3.82.71 \(\int \frac {e^4 (1+2592 x-864 x^2+64 x^3)+e^2 (576 x-96 x^2) \log (5)+32 x \log ^2(5)}{e^4} \, dx\)

Optimal. Leaf size=20 \[ -1+x+16 x^2 \left (-9+x-\frac {\log (5)}{e^2}\right )^2 \]

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Rubi [B]  time = 0.02, antiderivative size = 49, normalized size of antiderivative = 2.45, number of steps used = 4, number of rules used = 1, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {12} \begin {gather*} 16 x^4-288 x^3-\frac {32 x^3 \log (5)}{e^2}+1296 x^2+\frac {16 x^2 \log ^2(5)}{e^4}+\frac {288 x^2 \log (5)}{e^2}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^4*(1 + 2592*x - 864*x^2 + 64*x^3) + E^2*(576*x - 96*x^2)*Log[5] + 32*x*Log[5]^2)/E^4,x]

[Out]

x + 1296*x^2 - 288*x^3 + 16*x^4 + (288*x^2*Log[5])/E^2 - (32*x^3*Log[5])/E^2 + (16*x^2*Log[5]^2)/E^4

Rule 12

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \left (e^4 \left (1+2592 x-864 x^2+64 x^3\right )+e^2 \left (576 x-96 x^2\right ) \log (5)+32 x \log ^2(5)\right ) \, dx}{e^4}\\ &=\frac {16 x^2 \log ^2(5)}{e^4}+\frac {\log (5) \int \left (576 x-96 x^2\right ) \, dx}{e^2}+\int \left (1+2592 x-864 x^2+64 x^3\right ) \, dx\\ &=x+1296 x^2-288 x^3+16 x^4+\frac {288 x^2 \log (5)}{e^2}-\frac {32 x^3 \log (5)}{e^2}+\frac {16 x^2 \log ^2(5)}{e^4}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.03, size = 41, normalized size = 2.05 \begin {gather*} x+16 x^4-\frac {32 x^3 \left (9 e^2+\log (5)\right )}{e^2}+\frac {16 x^2 \left (9 e^2+\log (5)\right )^2}{e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^4*(1 + 2592*x - 864*x^2 + 64*x^3) + E^2*(576*x - 96*x^2)*Log[5] + 32*x*Log[5]^2)/E^4,x]

[Out]

x + 16*x^4 - (32*x^3*(9*E^2 + Log[5]))/E^2 + (16*x^2*(9*E^2 + Log[5])^2)/E^4

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fricas [B]  time = 0.72, size = 48, normalized size = 2.40 \begin {gather*} {\left (16 \, x^{2} \log \relax (5)^{2} - 32 \, {\left (x^{3} - 9 \, x^{2}\right )} e^{2} \log \relax (5) + {\left (16 \, x^{4} - 288 \, x^{3} + 1296 \, x^{2} + x\right )} e^{4}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((32*x*log(5)^2+(-96*x^2+576*x)*exp(2)*log(5)+(64*x^3-864*x^2+2592*x+1)*exp(2)^2)/exp(2)^2,x, algorit
hm="fricas")

[Out]

(16*x^2*log(5)^2 - 32*(x^3 - 9*x^2)*e^2*log(5) + (16*x^4 - 288*x^3 + 1296*x^2 + x)*e^4)*e^(-4)

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giac [B]  time = 0.15, size = 48, normalized size = 2.40 \begin {gather*} {\left (16 \, x^{2} \log \relax (5)^{2} - 32 \, {\left (x^{3} - 9 \, x^{2}\right )} e^{2} \log \relax (5) + {\left (16 \, x^{4} - 288 \, x^{3} + 1296 \, x^{2} + x\right )} e^{4}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((32*x*log(5)^2+(-96*x^2+576*x)*exp(2)*log(5)+(64*x^3-864*x^2+2592*x+1)*exp(2)^2)/exp(2)^2,x, algorit
hm="giac")

[Out]

(16*x^2*log(5)^2 - 32*(x^3 - 9*x^2)*e^2*log(5) + (16*x^4 - 288*x^3 + 1296*x^2 + x)*e^4)*e^(-4)

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maple [B]  time = 0.03, size = 47, normalized size = 2.35

method	result	size
risch	\(16 x^{4}-288 x^{3}-32 \,{\mathrm e}^{-2} \ln \relax (5) x^{3}+1296 x^{2}+288 \,{\mathrm e}^{-2} \ln \relax (5) x^{2}+16 \,{\mathrm e}^{-4} x^{2} \ln \relax (5)^{2}+x\)	\(47\)
default	\({\mathrm e}^{-4} \left (16 x^{2} \ln \relax (5)^{2}+{\mathrm e}^{2} \ln \relax (5) \left (-32 x^{3}+288 x^{2}\right )+{\mathrm e}^{4} \left (16 x^{4}-288 x^{3}+1296 x^{2}+x \right )\right )\)	\(54\)
norman	\(\left (\left (-288 \,{\mathrm e}^{2}-32 \ln \relax (5)\right ) x^{3}+{\mathrm e}^{2} x +16 x^{4} {\mathrm e}^{2}+16 \left (81 \,{\mathrm e}^{4}+18 \,{\mathrm e}^{2} \ln \relax (5)+\ln \relax (5)^{2}\right ) {\mathrm e}^{-2} x^{2}\right ) {\mathrm e}^{-2}\)	\(57\)
gosper	\(x \left (16 x^{3} {\mathrm e}^{4}-288 x^{2} {\mathrm e}^{4}-32 \,{\mathrm e}^{2} \ln \relax (5) x^{2}+1296 x \,{\mathrm e}^{4}+288 x \,{\mathrm e}^{2} \ln \relax (5)+16 x \ln \relax (5)^{2}+{\mathrm e}^{4}\right ) {\mathrm e}^{-4}\)	\(60\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((32*x*ln(5)^2+(-96*x^2+576*x)*exp(2)*ln(5)+(64*x^3-864*x^2+2592*x+1)*exp(2)^2)/exp(2)^2,x,method=_RETURNVE
RBOSE)

[Out]

16*x^4-288*x^3-32*exp(-2)*ln(5)*x^3+1296*x^2+288*exp(-2)*ln(5)*x^2+16*exp(-4)*x^2*ln(5)^2+x

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maxima [B]  time = 0.36, size = 48, normalized size = 2.40 \begin {gather*} {\left (16 \, x^{2} \log \relax (5)^{2} - 32 \, {\left (x^{3} - 9 \, x^{2}\right )} e^{2} \log \relax (5) + {\left (16 \, x^{4} - 288 \, x^{3} + 1296 \, x^{2} + x\right )} e^{4}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((32*x*log(5)^2+(-96*x^2+576*x)*exp(2)*log(5)+(64*x^3-864*x^2+2592*x+1)*exp(2)^2)/exp(2)^2,x, algorit
hm="maxima")

[Out]

(16*x^2*log(5)^2 - 32*(x^3 - 9*x^2)*e^2*log(5) + (16*x^4 - 288*x^3 + 1296*x^2 + x)*e^4)*e^(-4)

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mupad [B]  time = 0.16, size = 49, normalized size = 2.45 \begin {gather*} 16\,x^4-\frac {{\mathrm {e}}^{-4}\,\left (864\,{\mathrm {e}}^4+96\,{\mathrm {e}}^2\,\ln \relax (5)\right )\,x^3}{3}+\frac {{\mathrm {e}}^{-4}\,\left (2592\,{\mathrm {e}}^4+576\,{\mathrm {e}}^2\,\ln \relax (5)+32\,{\ln \relax (5)}^2\right )\,x^2}{2}+x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-4)*(exp(4)*(2592*x - 864*x^2 + 64*x^3 + 1) + 32*x*log(5)^2 + exp(2)*log(5)*(576*x - 96*x^2)),x)

[Out]

x + 16*x^4 + (x^2*exp(-4)*(2592*exp(4) + 576*exp(2)*log(5) + 32*log(5)^2))/2 - (x^3*exp(-4)*(864*exp(4) + 96*e
xp(2)*log(5)))/3

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sympy [B]  time = 0.07, size = 51, normalized size = 2.55 \begin {gather*} 16 x^{4} + \frac {x^{3} \left (- 288 e^{2} - 32 \log {\relax (5 )}\right )}{e^{2}} + \frac {x^{2} \left (16 \log {\relax (5 )}^{2} + 288 e^{2} \log {\relax (5 )} + 1296 e^{4}\right )}{e^{4}} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((32*x*ln(5)**2+(-96*x**2+576*x)*exp(2)*ln(5)+(64*x**3-864*x**2+2592*x+1)*exp(2)**2)/exp(2)**2,x)

[Out]

16*x**4 + x**3*(-288*exp(2) - 32*log(5))*exp(-2) + x**2*(16*log(5)**2 + 288*exp(2)*log(5) + 1296*exp(4))*exp(-
4) + x

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