\(\int \frac {(1+e^x) x}{\sqrt {e^x+x}} \, dx\) [747]

Optimal result

Integrand size = 16, antiderivative size = 16 \[ \int \frac {\left (1+e^x\right ) x}{\sqrt {e^x+x}} \, dx=2 x \sqrt {e^x+x}-2 \text {Int}\left (\sqrt {e^x+x},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.15 (sec) , antiderivative size = 16, 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 = {} \[ \int \frac {\left (1+e^x\right ) x}{\sqrt {e^x+x}} \, dx=\int \frac {\left (1+e^x\right ) x}{\sqrt {e^x+x}} \, dx \]

[In]

Int[((1 + E^x)*x)/Sqrt[E^x + x],x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.17 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \frac {\left (1+e^x\right ) x}{\sqrt {e^x+x}} \, dx=\int \frac {\left (1+e^x\right ) x}{\sqrt {e^x+x}} \, dx \]

[In]

Integrate[((1 + E^x)*x)/Sqrt[E^x + x],x]

[Out]

Integrate[((1 + E^x)*x)/Sqrt[E^x + x], x]

Maple [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75

\[\int \frac {\left (1+{\mathrm e}^{x}\right ) x}{\sqrt {{\mathrm e}^{x}+x}}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {\left (1+e^x\right ) x}{\sqrt {e^x+x}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((1+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 = 0.74 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \frac {\left (1+e^x\right ) x}{\sqrt {e^x+x}} \, dx=\int \frac {x \left (e^{x} + 1\right )}{\sqrt {x + e^{x}}}\, dx \]

[In]

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

[Out]

Integral(x*(exp(x) + 1)/sqrt(x + exp(x)), x)

Maxima [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {\left (1+e^x\right ) x}{\sqrt {e^x+x}} \, dx=\int { \frac {x {\left (e^{x} + 1\right )}}{\sqrt {x + e^{x}}} \,d x } \]

[In]

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

[Out]

integrate(x*(e^x + 1)/sqrt(x + e^x), x)

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {\left (1+e^x\right ) x}{\sqrt {e^x+x}} \, dx=\int { \frac {x {\left (e^{x} + 1\right )}}{\sqrt {x + e^{x}}} \,d x } \]

[In]

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

[Out]

integrate(x*(e^x + 1)/sqrt(x + e^x), x)

Mupad [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {\left (1+e^x\right ) x}{\sqrt {e^x+x}} \, dx=\int \frac {x\,\left ({\mathrm {e}}^x+1\right )}{\sqrt {x+{\mathrm {e}}^x}} \,d x \]

[In]

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

[Out]

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