3.234 \(\int \sqrt{a x^2+b x^3} \, dx\)

Optimal. Leaf size=52 \[ \frac{2 \left (a x^2+b x^3\right )^{3/2}}{5 b x^2}-\frac{4 a \left (a x^2+b x^3\right )^{3/2}}{15 b^2 x^3} \]

[Out]

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

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Rubi [A]  time = 0.0799932, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{2 \left (a x^2+b x^3\right )^{3/2}}{5 b x^2}-\frac{4 a \left (a x^2+b x^3\right )^{3/2}}{15 b^2 x^3} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a*x^2 + b*x^3],x]

[Out]

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

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Rubi in Sympy [A]  time = 8.54183, size = 46, normalized size = 0.88 \[ - \frac{4 a \left (a x^{2} + b x^{3}\right )^{\frac{3}{2}}}{15 b^{2} x^{3}} + \frac{2 \left (a x^{2} + b x^{3}\right )^{\frac{3}{2}}}{5 b x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0151346, size = 41, normalized size = 0.79 \[ \frac{2 \sqrt{x^2 (a+b x)} \left (-2 a^2+a b x+3 b^2 x^2\right )}{15 b^2 x} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a*x^2 + b*x^3],x]

[Out]

(2*Sqrt[x^2*(a + b*x)]*(-2*a^2 + a*b*x + 3*b^2*x^2))/(15*b^2*x)

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Maple [A]  time = 0.005, size = 35, normalized size = 0.7 \[ -{\frac{ \left ( 2\,bx+2\,a \right ) \left ( -3\,bx+2\,a \right ) }{15\,{b}^{2}x}\sqrt{b{x}^{3}+a{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a*x^2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.40548, size = 41, normalized size = 0.79 \[ \frac{2 \,{\left (3 \, b^{2} x^{2} + a b x - 2 \, a^{2}\right )} \sqrt{b x + a}}{15 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.216243, size = 53, normalized size = 1.02 \[ \frac{2 \,{\left (3 \, b^{2} x^{2} + a b x - 2 \, a^{2}\right )} \sqrt{b x^{3} + a x^{2}}}{15 \, b^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a*x**2)**(1/2),x)

[Out]

Integral(sqrt(a*x**2 + b*x**3), x)

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GIAC/XCAS [A]  time = 0.217484, size = 51, normalized size = 0.98 \[ \frac{4 \, a^{\frac{5}{2}}{\rm sign}\left (x\right )}{15 \, b^{2}} + \frac{2 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x + a\right )}^{\frac{3}{2}} a\right )}{\rm sign}\left (x\right )}{15 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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