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

Optimal. Leaf size=67 \[ \frac {2}{5} x^2 \sqrt {5 e^x+x^3}-\frac {3}{5} \text {Int}\left (\frac {x^4}{\sqrt {5 e^x+x^3}},x\right )-\frac {4}{5} \text {Int}\left (x \sqrt {5 e^x+x^3},x\right ) \]

[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)

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Rubi [A]
time = 0.11, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {e^x x^2}{\sqrt {5 e^x+x^3}} \, dx \end {gather*}

Verification is not applicable to the result.

[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*} \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} \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*}

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Mathematica [A]
time = 0.36, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^x x^2}{\sqrt {5 e^x+x^3}} \, dx \end {gather*}

Verification is not applicable to the result.

[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]

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Maple [A]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{x} x^{2}}{\sqrt {5 \,{\mathrm e}^{x}+x^{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} e^{x}}{\sqrt {x^{3} + 5 e^{x}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2\,{\mathrm {e}}^x}{\sqrt {5\,{\mathrm {e}}^x+x^3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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