3.565 \(\int \frac{1}{x^4 \left (a+b x^3\right )^{2/3}} \, dx\)

Optimal. Leaf size=110 \[ -\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{3 a^{5/3}}+\frac{2 b \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3}}+\frac{b \log (x)}{3 a^{5/3}}-\frac{\sqrt [3]{a+b x^3}}{3 a x^3} \]

[Out]

-(a + b*x^3)^(1/3)/(3*a*x^3) + (2*b*ArcTan[(a^(1/3) + 2*(a + b*x^3)^(1/3))/(Sqrt
[3]*a^(1/3))])/(3*Sqrt[3]*a^(5/3)) + (b*Log[x])/(3*a^(5/3)) - (b*Log[a^(1/3) - (
a + b*x^3)^(1/3)])/(3*a^(5/3))

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Rubi [A]  time = 0.153255, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{3 a^{5/3}}+\frac{2 b \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3}}+\frac{b \log (x)}{3 a^{5/3}}-\frac{\sqrt [3]{a+b x^3}}{3 a x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(a + b*x^3)^(2/3)),x]

[Out]

-(a + b*x^3)^(1/3)/(3*a*x^3) + (2*b*ArcTan[(a^(1/3) + 2*(a + b*x^3)^(1/3))/(Sqrt
[3]*a^(1/3))])/(3*Sqrt[3]*a^(5/3)) + (b*Log[x])/(3*a^(5/3)) - (b*Log[a^(1/3) - (
a + b*x^3)^(1/3)])/(3*a^(5/3))

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Rubi in Sympy [A]  time = 10.4717, size = 102, normalized size = 0.93 \[ - \frac{\sqrt [3]{a + b x^{3}}}{3 a x^{3}} + \frac{b \log{\left (x^{3} \right )}}{9 a^{\frac{5}{3}}} - \frac{b \log{\left (\sqrt [3]{a} - \sqrt [3]{a + b x^{3}} \right )}}{3 a^{\frac{5}{3}}} + \frac{2 \sqrt{3} b \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 \sqrt [3]{a + b x^{3}}}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{5}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(b*x**3+a)**(2/3),x)

[Out]

-(a + b*x**3)**(1/3)/(3*a*x**3) + b*log(x**3)/(9*a**(5/3)) - b*log(a**(1/3) - (a
+ b*x**3)**(1/3))/(3*a**(5/3)) + 2*sqrt(3)*b*atan(sqrt(3)*(a**(1/3)/3 + 2*(a +
b*x**3)**(1/3)/3)/a**(1/3))/(9*a**(5/3))

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Mathematica [C]  time = 0.0487433, size = 69, normalized size = 0.63 \[ \frac{b x^3 \left (\frac{a}{b x^3}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{a}{b x^3}\right )-a-b x^3}{3 a x^3 \left (a+b x^3\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*(a + b*x^3)^(2/3)),x]

[Out]

(-a - b*x^3 + b*(1 + a/(b*x^3))^(2/3)*x^3*Hypergeometric2F1[2/3, 2/3, 5/3, -(a/(
b*x^3))])/(3*a*x^3*(a + b*x^3)^(2/3))

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Maple [F]  time = 0.044, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{4}} \left ( b{x}^{3}+a \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(b*x^3+a)^(2/3),x)

[Out]

int(1/x^4/(b*x^3+a)^(2/3),x)

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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*x^3 + a)^(2/3)*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.251071, size = 212, normalized size = 1.93 \[ -\frac{\sqrt{3}{\left (\sqrt{3} b x^{3} \log \left (a^{2} -{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-a^{2}\right )^{\frac{1}{3}} a +{\left (b x^{3} + a\right )}^{\frac{2}{3}} \left (-a^{2}\right )^{\frac{2}{3}}\right ) - 2 \, \sqrt{3} b x^{3} \log \left (a +{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-a^{2}\right )^{\frac{1}{3}}\right ) - 6 \, b x^{3} \arctan \left (-\frac{\sqrt{3} a - 2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-a^{2}\right )^{\frac{1}{3}}}{3 \, a}\right ) + 3 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-a^{2}\right )^{\frac{1}{3}}\right )}}{27 \, \left (-a^{2}\right )^{\frac{1}{3}} a x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)^(2/3)*x^4),x, algorithm="fricas")

[Out]

-1/27*sqrt(3)*(sqrt(3)*b*x^3*log(a^2 - (b*x^3 + a)^(1/3)*(-a^2)^(1/3)*a + (b*x^3
+ a)^(2/3)*(-a^2)^(2/3)) - 2*sqrt(3)*b*x^3*log(a + (b*x^3 + a)^(1/3)*(-a^2)^(1/
3)) - 6*b*x^3*arctan(-1/3*(sqrt(3)*a - 2*sqrt(3)*(b*x^3 + a)^(1/3)*(-a^2)^(1/3))
/a) + 3*sqrt(3)*(b*x^3 + a)^(1/3)*(-a^2)^(1/3))/((-a^2)^(1/3)*a*x^3)

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Sympy [A]  time = 5.08161, size = 39, normalized size = 0.35 \[ - \frac{\Gamma \left (\frac{5}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{5}{3} \\ \frac{8}{3} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{3}}} \right )}}{3 b^{\frac{2}{3}} x^{5} \Gamma \left (\frac{8}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/(b*x**3+a)**(2/3),x)

[Out]

-gamma(5/3)*hyper((2/3, 5/3), (8/3,), a*exp_polar(I*pi)/(b*x**3))/(3*b**(2/3)*x*
*5*gamma(8/3))

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)^(2/3)*x^4),x, algorithm="giac")

[Out]

Timed out