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

Optimal. Leaf size=316 \[ -\frac{10 \left (a+b x^3\right )}{9 a^3 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{4}{9 a^2 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{6 a x^2 \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )}-\frac{20 b^{2/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{10 b^{2/3} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{11/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{20 b^{2/3} \left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{11/3} \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

[Out]

4/(9*a^2*x^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + 1/(6*a*x^2*(a + b*x^3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - (10*
(a + b*x^3))/(9*a^3*x^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (20*b^(2/3)*(a + b*x^3)*ArcTan[(a^(1/3) - 2*b^(1/3)
*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(11/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - (20*b^(2/3)*(a + b*x^3)*Log[a^(
1/3) + b^(1/3)*x])/(27*a^(11/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (10*b^(2/3)*(a + b*x^3)*Log[a^(2/3) - a^(1/
3)*b^(1/3)*x + b^(2/3)*x^2])/(27*a^(11/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])

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Rubi [A]  time = 0.158661, 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, 200, 31, 634, 617, 204, 628} \[ -\frac{10 \left (a+b x^3\right )}{9 a^3 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{4}{9 a^2 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{6 a x^2 \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )}-\frac{20 b^{2/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{10 b^{2/3} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{11/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{20 b^{2/3} \left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{11/3} \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

4/(9*a^2*x^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + 1/(6*a*x^2*(a + b*x^3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - (10*
(a + b*x^3))/(9*a^3*x^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (20*b^(2/3)*(a + b*x^3)*ArcTan[(a^(1/3) - 2*b^(1/3)
*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(11/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - (20*b^(2/3)*(a + b*x^3)*Log[a^(
1/3) + b^(1/3)*x])/(27*a^(11/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (10*b^(2/3)*(a + b*x^3)*Log[a^(2/3) - a^(1/
3)*b^(1/3)*x + b^(2/3)*x^2])/(27*a^(11/3)*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 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*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^3 \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^3 \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^2 \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (4 b \left (a b+b^2 x^3\right )\right ) \int \frac{1}{x^3 \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{4}{9 a^2 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{6 a x^2 \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (20 \left (a b+b^2 x^3\right )\right ) \int \frac{1}{x^3 \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{4}{9 a^2 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{6 a x^2 \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{10 \left (a+b x^3\right )}{9 a^3 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (20 b \left (a b+b^2 x^3\right )\right ) \int \frac{1}{a b+b^2 x^3} \, dx}{9 a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{4}{9 a^2 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{6 a x^2 \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{10 \left (a+b x^3\right )}{9 a^3 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (20 \sqrt [3]{b} \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^{11/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (20 \sqrt [3]{b} \left (a b+b^2 x^3\right )\right ) \int \frac{2 \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^{11/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{4}{9 a^2 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{6 a x^2 \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{10 \left (a+b x^3\right )}{9 a^3 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{20 b^{2/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (10 \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^{11/3} \sqrt [3]{b} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (10 b^{2/3} \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^{10/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{4}{9 a^2 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{6 a x^2 \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{10 \left (a+b x^3\right )}{9 a^3 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{20 b^{2/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{10 b^{2/3} \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^{11/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (20 \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^{11/3} \sqrt [3]{b} \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{4}{9 a^2 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{6 a x^2 \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{10 \left (a+b x^3\right )}{9 a^3 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{20 b^{2/3} \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^{11/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{20 b^{2/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{10 b^{2/3} \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^{11/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ \end{align*}

Mathematica [A]  time = 0.0877373, size = 266, normalized size = 0.84 \[ \frac{-60 a^{2/3} b^2 x^6+20 b^{8/3} x^8 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+40 a b^{5/3} x^5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+20 a^2 b^{2/3} x^2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-96 a^{5/3} b x^3-27 a^{8/3}-40 b^{2/3} x^2 \left (a+b x^3\right )^2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+40 \sqrt{3} b^{2/3} x^2 \left (a+b x^3\right )^2 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{54 a^{11/3} x^2 \left (a+b x^3\right ) \sqrt{\left (a+b x^3\right )^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-27*a^(8/3) - 96*a^(5/3)*b*x^3 - 60*a^(2/3)*b^2*x^6 + 40*Sqrt[3]*b^(2/3)*x^2*(a + b*x^3)^2*ArcTan[(1 - (2*b^(
1/3)*x)/a^(1/3))/Sqrt[3]] - 40*b^(2/3)*x^2*(a + b*x^3)^2*Log[a^(1/3) + b^(1/3)*x] + 20*a^2*b^(2/3)*x^2*Log[a^(
2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + 40*a*b^(5/3)*x^5*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + 20
*b^(8/3)*x^8*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(11/3)*x^2*(a + b*x^3)*Sqrt[(a + b*x^3)^2])

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Maple [A]  time = 0.015, size = 322, normalized size = 1. \begin{align*} -{\frac{b{x}^{3}+a}{54\,{x}^{2}{a}^{3}} \left ( -40\,\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}^{8}{b}^{2}+40\,\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){x}^{8}{b}^{2}-20\,\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{2/3} \right ){x}^{8}{b}^{2}+60\, \left ({\frac{a}{b}} \right ) ^{2/3}{x}^{6}{b}^{2}-80\,\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}^{5}ab+80\,\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){x}^{5}ab-40\,\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{2/3} \right ){x}^{5}ab+96\, \left ({\frac{a}{b}} \right ) ^{2/3}{x}^{3}ab-40\,\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}^{2}{a}^{2}+40\,\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){x}^{2}{a}^{2}-20\,\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{2/3} \right ){x}^{2}{a}^{2}+27\, \left ({\frac{a}{b}} \right ) ^{2/3}{a}^{2} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}} \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^3/(b^2*x^6+2*a*b*x^3+a^2)^(3/2),x)

[Out]

-1/54*(-40*3^(1/2)*arctan(1/3*3^(1/2)*(-2*x+(a/b)^(1/3))/(a/b)^(1/3))*x^8*b^2+40*ln(x+(a/b)^(1/3))*x^8*b^2-20*
ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))*x^8*b^2+60*(a/b)^(2/3)*x^6*b^2-80*3^(1/2)*arctan(1/3*3^(1/2)*(-2*x+(a/b)^(1/
3))/(a/b)^(1/3))*x^5*a*b+80*ln(x+(a/b)^(1/3))*x^5*a*b-40*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))*x^5*a*b+96*(a/b)^(2
/3)*x^3*a*b-40*3^(1/2)*arctan(1/3*3^(1/2)*(-2*x+(a/b)^(1/3))/(a/b)^(1/3))*x^2*a^2+40*ln(x+(a/b)^(1/3))*x^2*a^2
-20*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))*x^2*a^2+27*(a/b)^(2/3)*a^2)*(b*x^3+a)/(a/b)^(2/3)/x^2/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^3/(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 = 1.77107, size = 540, normalized size = 1.71 \begin{align*} -\frac{60 \, b^{2} x^{6} + 96 \, a b x^{3} - 40 \, \sqrt{3}{\left (b^{2} x^{8} + 2 \, a b x^{5} + a^{2} x^{2}\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} a x \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}} - \sqrt{3} b}{3 \, b}\right ) + 20 \,{\left (b^{2} x^{8} + 2 \, a b x^{5} + a^{2} x^{2}\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b^{2} x^{2} + a b x \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} + a^{2} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}\right ) - 40 \,{\left (b^{2} x^{8} + 2 \, a b x^{5} + a^{2} x^{2}\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x - a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) + 27 \, a^{2}}{54 \,{\left (a^{3} b^{2} x^{8} + 2 \, a^{4} b x^{5} + a^{5} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/54*(60*b^2*x^6 + 96*a*b*x^3 - 40*sqrt(3)*(b^2*x^8 + 2*a*b*x^5 + a^2*x^2)*(-b^2/a^2)^(1/3)*arctan(1/3*(2*sqr
t(3)*a*x*(-b^2/a^2)^(2/3) - sqrt(3)*b)/b) + 20*(b^2*x^8 + 2*a*b*x^5 + a^2*x^2)*(-b^2/a^2)^(1/3)*log(b^2*x^2 +
a*b*x*(-b^2/a^2)^(1/3) + a^2*(-b^2/a^2)^(2/3)) - 40*(b^2*x^8 + 2*a*b*x^5 + a^2*x^2)*(-b^2/a^2)^(1/3)*log(b*x -
a*(-b^2/a^2)^(1/3)) + 27*a^2)/(a^3*b^2*x^8 + 2*a^4*b*x^5 + a^5*x^2)

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

[Out]

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

[Out]

sage0*x