Optimal. Leaf size=66 \[ -\frac{\sqrt{\frac{d x^3}{c}+1} F_1\left (-\frac{2}{3};1,\frac{1}{2};\frac{1}{3};-\frac{d x^3}{4 c},-\frac{d x^3}{c}\right )}{8 c x^2 \sqrt{c+d x^3}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.203719, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{\sqrt{\frac{d x^3}{c}+1} F_1\left (-\frac{2}{3};1,\frac{1}{2};\frac{1}{3};-\frac{d x^3}{4 c},-\frac{d x^3}{c}\right )}{8 c x^2 \sqrt{c+d x^3}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^3*Sqrt[c + d*x^3]*(4*c + d*x^3)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 26.2291, size = 56, normalized size = 0.85 \[ - \frac{\sqrt{c + d x^{3}} \operatorname{appellf_{1}}{\left (- \frac{2}{3},\frac{1}{2},1,\frac{1}{3},- \frac{d x^{3}}{c},- \frac{d x^{3}}{4 c} \right )}}{8 c^{2} x^{2} \sqrt{1 + \frac{d x^{3}}{c}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**3/(d*x**3+4*c)/(d*x**3+c)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [B] time = 0.480912, size = 348, normalized size = 5.27 \[ \frac{\frac{-\frac{7 c d^2 x^6 F_1\left (\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )}{\left (4 c+d x^3\right ) \left (28 c F_1\left (\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )-3 d x^3 \left (F_1\left (\frac{7}{3};\frac{1}{2},2;\frac{10}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )+2 F_1\left (\frac{7}{3};\frac{3}{2},1;\frac{10}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )\right )\right )}-2 \left (c+d x^3\right )}{c^2}+\frac{128 d x^3 F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )}{\left (4 c+d x^3\right ) \left (3 d x^3 \left (F_1\left (\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )+2 F_1\left (\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )\right )-16 c F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},-\frac{d x^3}{4 c}\right )\right )}}{16 x^2 \sqrt{c+d x^3}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/(x^3*Sqrt[c + d*x^3]*(4*c + d*x^3)),x]
[Out]
_______________________________________________________________________________________
Maple [C] time = 0.031, size = 722, normalized size = 10.9 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^3/(d*x^3+4*c)/(d*x^3+c)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (d x^{3} + 4 \, c\right )} \sqrt{d x^{3} + c} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((d*x^3 + 4*c)*sqrt(d*x^3 + c)*x^3),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (d x^{6} + 4 \, c x^{3}\right )} \sqrt{d x^{3} + c}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((d*x^3 + 4*c)*sqrt(d*x^3 + c)*x^3),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \sqrt{c + d x^{3}} \left (4 c + d x^{3}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**3/(d*x**3+4*c)/(d*x**3+c)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (d x^{3} + 4 \, c\right )} \sqrt{d x^{3} + c} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((d*x^3 + 4*c)*sqrt(d*x^3 + c)*x^3),x, algorithm="giac")
[Out]