Optimal. Leaf size=53 \[ \frac{2 (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right )}{3 \sqrt{a} b^{3/2}}+\frac{2 B x^{3/2}}{3 b} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.108199, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{2 (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right )}{3 \sqrt{a} b^{3/2}}+\frac{2 B x^{3/2}}{3 b} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[x]*(A + B*x^3))/(a + b*x^3),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 13.5483, size = 48, normalized size = 0.91 \[ \frac{2 B x^{\frac{3}{2}}}{3 b} + \frac{2 \left (A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x^{\frac{3}{2}}}{\sqrt{a}} \right )}}{3 \sqrt{a} b^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**3+A)*x**(1/2)/(b*x**3+a),x)
[Out]
_______________________________________________________________________________________
Mathematica [B] time = 0.114264, size = 139, normalized size = 2.62 \[ \frac{2 \left ((a B-A b) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )+(A b-a B) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}+\sqrt{3}\right )-A b \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )+\sqrt{a} \sqrt{b} B x^{3/2}+a B \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )\right )}{3 \sqrt{a} b^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[x]*(A + B*x^3))/(a + b*x^3),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.01, size = 53, normalized size = 1. \[{\frac{2\,B}{3\,b}{x}^{{\frac{3}{2}}}}+{\frac{2\,A}{3}\arctan \left ({b{x}^{{\frac{3}{2}}}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{2\,Ba}{3\,b}\arctan \left ({b{x}^{{\frac{3}{2}}}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^3+A)*x^(1/2)/(b*x^3+a),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((B*x^3 + A)*sqrt(x)/(b*x^3 + a),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.245341, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, \sqrt{-a b} B x^{\frac{3}{2}} -{\left (B a - A b\right )} \log \left (\frac{2 \, a b x^{\frac{3}{2}} +{\left (b x^{3} - a\right )} \sqrt{-a b}}{b x^{3} + a}\right )}{3 \, \sqrt{-a b} b}, \frac{2 \,{\left (\sqrt{a b} B x^{\frac{3}{2}} -{\left (B a - A b\right )} \arctan \left (\frac{\sqrt{a b} x^{\frac{3}{2}}}{a}\right )\right )}}{3 \, \sqrt{a b} b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*sqrt(x)/(b*x^3 + a),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [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((B*x**3+A)*x**(1/2)/(b*x**3+a),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.21394, size = 53, normalized size = 1. \[ \frac{2 \, B x^{\frac{3}{2}}}{3 \, b} - \frac{2 \,{\left (B a - A b\right )} \arctan \left (\frac{b x^{\frac{3}{2}}}{\sqrt{a b}}\right )}{3 \, \sqrt{a b} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*sqrt(x)/(b*x^3 + a),x, algorithm="giac")
[Out]