3.210 \(\int \frac{x^9}{\left (b x^2+c x^4\right )^3} \, dx\)

Optimal. Leaf size=19 \[ \frac{x^4}{4 b \left (b+c x^2\right )^2} \]

[Out]

x^4/(4*b*(b + c*x^2)^2)

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Rubi [A]  time = 0.0248147, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{x^4}{4 b \left (b+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x^9/(b*x^2 + c*x^4)^3,x]

[Out]

x^4/(4*b*(b + c*x^2)^2)

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Rubi in Sympy [A]  time = 5.23147, size = 14, normalized size = 0.74 \[ \frac{x^{4}}{4 b \left (b + c x^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9/(c*x**4+b*x**2)**3,x)

[Out]

x**4/(4*b*(b + c*x**2)**2)

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Mathematica [A]  time = 0.0122253, size = 24, normalized size = 1.26 \[ -\frac{b+2 c x^2}{4 c^2 \left (b+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^9/(b*x^2 + c*x^4)^3,x]

[Out]

-(b + 2*c*x^2)/(4*c^2*(b + c*x^2)^2)

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Maple [A]  time = 0.011, size = 31, normalized size = 1.6 \[ -{\frac{1}{ \left ( 2\,c{x}^{2}+2\,b \right ){c}^{2}}}+{\frac{b}{4\,{c}^{2} \left ( c{x}^{2}+b \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9/(c*x^4+b*x^2)^3,x)

[Out]

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

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Maxima [A]  time = 0.708922, size = 49, normalized size = 2.58 \[ -\frac{2 \, c x^{2} + b}{4 \,{\left (c^{4} x^{4} + 2 \, b c^{3} x^{2} + b^{2} c^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(c*x^4 + b*x^2)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.250397, size = 49, normalized size = 2.58 \[ -\frac{2 \, c x^{2} + b}{4 \,{\left (c^{4} x^{4} + 2 \, b c^{3} x^{2} + b^{2} c^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(c*x^4 + b*x^2)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.6502, size = 36, normalized size = 1.89 \[ - \frac{b + 2 c x^{2}}{4 b^{2} c^{2} + 8 b c^{3} x^{2} + 4 c^{4} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9/(c*x**4+b*x**2)**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.269287, size = 30, normalized size = 1.58 \[ -\frac{2 \, c x^{2} + b}{4 \,{\left (c x^{2} + b\right )}^{2} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(c*x^4 + b*x^2)^3,x, algorithm="giac")

[Out]

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