3.41.38 \(\int \frac {468512 \log (5 x)+e^{2-2 x} x^2 (702768 \log (5 x)+(702768-702768 x) \log ^2(5 x))+e^{1-x} x (937024 \log (5 x)+(468512-468512 x) \log ^2(5 x))+e^{3-3 x} x^3 (234256 \log (5 x)+(351384-351384 x) \log ^2(5 x))+e^{4-4 x} x^4 (29282 \log (5 x)+(58564-58564 x) \log ^2(5 x))}{x} \, dx\)

Optimal. Leaf size=21 \[ 14641 \left (2+e^{1-x} x\right )^4 \log ^2(5 x) \]

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Rubi [B]  time = 0.32, antiderivative size = 79, normalized size of antiderivative = 3.76, number of steps used = 7, number of rules used = 3, integrand size = 129, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {14, 2301, 2288} \begin {gather*} 14641 e^{4-4 x} x^4 \log ^2(5 x)+117128 e^{3-3 x} x^3 \log ^2(5 x)+351384 e^{2-2 x} x^2 \log ^2(5 x)+468512 e^{1-x} x \log ^2(5 x)+234256 \log ^2(5 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(468512*Log[5*x] + E^(2 - 2*x)*x^2*(702768*Log[5*x] + (702768 - 702768*x)*Log[5*x]^2) + E^(1 - x)*x*(93702
4*Log[5*x] + (468512 - 468512*x)*Log[5*x]^2) + E^(3 - 3*x)*x^3*(234256*Log[5*x] + (351384 - 351384*x)*Log[5*x]
^2) + E^(4 - 4*x)*x^4*(29282*Log[5*x] + (58564 - 58564*x)*Log[5*x]^2))/x,x]

[Out]

234256*Log[5*x]^2 + 468512*E^(1 - x)*x*Log[5*x]^2 + 351384*E^(2 - 2*x)*x^2*Log[5*x]^2 + 117128*E^(3 - 3*x)*x^3
*Log[5*x]^2 + 14641*E^(4 - 4*x)*x^4*Log[5*x]^2

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {468512 \log (5 x)}{x}-468512 e^{1-x} \log (5 x) (-2-\log (5 x)+x \log (5 x))-702768 e^{2-2 x} x \log (5 x) (-1-\log (5 x)+x \log (5 x))-29282 e^{4-4 x} x^3 \log (5 x) (-1-2 \log (5 x)+2 x \log (5 x))-117128 e^{3-3 x} x^2 \log (5 x) (-2-3 \log (5 x)+3 x \log (5 x))\right ) \, dx\\ &=-\left (29282 \int e^{4-4 x} x^3 \log (5 x) (-1-2 \log (5 x)+2 x \log (5 x)) \, dx\right )-117128 \int e^{3-3 x} x^2 \log (5 x) (-2-3 \log (5 x)+3 x \log (5 x)) \, dx+468512 \int \frac {\log (5 x)}{x} \, dx-468512 \int e^{1-x} \log (5 x) (-2-\log (5 x)+x \log (5 x)) \, dx-702768 \int e^{2-2 x} x \log (5 x) (-1-\log (5 x)+x \log (5 x)) \, dx\\ &=234256 \log ^2(5 x)+468512 e^{1-x} x \log ^2(5 x)+351384 e^{2-2 x} x^2 \log ^2(5 x)+117128 e^{3-3 x} x^3 \log ^2(5 x)+14641 e^{4-4 x} x^4 \log ^2(5 x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.40, size = 24, normalized size = 1.14 \begin {gather*} 14641 e^{-4 x} \left (2 e^x+e x\right )^4 \log ^2(5 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(468512*Log[5*x] + E^(2 - 2*x)*x^2*(702768*Log[5*x] + (702768 - 702768*x)*Log[5*x]^2) + E^(1 - x)*x*
(937024*Log[5*x] + (468512 - 468512*x)*Log[5*x]^2) + E^(3 - 3*x)*x^3*(234256*Log[5*x] + (351384 - 351384*x)*Lo
g[5*x]^2) + E^(4 - 4*x)*x^4*(29282*Log[5*x] + (58564 - 58564*x)*Log[5*x]^2))/x,x]

[Out]

(14641*(2*E^x + E*x)^4*Log[5*x]^2)/E^(4*x)

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fricas [B]  time = 1.10, size = 119, normalized size = 5.67 \begin {gather*} 468512 \, {\left (\log \relax (5)^{2} + 2 \, \log \relax (5) \log \relax (x) + \log \relax (x)^{2}\right )} e^{\left (-x + \log \relax (x) + 1\right )} + 351384 \, {\left (\log \relax (5)^{2} + 2 \, \log \relax (5) \log \relax (x) + \log \relax (x)^{2}\right )} e^{\left (-2 \, x + 2 \, \log \relax (x) + 2\right )} + 117128 \, {\left (\log \relax (5)^{2} + 2 \, \log \relax (5) \log \relax (x) + \log \relax (x)^{2}\right )} e^{\left (-3 \, x + 3 \, \log \relax (x) + 3\right )} + 14641 \, {\left (\log \relax (5)^{2} + 2 \, \log \relax (5) \log \relax (x) + \log \relax (x)^{2}\right )} e^{\left (-4 \, x + 4 \, \log \relax (x) + 4\right )} + 468512 \, \log \relax (5) \log \relax (x) + 234256 \, \log \relax (x)^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-58564*x+58564)*log(5*x)^2+29282*log(5*x))*exp(1+log(x)-x)^4+((-351384*x+351384)*log(5*x)^2+23425
6*log(5*x))*exp(1+log(x)-x)^3+((-702768*x+702768)*log(5*x)^2+702768*log(5*x))*exp(1+log(x)-x)^2+((-468512*x+46
8512)*log(5*x)^2+937024*log(5*x))*exp(1+log(x)-x)+468512*log(5*x))/x,x, algorithm="fricas")

[Out]

468512*(log(5)^2 + 2*log(5)*log(x) + log(x)^2)*e^(-x + log(x) + 1) + 351384*(log(5)^2 + 2*log(5)*log(x) + log(
x)^2)*e^(-2*x + 2*log(x) + 2) + 117128*(log(5)^2 + 2*log(5)*log(x) + log(x)^2)*e^(-3*x + 3*log(x) + 3) + 14641
*(log(5)^2 + 2*log(5)*log(x) + log(x)^2)*e^(-4*x + 4*log(x) + 4) + 468512*log(5)*log(x) + 234256*log(x)^2

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giac [B]  time = 0.23, size = 187, normalized size = 8.90 \begin {gather*} 14641 \, x^{4} e^{\left (-4 \, x + 4\right )} \log \relax (5)^{2} + 29282 \, x^{4} e^{\left (-4 \, x + 4\right )} \log \relax (5) \log \relax (x) + 14641 \, x^{4} e^{\left (-4 \, x + 4\right )} \log \relax (x)^{2} + 117128 \, x^{3} e^{\left (-3 \, x + 3\right )} \log \relax (5)^{2} + 234256 \, x^{3} e^{\left (-3 \, x + 3\right )} \log \relax (5) \log \relax (x) + 117128 \, x^{3} e^{\left (-3 \, x + 3\right )} \log \relax (x)^{2} + 351384 \, x^{2} e^{\left (-2 \, x + 2\right )} \log \relax (5)^{2} + 702768 \, x^{2} e^{\left (-2 \, x + 2\right )} \log \relax (5) \log \relax (x) + 351384 \, x^{2} e^{\left (-2 \, x + 2\right )} \log \relax (x)^{2} + 468512 \, x e^{\left (-x + 1\right )} \log \relax (5)^{2} + 937024 \, x e^{\left (-x + 1\right )} \log \relax (5) \log \relax (x) + 468512 \, x e^{\left (-x + 1\right )} \log \relax (x)^{2} + 468512 \, \log \relax (5) \log \relax (x) + 234256 \, \log \relax (x)^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-58564*x+58564)*log(5*x)^2+29282*log(5*x))*exp(1+log(x)-x)^4+((-351384*x+351384)*log(5*x)^2+23425
6*log(5*x))*exp(1+log(x)-x)^3+((-702768*x+702768)*log(5*x)^2+702768*log(5*x))*exp(1+log(x)-x)^2+((-468512*x+46
8512)*log(5*x)^2+937024*log(5*x))*exp(1+log(x)-x)+468512*log(5*x))/x,x, algorithm="giac")

[Out]

14641*x^4*e^(-4*x + 4)*log(5)^2 + 29282*x^4*e^(-4*x + 4)*log(5)*log(x) + 14641*x^4*e^(-4*x + 4)*log(x)^2 + 117
128*x^3*e^(-3*x + 3)*log(5)^2 + 234256*x^3*e^(-3*x + 3)*log(5)*log(x) + 117128*x^3*e^(-3*x + 3)*log(x)^2 + 351
384*x^2*e^(-2*x + 2)*log(5)^2 + 702768*x^2*e^(-2*x + 2)*log(5)*log(x) + 351384*x^2*e^(-2*x + 2)*log(x)^2 + 468
512*x*e^(-x + 1)*log(5)^2 + 937024*x*e^(-x + 1)*log(5)*log(x) + 468512*x*e^(-x + 1)*log(x)^2 + 468512*log(5)*l
og(x) + 234256*log(x)^2

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maple [B]  time = 0.31, size = 128, normalized size = 6.10

method	result	size
risch	\(234256 \ln \relax (x )^{2}+468512 \ln \relax (5) \ln \relax (x )+\left (14641 \ln \relax (5)^{2}+29282 \ln \relax (5) \ln \relax (x )+14641 \ln \relax (x )^{2}\right ) x^{4} {\mathrm e}^{-4 x +4}+\left (117128 \ln \relax (5)^{2}+234256 \ln \relax (5) \ln \relax (x )+117128 \ln \relax (x )^{2}\right ) x^{3} {\mathrm e}^{-3 x +3}+\left (351384 \ln \relax (5)^{2}+702768 \ln \relax (5) \ln \relax (x )+351384 \ln \relax (x )^{2}\right ) x^{2} {\mathrm e}^{-2 x +2}+\left (468512 \ln \relax (5)^{2}+937024 \ln \relax (5) \ln \relax (x )+468512 \ln \relax (x )^{2}\right ) x \,{\mathrm e}^{1-x}\)	\(128\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-58564*x+58564)*ln(5*x)^2+29282*ln(5*x))*exp(1+ln(x)-x)^4+((-351384*x+351384)*ln(5*x)^2+234256*ln(5*x))
*exp(1+ln(x)-x)^3+((-702768*x+702768)*ln(5*x)^2+702768*ln(5*x))*exp(1+ln(x)-x)^2+((-468512*x+468512)*ln(5*x)^2
+937024*ln(5*x))*exp(1+ln(x)-x)+468512*ln(5*x))/x,x,method=_RETURNVERBOSE)

[Out]

234256*ln(x)^2+468512*ln(5)*ln(x)+(14641*ln(5)^2+29282*ln(5)*ln(x)+14641*ln(x)^2)*x^4*exp(-4*x+4)+(117128*ln(5
)^2+234256*ln(5)*ln(x)+117128*ln(x)^2)*x^3*exp(-3*x+3)+(351384*ln(5)^2+702768*ln(5)*ln(x)+351384*ln(x)^2)*x^2*
exp(-2*x+2)+(468512*ln(5)^2+937024*ln(5)*ln(x)+468512*ln(x)^2)*x*exp(1-x)

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maxima [B]  time = 0.55, size = 155, normalized size = 7.38 \begin {gather*} 468512 \, {\left (x e \log \relax (5)^{2} + 2 \, x e \log \relax (5) \log \relax (x) + x e \log \relax (x)^{2}\right )} e^{\left (-x\right )} + 351384 \, {\left (x^{2} e^{2} \log \relax (5)^{2} + 2 \, x^{2} e^{2} \log \relax (5) \log \relax (x) + x^{2} e^{2} \log \relax (x)^{2}\right )} e^{\left (-2 \, x\right )} + 117128 \, {\left (x^{3} e^{3} \log \relax (5)^{2} + 2 \, x^{3} e^{3} \log \relax (5) \log \relax (x) + x^{3} e^{3} \log \relax (x)^{2}\right )} e^{\left (-3 \, x\right )} + 14641 \, {\left (x^{4} e^{4} \log \relax (5)^{2} + 2 \, x^{4} e^{4} \log \relax (5) \log \relax (x) + x^{4} e^{4} \log \relax (x)^{2}\right )} e^{\left (-4 \, x\right )} + 234256 \, \log \left (5 \, x\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-58564*x+58564)*log(5*x)^2+29282*log(5*x))*exp(1+log(x)-x)^4+((-351384*x+351384)*log(5*x)^2+23425
6*log(5*x))*exp(1+log(x)-x)^3+((-702768*x+702768)*log(5*x)^2+702768*log(5*x))*exp(1+log(x)-x)^2+((-468512*x+46
8512)*log(5*x)^2+937024*log(5*x))*exp(1+log(x)-x)+468512*log(5*x))/x,x, algorithm="maxima")

[Out]

468512*(x*e*log(5)^2 + 2*x*e*log(5)*log(x) + x*e*log(x)^2)*e^(-x) + 351384*(x^2*e^2*log(5)^2 + 2*x^2*e^2*log(5
)*log(x) + x^2*e^2*log(x)^2)*e^(-2*x) + 117128*(x^3*e^3*log(5)^2 + 2*x^3*e^3*log(5)*log(x) + x^3*e^3*log(x)^2)
*e^(-3*x) + 14641*(x^4*e^4*log(5)^2 + 2*x^4*e^4*log(5)*log(x) + x^4*e^4*log(x)^2)*e^(-4*x) + 234256*log(5*x)^2

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mupad [B]  time = 4.11, size = 75, normalized size = 3.57 \begin {gather*} 234256\,{\ln \left (5\,x\right )}^2+468512\,x\,{\ln \left (5\,x\right )}^2\,{\mathrm {e}}^{1-x}+351384\,x^2\,{\ln \left (5\,x\right )}^2\,{\mathrm {e}}^{2-2\,x}+117128\,x^3\,{\ln \left (5\,x\right )}^2\,{\mathrm {e}}^{3-3\,x}+14641\,x^4\,{\ln \left (5\,x\right )}^2\,{\mathrm {e}}^{4-4\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((468512*log(5*x) + exp(log(x) - x + 1)*(937024*log(5*x) - log(5*x)^2*(468512*x - 468512)) + exp(4*log(x) -
4*x + 4)*(29282*log(5*x) - log(5*x)^2*(58564*x - 58564)) + exp(3*log(x) - 3*x + 3)*(234256*log(5*x) - log(5*x
)^2*(351384*x - 351384)) + exp(2*log(x) - 2*x + 2)*(702768*log(5*x) - log(5*x)^2*(702768*x - 702768)))/x,x)

[Out]

234256*log(5*x)^2 + 468512*x*log(5*x)^2*exp(1 - x) + 351384*x^2*log(5*x)^2*exp(2 - 2*x) + 117128*x^3*log(5*x)^
2*exp(3 - 3*x) + 14641*x^4*log(5*x)^2*exp(4 - 4*x)

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sympy [B]  time = 0.59, size = 78, normalized size = 3.71 \begin {gather*} 14641 x^{4} e^{4 - 4 x} \log {\left (5 x \right )}^{2} + 117128 x^{3} e^{3 - 3 x} \log {\left (5 x \right )}^{2} + 351384 x^{2} e^{2 - 2 x} \log {\left (5 x \right )}^{2} + 468512 x e^{1 - x} \log {\left (5 x \right )}^{2} + 234256 \log {\left (5 x \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-58564*x+58564)*ln(5*x)**2+29282*ln(5*x))*exp(1+ln(x)-x)**4+((-351384*x+351384)*ln(5*x)**2+234256
*ln(5*x))*exp(1+ln(x)-x)**3+((-702768*x+702768)*ln(5*x)**2+702768*ln(5*x))*exp(1+ln(x)-x)**2+((-468512*x+46851
2)*ln(5*x)**2+937024*ln(5*x))*exp(1+ln(x)-x)+468512*ln(5*x))/x,x)

[Out]

14641*x**4*exp(4 - 4*x)*log(5*x)**2 + 117128*x**3*exp(3 - 3*x)*log(5*x)**2 + 351384*x**2*exp(2 - 2*x)*log(5*x)
**2 + 468512*x*exp(1 - x)*log(5*x)**2 + 234256*log(5*x)**2

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