3.643 \(\int \frac{x^7}{a+c x^4} \, dx\)

Optimal. Leaf size=27 \[ \frac{x^4}{4 c}-\frac{a \log \left (a+c x^4\right )}{4 c^2} \]

[Out]

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

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Rubi [A]  time = 0.0465332, antiderivative size = 27, 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 \[ \frac{x^4}{4 c}-\frac{a \log \left (a+c x^4\right )}{4 c^2} \]

Antiderivative was successfully verified.

[In]  Int[x^7/(a + c*x^4),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*log(a + c*x**4)/(4*c**2) + Integral(1/c, (x, x**4))/4

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Mathematica [A]  time = 0.0104449, size = 27, normalized size = 1. \[ \frac{x^4}{4 c}-\frac{a \log \left (a+c x^4\right )}{4 c^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/(a + c*x^4),x]

[Out]

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

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Maple [A]  time = 0.004, size = 24, normalized size = 0.9 \[{\frac{{x}^{4}}{4\,c}}-{\frac{a\ln \left ( c{x}^{4}+a \right ) }{4\,{c}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(c*x^4+a),x)

[Out]

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

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Maxima [A]  time = 1.43794, size = 31, normalized size = 1.15 \[ \frac{x^{4}}{4 \, c} - \frac{a \log \left (c x^{4} + a\right )}{4 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*x^4/c - 1/4*a*log(c*x^4 + a)/c^2

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Fricas [A]  time = 0.22011, size = 30, normalized size = 1.11 \[ \frac{c x^{4} - a \log \left (c x^{4} + a\right )}{4 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(c*x^4 - a*log(c*x^4 + a))/c^2

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Sympy [A]  time = 1.33632, size = 20, normalized size = 0.74 \[ - \frac{a \log{\left (a + c x^{4} \right )}}{4 c^{2}} + \frac{x^{4}}{4 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(c*x**4+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.221271, size = 32, normalized size = 1.19 \[ \frac{x^{4}}{4 \, c} - \frac{a{\rm ln}\left ({\left | c x^{4} + a \right |}\right )}{4 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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