Optimal. Leaf size=66 \[ -\frac{\sqrt{\frac{d x^3}{c}+1} F_1\left (-\frac{5}{3};1,\frac{3}{2};-\frac{2}{3};\frac{d x^3}{8 c},-\frac{d x^3}{c}\right )}{40 c^2 x^5 \sqrt{c+d x^3}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.201058, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ -\frac{\sqrt{\frac{d x^3}{c}+1} F_1\left (-\frac{5}{3};1,\frac{3}{2};-\frac{2}{3};\frac{d x^3}{8 c},-\frac{d x^3}{c}\right )}{40 c^2 x^5 \sqrt{c+d x^3}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^6*(8*c - d*x^3)*(c + d*x^3)^(3/2)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 26.491, size = 56, normalized size = 0.85 \[ - \frac{\sqrt{c + d x^{3}} \operatorname{appellf_{1}}{\left (- \frac{5}{3},1,\frac{3}{2},- \frac{2}{3},\frac{d x^{3}}{8 c},- \frac{d x^{3}}{c} \right )}}{40 c^{3} x^{5} \sqrt{1 + \frac{d x^{3}}{c}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**6/(-d*x**3+8*c)/(d*x**3+c)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [B] time = 0.338136, size = 364, normalized size = 5.52 \[ \frac{\frac{382528 c^2 d^2 x^6 F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )}{\left (8 c-d x^3\right ) \left (3 d x^3 \left (F_1\left (\frac{4}{3};\frac{1}{2},2;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )-4 F_1\left (\frac{4}{3};\frac{3}{2},1;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )+32 c F_1\left (\frac{1}{3};\frac{1}{2},1;\frac{4}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )}-\frac{20867 c d^3 x^9 F_1\left (\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )}{\left (8 c-d x^3\right ) \left (3 d x^3 \left (F_1\left (\frac{7}{3};\frac{1}{2},2;\frac{10}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )-4 F_1\left (\frac{7}{3};\frac{3}{2},1;\frac{10}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )+56 c F_1\left (\frac{4}{3};\frac{1}{2},1;\frac{7}{3};-\frac{d x^3}{c},\frac{d x^3}{8 c}\right )\right )}-432 c^2+1269 c d x^3+2981 d^2 x^6}{17280 c^4 x^5 \sqrt{c+d x^3}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/(x^6*(8*c - d*x^3)*(c + d*x^3)^(3/2)),x]
[Out]
_______________________________________________________________________________________
Maple [C] time = 0.043, size = 1402, normalized size = 21.2 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^6/(-d*x^3+8*c)/(d*x^3+c)^(3/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{1}{{\left (d x^{3} + c\right )}^{\frac{3}{2}}{\left (d x^{3} - 8 \, c\right )} x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/((d*x^3 + c)^(3/2)*(d*x^3 - 8*c)*x^6),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(-1/((d*x^3 + c)^(3/2)*(d*x^3 - 8*c)*x^6),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**6/(-d*x**3+8*c)/(d*x**3+c)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{1}{{\left (d x^{3} + c\right )}^{\frac{3}{2}}{\left (d x^{3} - 8 \, c\right )} x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/((d*x^3 + c)^(3/2)*(d*x^3 - 8*c)*x^6),x, algorithm="giac")
[Out]