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

Optimal. Leaf size=145 \[ -\frac{a \log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{9 b^{2/3}}-\frac{a \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} b^{2/3}}+\frac{a \log \left (\frac{b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1\right )}{18 b^{2/3}}+\frac{1}{3} x^2 \sqrt [3]{a+b x^3} \]

[Out]

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

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Rubi [A]  time = 0.15952, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615 \[ -\frac{a \log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{9 b^{2/3}}-\frac{a \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} b^{2/3}}+\frac{a \log \left (\frac{b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1\right )}{18 b^{2/3}}+\frac{1}{3} x^2 \sqrt [3]{a+b x^3} \]

Antiderivative was successfully verified.

[In]  Int[x*(a + b*x^3)^(1/3),x]

[Out]

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

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Rubi in Sympy [A]  time = 21.2556, size = 134, normalized size = 0.92 \[ - \frac{a \log{\left (- \frac{\sqrt [3]{b} x}{\sqrt [3]{a + b x^{3}}} + 1 \right )}}{9 b^{\frac{2}{3}}} + \frac{a \log{\left (\frac{b^{\frac{2}{3}} x^{2}}{\left (a + b x^{3}\right )^{\frac{2}{3}}} + \frac{\sqrt [3]{b} x}{\sqrt [3]{a + b x^{3}}} + 1 \right )}}{18 b^{\frac{2}{3}}} - \frac{\sqrt{3} a \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{b} x}{3 \sqrt [3]{a + b x^{3}}} + \frac{1}{3}\right ) \right )}}{9 b^{\frac{2}{3}}} + \frac{x^{2} \sqrt [3]{a + b x^{3}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0450405, size = 63, normalized size = 0.43 \[ \frac{x^2 \left (a \left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right )+2 \left (a+b x^3\right )\right )}{6 \left (a+b x^3\right )^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

(x^2*(2*(a + b*x^3) + a*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, -
((b*x^3)/a)]))/(6*(a + b*x^3)^(2/3))

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Maple [F]  time = 0.033, size = 0, normalized size = 0. \[ \int x\sqrt [3]{b{x}^{3}+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x^3+a)^(1/3),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235518, size = 223, normalized size = 1.54 \[ \frac{\sqrt{3}{\left (6 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-b^{2}\right )^{\frac{1}{3}} x^{2} + 2 \, \sqrt{3} a \log \left (\frac{b x +{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-b^{2}\right )^{\frac{1}{3}}}{x}\right ) - \sqrt{3} a \log \left (\frac{b^{2} x^{2} -{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-b^{2}\right )^{\frac{1}{3}} b x +{\left (b x^{3} + a\right )}^{\frac{2}{3}} \left (-b^{2}\right )^{\frac{2}{3}}}{x^{2}}\right ) + 6 \, a \arctan \left (-\frac{\sqrt{3} b x - 2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-b^{2}\right )^{\frac{1}{3}}}{3 \, b x}\right )\right )}}{54 \, \left (-b^{2}\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.08574, size = 39, normalized size = 0.27 \[ \frac{\sqrt [3]{a} x^{2} \Gamma \left (\frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{5}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(1/3)*x**2*gamma(2/3)*hyper((-1/3, 2/3), (5/3,), b*x**3*exp_polar(I*pi)/a)/(3
*gamma(5/3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^3 + a)^(1/3)*x, x)