3.1967 \(\int \frac{x}{a+\frac{b}{x^3}} \, dx\)

Optimal. Leaf size=124 \[ -\frac{b^{2/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 a^{5/3}}+\frac{b^{2/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{5/3}}+\frac{b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} a^{5/3}}+\frac{x^2}{2 a} \]

[Out]

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

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Rubi [A]  time = 0.165952, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.727 \[ -\frac{b^{2/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 a^{5/3}}+\frac{b^{2/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{5/3}}+\frac{b^{2/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} a^{5/3}}+\frac{x^2}{2 a} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b/x^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 29.6992, size = 116, normalized size = 0.94 \[ \frac{x^{2}}{2 a} + \frac{b^{\frac{2}{3}} \log{\left (\sqrt [3]{a} x + \sqrt [3]{b} \right )}}{3 a^{\frac{5}{3}}} - \frac{b^{\frac{2}{3}} \log{\left (a^{\frac{2}{3}} x^{2} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} \right )}}{6 a^{\frac{5}{3}}} + \frac{\sqrt{3} b^{\frac{2}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (- \frac{2 \sqrt [3]{a} x}{3} + \frac{\sqrt [3]{b}}{3}\right )}{\sqrt [3]{b}} \right )}}{3 a^{\frac{5}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(a+b/x**3),x)

[Out]

x**2/(2*a) + b**(2/3)*log(a**(1/3)*x + b**(1/3))/(3*a**(5/3)) - b**(2/3)*log(a**
(2/3)*x**2 - a**(1/3)*b**(1/3)*x + b**(2/3))/(6*a**(5/3)) + sqrt(3)*b**(2/3)*ata
n(sqrt(3)*(-2*a**(1/3)*x/3 + b**(1/3)/3)/b**(1/3))/(3*a**(5/3))

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Mathematica [A]  time = 0.0340609, size = 111, normalized size = 0.9 \[ \frac{-b^{2/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )+3 a^{2/3} x^2+2 b^{2/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )+2 \sqrt{3} b^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt{3}}\right )}{6 a^{5/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(a + b/x^3),x]

[Out]

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

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Maple [A]  time = 0.004, size = 102, normalized size = 0.8 \[{\frac{{x}^{2}}{2\,a}}+{\frac{b}{3\,{a}^{2}}\ln \left ( x+\sqrt [3]{{\frac{b}{a}}} \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}-{\frac{b}{6\,{a}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{b}{a}}}+ \left ({\frac{b}{a}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}-{\frac{b\sqrt{3}}{3\,{a}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b/x^3),x)

[Out]

1/2*x^2/a+1/3*b/a^2/(b/a)^(1/3)*ln(x+(b/a)^(1/3))-1/6*b/a^2/(b/a)^(1/3)*ln(x^2-x
*(b/a)^(1/3)+(b/a)^(2/3))-1/3*b/a^2*3^(1/2)/(b/a)^(1/3)*arctan(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/(a + b/x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.231009, size = 190, normalized size = 1.53 \[ \frac{\sqrt{3}{\left (3 \, \sqrt{3} x^{2} - \sqrt{3} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x^{2} - a x \left (\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}} + b \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) + 2 \, \sqrt{3} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x + a \left (\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}\right ) + 6 \, \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} b x - \sqrt{3} a \left (\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}}{3 \, a \left (\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}}\right )\right )}}{18 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18*sqrt(3)*(3*sqrt(3)*x^2 - sqrt(3)*(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)) + 2*sqrt(3)*(b^2/a^2)^(1/3)*log(b*x + a*(b^2/a^2)^(2/
3)) + 6*(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))))/a

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Sympy [A]  time = 1.27921, size = 32, normalized size = 0.26 \[ \operatorname{RootSum}{\left (27 t^{3} a^{5} - b^{2}, \left ( t \mapsto t \log{\left (\frac{9 t^{2} a^{3}}{b} + x \right )} \right )\right )} + \frac{x^{2}}{2 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(a+b/x**3),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.229102, size = 154, normalized size = 1.24 \[ \frac{x^{2}}{2 \, a} + \frac{\left (-\frac{b}{a}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x - \left (-\frac{b}{a}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a} + \frac{\sqrt{3} \left (-a^{2} b\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{b}{a}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{b}{a}\right )^{\frac{1}{3}}}\right )}{3 \, a^{3}} - \frac{\left (-a^{2} b\right )^{\frac{2}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{b}{a}\right )^{\frac{1}{3}} + \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right )}{6 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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