3.140 \(\int \frac{1}{x^5 \left (a+b x^2\right )} \, dx\)

Optimal. Leaf size=49 \[ -\frac{b^2 \log \left (a+b x^2\right )}{2 a^3}+\frac{b^2 \log (x)}{a^3}+\frac{b}{2 a^2 x^2}-\frac{1}{4 a x^4} \]

[Out]

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

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Rubi [A]  time = 0.0739861, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{b^2 \log \left (a+b x^2\right )}{2 a^3}+\frac{b^2 \log (x)}{a^3}+\frac{b}{2 a^2 x^2}-\frac{1}{4 a x^4} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*(a + b*x^2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 11.9459, size = 48, normalized size = 0.98 \[ - \frac{1}{4 a x^{4}} + \frac{b}{2 a^{2} x^{2}} + \frac{b^{2} \log{\left (x^{2} \right )}}{2 a^{3}} - \frac{b^{2} \log{\left (a + b x^{2} \right )}}{2 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(b*x**2+a),x)

[Out]

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

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Mathematica [A]  time = 0.0127184, size = 49, normalized size = 1. \[ -\frac{b^2 \log \left (a+b x^2\right )}{2 a^3}+\frac{b^2 \log (x)}{a^3}+\frac{b}{2 a^2 x^2}-\frac{1}{4 a x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^5*(a + b*x^2)),x]

[Out]

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

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Maple [A]  time = 0.01, size = 44, normalized size = 0.9 \[ -{\frac{1}{4\,a{x}^{4}}}+{\frac{b}{2\,{a}^{2}{x}^{2}}}+{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{3}}}-{\frac{{b}^{2}\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(b*x^2+a),x)

[Out]

-1/4/a/x^4+1/2*b/a^2/x^2+b^2*ln(x)/a^3-1/2*b^2*ln(b*x^2+a)/a^3

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Maxima [A]  time = 1.35397, size = 63, normalized size = 1.29 \[ -\frac{b^{2} \log \left (b x^{2} + a\right )}{2 \, a^{3}} + \frac{b^{2} \log \left (x^{2}\right )}{2 \, a^{3}} + \frac{2 \, b x^{2} - a}{4 \, a^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*x^5),x, algorithm="maxima")

[Out]

-1/2*b^2*log(b*x^2 + a)/a^3 + 1/2*b^2*log(x^2)/a^3 + 1/4*(2*b*x^2 - a)/(a^2*x^4)

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Fricas [A]  time = 0.204541, size = 61, normalized size = 1.24 \[ -\frac{2 \, b^{2} x^{4} \log \left (b x^{2} + a\right ) - 4 \, b^{2} x^{4} \log \left (x\right ) - 2 \, a b x^{2} + a^{2}}{4 \, a^{3} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*x^5),x, algorithm="fricas")

[Out]

-1/4*(2*b^2*x^4*log(b*x^2 + a) - 4*b^2*x^4*log(x) - 2*a*b*x^2 + a^2)/(a^3*x^4)

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Sympy [A]  time = 1.78807, size = 42, normalized size = 0.86 \[ \frac{- a + 2 b x^{2}}{4 a^{2} x^{4}} + \frac{b^{2} \log{\left (x \right )}}{a^{3}} - \frac{b^{2} \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**5/(b*x**2+a),x)

[Out]

(-a + 2*b*x**2)/(4*a**2*x**4) + b**2*log(x)/a**3 - b**2*log(a/b + x**2)/(2*a**3)

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GIAC/XCAS [A]  time = 0.210879, size = 77, normalized size = 1.57 \[ \frac{b^{2}{\rm ln}\left (x^{2}\right )}{2 \, a^{3}} - \frac{b^{2}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{3}} - \frac{3 \, b^{2} x^{4} - 2 \, a b x^{2} + a^{2}}{4 \, a^{3} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*x^5),x, algorithm="giac")

[Out]

1/2*b^2*ln(x^2)/a^3 - 1/2*b^2*ln(abs(b*x^2 + a))/a^3 - 1/4*(3*b^2*x^4 - 2*a*b*x^
2 + a^2)/(a^3*x^4)