\(\int \frac {5 x+e^{5+e^x (-3+4 x) \log (x)} (-x+e^x (3 x-4 x^2+e (-3+4 x))+e^x (-x^2-4 x^3+e (x+4 x^2)) \log (x))}{x} \, dx\) [7469]

Optimal result

Integrand size = 74, antiderivative size = 26 \[ \int \frac {5 x+e^{5+e^x (-3+4 x) \log (x)} \left (-x+e^x \left (3 x-4 x^2+e (-3+4 x)\right )+e^x \left (-x^2-4 x^3+e \left (x+4 x^2\right )\right ) \log (x)\right )}{x} \, dx=\left (5-e^{5-e^x (3-4 x) \log (x)}\right ) (-e+x) \]

[Out]

(x-exp(1))*(5-exp(5-ln(x)*(3-4*x)*exp(x)))

Rubi [F]

\[ \int \frac {5 x+e^{5+e^x (-3+4 x) \log (x)} \left (-x+e^x \left (3 x-4 x^2+e (-3+4 x)\right )+e^x \left (-x^2-4 x^3+e \left (x+4 x^2\right )\right ) \log (x)\right )}{x} \, dx=\int \frac {5 x+e^{5+e^x (-3+4 x) \log (x)} \left (-x+e^x \left (3 x-4 x^2+e (-3+4 x)\right )+e^x \left (-x^2-4 x^3+e \left (x+4 x^2\right )\right ) \log (x)\right )}{x} \, dx \]

[In]

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

[Out]

5*x - E^5*Defer[Int][x^(E^x*(-3 + 4*x)), x] + 3*E^5*Defer[Int][E^x*x^(E^x*(-3 + 4*x)), x] + 4*E^5*Defer[Int][E
^(1 + x)*x^(E^x*(-3 + 4*x)), x] + E^5*Log[x]*Defer[Int][E^(1 + x)*x^(E^x*(-3 + 4*x)), x] - 3*E^5*Defer[Int][E^
(1 + x)*x^(-1 + E^x*(-3 + 4*x)), x] - 4*E^5*Defer[Int][E^x*x^(1 + E^x*(-3 + 4*x)), x] - (1 - 4*E)*E^5*Log[x]*D
efer[Int][E^x*x^(1 + E^x*(-3 + 4*x)), x] - 4*E^5*Log[x]*Defer[Int][E^x*x^(2 + E^x*(-3 + 4*x)), x] - E^5*Defer[
Int][Defer[Int][E^(1 + x)*x^(E^x*(-3 + 4*x)), x]/x, x] + (1 - 4*E)*E^5*Defer[Int][Defer[Int][E^x*x^(1 + E^x*(-
3 + 4*x)), x]/x, x] + 4*E^5*Defer[Int][Defer[Int][E^x*x^(2 + E^x*(-3 + 4*x)), x]/x, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (5+e^5 x^{-1+e^x (-3+4 x)} \left (-3 e^{1+x}-x+3 e^x \left (1+\frac {4 e}{3}\right ) x-4 e^x x^2+e^{1+x} x \log (x)-(1-4 e) e^x x^2 \log (x)-4 e^x x^3 \log (x)\right )\right ) \, dx \\ & = 5 x+e^5 \int x^{-1+e^x (-3+4 x)} \left (-3 e^{1+x}-x+3 e^x \left (1+\frac {4 e}{3}\right ) x-4 e^x x^2+e^{1+x} x \log (x)-(1-4 e) e^x x^2 \log (x)-4 e^x x^3 \log (x)\right ) \, dx \\ & = 5 x+e^5 \int x^{-1+e^x (-3+4 x)} \left (-x+e^{1+x} (-3+4 x)-e^x x (-3+4 x)+e^x (e-x) x (1+4 x) \log (x)\right ) \, dx \\ & = 5 x+e^5 \int \left (-x^{e^x (-3+4 x)}-e^x x^{e^x (-3+4 x)} (-3+4 x)+e^{1+x} x^{-1+e^x (-3+4 x)} (-3+4 x)+e^x (e-x) x^{e^x (-3+4 x)} (1+4 x) \log (x)\right ) \, dx \\ & = 5 x-e^5 \int x^{e^x (-3+4 x)} \, dx-e^5 \int e^x x^{e^x (-3+4 x)} (-3+4 x) \, dx+e^5 \int e^{1+x} x^{-1+e^x (-3+4 x)} (-3+4 x) \, dx+e^5 \int e^x (e-x) x^{e^x (-3+4 x)} (1+4 x) \log (x) \, dx \\ & = 5 x-e^5 \int x^{e^x (-3+4 x)} \, dx+e^5 \int \left (4 e^{1+x} x^{e^x (-3+4 x)}-3 e^{1+x} x^{-1+e^x (-3+4 x)}\right ) \, dx-e^5 \int \left (-3 e^x x^{e^x (-3+4 x)}+4 e^x x^{1+e^x (-3+4 x)}\right ) \, dx-e^5 \int \frac {\int e^{1+x} x^{e^x (-3+4 x)} \, dx+(-1+4 e) \int e^x x^{1+e^x (-3+4 x)} \, dx-4 \int e^x x^{2+e^x (-3+4 x)} \, dx}{x} \, dx+\left (e^5 \log (x)\right ) \int e^{1+x} x^{e^x (-3+4 x)} \, dx-\left (4 e^5 \log (x)\right ) \int e^x x^{2+e^x (-3+4 x)} \, dx-\left ((1-4 e) e^5 \log (x)\right ) \int e^x x^{1+e^x (-3+4 x)} \, dx \\ & = 5 x-e^5 \int x^{e^x (-3+4 x)} \, dx-e^5 \int \left (\frac {\int e^{1+x} x^{e^x (-3+4 x)} \, dx-(1-4 e) \int e^x x^{1+e^x (-3+4 x)} \, dx}{x}-\frac {4 \int e^x x^{2+e^x (-3+4 x)} \, dx}{x}\right ) \, dx+\left (3 e^5\right ) \int e^x x^{e^x (-3+4 x)} \, dx-\left (3 e^5\right ) \int e^{1+x} x^{-1+e^x (-3+4 x)} \, dx+\left (4 e^5\right ) \int e^{1+x} x^{e^x (-3+4 x)} \, dx-\left (4 e^5\right ) \int e^x x^{1+e^x (-3+4 x)} \, dx+\left (e^5 \log (x)\right ) \int e^{1+x} x^{e^x (-3+4 x)} \, dx-\left (4 e^5 \log (x)\right ) \int e^x x^{2+e^x (-3+4 x)} \, dx-\left ((1-4 e) e^5 \log (x)\right ) \int e^x x^{1+e^x (-3+4 x)} \, dx \\ & = 5 x-e^5 \int x^{e^x (-3+4 x)} \, dx-e^5 \int \frac {\int e^{1+x} x^{e^x (-3+4 x)} \, dx-(1-4 e) \int e^x x^{1+e^x (-3+4 x)} \, dx}{x} \, dx+\left (3 e^5\right ) \int e^x x^{e^x (-3+4 x)} \, dx-\left (3 e^5\right ) \int e^{1+x} x^{-1+e^x (-3+4 x)} \, dx+\left (4 e^5\right ) \int e^{1+x} x^{e^x (-3+4 x)} \, dx-\left (4 e^5\right ) \int e^x x^{1+e^x (-3+4 x)} \, dx+\left (4 e^5\right ) \int \frac {\int e^x x^{2+e^x (-3+4 x)} \, dx}{x} \, dx+\left (e^5 \log (x)\right ) \int e^{1+x} x^{e^x (-3+4 x)} \, dx-\left (4 e^5 \log (x)\right ) \int e^x x^{2+e^x (-3+4 x)} \, dx-\left ((1-4 e) e^5 \log (x)\right ) \int e^x x^{1+e^x (-3+4 x)} \, dx \\ & = 5 x-e^5 \int x^{e^x (-3+4 x)} \, dx-e^5 \int \left (\frac {\int e^{1+x} x^{e^x (-3+4 x)} \, dx}{x}+\frac {(-1+4 e) \int e^x x^{1+e^x (-3+4 x)} \, dx}{x}\right ) \, dx+\left (3 e^5\right ) \int e^x x^{e^x (-3+4 x)} \, dx-\left (3 e^5\right ) \int e^{1+x} x^{-1+e^x (-3+4 x)} \, dx+\left (4 e^5\right ) \int e^{1+x} x^{e^x (-3+4 x)} \, dx-\left (4 e^5\right ) \int e^x x^{1+e^x (-3+4 x)} \, dx+\left (4 e^5\right ) \int \frac {\int e^x x^{2+e^x (-3+4 x)} \, dx}{x} \, dx+\left (e^5 \log (x)\right ) \int e^{1+x} x^{e^x (-3+4 x)} \, dx-\left (4 e^5 \log (x)\right ) \int e^x x^{2+e^x (-3+4 x)} \, dx-\left ((1-4 e) e^5 \log (x)\right ) \int e^x x^{1+e^x (-3+4 x)} \, dx \\ & = 5 x-e^5 \int x^{e^x (-3+4 x)} \, dx-e^5 \int \frac {\int e^{1+x} x^{e^x (-3+4 x)} \, dx}{x} \, dx+\left (3 e^5\right ) \int e^x x^{e^x (-3+4 x)} \, dx-\left (3 e^5\right ) \int e^{1+x} x^{-1+e^x (-3+4 x)} \, dx+\left (4 e^5\right ) \int e^{1+x} x^{e^x (-3+4 x)} \, dx-\left (4 e^5\right ) \int e^x x^{1+e^x (-3+4 x)} \, dx+\left (4 e^5\right ) \int \frac {\int e^x x^{2+e^x (-3+4 x)} \, dx}{x} \, dx+\left ((1-4 e) e^5\right ) \int \frac {\int e^x x^{1+e^x (-3+4 x)} \, dx}{x} \, dx+\left (e^5 \log (x)\right ) \int e^{1+x} x^{e^x (-3+4 x)} \, dx-\left (4 e^5 \log (x)\right ) \int e^x x^{2+e^x (-3+4 x)} \, dx-\left ((1-4 e) e^5 \log (x)\right ) \int e^x x^{1+e^x (-3+4 x)} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.34 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {5 x+e^{5+e^x (-3+4 x) \log (x)} \left (-x+e^x \left (3 x-4 x^2+e (-3+4 x)\right )+e^x \left (-x^2-4 x^3+e \left (x+4 x^2\right )\right ) \log (x)\right )}{x} \, dx=5 x+e^5 (e-x) x^{e^x (-3+4 x)} \]

[In]

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

[Out]

5*x + E^5*(E - x)*x^(E^x*(-3 + 4*x))

Maple [A] (verified)

Time = 10.56 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92

method	result	size
risch	\(5 x +\left ({\mathrm e}-x \right ) x^{\left (-3+4 x \right ) {\mathrm e}^{x}} {\mathrm e}^{5}\)	\(24\)
parallelrisch	\({\mathrm e} \,{\mathrm e}^{\left (-3+4 x \right ) {\mathrm e}^{x} \ln \left (x \right )+5}-x \,{\mathrm e}^{\left (-3+4 x \right ) {\mathrm e}^{x} \ln \left (x \right )+5}+5 x\)	\(37\)

[In]

int(((((4*x^2+x)*exp(1)-4*x^3-x^2)*exp(x)*ln(x)+((-3+4*x)*exp(1)-4*x^2+3*x)*exp(x)-x)*exp((-3+4*x)*exp(x)*ln(x
)+5)+5*x)/x,x,method=_RETURNVERBOSE)

[Out]

5*x+(exp(1)-x)*x^((-3+4*x)*exp(x))*exp(5)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {5 x+e^{5+e^x (-3+4 x) \log (x)} \left (-x+e^x \left (3 x-4 x^2+e (-3+4 x)\right )+e^x \left (-x^2-4 x^3+e \left (x+4 x^2\right )\right ) \log (x)\right )}{x} \, dx=-{\left (x - e\right )} e^{\left ({\left (4 \, x - 3\right )} e^{x} \log \left (x\right ) + 5\right )} + 5 \, x \]

[In]

integrate(((((4*x^2+x)*exp(1)-4*x^3-x^2)*exp(x)*log(x)+((-3+4*x)*exp(1)-4*x^2+3*x)*exp(x)-x)*exp((-3+4*x)*exp(
x)*log(x)+5)+5*x)/x,x, algorithm="fricas")

[Out]

-(x - e)*e^((4*x - 3)*e^x*log(x) + 5) + 5*x

Sympy [A] (verification not implemented)

Time = 13.67 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {5 x+e^{5+e^x (-3+4 x) \log (x)} \left (-x+e^x \left (3 x-4 x^2+e (-3+4 x)\right )+e^x \left (-x^2-4 x^3+e \left (x+4 x^2\right )\right ) \log (x)\right )}{x} \, dx=5 x + \left (e - x\right ) e^{\left (4 x - 3\right ) e^{x} \log {\left (x \right )} + 5} \]

[In]

integrate(((((4*x**2+x)*exp(1)-4*x**3-x**2)*exp(x)*ln(x)+((-3+4*x)*exp(1)-4*x**2+3*x)*exp(x)-x)*exp((-3+4*x)*e
xp(x)*ln(x)+5)+5*x)/x,x)

[Out]

5*x + (E - x)*exp((4*x - 3)*exp(x)*log(x) + 5)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {5 x+e^{5+e^x (-3+4 x) \log (x)} \left (-x+e^x \left (3 x-4 x^2+e (-3+4 x)\right )+e^x \left (-x^2-4 x^3+e \left (x+4 x^2\right )\right ) \log (x)\right )}{x} \, dx=-{\left (x e^{5} - e^{6}\right )} e^{\left (4 \, x e^{x} \log \left (x\right ) - 3 \, e^{x} \log \left (x\right )\right )} + 5 \, x \]

[In]

integrate(((((4*x^2+x)*exp(1)-4*x^3-x^2)*exp(x)*log(x)+((-3+4*x)*exp(1)-4*x^2+3*x)*exp(x)-x)*exp((-3+4*x)*exp(
x)*log(x)+5)+5*x)/x,x, algorithm="maxima")

[Out]

-(x*e^5 - e^6)*e^(4*x*e^x*log(x) - 3*e^x*log(x)) + 5*x

Giac [F]

\[ \int \frac {5 x+e^{5+e^x (-3+4 x) \log (x)} \left (-x+e^x \left (3 x-4 x^2+e (-3+4 x)\right )+e^x \left (-x^2-4 x^3+e \left (x+4 x^2\right )\right ) \log (x)\right )}{x} \, dx=\int { -\frac {{\left ({\left (4 \, x^{3} + x^{2} - {\left (4 \, x^{2} + x\right )} e\right )} e^{x} \log \left (x\right ) + {\left (4 \, x^{2} - {\left (4 \, x - 3\right )} e - 3 \, x\right )} e^{x} + x\right )} e^{\left ({\left (4 \, x - 3\right )} e^{x} \log \left (x\right ) + 5\right )} - 5 \, x}{x} \,d x } \]

[In]

integrate(((((4*x^2+x)*exp(1)-4*x^3-x^2)*exp(x)*log(x)+((-3+4*x)*exp(1)-4*x^2+3*x)*exp(x)-x)*exp((-3+4*x)*exp(
x)*log(x)+5)+5*x)/x,x, algorithm="giac")

[Out]

integrate(-(((4*x^3 + x^2 - (4*x^2 + x)*e)*e^x*log(x) + (4*x^2 - (4*x - 3)*e - 3*x)*e^x + x)*e^((4*x - 3)*e^x*
log(x) + 5) - 5*x)/x, x)

Mupad [B] (verification not implemented)

Time = 12.32 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {5 x+e^{5+e^x (-3+4 x) \log (x)} \left (-x+e^x \left (3 x-4 x^2+e (-3+4 x)\right )+e^x \left (-x^2-4 x^3+e \left (x+4 x^2\right )\right ) \log (x)\right )}{x} \, dx=5\,x-\frac {x^{4\,x\,{\mathrm {e}}^x}\,{\mathrm {e}}^5\,\left (x-\mathrm {e}\right )}{x^{3\,{\mathrm {e}}^x}} \]

[In]

int((5*x - exp(exp(x)*log(x)*(4*x - 3) + 5)*(x - exp(x)*(3*x - 4*x^2 + exp(1)*(4*x - 3)) + exp(x)*log(x)*(x^2
- exp(1)*(x + 4*x^2) + 4*x^3)))/x,x)

[Out]

5*x - (x^(4*x*exp(x))*exp(5)*(x - exp(1)))/x^(3*exp(x))