b*x^3)*Log[a^(1/3)
+ b^(1/3)*x])/(27*a^(10/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - (7*b^(1/3)*(a + b*x^3)*Log[a^(2/3) - a^(1/3)*b^(
1/3)*x + b^(2/3)*x^2])/(27*a^(10/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])

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Rubi [A]  time = 0.160381, antiderivative size = 316, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346, Rules used = {1355, 290, 325, 292, 31, 634, 617, 204, 628} \[ -\frac{14 \left (a+b x^3\right )}{9 a^3 x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{7}{18 a^2 x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{6 a x \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )}+\frac{14 \sqrt [3]{b} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{7 \sqrt [3]{b} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{10/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{14 \sqrt [3]{b} \left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{10/3} \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)),x]

[Out]

7/(18*a^2*x*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + 1/(6*a*x*(a + b*x^3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - (14*(a
+ b*x^3))/(9*a^3*x*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (14*b^(1/3)*(a + b*x^3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(
Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(10/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (14*b^(1/3)*(a + b*x^3)*Log[a^(1/3)
+ b^(1/3)*x])/(27*a^(10/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - (7*b^(1/3)*(a + b*x^3)*Log[a^(2/3) - a^(1/3)*b^(
1/3)*x + b^(2/3)*x^2])/(27*a^(10/3)*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 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -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{1}{x^2 \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{1}{x^2 \left (a b+b^2 x^3\right )^3} \, dx}{\sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{1}{6 a x \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (7 b \left (a b+b^2 x^3\right )\right ) \int \frac{1}{x^2 \left (a b+b^2 x^3\right )^2} \, dx}{6 a \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{7}{18 a^2 x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{6 a x \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (14 \left (a b+b^2 x^3\right )\right ) \int \frac{1}{x^2 \left (a b+b^2 x^3\right )} \, dx}{9 a^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{7}{18 a^2 x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{6 a x \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{14 \left (a+b x^3\right )}{9 a^3 x \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (14 b \left (a b+b^2 x^3\right )\right ) \int \frac{x}{a b+b^2 x^3} \, dx}{9 a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{7}{18 a^2 x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{6 a x \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{14 \left (a+b x^3\right )}{9 a^3 x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (14 \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^{10/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (14 \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^{10/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{7}{18 a^2 x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{6 a x \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{14 \left (a+b x^3\right )}{9 a^3 x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{14 \sqrt [3]{b} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (7 \left (a b+b^2 x^3\right )\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^{10/3} b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (7 \sqrt [3]{b} \left (a b+b^2 x^3\right )\right ) \int \frac{1}{a^{2/3} b^{2/3}-\sqrt [3]{a} b x+b^{4/3} x^2} \, dx}{9 a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{7}{18 a^2 x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{6 a x \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{14 \left (a+b x^3\right )}{9 a^3 x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{14 \sqrt [3]{b} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{7 \sqrt [3]{b} \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^{10/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (14 \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^{10/3} b^{2/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{7}{18 a^2 x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{6 a x \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{14 \left (a+b x^3\right )}{9 a^3 x \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{14 \sqrt [3]{b} \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^{10/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{14 \sqrt [3]{b} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{7 \sqrt [3]{b} \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^{10/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ \end{align*}

Mathematica [A]  time = 0.0845271, size = 260, normalized size = 0.82 \[ \frac{-14 b^{7/3} x^7 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-28 a b^{4/3} x^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-14 a^2 \sqrt [3]{b} x \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-147 a^{4/3} b x^3-54 a^{7/3}-84 \sqrt [3]{a} b^2 x^6+28 \sqrt [3]{b} x \left (a+b x^3\right )^2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+28 \sqrt{3} \sqrt [3]{b} x \left (a+b x^3\right )^2 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{54 a^{10/3} x \left (a+b x^3\right ) \sqrt{\left (a+b x^3\right )^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a^2 + 2*a*b*x^3 + b^2*x^6)^(3/2)),x]

[Out]

(-54*a^(7/3) - 147*a^(4/3)*b*x^3 - 84*a^(1/3)*b^2*x^6 + 28*Sqrt[3]*b^(1/3)*x*(a + b*x^3)^2*ArcTan[(1 - (2*b^(1
/3)*x)/a^(1/3))/Sqrt[3]] + 28*b^(1/3)*x*(a + b*x^3)^2*Log[a^(1/3) + b^(1/3)*x] - 14*a^2*b^(1/3)*x*Log[a^(2/3)
- a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - 28*a*b^(4/3)*x^4*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - 14*b^(7
/3)*x^7*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(10/3)*x*(a + b*x^3)*Sqrt[(a + b*x^3)^2])

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Maple [A]  time = 0.016, size = 316, normalized size = 1. \begin{align*} -{\frac{b{x}^{3}+a}{54\,x{a}^{3}} \left ( -28\,\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}^{7}{b}^{2}-28\,\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){x}^{7}{b}^{2}+14\,\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{2/3} \right ){x}^{7}{b}^{2}+84\,\sqrt [3]{{\frac{a}{b}}}{x}^{6}{b}^{2}-56\,\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}^{4}ab-56\,\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){x}^{4}ab+28\,\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{2/3} \right ){x}^{4}ab+147\,\sqrt [3]{{\frac{a}{b}}}{x}^{3}ab-28\,\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{a}^{2}-28\,\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) x{a}^{2}+14\,\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{2/3} \right ) x{a}^{2}+54\,\sqrt [3]{{\frac{a}{b}}}{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(1/x^2/(b^2*x^6+2*a*b*x^3+a^2)^(3/2),x)

[Out]

-1/54*(-28*3^(1/2)*arctan(1/3*3^(1/2)*(-2*x+(a/b)^(1/3))/(a/b)^(1/3))*x^7*b^2-28*ln(x+(a/b)^(1/3))*x^7*b^2+14*
ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))*x^7*b^2+84*(a/b)^(1/3)*x^6*b^2-56*3^(1/2)*arctan(1/3*3^(1/2)*(-2*x+(a/b)^(1/
3))/(a/b)^(1/3))*x^4*a*b-56*ln(x+(a/b)^(1/3))*x^4*a*b+28*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))*x^4*a*b+147*(a/b)^(
1/3)*x^3*a*b-28*3^(1/2)*arctan(1/3*3^(1/2)*(-2*x+(a/b)^(1/3))/(a/b)^(1/3))*x*a^2-28*ln(x+(a/b)^(1/3))*x*a^2+14
*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))*x*a^2+54*(a/b)^(1/3)*a^2)*(b*x^3+a)/(a/b)^(1/3)/x/a^3/((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(1/x^2/(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 = 1.74604, size = 467, normalized size = 1.48 \begin{align*} -\frac{84 \, b^{2} x^{6} + 147 \, a b x^{3} + 28 \, \sqrt{3}{\left (b^{2} x^{7} + 2 \, a b x^{4} + a^{2} x\right )} \left (\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (\frac{2}{3} \, \sqrt{3} x \left (\frac{b}{a}\right )^{\frac{1}{3}} - \frac{1}{3} \, \sqrt{3}\right ) + 14 \,{\left (b^{2} x^{7} + 2 \, a b x^{4} + a^{2} x\right )} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x^{2} - a x \left (\frac{b}{a}\right )^{\frac{2}{3}} + a \left (\frac{b}{a}\right )^{\frac{1}{3}}\right ) - 28 \,{\left (b^{2} x^{7} + 2 \, a b x^{4} + a^{2} x\right )} \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (b x + a \left (\frac{b}{a}\right )^{\frac{2}{3}}\right ) + 54 \, a^{2}}{54 \,{\left (a^{3} b^{2} x^{7} + 2 \, a^{4} b x^{4} + a^{5} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/54*(84*b^2*x^6 + 147*a*b*x^3 + 28*sqrt(3)*(b^2*x^7 + 2*a*b*x^4 + a^2*x)*(b/a)^(1/3)*arctan(2/3*sqrt(3)*x*(b
/a)^(1/3) - 1/3*sqrt(3)) + 14*(b^2*x^7 + 2*a*b*x^4 + a^2*x)*(b/a)^(1/3)*log(b*x^2 - a*x*(b/a)^(2/3) + a*(b/a)^
(1/3)) - 28*(b^2*x^7 + 2*a*b*x^4 + a^2*x)*(b/a)^(1/3)*log(b*x + a*(b/a)^(2/3)) + 54*a^2)/(a^3*b^2*x^7 + 2*a^4*
b*x^4 + a^5*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \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(1/x**2/(b**2*x**6+2*a*b*x**3+a**2)**(3/2),x)

[Out]

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

[Out]

sage0*x