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

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

[Out]

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

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Rubi [A]  time = 0.2604, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0. \[ \text{Int}\left (\frac{e^x x^2}{\sqrt{5 e^x+x^3}},x\right ) \]

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*De
fer[Int][x*Sqrt[5*E^x + x^3], x])/5

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Rubi in Sympy [A]  time = 0., size = 0, normalized size = 0. \[ \frac{2 x^{2} \sqrt{x^{3} + 5 e^{x}}}{5} - \frac{4 \int x \sqrt{x^{3} + 5 e^{x}}\, dx}{5} - \frac{3 \int \frac{x^{4}}{\sqrt{x^{3} + 5 e^{x}}}\, dx}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(x)*x**2/(5*exp(x)+x**3)**(1/2),x)

[Out]

2*x**2*sqrt(x**3 + 5*exp(x))/5 - 4*Integral(x*sqrt(x**3 + 5*exp(x)), x)/5 - 3*In
tegral(x**4/sqrt(x**3 + 5*exp(x)), x)/5

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Mathematica [A]  time = 0.20717, size = 0, normalized size = 0. \[ \int \frac{e^x x^2}{\sqrt{5 e^x+x^3}} \, dx \]

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 = 0.029, size = 0, normalized size = 0. \[ \int{{{\rm e}^{x}}{x}^{2}{\frac{1}{\sqrt{5\,{{\rm e}^{x}}+{x}^{3}}}}}\, dx \]

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., size = 0, normalized size = 0. \[ \int \frac{x^{2} e^{x}}{\sqrt{x^{3} + 5 \, e^{x}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2*e^x/sqrt(x^3 + 5*e^x),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., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2*e^x/sqrt(x^3 + 5*e^x),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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

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/XCAS [A]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2} e^{x}}{\sqrt{x^{3} + 5 \, e^{x}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2*e^x/sqrt(x^3 + 5*e^x),x, algorithm="giac")

[Out]

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