3.670 \(\int \frac{x^{11}}{\left (a+c x^4\right )^3} \, dx\)

Optimal. Leaf size=52 \[ -\frac{a^2}{8 c^3 \left (a+c x^4\right )^2}+\frac{a}{2 c^3 \left (a+c x^4\right )}+\frac{\log \left (a+c x^4\right )}{4 c^3} \]

[Out]

-a^2/(8*c^3*(a + c*x^4)^2) + a/(2*c^3*(a + c*x^4)) + Log[a + c*x^4]/(4*c^3)

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Rubi [A]  time = 0.0832737, antiderivative size = 52, 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 \[ -\frac{a^2}{8 c^3 \left (a+c x^4\right )^2}+\frac{a}{2 c^3 \left (a+c x^4\right )}+\frac{\log \left (a+c x^4\right )}{4 c^3} \]

Antiderivative was successfully verified.

[In]  Int[x^11/(a + c*x^4)^3,x]

[Out]

-a^2/(8*c^3*(a + c*x^4)^2) + a/(2*c^3*(a + c*x^4)) + Log[a + c*x^4]/(4*c^3)

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Rubi in Sympy [A]  time = 11.6202, size = 42, normalized size = 0.81 \[ - \frac{a^{2}}{8 c^{3} \left (a + c x^{4}\right )^{2}} + \frac{a}{2 c^{3} \left (a + c x^{4}\right )} + \frac{\log{\left (a + c x^{4} \right )}}{4 c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11/(c*x**4+a)**3,x)

[Out]

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

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Mathematica [A]  time = 0.0324476, size = 39, normalized size = 0.75 \[ \frac{\frac{a \left (3 a+4 c x^4\right )}{\left (a+c x^4\right )^2}+2 \log \left (a+c x^4\right )}{8 c^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^11/(a + c*x^4)^3,x]

[Out]

((a*(3*a + 4*c*x^4))/(a + c*x^4)^2 + 2*Log[a + c*x^4])/(8*c^3)

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Maple [A]  time = 0.016, size = 47, normalized size = 0.9 \[ -{\frac{{a}^{2}}{8\,{c}^{3} \left ( c{x}^{4}+a \right ) ^{2}}}+{\frac{a}{2\,{c}^{3} \left ( c{x}^{4}+a \right ) }}+{\frac{\ln \left ( c{x}^{4}+a \right ) }{4\,{c}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11/(c*x^4+a)^3,x)

[Out]

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

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Maxima [A]  time = 1.45577, size = 74, normalized size = 1.42 \[ \frac{4 \, a c x^{4} + 3 \, a^{2}}{8 \,{\left (c^{5} x^{8} + 2 \, a c^{4} x^{4} + a^{2} c^{3}\right )}} + \frac{\log \left (c x^{4} + a\right )}{4 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(c*x^4 + a)^3,x, algorithm="maxima")

[Out]

1/8*(4*a*c*x^4 + 3*a^2)/(c^5*x^8 + 2*a*c^4*x^4 + a^2*c^3) + 1/4*log(c*x^4 + a)/c
^3

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Fricas [A]  time = 0.218644, size = 93, normalized size = 1.79 \[ \frac{4 \, a c x^{4} + 3 \, a^{2} + 2 \,{\left (c^{2} x^{8} + 2 \, a c x^{4} + a^{2}\right )} \log \left (c x^{4} + a\right )}{8 \,{\left (c^{5} x^{8} + 2 \, a c^{4} x^{4} + a^{2} c^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(c*x^4 + a)^3,x, algorithm="fricas")

[Out]

1/8*(4*a*c*x^4 + 3*a^2 + 2*(c^2*x^8 + 2*a*c*x^4 + a^2)*log(c*x^4 + a))/(c^5*x^8
+ 2*a*c^4*x^4 + a^2*c^3)

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Sympy [A]  time = 5.03825, size = 53, normalized size = 1.02 \[ \frac{3 a^{2} + 4 a c x^{4}}{8 a^{2} c^{3} + 16 a c^{4} x^{4} + 8 c^{5} x^{8}} + \frac{\log{\left (a + c x^{4} \right )}}{4 c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**11/(c*x**4+a)**3,x)

[Out]

(3*a**2 + 4*a*c*x**4)/(8*a**2*c**3 + 16*a*c**4*x**4 + 8*c**5*x**8) + log(a + c*x
**4)/(4*c**3)

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GIAC/XCAS [A]  time = 0.226214, size = 57, normalized size = 1.1 \[ \frac{{\rm ln}\left ({\left | c x^{4} + a \right |}\right )}{4 \, c^{3}} - \frac{3 \, c x^{8} + 2 \, a x^{4}}{8 \,{\left (c x^{4} + a\right )}^{2} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(c*x^4 + a)^3,x, algorithm="giac")

[Out]

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