3.1964 \(\int \frac{x^4}{a+\frac{b}{x^3}} \, dx\)

Optimal. Leaf size=136 \[ \frac{b^{5/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 a^{8/3}}-\frac{b^{5/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{8/3}}-\frac{b^{5/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} a^{8/3}}-\frac{b x^2}{2 a^2}+\frac{x^5}{5 a} \]

[Out]

-(b*x^2)/(2*a^2) + x^5/(5*a) - (b^(5/3)*ArcTan[(b^(1/3) - 2*a^(1/3)*x)/(Sqrt[3]*
b^(1/3))])/(Sqrt[3]*a^(8/3)) - (b^(5/3)*Log[b^(1/3) + a^(1/3)*x])/(3*a^(8/3)) +
(b^(5/3)*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(6*a^(8/3))

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Rubi [A]  time = 0.234764, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615 \[ \frac{b^{5/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 a^{8/3}}-\frac{b^{5/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{8/3}}-\frac{b^{5/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} a^{8/3}}-\frac{b x^2}{2 a^2}+\frac{x^5}{5 a} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a + b/x^3),x]

[Out]

-(b*x^2)/(2*a^2) + x^5/(5*a) - (b^(5/3)*ArcTan[(b^(1/3) - 2*a^(1/3)*x)/(Sqrt[3]*
b^(1/3))])/(Sqrt[3]*a^(8/3)) - (b^(5/3)*Log[b^(1/3) + a^(1/3)*x])/(3*a^(8/3)) +
(b^(5/3)*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(6*a^(8/3))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x^{5}}{5 a} - \frac{b \int x\, dx}{a^{2}} - \frac{b^{\frac{5}{3}} \log{\left (\sqrt [3]{a} x + \sqrt [3]{b} \right )}}{3 a^{\frac{8}{3}}} + \frac{b^{\frac{5}{3}} \log{\left (a^{\frac{2}{3}} x^{2} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} \right )}}{6 a^{\frac{8}{3}}} - \frac{\sqrt{3} b^{\frac{5}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (- \frac{2 \sqrt [3]{a} x}{3} + \frac{\sqrt [3]{b}}{3}\right )}{\sqrt [3]{b}} \right )}}{3 a^{\frac{8}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**5/(5*a) - b*Integral(x, x)/a**2 - b**(5/3)*log(a**(1/3)*x + b**(1/3))/(3*a**(
8/3)) + b**(5/3)*log(a**(2/3)*x**2 - a**(1/3)*b**(1/3)*x + b**(2/3))/(6*a**(8/3)
) - sqrt(3)*b**(5/3)*atan(sqrt(3)*(-2*a**(1/3)*x/3 + b**(1/3)/3)/b**(1/3))/(3*a*
*(8/3))

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Mathematica [A]  time = 0.0685256, size = 122, normalized size = 0.9 \[ \frac{5 b^{5/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )-15 a^{2/3} b x^2+6 a^{5/3} x^5-10 b^{5/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )-10 \sqrt{3} b^{5/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt{3}}\right )}{30 a^{8/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(a + b/x^3),x]

[Out]

(-15*a^(2/3)*b*x^2 + 6*a^(5/3)*x^5 - 10*Sqrt[3]*b^(5/3)*ArcTan[(1 - (2*a^(1/3)*x
)/b^(1/3))/Sqrt[3]] - 10*b^(5/3)*Log[b^(1/3) + a^(1/3)*x] + 5*b^(5/3)*Log[b^(2/3
) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(30*a^(8/3))

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Maple [A]  time = 0.008, size = 117, normalized size = 0.9 \[{\frac{{x}^{5}}{5\,a}}-{\frac{b{x}^{2}}{2\,{a}^{2}}}-{\frac{{b}^{2}}{3\,{a}^{3}}\ln \left ( x+\sqrt [3]{{\frac{b}{a}}} \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}+{\frac{{b}^{2}}{6\,{a}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{b}{a}}}+ \left ({\frac{b}{a}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}+{\frac{{b}^{2}\sqrt{3}}{3\,{a}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(a+b/x^3),x)

[Out]

1/5*x^5/a-1/2*b*x^2/a^2-1/3/a^3*b^2/(b/a)^(1/3)*ln(x+(b/a)^(1/3))+1/6/a^3*b^2/(b
/a)^(1/3)*ln(x^2-x*(b/a)^(1/3)+(b/a)^(2/3))+1/3/a^3*b^2*3^(1/2)/(b/a)^(1/3)*arct
an(1/3*3^(1/2)*(2/(b/a)^(1/3)*x-1))

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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^4/(a + b/x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23887, size = 220, normalized size = 1.62 \[ -\frac{\sqrt{3}{\left (5 \, \sqrt{3} b \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x^{2} - a x \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}} - b \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) - 10 \, \sqrt{3} b \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x + a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}\right ) - 30 \, b \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} b x - \sqrt{3} a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}}{3 \, a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}}\right ) - 3 \, \sqrt{3}{\left (2 \, a x^{5} - 5 \, b x^{2}\right )}\right )}}{90 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/90*sqrt(3)*(5*sqrt(3)*b*(-b^2/a^2)^(1/3)*log(b*x^2 - a*x*(-b^2/a^2)^(2/3) - b
*(-b^2/a^2)^(1/3)) - 10*sqrt(3)*b*(-b^2/a^2)^(1/3)*log(b*x + a*(-b^2/a^2)^(2/3))
- 30*b*(-b^2/a^2)^(1/3)*arctan(-1/3*(2*sqrt(3)*b*x - sqrt(3)*a*(-b^2/a^2)^(2/3)
)/(a*(-b^2/a^2)^(2/3))) - 3*sqrt(3)*(2*a*x^5 - 5*b*x^2))/a^2

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Sympy [A]  time = 1.35074, size = 44, normalized size = 0.32 \[ \operatorname{RootSum}{\left (27 t^{3} a^{8} + b^{5}, \left ( t \mapsto t \log{\left (\frac{9 t^{2} a^{5}}{b^{3}} + x \right )} \right )\right )} + \frac{x^{5}}{5 a} - \frac{b x^{2}}{2 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(a+b/x**3),x)

[Out]

RootSum(27*_t**3*a**8 + b**5, Lambda(_t, _t*log(9*_t**2*a**5/b**3 + x))) + x**5/
(5*a) - b*x**2/(2*a**2)

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GIAC/XCAS [A]  time = 0.230588, size = 178, normalized size = 1.31 \[ -\frac{b \left (-\frac{b}{a}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x - \left (-\frac{b}{a}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a^{2}} - \frac{\sqrt{3} \left (-a^{2} b\right )^{\frac{2}{3}} b \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{b}{a}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{b}{a}\right )^{\frac{1}{3}}}\right )}{3 \, a^{4}} + \frac{\left (-a^{2} b\right )^{\frac{2}{3}} b{\rm ln}\left (x^{2} + x \left (-\frac{b}{a}\right )^{\frac{1}{3}} + \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right )}{6 \, a^{4}} + \frac{2 \, a^{4} x^{5} - 5 \, a^{3} b x^{2}}{10 \, a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*b*(-b/a)^(2/3)*ln(abs(x - (-b/a)^(1/3)))/a^2 - 1/3*sqrt(3)*(-a^2*b)^(2/3)*b
*arctan(1/3*sqrt(3)*(2*x + (-b/a)^(1/3))/(-b/a)^(1/3))/a^4 + 1/6*(-a^2*b)^(2/3)*
b*ln(x^2 + x*(-b/a)^(1/3) + (-b/a)^(2/3))/a^4 + 1/10*(2*a^4*x^5 - 5*a^3*b*x^2)/a
^5