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

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

________________________________________________________________________________________

Rubi [A]  time = 0.0299033, 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, Rules used = {266, 43} \[ -\frac{x^2}{250}-\frac{6}{625 \left (2-5 x^2\right )}+\frac{2}{625 \left (2-5 x^2\right )^2}-\frac{3}{625} \log \left (2-5 x^2\right ) \]

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x^7}{\left (2-5 x^2\right )^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^3}{(2-5 x)^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{1}{125}-\frac{8}{125 (-2+5 x)^3}-\frac{12}{125 (-2+5 x)^2}-\frac{6}{125 (-2+5 x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{x^2}{250}+\frac{2}{625 \left (2-5 x^2\right )^2}-\frac{6}{625 \left (2-5 x^2\right )}-\frac{3}{625} \log \left (2-5 x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0134481, 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)

________________________________________________________________________________________

Maple [A]  time = 0.01, size = 39, normalized size = 0.9 \begin{align*} -{\frac{{x}^{2}}{250}}+{\frac{2}{625\, \left ( 5\,{x}^{2}-2 \right ) ^{2}}}-{\frac{3\,\ln \left ( 5\,{x}^{2}-2 \right ) }{625}}+{\frac{6}{3125\,{x}^{2}-1250}} \end{align*}

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

________________________________________________________________________________________

Maxima [A]  time = 0.92504, size = 53, normalized size = 1.15 \begin{align*} -\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 ) \end{align*}

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)

________________________________________________________________________________________

Fricas [A]  time = 1.70484, size = 143, normalized size = 3.11 \begin{align*} -\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 )}} \end{align*}

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) + 20)/(25*x^4 - 20*x^2 + 4)

________________________________________________________________________________________

Sympy [A]  time = 0.121358, size = 34, normalized size = 0.74 \begin{align*} - \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} \end{align*}

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

________________________________________________________________________________________

Giac [A]  time = 1.07179, size = 54, normalized size = 1.17 \begin{align*} -\frac{1}{250} \, x^{2} + \frac{225 \, x^{4} - 120 \, x^{2} + 16}{1250 \,{\left (5 \, x^{2} - 2\right )}^{2}} - \frac{3}{625} \, \log \left ({\left | 5 \, x^{2} - 2 \right |}\right ) \end{align*}

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*log(abs(5*x^2 - 2))