3.733 \(\int \left (\frac{x^2 \left (5 e^x+3 x^2\right )}{5 \sqrt{5 e^x+x^3}}+\frac{4}{5} x \sqrt{5 e^x+x^3}\right ) \, dx\)

Optimal. Leaf size=20 \[ \frac{2}{5} x^2 \sqrt{x^3+5 e^x} \]

[Out]

(2*x^2*Sqrt[5*E^x + x^3])/5

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Rubi [A]  time = 0.943667, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 50, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04 \[ \frac{2}{5} x^2 \sqrt{x^3+5 e^x} \]

Antiderivative was successfully verified.

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

[Out]

(2*x^2*Sqrt[5*E^x + x^3])/5

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/5*x**2*(5*exp(x)+3*x**2)/(5*exp(x)+x**3)**(1/2)+4/5*x*(5*exp(x)+x**3)**(1/2),x)

[Out]

Timed out

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Mathematica [A]  time = 0.0462046, size = 20, normalized size = 1. \[ \frac{2}{5} x^2 \sqrt{x^3+5 e^x} \]

Antiderivative was successfully verified.

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

[Out]

(2*x^2*Sqrt[5*E^x + x^3])/5

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Maple [A]  time = 0.05, size = 16, normalized size = 0.8 \[{\frac{2\,{x}^{2}}{5}\sqrt{5\,{{\rm e}^{x}}+{x}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/5*x^2*(5*exp(x)+3*x^2)/(5*exp(x)+x^3)^(1/2)+4/5*x*(5*exp(x)+x^3)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.920601, size = 31, normalized size = 1.55 \[ \frac{2 \,{\left (x^{5} + 5 \, x^{2} e^{x}\right )}}{5 \, \sqrt{x^{3} + 5 \, e^{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/5*(3*x^2 + 5*e^x)*x^2/sqrt(x^3 + 5*e^x) + 4/5*sqrt(x^3 + 5*e^x)*x,x, algorithm="maxima")

[Out]

2/5*(x^5 + 5*x^2*e^x)/sqrt(x^3 + 5*e^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(1/5*(3*x^2 + 5*e^x)*x^2/sqrt(x^3 + 5*e^x) + 4/5*sqrt(x^3 + 5*e^x)*x,x, algorithm="fricas")

[Out]

Exception raised: TypeError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int \frac{7 x^{4}}{\sqrt{x^{3} + 5 e^{x}}}\, dx + \int \frac{20 x e^{x}}{\sqrt{x^{3} + 5 e^{x}}}\, dx + \int \frac{5 x^{2} e^{x}}{\sqrt{x^{3} + 5 e^{x}}}\, dx}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/5*x**2*(5*exp(x)+3*x**2)/(5*exp(x)+x**3)**(1/2)+4/5*x*(5*exp(x)+x**3)**(1/2),x)

[Out]

(Integral(7*x**4/sqrt(x**3 + 5*exp(x)), x) + Integral(20*x*exp(x)/sqrt(x**3 + 5*
exp(x)), x) + Integral(5*x**2*exp(x)/sqrt(x**3 + 5*exp(x)), x))/5

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (3 \, x^{2} + 5 \, e^{x}\right )} x^{2}}{5 \, \sqrt{x^{3} + 5 \, e^{x}}} + \frac{4}{5} \, \sqrt{x^{3} + 5 \, e^{x}} x\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/5*(3*x^2 + 5*e^x)*x^2/sqrt(x^3 + 5*e^x) + 4/5*sqrt(x^3 + 5*e^x)*x,x, algorithm="giac")

[Out]

integrate(1/5*(3*x^2 + 5*e^x)*x^2/sqrt(x^3 + 5*e^x) + 4/5*sqrt(x^3 + 5*e^x)*x, x
)