3.536 \(\int x^3 \left (a+b x^3\right )^{2/3} \, dx\)

Optimal. Leaf size=117 \[ \frac{a^2 \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{18 b^{4/3}}-\frac{a^2 \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{9 \sqrt{3} b^{4/3}}+\frac{a x \left (a+b x^3\right )^{2/3}}{9 b}+\frac{1}{6} x^4 \left (a+b x^3\right )^{2/3} \]

[Out]

(a*x*(a + b*x^3)^(2/3))/(9*b) + (x^4*(a + b*x^3)^(2/3))/6 - (a^2*ArcTan[(1 + (2*
b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]])/(9*Sqrt[3]*b^(4/3)) + (a^2*Log[-(b^(1/3)
*x) + (a + b*x^3)^(1/3)])/(18*b^(4/3))

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Rubi [A]  time = 0.096407, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{a^2 \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{18 b^{4/3}}-\frac{a^2 \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{9 \sqrt{3} b^{4/3}}+\frac{a x \left (a+b x^3\right )^{2/3}}{9 b}+\frac{1}{6} x^4 \left (a+b x^3\right )^{2/3} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(a + b*x^3)^(2/3),x]

[Out]

(a*x*(a + b*x^3)^(2/3))/(9*b) + (x^4*(a + b*x^3)^(2/3))/6 - (a^2*ArcTan[(1 + (2*
b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]])/(9*Sqrt[3]*b^(4/3)) + (a^2*Log[-(b^(1/3)
*x) + (a + b*x^3)^(1/3)])/(18*b^(4/3))

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Rubi in Sympy [A]  time = 23.1416, size = 156, normalized size = 1.33 \[ \frac{a^{2} \log{\left (- \frac{\sqrt [3]{b} x}{\sqrt [3]{a + b x^{3}}} + 1 \right )}}{27 b^{\frac{4}{3}}} - \frac{a^{2} \log{\left (\frac{b^{\frac{2}{3}} x^{2}}{\left (a + b x^{3}\right )^{\frac{2}{3}}} + \frac{\sqrt [3]{b} x}{\sqrt [3]{a + b x^{3}}} + 1 \right )}}{54 b^{\frac{4}{3}}} - \frac{\sqrt{3} a^{2} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{b} x}{3 \sqrt [3]{a + b x^{3}}} + \frac{1}{3}\right ) \right )}}{27 b^{\frac{4}{3}}} + \frac{a x \left (a + b x^{3}\right )^{\frac{2}{3}}}{9 b} + \frac{x^{4} \left (a + b x^{3}\right )^{\frac{2}{3}}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*log(-b**(1/3)*x/(a + b*x**3)**(1/3) + 1)/(27*b**(4/3)) - a**2*log(b**(2/3)*
x**2/(a + b*x**3)**(2/3) + b**(1/3)*x/(a + b*x**3)**(1/3) + 1)/(54*b**(4/3)) - s
qrt(3)*a**2*atan(sqrt(3)*(2*b**(1/3)*x/(3*(a + b*x**3)**(1/3)) + 1/3))/(27*b**(4
/3)) + a*x*(a + b*x**3)**(2/3)/(9*b) + x**4*(a + b*x**3)**(2/3)/6

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Mathematica [A]  time = 0.187884, size = 143, normalized size = 1.22 \[ \left (a+b x^3\right )^{2/3} \left (\frac{a x}{9 b}+\frac{x^4}{6}\right )-\frac{a^2 \left (\log \left (\frac{b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1\right )-2 \log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )\right )}{54 b^{4/3}} \]

Antiderivative was successfully verified.

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

[Out]

(a + b*x^3)^(2/3)*((a*x)/(9*b) + x^4/6) - (a^2*(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)]))/(54*b^(4
/3))

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Maple [F]  time = 0.041, size = 0, normalized size = 0. \[ \int{x}^{3} \left ( b{x}^{3}+a \right ) ^{{\frac{2}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b*x^3+a)^(2/3),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.260401, size = 207, normalized size = 1.77 \[ \frac{\sqrt{3}{\left (2 \, \sqrt{3} a^{2} \log \left (-\frac{b x -{\left (b x^{3} + a\right )}^{\frac{1}{3}} b^{\frac{2}{3}}}{x}\right ) - \sqrt{3} a^{2} \log \left (\frac{b x^{2} +{\left (b x^{3} + a\right )}^{\frac{1}{3}} b^{\frac{2}{3}} x +{\left (b x^{3} + a\right )}^{\frac{2}{3}} b^{\frac{1}{3}}}{x^{2}}\right ) + 6 \, a^{2} \arctan \left (\frac{\sqrt{3} b x + 2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}} b^{\frac{2}{3}}}{3 \, b x}\right ) + 3 \, \sqrt{3}{\left (3 \, b x^{4} + 2 \, a x\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}} b^{\frac{1}{3}}\right )}}{162 \, b^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/162*sqrt(3)*(2*sqrt(3)*a^2*log(-(b*x - (b*x^3 + a)^(1/3)*b^(2/3))/x) - sqrt(3)
*a^2*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^2*arctan(1/3*(sqrt(3)*b*x + 2*sqrt(3)*(b*x^3 + a)^(1/3)*b^(2/3))/(b*x)) +
3*sqrt(3)*(3*b*x^4 + 2*a*x)*(b*x^3 + a)^(2/3)*b^(1/3))/b^(4/3)

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Sympy [A]  time = 5.22199, size = 39, normalized size = 0.33 \[ \frac{a^{\frac{2}{3}} x^{4} \Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{7}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(2/3)*x**4*gamma(4/3)*hyper((-2/3, 4/3), (7/3,), b*x**3*exp_polar(I*pi)/a)/(3
*gamma(7/3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{3} + a\right )}^{\frac{2}{3}} x^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^3 + a)^(2/3)*x^3, x)