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

Optimal. Leaf size=173 \[ \frac{a^2 \log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{27 b^{5/3}}+\frac{a^2 \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{9 \sqrt{3} b^{5/3}}-\frac{a^2 \log \left (\frac{b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1\right )}{54 b^{5/3}}+\frac{1}{6} x^5 \sqrt [3]{a+b x^3}+\frac{a x^2 \sqrt [3]{a+b x^3}}{18 b} \]

[Out]

(a*x^2*(a + b*x^3)^(1/3))/(18*b) + (x^5*(a + b*x^3)^(1/3))/6 + (a^2*ArcTan[(1 +
(2*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]])/(9*Sqrt[3]*b^(5/3)) + (a^2*Log[1 - (b
^(1/3)*x)/(a + b*x^3)^(1/3)])/(27*b^(5/3)) - (a^2*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^(5/3))

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Rubi [A]  time = 0.238675, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6 \[ \frac{a^2 \log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{27 b^{5/3}}+\frac{a^2 \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{9 \sqrt{3} b^{5/3}}-\frac{a^2 \log \left (\frac{b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1\right )}{54 b^{5/3}}+\frac{1}{6} x^5 \sqrt [3]{a+b x^3}+\frac{a x^2 \sqrt [3]{a+b x^3}}{18 b} \]

Antiderivative was successfully verified.

[In]  Int[x^4*(a + b*x^3)^(1/3),x]

[Out]

(a*x^2*(a + b*x^3)^(1/3))/(18*b) + (x^5*(a + b*x^3)^(1/3))/6 + (a^2*ArcTan[(1 +
(2*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]])/(9*Sqrt[3]*b^(5/3)) + (a^2*Log[1 - (b
^(1/3)*x)/(a + b*x^3)^(1/3)])/(27*b^(5/3)) - (a^2*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^(5/3))

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Rubi in Sympy [A]  time = 26.3248, size = 158, normalized size = 0.91 \[ \frac{a^{2} \log{\left (- \frac{\sqrt [3]{b} x}{\sqrt [3]{a + b x^{3}}} + 1 \right )}}{27 b^{\frac{5}{3}}} - \frac{a^{2} \log{\left (\frac{b^{\frac{2}{3}} x^{2}}{\left (a + b x^{3}\right )^{\frac{2}{3}}} + \frac{\sqrt [3]{b} x}{\sqrt [3]{a + b x^{3}}} + 1 \right )}}{54 b^{\frac{5}{3}}} + \frac{\sqrt{3} a^{2} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{b} x}{3 \sqrt [3]{a + b x^{3}}} + \frac{1}{3}\right ) \right )}}{27 b^{\frac{5}{3}}} + \frac{a x^{2} \sqrt [3]{a + b x^{3}}}{18 b} + \frac{x^{5} \sqrt [3]{a + b x^{3}}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*log(-b**(1/3)*x/(a + b*x**3)**(1/3) + 1)/(27*b**(5/3)) - 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**(5/3)) + s
qrt(3)*a**2*atan(sqrt(3)*(2*b**(1/3)*x/(3*(a + b*x**3)**(1/3)) + 1/3))/(27*b**(5
/3)) + a*x**2*(a + b*x**3)**(1/3)/(18*b) + x**5*(a + b*x**3)**(1/3)/6

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Mathematica [C]  time = 0.0586426, size = 78, normalized size = 0.45 \[ \frac{x^2 \left (-a^2 \left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right )+a^2+4 a b x^3+3 b^2 x^6\right )}{18 b \left (a+b x^3\right )^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

(x^2*(a^2 + 4*a*b*x^3 + 3*b^2*x^6 - a^2*(1 + (b*x^3)/a)^(2/3)*Hypergeometric2F1[
2/3, 2/3, 5/3, -((b*x^3)/a)]))/(18*b*(a + b*x^3)^(2/3))

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Maple [F]  time = 0.042, size = 0, normalized size = 0. \[ \int{x}^{4}\sqrt [3]{b{x}^{3}+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(b*x^3+a)^(1/3),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.242345, size = 232, normalized size = 1.34 \[ \frac{\sqrt{3}{\left (2 \, \sqrt{3} a^{2} \log \left (-\frac{b x -{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (b^{2}\right )}^{\frac{1}{3}}}{x}\right ) - \sqrt{3} a^{2} \log \left (\frac{b^{2} x^{2} +{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (b^{2}\right )}^{\frac{1}{3}} b x +{\left (b x^{3} + a\right )}^{\frac{2}{3}}{\left (b^{2}\right )}^{\frac{2}{3}}}{x^{2}}\right ) - 6 \, a^{2} \arctan \left (\frac{\sqrt{3} b x + 2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (b^{2}\right )}^{\frac{1}{3}}}{3 \, b x}\right ) + 3 \, \sqrt{3}{\left (3 \, b x^{5} + a x^{2}\right )}{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (b^{2}\right )}^{\frac{1}{3}}\right )}}{162 \,{\left (b^{2}\right )}^{\frac{1}{3}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/162*sqrt(3)*(2*sqrt(3)*a^2*log(-(b*x - (b*x^3 + a)^(1/3)*(b^2)^(1/3))/x) - sqr
t(3)*a^2*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^2*arctan(1/3*(sqrt(3)*b*x + 2*sqrt(3)*(b*x^3 + a)^(1/3)*(b
^2)^(1/3))/(b*x)) + 3*sqrt(3)*(3*b*x^5 + a*x^2)*(b*x^3 + a)^(1/3)*(b^2)^(1/3))/(
(b^2)^(1/3)*b)

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Sympy [A]  time = 5.09841, size = 39, normalized size = 0.23 \[ \frac{\sqrt [3]{a} x^{5} \Gamma \left (\frac{5}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, \frac{5}{3} \\ \frac{8}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{8}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(1/3)*x**5*gamma(5/3)*hyper((-1/3, 5/3), (8/3,), b*x**3*exp_polar(I*pi)/a)/(3
*gamma(8/3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{3} + a\right )}^{\frac{1}{3}} x^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^3 + a)^(1/3)*x^4, x)