Optimal. Leaf size=118 \[ \frac{(a B+2 A b) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{3 \sqrt{b} e^{5/2}}+\frac{(e x)^{3/2} \sqrt{a+b x^3} (a B+2 A b)}{3 a e^4}-\frac{2 A \left (a+b x^3\right )^{3/2}}{3 a e (e x)^{3/2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.247429, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{(a B+2 A b) \tanh ^{-1}\left (\frac{\sqrt{b} (e x)^{3/2}}{e^{3/2} \sqrt{a+b x^3}}\right )}{3 \sqrt{b} e^{5/2}}+\frac{(e x)^{3/2} \sqrt{a+b x^3} (a B+2 A b)}{3 a e^4}-\frac{2 A \left (a+b x^3\right )^{3/2}}{3 a e (e x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[a + b*x^3]*(A + B*x^3))/(e*x)^(5/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 24.0637, size = 109, normalized size = 0.92 \[ - \frac{2 A \left (a + b x^{3}\right )^{\frac{3}{2}}}{3 a e \left (e x\right )^{\frac{3}{2}}} + \frac{2 \left (A b + \frac{B a}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \left (e x\right )^{\frac{3}{2}}}{e^{\frac{3}{2}} \sqrt{a + b x^{3}}} \right )}}{3 \sqrt{b} e^{\frac{5}{2}}} + \frac{2 \left (e x\right )^{\frac{3}{2}} \sqrt{a + b x^{3}} \left (A b + \frac{B a}{2}\right )}{3 a e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**3+A)*(b*x**3+a)**(1/2)/(e*x)**(5/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.246398, size = 92, normalized size = 0.78 \[ \frac{x \left (\sqrt{b} \left (a+b x^3\right ) \left (B x^3-2 A\right )+x^3 \sqrt{\frac{a}{x^3}+b} (a B+2 A b) \tanh ^{-1}\left (\frac{\sqrt{\frac{a}{x^3}+b}}{\sqrt{b}}\right )\right )}{3 \sqrt{b} (e x)^{5/2} \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[a + b*x^3]*(A + B*x^3))/(e*x)^(5/2),x]
[Out]
_______________________________________________________________________________________
Maple [C] time = 0.07, size = 6668, normalized size = 56.5 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^3+A)*(b*x^3+a)^(1/2)/(e*x)^(5/2),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((B*x^3 + A)*sqrt(b*x^3 + a)/(e*x)^(5/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.677675, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (B a + 2 \, A b\right )} e x^{2} \log \left (-4 \,{\left (2 \, b^{2} x^{4} + a b x\right )} \sqrt{b x^{3} + a} \sqrt{e x} -{\left (8 \, b^{2} x^{6} + 8 \, a b x^{3} + a^{2}\right )} \sqrt{b e}\right ) + 4 \,{\left (B x^{3} - 2 \, A\right )} \sqrt{b x^{3} + a} \sqrt{b e} \sqrt{e x}}{12 \, \sqrt{b e} e^{3} x^{2}}, \frac{{\left (B a + 2 \, A b\right )} e x^{2} \arctan \left (\frac{2 \, \sqrt{b x^{3} + a} \sqrt{-b e} \sqrt{e x} x}{2 \, b e x^{3} + a e}\right ) + 2 \,{\left (B x^{3} - 2 \, A\right )} \sqrt{b x^{3} + a} \sqrt{-b e} \sqrt{e x}}{6 \, \sqrt{-b e} e^{3} x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*sqrt(b*x^3 + a)/(e*x)^(5/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 31.6847, size = 160, normalized size = 1.36 \[ - \frac{2 A \sqrt{a}}{3 e^{\frac{5}{2}} x^{\frac{3}{2}} \sqrt{1 + \frac{b x^{3}}{a}}} + \frac{2 A \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b} x^{\frac{3}{2}}}{\sqrt{a}} \right )}}{3 e^{\frac{5}{2}}} - \frac{2 A b x^{\frac{3}{2}}}{3 \sqrt{a} e^{\frac{5}{2}} \sqrt{1 + \frac{b x^{3}}{a}}} + \frac{B \sqrt{a} x^{\frac{3}{2}} \sqrt{1 + \frac{b x^{3}}{a}}}{3 e^{\frac{5}{2}}} + \frac{B a \operatorname{asinh}{\left (\frac{\sqrt{b} x^{\frac{3}{2}}}{\sqrt{a}} \right )}}{3 \sqrt{b} e^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**3+A)*(b*x**3+a)**(1/2)/(e*x)**(5/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{3} + A\right )} \sqrt{b x^{3} + a}}{\left (e x\right )^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^3 + A)*sqrt(b*x^3 + a)/(e*x)^(5/2),x, algorithm="giac")
[Out]