\(\int \frac {1}{5} e^{\frac {1}{5} (5-7 x)} x^{e^{\frac {1}{5} (5-7 x)} (x+x^2)} (5+5 x+(5+3 x-7 x^2) \log (x)) \, dx\) [7668]

Optimal result

Integrand size = 52, antiderivative size = 24 \[ \int \frac {1}{5} e^{\frac {1}{5} (5-7 x)} x^{e^{\frac {1}{5} (5-7 x)} \left (x+x^2\right )} \left (5+5 x+\left (5+3 x-7 x^2\right ) \log (x)\right ) \, dx=5+x^{e^{\frac {1}{5} (5-2 x)-x} x (1+x)} \]

[Out]

5+exp((1+x)/exp(7/5*x-1)*x*ln(x))

Rubi [F]

\[ \int \frac {1}{5} e^{\frac {1}{5} (5-7 x)} x^{e^{\frac {1}{5} (5-7 x)} \left (x+x^2\right )} \left (5+5 x+\left (5+3 x-7 x^2\right ) \log (x)\right ) \, dx=\int \frac {1}{5} e^{\frac {1}{5} (5-7 x)} x^{e^{\frac {1}{5} (5-7 x)} \left (x+x^2\right )} \left (5+5 x+\left (5+3 x-7 x^2\right ) \log (x)\right ) \, dx \]

[In]

Int[(E^((5 - 7*x)/5)*x^(E^((5 - 7*x)/5)*(x + x^2))*(5 + 5*x + (5 + 3*x - 7*x^2)*Log[x]))/5,x]

[Out]

Defer[Int][E^(1 - (7*x)/5)*x^(E^(1 - (7*x)/5)*x*(1 + x)), x] + Log[x]*Defer[Int][E^(1 - (7*x)/5)*x^(E^(1 - (7*
x)/5)*x*(1 + x)), x] + Defer[Int][E^(1 - (7*x)/5)*x^(1 + E^(1 - (7*x)/5)*x*(1 + x)), x] + (3*Log[x]*Defer[Int]
[E^(1 - (7*x)/5)*x^(1 + E^(1 - (7*x)/5)*x*(1 + x)), x])/5 - (7*Log[x]*Defer[Int][E^(1 - (7*x)/5)*x^(2 + E^(1 -
(7*x)/5)*x*(1 + x)), x])/5 - Defer[Int][Defer[Int][E^(1 - (7*x)/5)*x^(E^(1 - (7*x)/5)*x*(1 + x)), x]/x, x] -
(3*Defer[Int][Defer[Int][E^(1 - (7*x)/5)*x^(1 + E^(1 - (7*x)/5)*x*(1 + x)), x]/x, x])/5 + (7*Defer[Int][Defer[
Int][E^(1 - (7*x)/5)*x^(2 + E^(1 - (7*x)/5)*x*(1 + x)), x]/x, x])/5

Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} \int e^{\frac {1}{5} (5-7 x)} x^{e^{\frac {1}{5} (5-7 x)} \left (x+x^2\right )} \left (5+5 x+\left (5+3 x-7 x^2\right ) \log (x)\right ) \, dx \\ & = \frac {1}{5} \int e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)} \left (5+5 x+\left (5+3 x-7 x^2\right ) \log (x)\right ) \, dx \\ & = \frac {1}{5} \int \left (5 e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)}+5 e^{1-\frac {7 x}{5}} x^{1+e^{1-\frac {7 x}{5}} x (1+x)}-e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)} \left (-5-3 x+7 x^2\right ) \log (x)\right ) \, dx \\ & = -\left (\frac {1}{5} \int e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)} \left (-5-3 x+7 x^2\right ) \log (x) \, dx\right )+\int e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)} \, dx+\int e^{1-\frac {7 x}{5}} x^{1+e^{1-\frac {7 x}{5}} x (1+x)} \, dx \\ & = \frac {1}{5} \int \frac {-5 \int e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)} \, dx-3 \int e^{1-\frac {7 x}{5}} x^{1+e^{1-\frac {7 x}{5}} x (1+x)} \, dx+7 \int e^{1-\frac {7 x}{5}} x^{2+e^{1-\frac {7 x}{5}} x (1+x)} \, dx}{x} \, dx+\frac {1}{5} (3 \log (x)) \int e^{1-\frac {7 x}{5}} x^{1+e^{1-\frac {7 x}{5}} x (1+x)} \, dx+\log (x) \int e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)} \, dx-\frac {1}{5} (7 \log (x)) \int e^{1-\frac {7 x}{5}} x^{2+e^{1-\frac {7 x}{5}} x (1+x)} \, dx+\int e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)} \, dx+\int e^{1-\frac {7 x}{5}} x^{1+e^{1-\frac {7 x}{5}} x (1+x)} \, dx \\ & = \frac {1}{5} \int \left (\frac {-5 \int e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)} \, dx-3 \int e^{1-\frac {7 x}{5}} x^{1+e^{1-\frac {7 x}{5}} x (1+x)} \, dx}{x}+\frac {7 \int e^{1-\frac {7 x}{5}} x^{2+e^{1-\frac {7 x}{5}} x (1+x)} \, dx}{x}\right ) \, dx+\frac {1}{5} (3 \log (x)) \int e^{1-\frac {7 x}{5}} x^{1+e^{1-\frac {7 x}{5}} x (1+x)} \, dx+\log (x) \int e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)} \, dx-\frac {1}{5} (7 \log (x)) \int e^{1-\frac {7 x}{5}} x^{2+e^{1-\frac {7 x}{5}} x (1+x)} \, dx+\int e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)} \, dx+\int e^{1-\frac {7 x}{5}} x^{1+e^{1-\frac {7 x}{5}} x (1+x)} \, dx \\ & = \frac {1}{5} \int \frac {-5 \int e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)} \, dx-3 \int e^{1-\frac {7 x}{5}} x^{1+e^{1-\frac {7 x}{5}} x (1+x)} \, dx}{x} \, dx+\frac {7}{5} \int \frac {\int e^{1-\frac {7 x}{5}} x^{2+e^{1-\frac {7 x}{5}} x (1+x)} \, dx}{x} \, dx+\frac {1}{5} (3 \log (x)) \int e^{1-\frac {7 x}{5}} x^{1+e^{1-\frac {7 x}{5}} x (1+x)} \, dx+\log (x) \int e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)} \, dx-\frac {1}{5} (7 \log (x)) \int e^{1-\frac {7 x}{5}} x^{2+e^{1-\frac {7 x}{5}} x (1+x)} \, dx+\int e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)} \, dx+\int e^{1-\frac {7 x}{5}} x^{1+e^{1-\frac {7 x}{5}} x (1+x)} \, dx \\ & = \frac {1}{5} \int \left (-\frac {5 \int e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)} \, dx}{x}-\frac {3 \int e^{1-\frac {7 x}{5}} x^{1+e^{1-\frac {7 x}{5}} x (1+x)} \, dx}{x}\right ) \, dx+\frac {7}{5} \int \frac {\int e^{1-\frac {7 x}{5}} x^{2+e^{1-\frac {7 x}{5}} x (1+x)} \, dx}{x} \, dx+\frac {1}{5} (3 \log (x)) \int e^{1-\frac {7 x}{5}} x^{1+e^{1-\frac {7 x}{5}} x (1+x)} \, dx+\log (x) \int e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)} \, dx-\frac {1}{5} (7 \log (x)) \int e^{1-\frac {7 x}{5}} x^{2+e^{1-\frac {7 x}{5}} x (1+x)} \, dx+\int e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)} \, dx+\int e^{1-\frac {7 x}{5}} x^{1+e^{1-\frac {7 x}{5}} x (1+x)} \, dx \\ & = -\left (\frac {3}{5} \int \frac {\int e^{1-\frac {7 x}{5}} x^{1+e^{1-\frac {7 x}{5}} x (1+x)} \, dx}{x} \, dx\right )+\frac {7}{5} \int \frac {\int e^{1-\frac {7 x}{5}} x^{2+e^{1-\frac {7 x}{5}} x (1+x)} \, dx}{x} \, dx+\frac {1}{5} (3 \log (x)) \int e^{1-\frac {7 x}{5}} x^{1+e^{1-\frac {7 x}{5}} x (1+x)} \, dx+\log (x) \int e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)} \, dx-\frac {1}{5} (7 \log (x)) \int e^{1-\frac {7 x}{5}} x^{2+e^{1-\frac {7 x}{5}} x (1+x)} \, dx+\int e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)} \, dx+\int e^{1-\frac {7 x}{5}} x^{1+e^{1-\frac {7 x}{5}} x (1+x)} \, dx-\int \frac {\int e^{1-\frac {7 x}{5}} x^{e^{1-\frac {7 x}{5}} x (1+x)} \, dx}{x} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.85 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.67 \[ \int \frac {1}{5} e^{\frac {1}{5} (5-7 x)} x^{e^{\frac {1}{5} (5-7 x)} \left (x+x^2\right )} \left (5+5 x+\left (5+3 x-7 x^2\right ) \log (x)\right ) \, dx=x^{e^{1-\frac {7 x}{5}} x (1+x)} \]

[In]

Integrate[(E^((5 - 7*x)/5)*x^(E^((5 - 7*x)/5)*(x + x^2))*(5 + 5*x + (5 + 3*x - 7*x^2)*Log[x]))/5,x]

[Out]

x^(E^(1 - (7*x)/5)*x*(1 + x))

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.58

method	result	size
risch	\(x^{\left (1+x \right ) x \,{\mathrm e}^{-\frac {7 x}{5}+1}}\)	\(14\)
parallelrisch	\({\mathrm e}^{\left (1+x \right ) x \ln \left (x \right ) {\mathrm e}^{-\frac {7 x}{5}+1}}\)	\(17\)

[In]

int(1/5*((-7*x^2+3*x+5)*ln(x)+5*x+5)*exp((x^2+x)*ln(x)/exp(7/5*x-1))/exp(7/5*x-1),x,method=_RETURNVERBOSE)

[Out]

x^((1+x)*x*exp(-7/5*x+1))

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.58 \[ \int \frac {1}{5} e^{\frac {1}{5} (5-7 x)} x^{e^{\frac {1}{5} (5-7 x)} \left (x+x^2\right )} \left (5+5 x+\left (5+3 x-7 x^2\right ) \log (x)\right ) \, dx=x^{{\left (x^{2} + x\right )} e^{\left (-\frac {7}{5} \, x + 1\right )}} \]

[In]

integrate(1/5*((-7*x^2+3*x+5)*log(x)+5*x+5)*exp((x^2+x)*log(x)/exp(7/5*x-1))/exp(7/5*x-1),x, algorithm="fricas
")

[Out]

x^((x^2 + x)*e^(-7/5*x + 1))

Sympy [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.71 \[ \int \frac {1}{5} e^{\frac {1}{5} (5-7 x)} x^{e^{\frac {1}{5} (5-7 x)} \left (x+x^2\right )} \left (5+5 x+\left (5+3 x-7 x^2\right ) \log (x)\right ) \, dx=e^{\left (x^{2} + x\right ) e^{1 - \frac {7 x}{5}} \log {\left (x \right )}} \]

[In]

integrate(1/5*((-7*x**2+3*x+5)*ln(x)+5*x+5)*exp((x**2+x)*ln(x)/exp(7/5*x-1))/exp(7/5*x-1),x)

[Out]

exp((x**2 + x)*exp(1 - 7*x/5)*log(x))

Maxima [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{5} e^{\frac {1}{5} (5-7 x)} x^{e^{\frac {1}{5} (5-7 x)} \left (x+x^2\right )} \left (5+5 x+\left (5+3 x-7 x^2\right ) \log (x)\right ) \, dx=e^{\left (x^{2} e^{\left (-\frac {7}{5} \, x + 1\right )} \log \left (x\right ) + x e^{\left (-\frac {7}{5} \, x + 1\right )} \log \left (x\right )\right )} \]

[In]

integrate(1/5*((-7*x^2+3*x+5)*log(x)+5*x+5)*exp((x^2+x)*log(x)/exp(7/5*x-1))/exp(7/5*x-1),x, algorithm="maxima
")

[Out]

e^(x^2*e^(-7/5*x + 1)*log(x) + x*e^(-7/5*x + 1)*log(x))

Giac [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88 \[ \int \frac {1}{5} e^{\frac {1}{5} (5-7 x)} x^{e^{\frac {1}{5} (5-7 x)} \left (x+x^2\right )} \left (5+5 x+\left (5+3 x-7 x^2\right ) \log (x)\right ) \, dx=x^{x^{2} e^{\left (-\frac {7}{5} \, x + 1\right )} + x e^{\left (-\frac {7}{5} \, x + 1\right )}} \]

[In]

integrate(1/5*((-7*x^2+3*x+5)*log(x)+5*x+5)*exp((x^2+x)*log(x)/exp(7/5*x-1))/exp(7/5*x-1),x, algorithm="giac")

[Out]

x^(x^2*e^(-7/5*x + 1) + x*e^(-7/5*x + 1))

Mupad [B] (verification not implemented)

Time = 13.21 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.58 \[ \int \frac {1}{5} e^{\frac {1}{5} (5-7 x)} x^{e^{\frac {1}{5} (5-7 x)} \left (x+x^2\right )} \left (5+5 x+\left (5+3 x-7 x^2\right ) \log (x)\right ) \, dx=x^{{\mathrm {e}}^{1-\frac {7\,x}{5}}\,\left (x^2+x\right )} \]

[In]

int((exp(1 - (7*x)/5)*exp(exp(1 - (7*x)/5)*log(x)*(x + x^2))*(5*x + log(x)*(3*x - 7*x^2 + 5) + 5))/5,x)

[Out]

x^(exp(1 - (7*x)/5)*(x + x^2))