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

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

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Rubi [A]  time = 0.03, antiderivative size = 46, normalized size of antiderivative = 1.00, 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 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])

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]]

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*}

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Mathematica [A]  time = 0.02, 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]

-1/1250*(12 - 150*x^4 + 125*x^6 + 6*(2 - 5*x^2)^2*Log[-2 + 5*x^2])/(2 - 5*x^2)^2

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IntegrateAlgebraic [A]  time = 0.02, size = 38, normalized size = 0.83 \[ \frac {-125 x^6+150 x^4-12}{1250 \left (5 x^2-2\right )^2}-\frac {3}{625} \log \left (5 x^2-2\right ) \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^7/(2 - 5*x^2)^3,x]

[Out]

(-12 + 150*x^4 - 125*x^6)/(1250*(-2 + 5*x^2)^2) - (3*Log[-2 + 5*x^2])/625

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fricas [A]  time = 1.21, size = 55, normalized size = 1.20 \[ -\frac {125 \, x^{6} - 100 \, x^{4} - 40 \, x^{2} + 6 \, {\left (25 \, x^{4} - 20 \, x^{2} + 4\right )} \log \left (5 \, x^{2} - 2\right ) + 20}{1250 \, {\left (25 \, x^{4} - 20 \, x^{2} + 4\right )}} \]

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) + 20)/(25*x^4 - 20*x^2 + 4)

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giac [A]  time = 1.03, size = 40, normalized size = 0.87 \[ -\frac {1}{250} \, x^{2} + \frac {225 \, x^{4} - 120 \, x^{2} + 16}{1250 \, {\left (5 \, x^{2} - 2\right )}^{2}} - \frac {3}{625} \, \log \left ({\left | 5 \, x^{2} - 2 \right |}\right ) \]

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))

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maple [A]  time = 0.28, size = 35, normalized size = 0.76

method	result	size
risch	\(-\frac {x^{2}}{250}+\frac {\frac {6 x^{2}}{125}-\frac {2}{125}}{\left (5 x^{2}-2\right )^{2}}-\frac {3 \ln \left (5 x^{2}-2\right )}{625}\)	\(35\)
norman	\(\frac {-\frac {6}{125} x^{2}+\frac {9}{50} x^{4}-\frac {1}{10} x^{6}}{\left (5 x^{2}-2\right )^{2}}-\frac {3 \ln \left (5 x^{2}-2\right )}{625}\)	\(38\)
meijerg	\(-\frac {x^{2} \left (25 x^{4}-45 x^{2}+12\right )}{1000 \left (1-\frac {5 x^{2}}{2}\right )^{2}}-\frac {3 \ln \left (1-\frac {5 x^{2}}{2}\right )}{625}\)	\(38\)
default	\(-\frac {x^{2}}{250}+\frac {6}{625 \left (5 x^{2}-2\right )}+\frac {2}{625 \left (5 x^{2}-2\right )^{2}}-\frac {3 \ln \left (5 x^{2}-2\right )}{625}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(-5*x^2+2)^3,x,method=_RETURNVERBOSE)

[Out]

-1/250*x^2+25*(6/3125*x^2-2/3125)/(5*x^2-2)^2-3/625*ln(5*x^2-2)

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maxima [A]  time = 0.53, size = 39, normalized size = 0.85 \[ -\frac {1}{250} \, x^{2} + \frac {2 \, {\left (3 \, x^{2} - 1\right )}}{125 \, {\left (25 \, x^{4} - 20 \, x^{2} + 4\right )}} - \frac {3}{625} \, \log \left (5 \, x^{2} - 2\right ) \]

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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mupad [B]  time = 0.19, size = 34, normalized size = 0.74 \[ \frac {\frac {6\,x^2}{3125}-\frac {2}{3125}}{x^4-\frac {4\,x^2}{5}+\frac {4}{25}}-\frac {3\,\ln \left (x^2-\frac {2}{5}\right )}{625}-\frac {x^2}{250} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-x^7/(5*x^2 - 2)^3,x)

[Out]

((6*x^2)/3125 - 2/3125)/(x^4 - (4*x^2)/5 + 4/25) - (3*log(x^2 - 2/5))/625 - x^2/250

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sympy [A]  time = 0.13, size = 36, normalized size = 0.78 \[ - \frac {x^{2}}{250} - \frac {2 - 6 x^{2}}{3125 x^{4} - 2500 x^{2} + 500} - \frac {3 \log {\left (5 x^{2} - 2 \right )}}{625} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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