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

Optimal. Leaf size=46 \[ -\frac{a^2}{4 c^3 \left (a+c x^4\right )}-\frac{a \log \left (a+c x^4\right )}{2 c^3}+\frac{x^4}{4 c^2} \]

[Out]

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

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Rubi [A]  time = 0.0758085, antiderivative size = 46, 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 \[ -\frac{a^2}{4 c^3 \left (a+c x^4\right )}-\frac{a \log \left (a+c x^4\right )}{2 c^3}+\frac{x^4}{4 c^2} \]

Antiderivative was successfully verified.

[In]  Int[x^11/(a + c*x^4)^2,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2}}{4 c^{3} \left (a + c x^{4}\right )} - \frac{a \log{\left (a + c x^{4} \right )}}{2 c^{3}} + \frac{\int ^{x^{4}} \frac{1}{c^{2}}\, dx}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**2/(4*c**3*(a + c*x**4)) - a*log(a + c*x**4)/(2*c**3) + Integral(c**(-2), (x,
x**4))/4

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Mathematica [A]  time = 0.0344519, size = 38, normalized size = 0.83 \[ \frac{-\frac{a^2}{a+c x^4}-2 a \log \left (a+c x^4\right )+c x^4}{4 c^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^11/(a + c*x^4)^2,x]

[Out]

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

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Maple [A]  time = 0.019, size = 41, normalized size = 0.9 \[{\frac{{x}^{4}}{4\,{c}^{2}}}-{\frac{{a}^{2}}{4\,{c}^{3} \left ( c{x}^{4}+a \right ) }}-{\frac{a\ln \left ( c{x}^{4}+a \right ) }{2\,{c}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11/(c*x^4+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.43671, size = 58, normalized size = 1.26 \[ \frac{x^{4}}{4 \, c^{2}} - \frac{a^{2}}{4 \,{\left (c^{4} x^{4} + a c^{3}\right )}} - \frac{a \log \left (c x^{4} + a\right )}{2 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.224977, size = 76, normalized size = 1.65 \[ \frac{c^{2} x^{8} + a c x^{4} - a^{2} - 2 \,{\left (a c x^{4} + a^{2}\right )} \log \left (c x^{4} + a\right )}{4 \,{\left (c^{4} x^{4} + a c^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(c^2*x^8 + a*c*x^4 - a^2 - 2*(a*c*x^4 + a^2)*log(c*x^4 + a))/(c^4*x^4 + a*c^
3)

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Sympy [A]  time = 2.04735, size = 41, normalized size = 0.89 \[ - \frac{a^{2}}{4 a c^{3} + 4 c^{4} x^{4}} - \frac{a \log{\left (a + c x^{4} \right )}}{2 c^{3}} + \frac{x^{4}}{4 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.22016, size = 66, normalized size = 1.43 \[ \frac{x^{4}}{4 \, c^{2}} - \frac{a{\rm ln}\left ({\left | c x^{4} + a \right |}\right )}{2 \, c^{3}} + \frac{2 \, a c x^{4} + a^{2}}{4 \,{\left (c x^{4} + a\right )} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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