\(\int \frac {e^x x}{\sqrt {e^x+x}} \, dx\) [749]

Optimal result

Integrand size = 14, antiderivative size = 14 \[ \int \frac {e^x x}{\sqrt {e^x+x}} \, dx=2 \sqrt {e^x+x}+2 x \sqrt {e^x+x}-\text {Int}\left (\frac {1}{\sqrt {e^x+x}},x\right )-3 \text {Int}\left (\sqrt {e^x+x},x\right ) \]

[Out]

-CannotIntegrate(1/(exp(x)+x)^(1/2),x)-3*CannotIntegrate((exp(x)+x)^(1/2),x)+2*(exp(x)+x)^(1/2)+2*x*(exp(x)+x)
^(1/2)

Rubi [N/A]

Not integrable

Time = 0.07 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {e^x x}{\sqrt {e^x+x}} \, dx=\int \frac {e^x x}{\sqrt {e^x+x}} \, dx \]

[In]

Int[(E^x*x)/Sqrt[E^x + x],x]

[Out]

2*Sqrt[E^x + x] + 2*x*Sqrt[E^x + x] - Defer[Int][1/Sqrt[E^x + x], x] - 3*Defer[Int][Sqrt[E^x + x], x]

Rubi steps \begin{align*} \text {integral}& = 2 x \sqrt {e^x+x}-2 \int \sqrt {e^x+x} \, dx-\int \frac {x}{\sqrt {e^x+x}} \, dx \\ & = 2 \sqrt {e^x+x}+2 x \sqrt {e^x+x}-2 \int \sqrt {e^x+x} \, dx-\int \frac {1}{\sqrt {e^x+x}} \, dx-\int \sqrt {e^x+x} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.14 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {e^x x}{\sqrt {e^x+x}} \, dx=\int \frac {e^x x}{\sqrt {e^x+x}} \, dx \]

[In]

Integrate[(E^x*x)/Sqrt[E^x + x],x]

[Out]

Integrate[(E^x*x)/Sqrt[E^x + x], x]

Maple [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71

\[\int \frac {{\mathrm e}^{x} x}{\sqrt {{\mathrm e}^{x}+x}}d x\]

[In]

int(exp(x)*x/(exp(x)+x)^(1/2),x)

[Out]

int(exp(x)*x/(exp(x)+x)^(1/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {e^x x}{\sqrt {e^x+x}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(exp(x)*x/(exp(x)+x)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

Sympy [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {e^x x}{\sqrt {e^x+x}} \, dx=\int \frac {x e^{x}}{\sqrt {x + e^{x}}}\, dx \]

[In]

integrate(exp(x)*x/(exp(x)+x)**(1/2),x)

[Out]

Integral(x*exp(x)/sqrt(x + exp(x)), x)

Maxima [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {e^x x}{\sqrt {e^x+x}} \, dx=\int { \frac {x e^{x}}{\sqrt {x + e^{x}}} \,d x } \]

[In]

integrate(exp(x)*x/(exp(x)+x)^(1/2),x, algorithm="maxima")

[Out]

integrate(x*e^x/sqrt(x + e^x), x)

Giac [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {e^x x}{\sqrt {e^x+x}} \, dx=\int { \frac {x e^{x}}{\sqrt {x + e^{x}}} \,d x } \]

[In]

integrate(exp(x)*x/(exp(x)+x)^(1/2),x, algorithm="giac")

[Out]

integrate(x*e^x/sqrt(x + e^x), x)

Mupad [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {e^x x}{\sqrt {e^x+x}} \, dx=\int \frac {x\,{\mathrm {e}}^x}{\sqrt {x+{\mathrm {e}}^x}} \,d x \]

[In]

int((x*exp(x))/(x + exp(x))^(1/2),x)

[Out]

int((x*exp(x))/(x + exp(x))^(1/2), x)