3.818 \(\int \sqrt{x^2+x^3} \, dx\)

Optimal. Leaf size=37 \[ \frac{2 \left (x^3+x^2\right )^{3/2}}{5 x^2}-\frac{4 \left (x^3+x^2\right )^{3/2}}{15 x^3} \]

[Out]

(-4*(x^2 + x^3)^(3/2))/(15*x^3) + (2*(x^2 + x^3)^(3/2))/(5*x^2)

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Rubi [A]  time = 0.0455966, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{2 \left (x^3+x^2\right )^{3/2}}{5 x^2}-\frac{4 \left (x^3+x^2\right )^{3/2}}{15 x^3} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x^2 + x^3],x]

[Out]

(-4*(x^2 + x^3)^(3/2))/(15*x^3) + (2*(x^2 + x^3)^(3/2))/(5*x^2)

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Rubi in Sympy [A]  time = 3.03249, size = 32, normalized size = 0.86 \[ \frac{2 \left (x^{3} + x^{2}\right )^{\frac{3}{2}}}{5 x^{2}} - \frac{4 \left (x^{3} + x^{2}\right )^{\frac{3}{2}}}{15 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(x**3 + x**2)**(3/2)/(5*x**2) - 4*(x**3 + x**2)**(3/2)/(15*x**3)

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Mathematica [A]  time = 0.0110324, size = 23, normalized size = 0.62 \[ \frac{2 \left (x^2 (x+1)\right )^{3/2} (3 x-2)}{15 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x^2 + x^3],x]

[Out]

(2*(x^2*(1 + x))^(3/2)*(-2 + 3*x))/(15*x^3)

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Maple [A]  time = 0.003, size = 23, normalized size = 0.6 \[{\frac{ \left ( 2+2\,x \right ) \left ( 3\,x-2 \right ) }{15\,x}\sqrt{{x}^{3}+{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^3+x^2)^(1/2),x)

[Out]

2/15*(1+x)*(3*x-2)*(x^3+x^2)^(1/2)/x

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Maxima [A]  time = 0.721156, size = 20, normalized size = 0.54 \[ \frac{2}{15} \,{\left (3 \, x^{2} + x - 2\right )} \sqrt{x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^3 + x^2),x, algorithm="maxima")

[Out]

2/15*(3*x^2 + x - 2)*sqrt(x + 1)

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Fricas [A]  time = 0.265747, size = 30, normalized size = 0.81 \[ \frac{2 \, \sqrt{x^{3} + x^{2}}{\left (3 \, x^{2} + x - 2\right )}}{15 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*sqrt(x^3 + x^2)*(3*x^2 + x - 2)/x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**3+x**2)**(1/2),x)

[Out]

Integral(sqrt(x**3 + x**2), x)

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GIAC/XCAS [A]  time = 0.262637, size = 32, normalized size = 0.86 \[ \frac{2}{15} \,{\left (3 \,{\left (x + 1\right )}^{\frac{5}{2}} - 5 \,{\left (x + 1\right )}^{\frac{3}{2}}\right )}{\rm sign}\left (x\right ) + \frac{4}{15} \,{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(3*(x + 1)^(5/2) - 5*(x + 1)^(3/2))*sign(x) + 4/15*sign(x)