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

Optimal. Leaf size=43 \[ \frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2}}+\frac{b}{a^2 x}-\frac{1}{3 a x^3} \]

[Out]

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

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Rubi [A]  time = 0.0517105, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2}}+\frac{b}{a^2 x}-\frac{1}{3 a x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(a*x + b*x^3)),x]

[Out]

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

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Rubi in Sympy [A]  time = 10.7657, size = 37, normalized size = 0.86 \[ - \frac{1}{3 a x^{3}} + \frac{b}{a^{2} x} + \frac{b^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(3*a*x**3) + b/(a**2*x) + b**(3/2)*atan(sqrt(b)*x/sqrt(a))/a**(5/2)

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Mathematica [A]  time = 0.0337876, size = 43, normalized size = 1. \[ \frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2}}+\frac{b}{a^2 x}-\frac{1}{3 a x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 39, normalized size = 0.9 \[ -{\frac{1}{3\,a{x}^{3}}}+{\frac{b}{{a}^{2}x}}+{\frac{{b}^{2}}{{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(b*x^3+a*x),x)

[Out]

-1/3/a/x^3+b/a^2/x+b^2/a^2/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.21424, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, b x^{3} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) + 6 \, b x^{2} - 2 \, a}{6 \, a^{2} x^{3}}, \frac{3 \, b x^{3} \sqrt{\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{\frac{b}{a}}}\right ) + 3 \, b x^{2} - a}{3 \, a^{2} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.52177, size = 87, normalized size = 2.02 \[ - \frac{\sqrt{- \frac{b^{3}}{a^{5}}} \log{\left (- \frac{a^{3} \sqrt{- \frac{b^{3}}{a^{5}}}}{b^{2}} + x \right )}}{2} + \frac{\sqrt{- \frac{b^{3}}{a^{5}}} \log{\left (\frac{a^{3} \sqrt{- \frac{b^{3}}{a^{5}}}}{b^{2}} + x \right )}}{2} + \frac{- a + 3 b x^{2}}{3 a^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-b**3/a**5)*log(-a**3*sqrt(-b**3/a**5)/b**2 + x)/2 + sqrt(-b**3/a**5)*log(
a**3*sqrt(-b**3/a**5)/b**2 + x)/2 + (-a + 3*b*x**2)/(3*a**2*x**3)

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GIAC/XCAS [A]  time = 0.217859, size = 54, normalized size = 1.26 \[ \frac{b^{2} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a^{2}} + \frac{3 \, b x^{2} - a}{3 \, a^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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