3.88.76 \(\int \frac {e^x x \log (x)+(2 e^{2 x} x+e^x (1+2 x^2)) \log ^2(x)+(-e^x x+e^x (x+x^2) \log (x)+(e^{2 x} (2 x+4 x^2)+e^x (x+4 x^2+2 x^3)) \log ^2(x)) \log (25 x)}{x \log ^2(x)} \, dx\)

Optimal. Leaf size=24 \[ e^x \left (1+2 x \left (e^x+x\right )+\frac {x}{\log (x)}\right ) \log (25 x) \]

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Rubi [B]  time = 1.62, antiderivative size = 58, normalized size of antiderivative = 2.42, number of steps used = 5, number of rules used = 3, integrand size = 101, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {6688, 6742, 2288} \begin {gather*} \frac {e^x \left (2 x^3 \log ^2(x) \log (25 x)+x^2 \log (x) \log (25 x)+x \log ^2(x) \log (25 x)\right )}{x \log ^2(x)}+2 e^{2 x} x \log (25 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^x*x*Log[x] + (2*E^(2*x)*x + E^x*(1 + 2*x^2))*Log[x]^2 + (-(E^x*x) + E^x*(x + x^2)*Log[x] + (E^(2*x)*(2*
x + 4*x^2) + E^x*(x + 4*x^2 + 2*x^3))*Log[x]^2)*Log[25*x])/(x*Log[x]^2),x]

[Out]

2*E^(2*x)*x*Log[25*x] + (E^x*(x^2*Log[x]*Log[25*x] + x*Log[x]^2*Log[25*x] + 2*x^3*Log[x]^2*Log[25*x]))/(x*Log[
x]^2)

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 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^x \left (-x \log (25 x)+\log (x) (x+x (1+x) \log (25 x))+\log ^2(x) \left (1+2 e^x x+2 x^2+x \left (1+4 x+2 x^2+e^x (2+4 x)\right ) \log (25 x)\right )\right )}{x \log ^2(x)} \, dx\\ &=\int \left (2 e^{2 x} (1+\log (25 x)+2 x \log (25 x))+\frac {e^x \left (x \log (x)+\log ^2(x)+2 x^2 \log ^2(x)-x \log (25 x)+x \log (x) \log (25 x)+x^2 \log (x) \log (25 x)+x \log ^2(x) \log (25 x)+4 x^2 \log ^2(x) \log (25 x)+2 x^3 \log ^2(x) \log (25 x)\right )}{x \log ^2(x)}\right ) \, dx\\ &=2 \int e^{2 x} (1+\log (25 x)+2 x \log (25 x)) \, dx+\int \frac {e^x \left (x \log (x)+\log ^2(x)+2 x^2 \log ^2(x)-x \log (25 x)+x \log (x) \log (25 x)+x^2 \log (x) \log (25 x)+x \log ^2(x) \log (25 x)+4 x^2 \log ^2(x) \log (25 x)+2 x^3 \log ^2(x) \log (25 x)\right )}{x \log ^2(x)} \, dx\\ &=2 e^{2 x} x \log (25 x)+\frac {e^x \left (x^2 \log (x) \log (25 x)+x \log ^2(x) \log (25 x)+2 x^3 \log ^2(x) \log (25 x)\right )}{x \log ^2(x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.71, size = 30, normalized size = 1.25 \begin {gather*} \frac {e^x \left (x+\left (1+2 e^x x+2 x^2\right ) \log (x)\right ) \log (25 x)}{\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*x*Log[x] + (2*E^(2*x)*x + E^x*(1 + 2*x^2))*Log[x]^2 + (-(E^x*x) + E^x*(x + x^2)*Log[x] + (E^(2*
x)*(2*x + 4*x^2) + E^x*(x + 4*x^2 + 2*x^3))*Log[x]^2)*Log[25*x])/(x*Log[x]^2),x]

[Out]

(E^x*(x + (1 + 2*E^x*x + 2*x^2)*Log[x])*Log[25*x])/Log[x]

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fricas [B]  time = 0.64, size = 65, normalized size = 2.71 \begin {gather*} \frac {2 \, x e^{x} \log \relax (5) + {\left (2 \, x e^{\left (2 \, x\right )} + {\left (2 \, x^{2} + 1\right )} e^{x}\right )} \log \relax (x)^{2} + {\left (4 \, x e^{\left (2 \, x\right )} \log \relax (5) + {\left (2 \, {\left (2 \, x^{2} + 1\right )} \log \relax (5) + x\right )} e^{x}\right )} \log \relax (x)}{\log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((4*x^2+2*x)*exp(x)^2+(2*x^3+4*x^2+x)*exp(x))*log(x)^2+(x^2+x)*exp(x)*log(x)-exp(x)*x)*log(25*x)+(
2*x*exp(x)^2+(2*x^2+1)*exp(x))*log(x)^2+x*exp(x)*log(x))/x/log(x)^2,x, algorithm="fricas")

[Out]

(2*x*e^x*log(5) + (2*x*e^(2*x) + (2*x^2 + 1)*e^x)*log(x)^2 + (4*x*e^(2*x)*log(5) + (2*(2*x^2 + 1)*log(5) + x)*
e^x)*log(x))/log(x)

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giac [B]  time = 0.14, size = 78, normalized size = 3.25 \begin {gather*} \frac {4 \, x^{2} e^{x} \log \relax (5) \log \relax (x) + 2 \, x^{2} e^{x} \log \relax (x)^{2} + 4 \, x e^{\left (2 \, x\right )} \log \relax (5) \log \relax (x) + 2 \, x e^{\left (2 \, x\right )} \log \relax (x)^{2} + 2 \, x e^{x} \log \relax (5) + x e^{x} \log \relax (x) + 2 \, e^{x} \log \relax (5) \log \relax (x) + e^{x} \log \relax (x)^{2}}{\log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((4*x^2+2*x)*exp(x)^2+(2*x^3+4*x^2+x)*exp(x))*log(x)^2+(x^2+x)*exp(x)*log(x)-exp(x)*x)*log(25*x)+(
2*x*exp(x)^2+(2*x^2+1)*exp(x))*log(x)^2+x*exp(x)*log(x))/x/log(x)^2,x, algorithm="giac")

[Out]

(4*x^2*e^x*log(5)*log(x) + 2*x^2*e^x*log(x)^2 + 4*x*e^(2*x)*log(5)*log(x) + 2*x*e^(2*x)*log(x)^2 + 2*x*e^x*log
(5) + x*e^x*log(x) + 2*e^x*log(5)*log(x) + e^x*log(x)^2)/log(x)

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maple [B]  time = 0.18, size = 61, normalized size = 2.54

method	result	size
risch	\(\left (2 \,{\mathrm e}^{x} x^{2}+2 x \,{\mathrm e}^{2 x}+{\mathrm e}^{x}\right ) \ln \relax (x )+4 x \ln \relax (5) {\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} \ln \relax (5)+{\mathrm e}^{x} x +4 x^{2} \ln \relax (5) {\mathrm e}^{x}+\frac {2 x \,{\mathrm e}^{x} \ln \relax (5)}{\ln \relax (x )}\)	\(61\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((((4*x^2+2*x)*exp(x)^2+(2*x^3+4*x^2+x)*exp(x))*ln(x)^2+(x^2+x)*exp(x)*ln(x)-exp(x)*x)*ln(25*x)+(2*x*exp(x
)^2+(2*x^2+1)*exp(x))*ln(x)^2+x*exp(x)*ln(x))/x/ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

(2*exp(x)*x^2+2*x*exp(2*x)+exp(x))*ln(x)+4*x*ln(5)*exp(2*x)+2*exp(x)*ln(5)+exp(x)*x+4*x^2*ln(5)*exp(x)+2*x*exp
(x)*ln(5)/ln(x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} {\left (2 \, x^{2} + 1\right )} e^{x} \log \relax (x) + {\left (4 \, x \log \relax (5) + 2 \, x \log \relax (x) - 1\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x - 1\right )} e^{x} + {\rm Ei}\relax (x) + e^{\left (2 \, x\right )} + \int \frac {{\left ({\left (4 \, x^{3} \log \relax (5) + x^{2} {\left (8 \, \log \relax (5) - 1\right )} + x {\left (2 \, \log \relax (5) + 1\right )} - 1\right )} \log \relax (x)^{2} - 2 \, x \log \relax (5) + 2 \, {\left (x^{2} \log \relax (5) + x \log \relax (5)\right )} \log \relax (x)\right )} e^{x}}{x \log \relax (x)^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((4*x^2+2*x)*exp(x)^2+(2*x^3+4*x^2+x)*exp(x))*log(x)^2+(x^2+x)*exp(x)*log(x)-exp(x)*x)*log(25*x)+(
2*x*exp(x)^2+(2*x^2+1)*exp(x))*log(x)^2+x*exp(x)*log(x))/x/log(x)^2,x, algorithm="maxima")

[Out]

(2*x^2 + 1)*e^x*log(x) + (4*x*log(5) + 2*x*log(x) - 1)*e^(2*x) + 2*(x - 1)*e^x + Ei(x) + e^(2*x) + integrate((
(4*x^3*log(5) + x^2*(8*log(5) - 1) + x*(2*log(5) + 1) - 1)*log(x)^2 - 2*x*log(5) + 2*(x^2*log(5) + x*log(5))*l
og(x))*e^x/(x*log(x)^2), x)

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mupad [B]  time = 5.51, size = 63, normalized size = 2.62 \begin {gather*} {\mathrm {e}}^x\,\ln \relax (x)+2\,{\mathrm {e}}^x\,\ln \relax (5)+x\,{\mathrm {e}}^x+4\,x\,{\mathrm {e}}^{2\,x}\,\ln \relax (5)+4\,x^2\,{\mathrm {e}}^x\,\ln \relax (5)+2\,x\,{\mathrm {e}}^{2\,x}\,\ln \relax (x)+2\,x^2\,{\mathrm {e}}^x\,\ln \relax (x)+\frac {2\,x\,{\mathrm {e}}^x\,\ln \relax (5)}{\ln \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(25*x)*(log(x)^2*(exp(2*x)*(2*x + 4*x^2) + exp(x)*(x + 4*x^2 + 2*x^3)) - x*exp(x) + exp(x)*log(x)*(x +
x^2)) + log(x)^2*(2*x*exp(2*x) + exp(x)*(2*x^2 + 1)) + x*exp(x)*log(x))/(x*log(x)^2),x)

[Out]

exp(x)*log(x) + 2*exp(x)*log(5) + x*exp(x) + 4*x*exp(2*x)*log(5) + 4*x^2*exp(x)*log(5) + 2*x*exp(2*x)*log(x) +
2*x^2*exp(x)*log(x) + (2*x*exp(x)*log(5))/log(x)

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sympy [B]  time = 0.43, size = 76, normalized size = 3.17 \begin {gather*} \frac {\left (2 x \log {\relax (x )}^{2} + 4 x \log {\relax (5 )} \log {\relax (x )}\right ) e^{2 x} + \left (2 x^{2} \log {\relax (x )}^{2} + 4 x^{2} \log {\relax (5 )} \log {\relax (x )} + x \log {\relax (x )} + 2 x \log {\relax (5 )} + \log {\relax (x )}^{2} + 2 \log {\relax (5 )} \log {\relax (x )}\right ) e^{x}}{\log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((4*x**2+2*x)*exp(x)**2+(2*x**3+4*x**2+x)*exp(x))*ln(x)**2+(x**2+x)*exp(x)*ln(x)-exp(x)*x)*ln(25*x
)+(2*x*exp(x)**2+(2*x**2+1)*exp(x))*ln(x)**2+x*exp(x)*ln(x))/x/ln(x)**2,x)

[Out]

((2*x*log(x)**2 + 4*x*log(5)*log(x))*exp(2*x) + (2*x**2*log(x)**2 + 4*x**2*log(5)*log(x) + x*log(x) + 2*x*log(
5) + log(x)**2 + 2*log(5)*log(x))*exp(x))/log(x)

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