3.548 \(\int \frac{1}{x^4 \sqrt [3]{a+b x^3}} \, dx\)

Optimal. Leaf size=110 \[ -\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{6 a^{4/3}}-\frac{b \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{4/3}}+\frac{b \log (x)}{6 a^{4/3}}-\frac{\left (a+b x^3\right )^{2/3}}{3 a x^3} \]

[Out]

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

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Rubi [A]  time = 0.148132, 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 \[ -\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{6 a^{4/3}}-\frac{b \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{4/3}}+\frac{b \log (x)}{6 a^{4/3}}-\frac{\left (a+b x^3\right )^{2/3}}{3 a x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 10.4411, size = 100, normalized size = 0.91 \[ - \frac{\left (a + b x^{3}\right )^{\frac{2}{3}}}{3 a x^{3}} + \frac{b \log{\left (x^{3} \right )}}{18 a^{\frac{4}{3}}} - \frac{b \log{\left (\sqrt [3]{a} - \sqrt [3]{a + b x^{3}} \right )}}{6 a^{\frac{4}{3}}} - \frac{\sqrt{3} b \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 \sqrt [3]{a + b x^{3}}}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a + b*x**3)**(2/3)/(3*a*x**3) + b*log(x**3)/(18*a**(4/3)) - b*log(a**(1/3) - (
a + b*x**3)**(1/3))/(6*a**(4/3)) - sqrt(3)*b*atan(sqrt(3)*(a**(1/3)/3 + 2*(a + b
*x**3)**(1/3)/3)/a**(1/3))/(9*a**(4/3))

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Mathematica [C]  time = 0.0523134, size = 69, normalized size = 0.63 \[ \frac{b x^3 \sqrt [3]{\frac{a}{b x^3}+1} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};-\frac{a}{b x^3}\right )-a-b x^3}{3 a x^3 \sqrt [3]{a+b x^3}} \]

Antiderivative was successfully verified.

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

[Out]

(-a - b*x^3 + b*(1 + a/(b*x^3))^(1/3)*x^3*Hypergeometric2F1[1/3, 1/3, 4/3, -(a/(
b*x^3))])/(3*a*x^3*(a + b*x^3)^(1/3))

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Maple [F]  time = 0.046, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{4}}{\frac{1}{\sqrt [3]{b{x}^{3}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(b*x^3+a)^(1/3),x)

[Out]

int(1/x^4/(b*x^3+a)^(1/3),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)^(1/3)*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.256902, size = 197, normalized size = 1.79 \[ -\frac{\sqrt{3}{\left (\sqrt{3} b x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac{2}{3}} \left (-a\right )^{\frac{1}{3}} -{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{2}{3}} - a\right ) - 2 \, \sqrt{3} b x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{2}{3}} - a\right ) - 6 \, b x^{3} \arctan \left (\frac{2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{2}{3}} + \sqrt{3} a}{3 \, a}\right ) + 6 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{2}{3}} \left (-a\right )^{\frac{1}{3}}\right )}}{54 \, \left (-a\right )^{\frac{1}{3}} a x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/54*sqrt(3)*(sqrt(3)*b*x^3*log((b*x^3 + a)^(2/3)*(-a)^(1/3) - (b*x^3 + a)^(1/3
)*(-a)^(2/3) - a) - 2*sqrt(3)*b*x^3*log((b*x^3 + a)^(1/3)*(-a)^(2/3) - a) - 6*b*
x^3*arctan(1/3*(2*sqrt(3)*(b*x^3 + a)^(1/3)*(-a)^(2/3) + sqrt(3)*a)/a) + 6*sqrt(
3)*(b*x^3 + a)^(2/3)*(-a)^(1/3))/((-a)^(1/3)*a*x^3)

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Sympy [A]  time = 4.67781, size = 39, normalized size = 0.35 \[ - \frac{\Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{3}}} \right )}}{3 \sqrt [3]{b} x^{4} \Gamma \left (\frac{7}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-gamma(4/3)*hyper((1/3, 4/3), (7/3,), a*exp_polar(I*pi)/(b*x**3))/(3*b**(1/3)*x*
*4*gamma(7/3))

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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)^(1/3)*x^4),x, algorithm="giac")

[Out]

Timed out