3.195 \(\int \frac{x^7}{\left (b x^2+c x^4\right )^2} \, dx\)

Optimal. Leaf size=33 \[ \frac{b}{2 c^2 \left (b+c x^2\right )}+\frac{\log \left (b+c x^2\right )}{2 c^2} \]

[Out]

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

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Rubi [A]  time = 0.0667373, antiderivative size = 33, 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 \[ \frac{b}{2 c^2 \left (b+c x^2\right )}+\frac{\log \left (b+c x^2\right )}{2 c^2} \]

Antiderivative was successfully verified.

[In]  Int[x^7/(b*x^2 + c*x^4)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 10.7874, size = 26, normalized size = 0.79 \[ \frac{b}{2 c^{2} \left (b + c x^{2}\right )} + \frac{\log{\left (b + c x^{2} \right )}}{2 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(c*x**4+b*x**2)**2,x)

[Out]

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

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Mathematica [A]  time = 0.0132495, size = 27, normalized size = 0.82 \[ \frac{\frac{b}{b+c x^2}+\log \left (b+c x^2\right )}{2 c^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/(b*x^2 + c*x^4)^2,x]

[Out]

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

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Maple [A]  time = 0.012, size = 30, normalized size = 0.9 \[{\frac{b}{2\,{c}^{2} \left ( c{x}^{2}+b \right ) }}+{\frac{\ln \left ( c{x}^{2}+b \right ) }{2\,{c}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(c*x^4+b*x^2)^2,x)

[Out]

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

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Maxima [A]  time = 0.699619, size = 43, normalized size = 1.3 \[ \frac{b}{2 \,{\left (c^{3} x^{2} + b c^{2}\right )}} + \frac{\log \left (c x^{2} + b\right )}{2 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(c*x^4 + b*x^2)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.251406, size = 47, normalized size = 1.42 \[ \frac{{\left (c x^{2} + b\right )} \log \left (c x^{2} + b\right ) + b}{2 \,{\left (c^{3} x^{2} + b c^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(c*x^4 + b*x^2)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.34337, size = 29, normalized size = 0.88 \[ \frac{b}{2 b c^{2} + 2 c^{3} x^{2}} + \frac{\log{\left (b + c x^{2} \right )}}{2 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(c*x**4+b*x**2)**2,x)

[Out]

b/(2*b*c**2 + 2*c**3*x**2) + log(b + c*x**2)/(2*c**2)

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GIAC/XCAS [A]  time = 0.270413, size = 43, normalized size = 1.3 \[ -\frac{x^{2}}{2 \,{\left (c x^{2} + b\right )} c} + \frac{{\rm ln}\left ({\left | c x^{2} + b \right |}\right )}{2 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(c*x^4 + b*x^2)^2,x, algorithm="giac")

[Out]

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