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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]