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

Optimal. Leaf size=94 \[ \frac{a \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{6 b^{4/3}}-\frac{a \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} b^{4/3}}+\frac{x \left (a+b x^3\right )^{2/3}}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.0599741, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{a \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{6 b^{4/3}}-\frac{a \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} b^{4/3}}+\frac{x \left (a+b x^3\right )^{2/3}}{3 b} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a + b*x^3)^(1/3),x]

[Out]

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

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Rubi in Sympy [A]  time = 18.4935, size = 134, normalized size = 1.43 \[ \frac{a \log{\left (- \frac{\sqrt [3]{b} x}{\sqrt [3]{a + b x^{3}}} + 1 \right )}}{9 b^{\frac{4}{3}}} - \frac{a \log{\left (\frac{b^{\frac{2}{3}} x^{2}}{\left (a + b x^{3}\right )^{\frac{2}{3}}} + \frac{\sqrt [3]{b} x}{\sqrt [3]{a + b x^{3}}} + 1 \right )}}{18 b^{\frac{4}{3}}} - \frac{\sqrt{3} a \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{b} x}{3 \sqrt [3]{a + b x^{3}}} + \frac{1}{3}\right ) \right )}}{9 b^{\frac{4}{3}}} + \frac{x \left (a + b x^{3}\right )^{\frac{2}{3}}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.122947, size = 131, normalized size = 1.39 \[ \frac{x \left (a+b x^3\right )^{2/3}}{3 b}-\frac{a \left (\log \left (\frac{b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1\right )-2 \log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )\right )}{18 b^{4/3}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(a + b*x^3)^(2/3))/(3*b) - (a*(2*Sqrt[3]*ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3
)^(1/3))/Sqrt[3]] - 2*Log[1 - (b^(1/3)*x)/(a + b*x^3)^(1/3)] + Log[1 + (b^(2/3)*
x^2)/(a + b*x^3)^(2/3) + (b^(1/3)*x)/(a + b*x^3)^(1/3)]))/(18*b^(4/3))

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Maple [F]  time = 0.035, size = 0, normalized size = 0. \[ \int{{x}^{3}{\frac{1}{\sqrt [3]{b{x}^{3}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x^3+a)^(1/3),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.243867, size = 185, normalized size = 1.97 \[ \frac{\sqrt{3}{\left (6 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{2}{3}} b^{\frac{1}{3}} x + 2 \, \sqrt{3} a \log \left (-\frac{b x -{\left (b x^{3} + a\right )}^{\frac{1}{3}} b^{\frac{2}{3}}}{x}\right ) - \sqrt{3} a \log \left (\frac{b x^{2} +{\left (b x^{3} + a\right )}^{\frac{1}{3}} b^{\frac{2}{3}} x +{\left (b x^{3} + a\right )}^{\frac{2}{3}} b^{\frac{1}{3}}}{x^{2}}\right ) + 6 \, a \arctan \left (\frac{\sqrt{3} b x + 2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}} b^{\frac{2}{3}}}{3 \, b x}\right )\right )}}{54 \, b^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/54*sqrt(3)*(6*sqrt(3)*(b*x^3 + a)^(2/3)*b^(1/3)*x + 2*sqrt(3)*a*log(-(b*x - (b
*x^3 + a)^(1/3)*b^(2/3))/x) - sqrt(3)*a*log((b*x^2 + (b*x^3 + a)^(1/3)*b^(2/3)*x
+ (b*x^3 + a)^(2/3)*b^(1/3))/x^2) + 6*a*arctan(1/3*(sqrt(3)*b*x + 2*sqrt(3)*(b*
x^3 + a)^(1/3)*b^(2/3))/(b*x)))/b^(4/3)

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Sympy [A]  time = 4.11645, size = 37, normalized size = 0.39 \[ \frac{x^{4} \Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [3]{a} \Gamma \left (\frac{7}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{{\left (b x^{3} + a\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^3/(b*x^3 + a)^(1/3), x)