\(\int \frac {e^x x^2}{\sqrt {5 e^x+x^3}} \, dx\) [751]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {e^x x^2}{\sqrt {5 e^x+x^3}} \, dx=\frac {2}{5} x^2 \sqrt {5 e^x+x^3}-\frac {3}{5} \text {Int}\left (\frac {x^4}{\sqrt {5 e^x+x^3}},x\right )-\frac {4}{5} \text {Int}\left (x \sqrt {5 e^x+x^3},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.11 (sec) , antiderivative size = 20, 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 = {} \[ \int \frac {e^x x^2}{\sqrt {5 e^x+x^3}} \, dx=\int \frac {e^x x^2}{\sqrt {5 e^x+x^3}} \, dx \]

[In]

Int[(E^x*x^2)/Sqrt[5*E^x + x^3],x]

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \frac {2}{5} x^2 \sqrt {5 e^x+x^3}-\frac {3}{5} \int \frac {x^4}{\sqrt {5 e^x+x^3}} \, dx-\frac {4}{5} \int x \sqrt {5 e^x+x^3} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.38 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {e^x x^2}{\sqrt {5 e^x+x^3}} \, dx=\int \frac {e^x x^2}{\sqrt {5 e^x+x^3}} \, dx \]

[In]

Integrate[(E^x*x^2)/Sqrt[5*E^x + x^3],x]

[Out]

Integrate[(E^x*x^2)/Sqrt[5*E^x + x^3], x]

Maple [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80

\[\int \frac {{\mathrm e}^{x} x^{2}}{\sqrt {5 \,{\mathrm e}^{x}+x^{3}}}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {e^x x^2}{\sqrt {5 e^x+x^3}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(exp(x)*x^2/(5*exp(x)+x^3)^(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 = 0.44 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {e^x x^2}{\sqrt {5 e^x+x^3}} \, dx=\int \frac {x^{2} e^{x}}{\sqrt {x^{3} + 5 e^{x}}}\, dx \]

[In]

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

[Out]

Integral(x**2*exp(x)/sqrt(x**3 + 5*exp(x)), x)

Maxima [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {e^x x^2}{\sqrt {5 e^x+x^3}} \, dx=\int { \frac {x^{2} e^{x}}{\sqrt {x^{3} + 5 \, e^{x}}} \,d x } \]

[In]

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

[Out]

integrate(x^2*e^x/sqrt(x^3 + 5*e^x), x)

Giac [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {e^x x^2}{\sqrt {5 e^x+x^3}} \, dx=\int { \frac {x^{2} e^{x}}{\sqrt {x^{3} + 5 \, e^{x}}} \,d x } \]

[In]

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

[Out]

integrate(x^2*e^x/sqrt(x^3 + 5*e^x), x)

Mupad [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {e^x x^2}{\sqrt {5 e^x+x^3}} \, dx=\int \frac {x^2\,{\mathrm {e}}^x}{\sqrt {5\,{\mathrm {e}}^x+x^3}} \,d x \]

[In]

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

[Out]

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