3.199 \(\int \frac{x^3}{\left (b x^2+c x^4\right )^2} \, dx\)

Optimal. Leaf size=38 \[ -\frac{\log \left (b+c x^2\right )}{2 b^2}+\frac{\log (x)}{b^2}+\frac{1}{2 b \left (b+c x^2\right )} \]

[Out]

1/(2*b*(b + c*x^2)) + Log[x]/b^2 - Log[b + c*x^2]/(2*b^2)

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Rubi [A]  time = 0.0718247, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{\log \left (b+c x^2\right )}{2 b^2}+\frac{\log (x)}{b^2}+\frac{1}{2 b \left (b+c x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(b*x^2 + c*x^4)^2,x]

[Out]

1/(2*b*(b + c*x^2)) + Log[x]/b^2 - Log[b + c*x^2]/(2*b^2)

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Rubi in Sympy [A]  time = 11.9673, size = 34, normalized size = 0.89 \[ \frac{1}{2 b \left (b + c x^{2}\right )} + \frac{\log{\left (x^{2} \right )}}{2 b^{2}} - \frac{\log{\left (b + c x^{2} \right )}}{2 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(c*x**4+b*x**2)**2,x)

[Out]

1/(2*b*(b + c*x**2)) + log(x**2)/(2*b**2) - log(b + c*x**2)/(2*b**2)

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Mathematica [A]  time = 0.0219342, size = 33, normalized size = 0.87 \[ \frac{\frac{b}{b+c x^2}-\log \left (b+c x^2\right )+2 \log (x)}{2 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(b*x^2 + c*x^4)^2,x]

[Out]

(b/(b + c*x^2) + 2*Log[x] - Log[b + c*x^2])/(2*b^2)

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Maple [A]  time = 0.017, size = 35, normalized size = 0.9 \[{\frac{1}{2\,b \left ( c{x}^{2}+b \right ) }}+{\frac{\ln \left ( x \right ) }{{b}^{2}}}-{\frac{\ln \left ( c{x}^{2}+b \right ) }{2\,{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(c*x^4+b*x^2)^2,x)

[Out]

1/2/b/(c*x^2+b)+ln(x)/b^2-1/2*ln(c*x^2+b)/b^2

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Maxima [A]  time = 0.702543, size = 50, normalized size = 1.32 \[ \frac{1}{2 \,{\left (b c x^{2} + b^{2}\right )}} - \frac{\log \left (c x^{2} + b\right )}{2 \, b^{2}} + \frac{\log \left (x^{2}\right )}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(c*x^4 + b*x^2)^2,x, algorithm="maxima")

[Out]

1/2/(b*c*x^2 + b^2) - 1/2*log(c*x^2 + b)/b^2 + 1/2*log(x^2)/b^2

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Fricas [A]  time = 0.260859, size = 63, normalized size = 1.66 \[ -\frac{{\left (c x^{2} + b\right )} \log \left (c x^{2} + b\right ) - 2 \,{\left (c x^{2} + b\right )} \log \left (x\right ) - b}{2 \,{\left (b^{2} c x^{2} + b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(c*x^4 + b*x^2)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.65951, size = 34, normalized size = 0.89 \[ \frac{1}{2 b^{2} + 2 b c x^{2}} + \frac{\log{\left (x \right )}}{b^{2}} - \frac{\log{\left (\frac{b}{c} + x^{2} \right )}}{2 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(c*x**4+b*x**2)**2,x)

[Out]

1/(2*b**2 + 2*b*c*x**2) + log(x)/b**2 - log(b/c + x**2)/(2*b**2)

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GIAC/XCAS [A]  time = 0.271521, size = 49, normalized size = 1.29 \[ -\frac{{\rm ln}\left ({\left | c x^{2} + b \right |}\right )}{2 \, b^{2}} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{b^{2}} + \frac{1}{2 \,{\left (c x^{2} + b\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(c*x^4 + b*x^2)^2,x, algorithm="giac")

[Out]

-1/2*ln(abs(c*x^2 + b))/b^2 + ln(abs(x))/b^2 + 1/2/((c*x^2 + b)*b)