3.212 \(\int \frac{x^7}{\left (b x^2+c x^4\right )^3} \, dx\)

Optimal. Leaf size=23 \[ -\frac{x^4}{4 c \left (b x^2+c x^4\right )^2} \]

[Out]

-x^4/(4*c*(b*x^2 + c*x^4)^2)

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Rubi [A]  time = 0.0135321, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ -\frac{x^4}{4 c \left (b x^2+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x^7/(b*x^2 + c*x^4)^3,x]

[Out]

-x^4/(4*c*(b*x^2 + c*x^4)^2)

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Rubi in Sympy [A]  time = 4.07219, size = 14, normalized size = 0.61 \[ - \frac{1}{4 c \left (b + c x^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(c*x**4+b*x**2)**3,x)

[Out]

-1/(4*c*(b + c*x**2)**2)

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Mathematica [A]  time = 0.00520676, size = 16, normalized size = 0.7 \[ -\frac{1}{4 c \left (b+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/(b*x^2 + c*x^4)^3,x]

[Out]

-1/(4*c*(b + c*x^2)^2)

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Maple [A]  time = 0.001, size = 15, normalized size = 0.7 \[ -{\frac{1}{4\,c \left ( c{x}^{2}+b \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(c*x^4+b*x^2)^3,x)

[Out]

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

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Maxima [A]  time = 0.689434, size = 35, normalized size = 1.52 \[ -\frac{1}{4 \,{\left (c^{3} x^{4} + 2 \, b c^{2} x^{2} + b^{2} c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(c*x^4 + b*x^2)^3,x, algorithm="maxima")

[Out]

-1/4/(c^3*x^4 + 2*b*c^2*x^2 + b^2*c)

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Fricas [A]  time = 0.247427, size = 35, normalized size = 1.52 \[ -\frac{1}{4 \,{\left (c^{3} x^{4} + 2 \, b c^{2} x^{2} + b^{2} c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(c*x^4 + b*x^2)^3,x, algorithm="fricas")

[Out]

-1/4/(c^3*x^4 + 2*b*c^2*x^2 + b^2*c)

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Sympy [A]  time = 1.53875, size = 27, normalized size = 1.17 \[ - \frac{1}{4 b^{2} c + 8 b c^{2} x^{2} + 4 c^{3} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(c*x**4+b*x**2)**3,x)

[Out]

-1/(4*b**2*c + 8*b*c**2*x**2 + 4*c**3*x**4)

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GIAC/XCAS [A]  time = 0.271606, size = 19, normalized size = 0.83 \[ -\frac{1}{4 \,{\left (c x^{2} + b\right )}^{2} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(c*x^4 + b*x^2)^3,x, algorithm="giac")

[Out]

-1/4/((c*x^2 + b)^2*c)