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

Optimal. Leaf size=46 \[ \frac{x^2}{2}+\frac{75}{2 \left (5-x^2\right )}-\frac{125}{4 \left (5-x^2\right )^2}+\frac{15}{2} \log \left (5-x^2\right ) \]

[Out]

x^2/2 - 125/(4*(5 - x^2)^2) + 75/(2*(5 - x^2)) + (15*Log[5 - x^2])/2

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Rubi [A]  time = 0.0237345, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {266, 43} \[ \frac{x^2}{2}+\frac{75}{2 \left (5-x^2\right )}-\frac{125}{4 \left (5-x^2\right )^2}+\frac{15}{2} \log \left (5-x^2\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^7/(-5 + x^2)^3,x]

[Out]

x^2/2 - 125/(4*(5 - x^2)^2) + 75/(2*(5 - x^2)) + (15*Log[5 - x^2])/2

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 (-5+x^2\right )^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^3}{(-5+x)^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (1+\frac{125}{(-5+x)^3}+\frac{75}{(-5+x)^2}+\frac{15}{-5+x}\right ) \, dx,x,x^2\right )\\ &=\frac{x^2}{2}-\frac{125}{4 \left (5-x^2\right )^2}+\frac{75}{2 \left (5-x^2\right )}+\frac{15}{2} \log \left (5-x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0127481, size = 36, normalized size = 0.78 \[ \frac{1}{4} \left (2 x^2-\frac{150}{x^2-5}-\frac{125}{\left (x^2-5\right )^2}+30 \log \left (x^2-5\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^7/(-5 + x^2)^3,x]

[Out]

(2*x^2 - 125/(-5 + x^2)^2 - 150/(-5 + x^2) + 30*Log[-5 + x^2])/4

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Maple [A]  time = 0.009, size = 33, normalized size = 0.7 \begin{align*}{\frac{{x}^{2}}{2}}-{\frac{75}{2\,{x}^{2}-10}}+{\frac{15\,\ln \left ({x}^{2}-5 \right ) }{2}}-{\frac{125}{4\, \left ({x}^{2}-5 \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(x^2-5)^3,x)

[Out]

1/2*x^2-75/2/(x^2-5)+15/2*ln(x^2-5)-125/4/(x^2-5)^2

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Maxima [A]  time = 0.947302, size = 47, normalized size = 1.02 \begin{align*} \frac{1}{2} \, x^{2} - \frac{25 \,{\left (6 \, x^{2} - 25\right )}}{4 \,{\left (x^{4} - 10 \, x^{2} + 25\right )}} + \frac{15}{2} \, \log \left (x^{2} - 5\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 - 25/4*(6*x^2 - 25)/(x^4 - 10*x^2 + 25) + 15/2*log(x^2 - 5)

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Fricas [A]  time = 1.86473, size = 130, normalized size = 2.83 \begin{align*} \frac{2 \, x^{6} - 20 \, x^{4} - 100 \, x^{2} + 30 \,{\left (x^{4} - 10 \, x^{2} + 25\right )} \log \left (x^{2} - 5\right ) + 625}{4 \,{\left (x^{4} - 10 \, x^{2} + 25\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*x^6 - 20*x^4 - 100*x^2 + 30*(x^4 - 10*x^2 + 25)*log(x^2 - 5) + 625)/(x^4 - 10*x^2 + 25)

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Sympy [A]  time = 0.111764, size = 32, normalized size = 0.7 \begin{align*} \frac{x^{2}}{2} - \frac{150 x^{2} - 625}{4 x^{4} - 40 x^{2} + 100} + \frac{15 \log{\left (x^{2} - 5 \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7/(x**2-5)**3,x)

[Out]

x**2/2 - (150*x**2 - 625)/(4*x**4 - 40*x**2 + 100) + 15*log(x**2 - 5)/2

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Giac [A]  time = 1.05348, size = 49, normalized size = 1.07 \begin{align*} \frac{1}{2} \, x^{2} - \frac{5 \,{\left (9 \, x^{4} - 60 \, x^{2} + 100\right )}}{4 \,{\left (x^{2} - 5\right )}^{2}} + \frac{15}{2} \, \log \left ({\left | x^{2} - 5 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 - 5/4*(9*x^4 - 60*x^2 + 100)/(x^2 - 5)^2 + 15/2*log(abs(x^2 - 5))