3.37.96 \(\int x^{-1+e^x} (8 x+8 x^2+e^x (4+8 x+4 x^2)+e^x (4 x+8 x^2+4 x^3) \log (x)) \, dx\)

Optimal. Leaf size=13 \[ x^{e^x} (2+2 x)^2 \]

________________________________________________________________________________________

Rubi [F]  time = 0.78, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int x^{-1+e^x} \left (8 x+8 x^2+e^x \left (4+8 x+4 x^2\right )+e^x \left (4 x+8 x^2+4 x^3\right ) \log (x)\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[x^(-1 + E^x)*(8*x + 8*x^2 + E^x*(4 + 8*x + 4*x^2) + E^x*(4*x + 8*x^2 + 4*x^3)*Log[x]),x]

[Out]

(8*x^(1 + E^x)*(1 + x))/(2 + E^x) + (8*Defer[Int][x^E^x, x])/(2 + E^x) + 8*Defer[Int][E^x*x^E^x, x] + 4*Log[x]
*Defer[Int][E^x*x^E^x, x] + 4*Defer[Int][E^x*x^(-1 + E^x), x] + 4*Defer[Int][E^x*x^(1 + E^x), x] + 8*Log[x]*De
fer[Int][E^x*x^(1 + E^x), x] + 4*Log[x]*Defer[Int][E^x*x^(2 + E^x), x] - 4*Defer[Int][Defer[Int][E^x*x^E^x, x]
/x, x] - 8*Defer[Int][Defer[Int][E^x*x^(1 + E^x), x]/x, x] - 4*Defer[Int][Defer[Int][E^x*x^(2 + E^x), x]/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int 4 x^{-1+e^x} (1+x) \left (2 x+e^x (1+x)+e^x x (1+x) \log (x)\right ) \, dx\\ &=4 \int x^{-1+e^x} (1+x) \left (2 x+e^x (1+x)+e^x x (1+x) \log (x)\right ) \, dx\\ &=4 \int \left (2 x^{e^x} (1+x)+e^x x^{-1+e^x} (1+x)^2 (1+x \log (x))\right ) \, dx\\ &=4 \int e^x x^{-1+e^x} (1+x)^2 (1+x \log (x)) \, dx+8 \int x^{e^x} (1+x) \, dx\\ &=\frac {8 x^{1+e^x} (1+x)}{2+e^x}+4 \int \left (e^x x^{-1+e^x} (1+x)^2+e^x x^{e^x} (1+x)^2 \log (x)\right ) \, dx+\frac {8 \int x^{e^x} \, dx}{2+e^x}\\ &=\frac {8 x^{1+e^x} (1+x)}{2+e^x}+4 \int e^x x^{-1+e^x} (1+x)^2 \, dx+4 \int e^x x^{e^x} (1+x)^2 \log (x) \, dx+\frac {8 \int x^{e^x} \, dx}{2+e^x}\\ &=\frac {8 x^{1+e^x} (1+x)}{2+e^x}+4 \int \left (2 e^x x^{e^x}+e^x x^{-1+e^x}+e^x x^{1+e^x}\right ) \, dx-4 \int \frac {\int e^x x^{e^x} \, dx+2 \int e^x x^{1+e^x} \, dx+\int e^x x^{2+e^x} \, dx}{x} \, dx+\frac {8 \int x^{e^x} \, dx}{2+e^x}+(4 \log (x)) \int e^x x^{e^x} \, dx+(4 \log (x)) \int e^x x^{2+e^x} \, dx+(8 \log (x)) \int e^x x^{1+e^x} \, dx\\ &=\frac {8 x^{1+e^x} (1+x)}{2+e^x}+4 \int e^x x^{-1+e^x} \, dx+4 \int e^x x^{1+e^x} \, dx-4 \int \left (\frac {\int e^x x^{e^x} \, dx+2 \int e^x x^{1+e^x} \, dx}{x}+\frac {\int e^x x^{2+e^x} \, dx}{x}\right ) \, dx+8 \int e^x x^{e^x} \, dx+\frac {8 \int x^{e^x} \, dx}{2+e^x}+(4 \log (x)) \int e^x x^{e^x} \, dx+(4 \log (x)) \int e^x x^{2+e^x} \, dx+(8 \log (x)) \int e^x x^{1+e^x} \, dx\\ &=\frac {8 x^{1+e^x} (1+x)}{2+e^x}+4 \int e^x x^{-1+e^x} \, dx+4 \int e^x x^{1+e^x} \, dx-4 \int \frac {\int e^x x^{e^x} \, dx+2 \int e^x x^{1+e^x} \, dx}{x} \, dx-4 \int \frac {\int e^x x^{2+e^x} \, dx}{x} \, dx+8 \int e^x x^{e^x} \, dx+\frac {8 \int x^{e^x} \, dx}{2+e^x}+(4 \log (x)) \int e^x x^{e^x} \, dx+(4 \log (x)) \int e^x x^{2+e^x} \, dx+(8 \log (x)) \int e^x x^{1+e^x} \, dx\\ &=\frac {8 x^{1+e^x} (1+x)}{2+e^x}+4 \int e^x x^{-1+e^x} \, dx+4 \int e^x x^{1+e^x} \, dx-4 \int \left (\frac {\int e^x x^{e^x} \, dx}{x}+\frac {2 \int e^x x^{1+e^x} \, dx}{x}\right ) \, dx-4 \int \frac {\int e^x x^{2+e^x} \, dx}{x} \, dx+8 \int e^x x^{e^x} \, dx+\frac {8 \int x^{e^x} \, dx}{2+e^x}+(4 \log (x)) \int e^x x^{e^x} \, dx+(4 \log (x)) \int e^x x^{2+e^x} \, dx+(8 \log (x)) \int e^x x^{1+e^x} \, dx\\ &=\frac {8 x^{1+e^x} (1+x)}{2+e^x}+4 \int e^x x^{-1+e^x} \, dx+4 \int e^x x^{1+e^x} \, dx-4 \int \frac {\int e^x x^{e^x} \, dx}{x} \, dx-4 \int \frac {\int e^x x^{2+e^x} \, dx}{x} \, dx+8 \int e^x x^{e^x} \, dx-8 \int \frac {\int e^x x^{1+e^x} \, dx}{x} \, dx+\frac {8 \int x^{e^x} \, dx}{2+e^x}+(4 \log (x)) \int e^x x^{e^x} \, dx+(4 \log (x)) \int e^x x^{2+e^x} \, dx+(8 \log (x)) \int e^x x^{1+e^x} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.20, size = 12, normalized size = 0.92 \begin {gather*} 4 x^{e^x} (1+x)^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 + E^x)*(8*x + 8*x^2 + E^x*(4 + 8*x + 4*x^2) + E^x*(4*x + 8*x^2 + 4*x^3)*Log[x]),x]

[Out]

4*x^E^x*(1 + x)^2

________________________________________________________________________________________

fricas [A]  time = 0.61, size = 14, normalized size = 1.08 \begin {gather*} 4 \, {\left (x^{2} + 2 \, x + 1\right )} x^{e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3+8*x^2+4*x)*exp(x)*log(x)+(4*x^2+8*x+4)*exp(x)+8*x^2+8*x)*exp(exp(x)*log(x))/x,x, algorithm="
fricas")

[Out]

4*(x^2 + 2*x + 1)*x^e^x

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3+8*x^2+4*x)*exp(x)*log(x)+(4*x^2+8*x+4)*exp(x)+8*x^2+8*x)*exp(exp(x)*log(x))/x,x, algorithm="
giac")

[Out]

integrate(4*((x^3 + 2*x^2 + x)*e^x*log(x) + 2*x^2 + (x^2 + 2*x + 1)*e^x + 2*x)*x^e^x/x, x)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 16, normalized size = 1.23

method	result	size
risch	\(\left (4 x^{2}+8 x +4\right ) x^{{\mathrm e}^{x}}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^3+8*x^2+4*x)*exp(x)*ln(x)+(4*x^2+8*x+4)*exp(x)+8*x^2+8*x)*exp(exp(x)*ln(x))/x,x,method=_RETURNVERBOS
E)

[Out]

(4*x^2+8*x+4)*x^exp(x)

________________________________________________________________________________________

maxima [A]  time = 0.40, size = 14, normalized size = 1.08 \begin {gather*} 4 \, {\left (x^{2} + 2 \, x + 1\right )} x^{e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3+8*x^2+4*x)*exp(x)*log(x)+(4*x^2+8*x+4)*exp(x)+8*x^2+8*x)*exp(exp(x)*log(x))/x,x, algorithm="
maxima")

[Out]

4*(x^2 + 2*x + 1)*x^e^x

________________________________________________________________________________________

mupad [B]  time = 2.24, size = 15, normalized size = 1.15 \begin {gather*} x^{{\mathrm {e}}^x}\,\left (4\,x^2+8\,x+4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(x)*log(x))*(8*x + exp(x)*(8*x + 4*x^2 + 4) + 8*x^2 + exp(x)*log(x)*(4*x + 8*x^2 + 4*x^3)))/x,x)

[Out]

x^exp(x)*(8*x + 4*x^2 + 4)

________________________________________________________________________________________

sympy [A]  time = 0.49, size = 17, normalized size = 1.31 \begin {gather*} \left (4 x^{2} + 8 x + 4\right ) e^{e^{x} \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**3+8*x**2+4*x)*exp(x)*ln(x)+(4*x**2+8*x+4)*exp(x)+8*x**2+8*x)*exp(exp(x)*ln(x))/x,x)

[Out]

(4*x**2 + 8*x + 4)*exp(exp(x)*log(x))

________________________________________________________________________________________