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

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

[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 = 0.356949, 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 \[ \frac{4}{9 a^2 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{6 a x^2 \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )}-\frac{20 b^{2/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{20 b^{2/3} \left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{11/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{10 b^{2/3} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{11/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{10 \left (a+b x^3\right )}{9 a^3 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

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])

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Rubi in Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(b**2*x**6+2*a*b*x**3+a**2)**(3/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.197154, size = 266, normalized size = 0.84 \[ \frac{20 b^{8/3} x^8 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+40 a b^{5/3} x^5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-60 a^{2/3} b^2 x^6-96 a^{5/3} b x^3-27 a^{8/3}+20 a^2 b^{2/3} x^2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-40 b^{2/3} x^2 \left (a+b x^3\right )^2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+40 \sqrt{3} b^{2/3} x^2 \left (a+b x^3\right )^2 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{54 a^{11/3} x^2 \left (a+b x^3\right ) \sqrt{\left (a+b x^3\right )^2}} \]

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 = 0.027, size = 322, normalized size = 1. \[ -{\frac{b{x}^{3}+a}{54\,{x}^{2}{a}^{3}} \left ( -40\,\arctan \left ( 1/3\,{\sqrt{3} \left ( -2\,x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \right ) \sqrt{3}{x}^{8}{b}^{2}+40\,\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){x}^{8}{b}^{2}-20\,\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{2/3} \right ){x}^{8}{b}^{2}+60\, \left ({\frac{a}{b}} \right ) ^{2/3}{x}^{6}{b}^{2}-80\,\arctan \left ( 1/3\,{\sqrt{3} \left ( -2\,x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \right ) \sqrt{3}{x}^{5}ab+80\,\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){x}^{5}ab-40\,\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{2/3} \right ){x}^{5}ab+96\, \left ({\frac{a}{b}} \right ) ^{2/3}{x}^{3}ab-40\,\arctan \left ( 1/3\,{\sqrt{3} \left ( -2\,x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \right ) \sqrt{3}{x}^{2}{a}^{2}+40\,\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){x}^{2}{a}^{2}-20\,\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{2/3} \right ){x}^{2}{a}^{2}+27\, \left ({\frac{a}{b}} \right ) ^{2/3}{a}^{2} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

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*arctan(1/3*(-2*x+(a/b)^(1/3))*3^(1/2)/(a/b)^(1/3))*3^(1/2)*x^8*b^2+40
*ln(x+(a/b)^(1/3))*x^8*b^2-20*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*x^8*b^2+60*(a/b)
^(2/3)*x^6*b^2-80*arctan(1/3*(-2*x+(a/b)^(1/3))*3^(1/2)/(a/b)^(1/3))*3^(1/2)*x^5
*a*b+80*ln(x+(a/b)^(1/3))*x^5*a*b-40*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*x^5*a*b+9
6*(a/b)^(2/3)*x^3*a*b-40*arctan(1/3*(-2*x+(a/b)^(1/3))*3^(1/2)/(a/b)^(1/3))*3^(1
/2)*x^2*a^2+40*ln(x+(a/b)^(1/3))*x^2*a^2-20*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*x^
2*a^2+27*(a/b)^(2/3)*a^2)*(b*x^3+a)/x^2/(a/b)^(2/3)/a^3/((b*x^3+a)^2)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^6 + 2*a*b*x^3 + a^2)^(3/2)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.273772, size = 355, normalized size = 1.12 \[ -\frac{\sqrt{3}{\left (20 \, \sqrt{3}{\left (b^{2} x^{8} + 2 \, a b x^{5} + a^{2} x^{2}\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b^{2} x^{2} + a b x \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} + a^{2} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}\right ) - 40 \, \sqrt{3}{\left (b^{2} x^{8} + 2 \, a b x^{5} + a^{2} x^{2}\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x - a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) + 120 \,{\left (b^{2} x^{8} + 2 \, a b x^{5} + a^{2} x^{2}\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} b x + \sqrt{3} a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}{3 \, a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}\right ) + 3 \, \sqrt{3}{\left (20 \, b^{2} x^{6} + 32 \, a b x^{3} + 9 \, a^{2}\right )}\right )}}{162 \,{\left (a^{3} b^{2} x^{8} + 2 \, a^{4} b x^{5} + a^{5} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/162*sqrt(3)*(20*sqrt(3)*(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*sqrt(3)*(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)) + 120*(b^2*
x^8 + 2*a*b*x^5 + a^2*x^2)*(-b^2/a^2)^(1/3)*arctan(1/3*(2*sqrt(3)*b*x + sqrt(3)*
a*(-b^2/a^2)^(1/3))/(a*(-b^2/a^2)^(1/3))) + 3*sqrt(3)*(20*b^2*x^6 + 32*a*b*x^3 +
9*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. \[ \int \frac{1}{x^{3} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac{3}{2}}}\, dx \]

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/XCAS [A]  time = 0.731587, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^6 + 2*a*b*x^3 + a^2)^(3/2)*x^3),x, algorithm="giac")

[Out]

sage0*x