Optimal. Leaf size=36 \[ \frac{\left (a+b x^2\right )^{3/2}}{3 b^2}-\frac{a \sqrt{a+b x^2}}{b^2} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0669123, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{\left (a+b x^2\right )^{3/2}}{3 b^2}-\frac{a \sqrt{a+b x^2}}{b^2} \]
Antiderivative was successfully verified.
[In] Int[x^3/Sqrt[a + b*x^2],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 7.92724, size = 29, normalized size = 0.81 \[ - \frac{a \sqrt{a + b x^{2}}}{b^{2}} + \frac{\left (a + b x^{2}\right )^{\frac{3}{2}}}{3 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3/(b*x**2+a)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0199465, size = 27, normalized size = 0.75 \[ \frac{\left (b x^2-2 a\right ) \sqrt{a+b x^2}}{3 b^2} \]
Antiderivative was successfully verified.
[In] Integrate[x^3/Sqrt[a + b*x^2],x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.006, size = 25, normalized size = 0.7 \[ -{\frac{-b{x}^{2}+2\,a}{3\,{b}^{2}}\sqrt{b{x}^{2}+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/(b*x^2+a)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/sqrt(b*x^2 + a),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.226963, size = 31, normalized size = 0.86 \[ \frac{\sqrt{b x^{2} + a}{\left (b x^{2} - 2 \, a\right )}}{3 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/sqrt(b*x^2 + a),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 1.74897, size = 44, normalized size = 1.22 \[ \begin{cases} - \frac{2 a \sqrt{a + b x^{2}}}{3 b^{2}} + \frac{x^{2} \sqrt{a + b x^{2}}}{3 b} & \text{for}\: b \neq 0 \\\frac{x^{4}}{4 \sqrt{a}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3/(b*x**2+a)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.200588, size = 36, normalized size = 1. \[ \frac{{\left (b x^{2} + a\right )}^{\frac{3}{2}} - 3 \, \sqrt{b x^{2} + a} a}{3 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/sqrt(b*x^2 + a),x, algorithm="giac")
[Out]