3.1973 \(\int \frac{1}{\left (a+\frac{b}{x^3}\right ) x^5} \, dx\)

Optimal. Leaf size=122 \[ -\frac{\sqrt [3]{a} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 b^{4/3}}+\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 b^{4/3}}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} b^{4/3}}-\frac{1}{b x} \]

[Out]

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

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Rubi [A]  time = 0.162251, antiderivative size = 122, 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 \[ -\frac{\sqrt [3]{a} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 b^{4/3}}+\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 b^{4/3}}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} b^{4/3}}-\frac{1}{b x} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^3)*x^5),x]

[Out]

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

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Rubi in Sympy [A]  time = 29.7755, size = 114, normalized size = 0.93 \[ \frac{\sqrt [3]{a} \log{\left (\sqrt [3]{a} x + \sqrt [3]{b} \right )}}{3 b^{\frac{4}{3}}} - \frac{\sqrt [3]{a} \log{\left (a^{\frac{2}{3}} x^{2} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} \right )}}{6 b^{\frac{4}{3}}} + \frac{\sqrt{3} \sqrt [3]{a} \operatorname{atan}{\left (\frac{\sqrt{3} \left (- \frac{2 \sqrt [3]{a} x}{3} + \frac{\sqrt [3]{b}}{3}\right )}{\sqrt [3]{b}} \right )}}{3 b^{\frac{4}{3}}} - \frac{1}{b x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**3)/x**5,x)

[Out]

a**(1/3)*log(a**(1/3)*x + b**(1/3))/(3*b**(4/3)) - a**(1/3)*log(a**(2/3)*x**2 -
a**(1/3)*b**(1/3)*x + b**(2/3))/(6*b**(4/3)) + sqrt(3)*a**(1/3)*atan(sqrt(3)*(-2
*a**(1/3)*x/3 + b**(1/3)/3)/b**(1/3))/(3*b**(4/3)) - 1/(b*x)

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Mathematica [A]  time = 0.0418416, size = 114, normalized size = 0.93 \[ \frac{-\sqrt [3]{a} x \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )+2 \sqrt [3]{a} x \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )+2 \sqrt{3} \sqrt [3]{a} x \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt{3}}\right )-6 \sqrt [3]{b}}{6 b^{4/3} x} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^3)*x^5),x]

[Out]

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

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Maple [A]  time = 0.006, size = 99, normalized size = 0.8 \[{\frac{1}{3\,b}\ln \left ( x+\sqrt [3]{{\frac{b}{a}}} \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}-{\frac{1}{6\,b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{b}{a}}}+ \left ({\frac{b}{a}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}-{\frac{\sqrt{3}}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}-{\frac{1}{bx}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^3)/x^5,x)

[Out]

1/3/b/(b/a)^(1/3)*ln(x+(b/a)^(1/3))-1/6/b/(b/a)^(1/3)*ln(x^2-x*(b/a)^(1/3)+(b/a)
^(2/3))-1/3/b*3^(1/2)/(b/a)^(1/3)*arctan(1/3*3^(1/2)*(2/(b/a)^(1/3)*x-1))-1/b/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(1/((a + b/x^3)*x^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.229223, size = 171, normalized size = 1.4 \[ -\frac{\sqrt{3}{\left (\sqrt{3} x \left (\frac{a}{b}\right )^{\frac{1}{3}} \log \left (a x^{2} - b x \left (\frac{a}{b}\right )^{\frac{2}{3}} + b \left (\frac{a}{b}\right )^{\frac{1}{3}}\right ) - 2 \, \sqrt{3} x \left (\frac{a}{b}\right )^{\frac{1}{3}} \log \left (a x + b \left (\frac{a}{b}\right )^{\frac{2}{3}}\right ) - 6 \, x \left (\frac{a}{b}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} a x - \sqrt{3} b \left (\frac{a}{b}\right )^{\frac{2}{3}}}{3 \, b \left (\frac{a}{b}\right )^{\frac{2}{3}}}\right ) + 6 \, \sqrt{3}\right )}}{18 \, b x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^3)*x^5),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.34028, size = 29, normalized size = 0.24 \[ \operatorname{RootSum}{\left (27 t^{3} b^{4} - a, \left ( t \mapsto t \log{\left (\frac{9 t^{2} b^{3}}{a} + x \right )} \right )\right )} - \frac{1}{b x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**3)/x**5,x)

[Out]

RootSum(27*_t**3*b**4 - a, Lambda(_t, _t*log(9*_t**2*b**3/a + x))) - 1/(b*x)

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GIAC/XCAS [A]  time = 0.226631, size = 163, normalized size = 1.34 \[ \frac{a \left (-\frac{b}{a}\right )^{\frac{2}{3}}{\rm ln}\left ({\left | x - \left (-\frac{b}{a}\right )^{\frac{1}{3}} \right |}\right )}{3 \, b^{2}} + \frac{\sqrt{3} \left (-a^{2} b\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{b}{a}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{b}{a}\right )^{\frac{1}{3}}}\right )}{3 \, a b^{2}} - \frac{\left (-a^{2} b\right )^{\frac{2}{3}}{\rm ln}\left (x^{2} + x \left (-\frac{b}{a}\right )^{\frac{1}{3}} + \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right )}{6 \, a b^{2}} - \frac{1}{b x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^3)*x^5),x, algorithm="giac")

[Out]

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