3.26.13 \(\int \frac {e^{4 \log ^2(2)} (40-24 x^2) (100 x^2-40 x^4+4 x^6)^4}{-5 x+x^3} \, dx\)

Optimal. Leaf size=28 \[ e^3-e^{4 \left (\log ^2(2)+\log \left (4 x^2 \left (-5+x^2\right )^2\right )\right )} \]

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Rubi [A]  time = 0.06, antiderivative size = 21, normalized size of antiderivative = 0.75, number of steps used = 3, number of rules used = 3, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.070, Rules used = {12, 1586, 1588} \begin {gather*} -256 \left (5 x-x^3\right )^8 e^{4 \log ^2(2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(4*Log[2]^2)*(40 - 24*x^2)*(100*x^2 - 40*x^4 + 4*x^6)^4)/(-5*x + x^3),x]

[Out]

-256*E^(4*Log[2]^2)*(5*x - x^3)^8

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 1588

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*x^(p - q
+ 1)*Qq^(m + 1))/((p + m*q + 1)*Coeff[Qq, x, q]), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=e^{4 \log ^2(2)} \int \frac {\left (40-24 x^2\right ) \left (100 x^2-40 x^4+4 x^6\right )^4}{-5 x+x^3} \, dx\\ &=e^{4 \log ^2(2)} \int \left (40-24 x^2\right ) \left (-5 x+x^3\right )^3 \left (-20 x+4 x^3\right )^4 \, dx\\ &=-256 e^{4 \log ^2(2)} \left (5 x-x^3\right )^8\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.01, size = 66, normalized size = 2.36 \begin {gather*} -2048 e^{4 \log ^2(2)} \left (\frac {390625 x^8}{8}-78125 x^{10}+\frac {109375 x^{12}}{2}-21875 x^{14}+\frac {21875 x^{16}}{4}-875 x^{18}+\frac {175 x^{20}}{2}-5 x^{22}+\frac {x^{24}}{8}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(4*Log[2]^2)*(40 - 24*x^2)*(100*x^2 - 40*x^4 + 4*x^6)^4)/(-5*x + x^3),x]

[Out]

-2048*E^(4*Log[2]^2)*((390625*x^8)/8 - 78125*x^10 + (109375*x^12)/2 - 21875*x^14 + (21875*x^16)/4 - 875*x^18 +
(175*x^20)/2 - 5*x^22 + x^24/8)

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fricas [A]  time = 0.63, size = 53, normalized size = 1.89 \begin {gather*} -256 \, {\left (x^{24} - 40 \, x^{22} + 700 \, x^{20} - 7000 \, x^{18} + 43750 \, x^{16} - 175000 \, x^{14} + 437500 \, x^{12} - 625000 \, x^{10} + 390625 \, x^{8}\right )} e^{\left (4 \, \log \relax (2)^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-24*x^2+40)*exp(4*log(4*x^6-40*x^4+100*x^2)+4*log(2)^2)/(x^3-5*x),x, algorithm="fricas")

[Out]

-256*(x^24 - 40*x^22 + 700*x^20 - 7000*x^18 + 43750*x^16 - 175000*x^14 + 437500*x^12 - 625000*x^10 + 390625*x^
8)*e^(4*log(2)^2)

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giac [B]  time = 0.13, size = 109, normalized size = 3.89 \begin {gather*} -256 \, x^{24} e^{\left (4 \, \log \relax (2)^{2}\right )} + 10240 \, x^{22} e^{\left (4 \, \log \relax (2)^{2}\right )} - 179200 \, x^{20} e^{\left (4 \, \log \relax (2)^{2}\right )} + 1792000 \, x^{18} e^{\left (4 \, \log \relax (2)^{2}\right )} - 11200000 \, x^{16} e^{\left (4 \, \log \relax (2)^{2}\right )} + 44800000 \, x^{14} e^{\left (4 \, \log \relax (2)^{2}\right )} - 112000000 \, x^{12} e^{\left (4 \, \log \relax (2)^{2}\right )} + 160000000 \, x^{10} e^{\left (4 \, \log \relax (2)^{2}\right )} - 100000000 \, x^{8} e^{\left (4 \, \log \relax (2)^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-24*x^2+40)*exp(4*log(4*x^6-40*x^4+100*x^2)+4*log(2)^2)/(x^3-5*x),x, algorithm="giac")

[Out]

-256*x^24*e^(4*log(2)^2) + 10240*x^22*e^(4*log(2)^2) - 179200*x^20*e^(4*log(2)^2) + 1792000*x^18*e^(4*log(2)^2
) - 11200000*x^16*e^(4*log(2)^2) + 44800000*x^14*e^(4*log(2)^2) - 112000000*x^12*e^(4*log(2)^2) + 160000000*x^
10*e^(4*log(2)^2) - 100000000*x^8*e^(4*log(2)^2)

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maple [A]  time = 0.38, size = 30, normalized size = 1.07

method	result	size
gosper	\(-{\mathrm e}^{4 \ln \left (4 x^{6}-40 x^{4}+100 x^{2}\right )+4 \ln \relax (2)^{2}}\)	\(30\)
default	\(2048 \,{\mathrm e}^{4 \ln \relax (2)^{2}} \left (-\frac {1}{8} x^{24}+5 x^{22}-\frac {175}{2} x^{20}+875 x^{18}-\frac {21875}{4} x^{16}+21875 x^{14}-\frac {109375}{2} x^{12}+78125 x^{10}-\frac {390625}{8} x^{8}\right )\)	\(56\)
norman	\(-100000000 \,{\mathrm e}^{4 \ln \relax (2)^{2}} x^{8}[phone] \,{\mathrm e}^{4 \ln \relax (2)^{2}} x^{10}-112000000 \,{\mathrm e}^{4 \ln \relax (2)^{2}} x^{12}[phone] \,{\mathrm e}^{4 \ln \relax (2)^{2}} x^{14}-11200000 \,{\mathrm e}^{4 \ln \relax (2)^{2}} x^{16}+1792000 \,{\mathrm e}^{4 \ln \relax (2)^{2}} x^{18}-179200 \,{\mathrm e}^{4 \ln \relax (2)^{2}} x^{20}+10240 \,{\mathrm e}^{4 \ln \relax (2)^{2}} x^{22}-256 \,{\mathrm e}^{4 \ln \relax (2)^{2}} x^{24}\)	\(110\)
risch	\(-100000000 \,{\mathrm e}^{4 \ln \relax (2)^{2}} x^{8}[phone] \,{\mathrm e}^{4 \ln \relax (2)^{2}} x^{10}-112000000 \,{\mathrm e}^{4 \ln \relax (2)^{2}} x^{12}[phone] \,{\mathrm e}^{4 \ln \relax (2)^{2}} x^{14}-11200000 \,{\mathrm e}^{4 \ln \relax (2)^{2}} x^{16}+1792000 \,{\mathrm e}^{4 \ln \relax (2)^{2}} x^{18}-179200 \,{\mathrm e}^{4 \ln \relax (2)^{2}} x^{20}+10240 \,{\mathrm e}^{4 \ln \relax (2)^{2}} x^{22}-256 \,{\mathrm e}^{4 \ln \relax (2)^{2}} x^{24}\)	\(110\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-24*x^2+40)*exp(4*ln(4*x^6-40*x^4+100*x^2)+4*ln(2)^2)/(x^3-5*x),x,method=_RETURNVERBOSE)

[Out]

-exp(4*ln(4*x^6-40*x^4+100*x^2)+4*ln(2)^2)

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maxima [A]  time = 0.38, size = 53, normalized size = 1.89 \begin {gather*} -256 \, {\left (x^{24} - 40 \, x^{22} + 700 \, x^{20} - 7000 \, x^{18} + 43750 \, x^{16} - 175000 \, x^{14} + 437500 \, x^{12} - 625000 \, x^{10} + 390625 \, x^{8}\right )} e^{\left (4 \, \log \relax (2)^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-24*x^2+40)*exp(4*log(4*x^6-40*x^4+100*x^2)+4*log(2)^2)/(x^3-5*x),x, algorithm="maxima")

[Out]

-256*(x^24 - 40*x^22 + 700*x^20 - 7000*x^18 + 43750*x^16 - 175000*x^14 + 437500*x^12 - 625000*x^10 + 390625*x^
8)*e^(4*log(2)^2)

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mupad [B]  time = 0.14, size = 19, normalized size = 0.68 \begin {gather*} -256\,x^8\,{\mathrm {e}}^{4\,{\ln \relax (2)}^2}\,{\left (x^2-5\right )}^8 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(4*log(100*x^2 - 40*x^4 + 4*x^6) + 4*log(2)^2)*(24*x^2 - 40))/(5*x - x^3),x)

[Out]

-256*x^8*exp(4*log(2)^2)*(x^2 - 5)^8

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sympy [B]  time = 0.12, size = 121, normalized size = 4.32 \begin {gather*} - 256 x^{24} e^{4 \log {\relax (2 )}^{2}} + 10240 x^{22} e^{4 \log {\relax (2 )}^{2}} - 179200 x^{20} e^{4 \log {\relax (2 )}^{2}} + 1792000 x^{18} e^{4 \log {\relax (2 )}^{2}} - 11200000 x^{16} e^{4 \log {\relax (2 )}^{2}} + 44800000 x^{14} e^{4 \log {\relax (2 )}^{2}} - 112000000 x^{12} e^{4 \log {\relax (2 )}^{2}} + 160000000 x^{10} e^{4 \log {\relax (2 )}^{2}} - 100000000 x^{8} e^{4 \log {\relax (2 )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-24*x**2+40)*exp(4*ln(4*x**6-40*x**4+100*x**2)+4*ln(2)**2)/(x**3-5*x),x)

[Out]

-256*x**24*exp(4*log(2)**2) + 10240*x**22*exp(4*log(2)**2) - 179200*x**20*exp(4*log(2)**2) + 1792000*x**18*exp
(4*log(2)**2) - 11200000*x**16*exp(4*log(2)**2) + 44800000*x**14*exp(4*log(2)**2) - 112000000*x**12*exp(4*log(
2)**2) + 160000000*x**10*exp(4*log(2)**2) - 100000000*x**8*exp(4*log(2)**2)

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