Optimal. Leaf size=64 \[ \frac{x^2 \sqrt{c+d x^3} F_1\left (\frac{2}{3};2,-\frac{1}{2};\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{2 a^2 \sqrt{\frac{d x^3}{c}+1}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.155973, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{x^2 \sqrt{c+d x^3} F_1\left (\frac{2}{3};2,-\frac{1}{2};\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{2 a^2 \sqrt{\frac{d x^3}{c}+1}} \]
Antiderivative was successfully verified.
[In] Int[(x*Sqrt[c + d*x^3])/(a + b*x^3)^2,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 18.5954, size = 53, normalized size = 0.83 \[ \frac{x^{2} \sqrt{c + d x^{3}} \operatorname{appellf_{1}}{\left (\frac{2}{3},- \frac{1}{2},2,\frac{5}{3},- \frac{d x^{3}}{c},- \frac{b x^{3}}{a} \right )}}{2 a^{2} \sqrt{1 + \frac{d x^{3}}{c}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(d*x**3+c)**(1/2)/(b*x**3+a)**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [B] time = 0.338729, size = 324, normalized size = 5.06 \[ \frac{x^2 \left (\frac{25 c^2 F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}{10 a c F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )-3 x^3 \left (2 b c F_1\left (\frac{5}{3};\frac{1}{2},2;\frac{8}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (\frac{5}{3};\frac{3}{2},1;\frac{8}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )}+\frac{8 c d x^3 F_1\left (\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}{3 x^3 \left (2 b c F_1\left (\frac{8}{3};\frac{1}{2},2;\frac{11}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (\frac{8}{3};\frac{3}{2},1;\frac{11}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )-16 a c F_1\left (\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}+\frac{5 \left (c+d x^3\right )}{a}\right )}{15 \left (a+b x^3\right ) \sqrt{c+d x^3}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(x*Sqrt[c + d*x^3])/(a + b*x^3)^2,x]
[Out]
_______________________________________________________________________________________
Maple [C] time = 0.055, size = 908, normalized size = 14.2 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(d*x^3+c)^(1/2)/(b*x^3+a)^2,x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x^{3} + c} x}{{\left (b x^{3} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^3 + c)*x/(b*x^3 + a)^2,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^3 + c)*x/(b*x^3 + a)^2,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x \sqrt{c + d x^{3}}}{\left (a + b x^{3}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(d*x**3+c)**(1/2)/(b*x**3+a)**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x^{3} + c} x}{{\left (b x^{3} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(d*x^3 + c)*x/(b*x^3 + a)^2,x, algorithm="giac")
[Out]