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

Optimal. Leaf size=33 \[ \frac{a}{4 c^2 \left (a+c x^4\right )}+\frac{\log \left (a+c x^4\right )}{4 c^2} \]

[Out]

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

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Rubi [A]  time = 0.0556498, antiderivative size = 33, 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 \[ \frac{a}{4 c^2 \left (a+c x^4\right )}+\frac{\log \left (a+c x^4\right )}{4 c^2} \]

Antiderivative was successfully verified.

[In]  Int[x^7/(a + c*x^4)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 7.45886, size = 26, normalized size = 0.79 \[ \frac{a}{4 c^{2} \left (a + c x^{4}\right )} + \frac{\log{\left (a + c x^{4} \right )}}{4 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0185315, size = 27, normalized size = 0.82 \[ \frac{\frac{a}{a+c x^4}+\log \left (a+c x^4\right )}{4 c^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/(a + c*x^4)^2,x]

[Out]

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

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Maple [A]  time = 0.014, size = 30, normalized size = 0.9 \[{\frac{a}{4\,{c}^{2} \left ( c{x}^{4}+a \right ) }}+{\frac{\ln \left ( c{x}^{4}+a \right ) }{4\,{c}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(c*x^4+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.44569, size = 43, normalized size = 1.3 \[ \frac{a}{4 \,{\left (c^{3} x^{4} + a c^{2}\right )}} + \frac{\log \left (c x^{4} + a\right )}{4 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.243661, size = 47, normalized size = 1.42 \[ \frac{{\left (c x^{4} + a\right )} \log \left (c x^{4} + a\right ) + a}{4 \,{\left (c^{3} x^{4} + a c^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.8564, size = 29, normalized size = 0.88 \[ \frac{a}{4 a c^{2} + 4 c^{3} x^{4}} + \frac{\log{\left (a + c x^{4} \right )}}{4 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(c*x**4+a)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.221555, size = 65, normalized size = 1.97 \[ -\frac{\frac{{\rm ln}\left (\frac{{\left | c x^{4} + a \right |}}{{\left (c x^{4} + a\right )}^{2}{\left | c \right |}}\right )}{c} - \frac{a}{{\left (c x^{4} + a\right )} c}}{4 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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