3.191 \(\int \frac{x^{11}}{\left (b x^2+c x^4\right )^2} \, dx\)

Optimal. Leaf size=57 \[ \frac{b^3}{2 c^4 \left (b+c x^2\right )}+\frac{3 b^2 \log \left (b+c x^2\right )}{2 c^4}-\frac{b x^2}{c^3}+\frac{x^4}{4 c^2} \]

[Out]

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

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Rubi [A]  time = 0.113125, antiderivative size = 57, 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 \[ \frac{b^3}{2 c^4 \left (b+c x^2\right )}+\frac{3 b^2 \log \left (b+c x^2\right )}{2 c^4}-\frac{b x^2}{c^3}+\frac{x^4}{4 c^2} \]

Antiderivative was successfully verified.

[In]  Int[x^11/(b*x^2 + c*x^4)^2,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{b^{3}}{2 c^{4} \left (b + c x^{2}\right )} + \frac{3 b^{2} \log{\left (b + c x^{2} \right )}}{2 c^{4}} - \frac{b x^{2}}{c^{3}} + \frac{\int ^{x^{2}} x\, dx}{2 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11/(c*x**4+b*x**2)**2,x)

[Out]

b**3/(2*c**4*(b + c*x**2)) + 3*b**2*log(b + c*x**2)/(2*c**4) - b*x**2/c**3 + Int
egral(x, (x, x**2))/(2*c**2)

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Mathematica [A]  time = 0.0301901, size = 49, normalized size = 0.86 \[ \frac{\frac{2 b^3}{b+c x^2}+6 b^2 \log \left (b+c x^2\right )-4 b c x^2+c^2 x^4}{4 c^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^11/(b*x^2 + c*x^4)^2,x]

[Out]

(-4*b*c*x^2 + c^2*x^4 + (2*b^3)/(b + c*x^2) + 6*b^2*Log[b + c*x^2])/(4*c^4)

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Maple [A]  time = 0.014, size = 52, normalized size = 0.9 \[ -{\frac{b{x}^{2}}{{c}^{3}}}+{\frac{{x}^{4}}{4\,{c}^{2}}}+{\frac{{b}^{3}}{2\,{c}^{4} \left ( c{x}^{2}+b \right ) }}+{\frac{3\,{b}^{2}\ln \left ( c{x}^{2}+b \right ) }{2\,{c}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11/(c*x^4+b*x^2)^2,x)

[Out]

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

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Maxima [A]  time = 0.696912, size = 73, normalized size = 1.28 \[ \frac{b^{3}}{2 \,{\left (c^{5} x^{2} + b c^{4}\right )}} + \frac{3 \, b^{2} \log \left (c x^{2} + b\right )}{2 \, c^{4}} + \frac{c x^{4} - 4 \, b x^{2}}{4 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(c*x^4 + b*x^2)^2,x, algorithm="maxima")

[Out]

1/2*b^3/(c^5*x^2 + b*c^4) + 3/2*b^2*log(c*x^2 + b)/c^4 + 1/4*(c*x^4 - 4*b*x^2)/c
^3

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Fricas [A]  time = 0.250712, size = 95, normalized size = 1.67 \[ \frac{c^{3} x^{6} - 3 \, b c^{2} x^{4} - 4 \, b^{2} c x^{2} + 2 \, b^{3} + 6 \,{\left (b^{2} c x^{2} + b^{3}\right )} \log \left (c x^{2} + b\right )}{4 \,{\left (c^{5} x^{2} + b c^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(c*x^4 + b*x^2)^2,x, algorithm="fricas")

[Out]

1/4*(c^3*x^6 - 3*b*c^2*x^4 - 4*b^2*c*x^2 + 2*b^3 + 6*(b^2*c*x^2 + b^3)*log(c*x^2
+ b))/(c^5*x^2 + b*c^4)

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Sympy [A]  time = 1.57398, size = 53, normalized size = 0.93 \[ \frac{b^{3}}{2 b c^{4} + 2 c^{5} x^{2}} + \frac{3 b^{2} \log{\left (b + c x^{2} \right )}}{2 c^{4}} - \frac{b x^{2}}{c^{3}} + \frac{x^{4}}{4 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**11/(c*x**4+b*x**2)**2,x)

[Out]

b**3/(2*b*c**4 + 2*c**5*x**2) + 3*b**2*log(b + c*x**2)/(2*c**4) - b*x**2/c**3 +
x**4/(4*c**2)

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GIAC/XCAS [A]  time = 0.27074, size = 90, normalized size = 1.58 \[ \frac{3 \, b^{2}{\rm ln}\left ({\left | c x^{2} + b \right |}\right )}{2 \, c^{4}} + \frac{c^{2} x^{4} - 4 \, b c x^{2}}{4 \, c^{4}} - \frac{3 \, b^{2} c x^{2} + 2 \, b^{3}}{2 \,{\left (c x^{2} + b\right )} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(c*x^4 + b*x^2)^2,x, algorithm="giac")

[Out]

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