3.185 \(\int \frac{x^7}{\left (2-5 x^2\right )^3} \, dx\)

Optimal. Leaf size=46 \[ -\frac{x^2}{250}-\frac{6}{625 \left (2-5 x^2\right )}+\frac{2}{625 \left (2-5 x^2\right )^2}-\frac{3}{625} \log \left (2-5 x^2\right ) \]

[Out]

-x^2/250 + 2/(625*(2 - 5*x^2)^2) - 6/(625*(2 - 5*x^2)) - (3*Log[2 - 5*x^2])/625

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Rubi [A]  time = 0.0608665, antiderivative size = 46, 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 \[ -\frac{x^2}{250}-\frac{6}{625 \left (2-5 x^2\right )}+\frac{2}{625 \left (2-5 x^2\right )^2}-\frac{3}{625} \log \left (2-5 x^2\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^7/(2 - 5*x^2)^3,x]

[Out]

-x^2/250 + 2/(625*(2 - 5*x^2)^2) - 6/(625*(2 - 5*x^2)) - (3*Log[2 - 5*x^2])/625

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{3 \log{\left (- 5 x^{2} + 2 \right )}}{625} + \frac{\int ^{x^{2}} \left (- \frac{1}{125}\right )\, dx}{2} - \frac{6}{625 \left (- 5 x^{2} + 2\right )} + \frac{2}{625 \left (- 5 x^{2} + 2\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(-5*x**2+2)**3,x)

[Out]

-3*log(-5*x**2 + 2)/625 + Integral(-1/125, (x, x**2))/2 - 6/(625*(-5*x**2 + 2))
+ 2/(625*(-5*x**2 + 2)**2)

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Mathematica [A]  time = 0.0210744, size = 44, normalized size = 0.96 \[ -\frac{125 x^6-150 x^4+6 \left (2-5 x^2\right )^2 \log \left (5 x^2-2\right )+12}{1250 \left (2-5 x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/(2 - 5*x^2)^3,x]

[Out]

-(12 - 150*x^4 + 125*x^6 + 6*(2 - 5*x^2)^2*Log[-2 + 5*x^2])/(1250*(2 - 5*x^2)^2)

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Maple [A]  time = 0.015, size = 39, normalized size = 0.9 \[ -{\frac{{x}^{2}}{250}}-{\frac{3\,\ln \left ( 5\,{x}^{2}-2 \right ) }{625}}+{\frac{2}{625\, \left ( 5\,{x}^{2}-2 \right ) ^{2}}}+{\frac{6}{3125\,{x}^{2}-1250}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(-5*x^2+2)^3,x)

[Out]

-1/250*x^2-3/625*ln(5*x^2-2)+2/625/(5*x^2-2)^2+6/625/(5*x^2-2)

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Maxima [A]  time = 1.43499, size = 53, normalized size = 1.15 \[ -\frac{1}{250} \, x^{2} + \frac{2 \,{\left (3 \, x^{2} - 1\right )}}{125 \,{\left (25 \, x^{4} - 20 \, x^{2} + 4\right )}} - \frac{3}{625} \, \log \left (5 \, x^{2} - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^7/(5*x^2 - 2)^3,x, algorithm="maxima")

[Out]

-1/250*x^2 + 2/125*(3*x^2 - 1)/(25*x^4 - 20*x^2 + 4) - 3/625*log(5*x^2 - 2)

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Fricas [A]  time = 0.198086, size = 74, normalized size = 1.61 \[ -\frac{125 \, x^{6} - 100 \, x^{4} - 40 \, x^{2} + 6 \,{\left (25 \, x^{4} - 20 \, x^{2} + 4\right )} \log \left (5 \, x^{2} - 2\right ) + 20}{1250 \,{\left (25 \, x^{4} - 20 \, x^{2} + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^7/(5*x^2 - 2)^3,x, algorithm="fricas")

[Out]

-1/1250*(125*x^6 - 100*x^4 - 40*x^2 + 6*(25*x^4 - 20*x^2 + 4)*log(5*x^2 - 2) + 2
0)/(25*x^4 - 20*x^2 + 4)

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Sympy [A]  time = 0.158108, size = 34, normalized size = 0.74 \[ - \frac{x^{2}}{250} + \frac{6 x^{2} - 2}{3125 x^{4} - 2500 x^{2} + 500} - \frac{3 \log{\left (5 x^{2} - 2 \right )}}{625} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(-5*x**2+2)**3,x)

[Out]

-x**2/250 + (6*x**2 - 2)/(3125*x**4 - 2500*x**2 + 500) - 3*log(5*x**2 - 2)/625

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GIAC/XCAS [A]  time = 0.202203, size = 54, normalized size = 1.17 \[ -\frac{1}{250} \, x^{2} + \frac{225 \, x^{4} - 120 \, x^{2} + 16}{1250 \,{\left (5 \, x^{2} - 2\right )}^{2}} - \frac{3}{625} \,{\rm ln}\left ({\left | 5 \, x^{2} - 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^7/(5*x^2 - 2)^3,x, algorithm="giac")

[Out]

-1/250*x^2 + 1/1250*(225*x^4 - 120*x^2 + 16)/(5*x^2 - 2)^2 - 3/625*ln(abs(5*x^2
- 2))