Optimal. Leaf size=62 \[ -\frac{\sqrt{\frac{d x^6}{c}+1} F_1\left (-\frac{1}{6};1,\frac{1}{2};\frac{5}{6};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{a x \sqrt{c+d x^6}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.189551, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{\sqrt{\frac{d x^6}{c}+1} F_1\left (-\frac{1}{6};1,\frac{1}{2};\frac{5}{6};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{a x \sqrt{c+d x^6}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^2*(a + b*x^6)*Sqrt[c + d*x^6]),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 25.2079, size = 51, normalized size = 0.82 \[ - \frac{\sqrt{c + d x^{6}} \operatorname{appellf_{1}}{\left (- \frac{1}{6},\frac{1}{2},1,\frac{5}{6},- \frac{d x^{6}}{c},- \frac{b x^{6}}{a} \right )}}{a c x \sqrt{1 + \frac{d x^{6}}{c}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**2/(b*x**6+a)/(d*x**6+c)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [B] time = 0.721137, size = 344, normalized size = 5.55 \[ \frac{\frac{121 x^6 (b c-2 a d) F_1\left (\frac{5}{6};\frac{1}{2},1;\frac{11}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )}{\left (a+b x^6\right ) \left (3 x^6 \left (2 b c F_1\left (\frac{11}{6};\frac{1}{2},2;\frac{17}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )+a d F_1\left (\frac{11}{6};\frac{3}{2},1;\frac{17}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )\right )-11 a c F_1\left (\frac{5}{6};\frac{1}{2},1;\frac{11}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )\right )}-\frac{170 b d x^{12} F_1\left (\frac{11}{6};\frac{1}{2},1;\frac{17}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )}{\left (a+b x^6\right ) \left (3 x^6 \left (2 b c F_1\left (\frac{17}{6};\frac{1}{2},2;\frac{23}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )+a d F_1\left (\frac{17}{6};\frac{3}{2},1;\frac{23}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )\right )-17 a c F_1\left (\frac{11}{6};\frac{1}{2},1;\frac{17}{6};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )\right )}-\frac{55 \left (c+d x^6\right )}{a c}}{55 x \sqrt{c+d x^6}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/(x^2*(a + b*x^6)*Sqrt[c + d*x^6]),x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.11, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2} \left ( b{x}^{6}+a \right ) }{\frac{1}{\sqrt{d{x}^{6}+c}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^2/(b*x^6+a)/(d*x^6+c)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{6} + a\right )} \sqrt{d x^{6} + c} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^6 + a)*sqrt(d*x^6 + c)*x^2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{8} + a x^{2}\right )} \sqrt{d x^{6} + c}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^6 + a)*sqrt(d*x^6 + c)*x^2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \left (a + b x^{6}\right ) \sqrt{c + d x^{6}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**2/(b*x**6+a)/(d*x**6+c)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{6} + a\right )} \sqrt{d x^{6} + c} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^6 + a)*sqrt(d*x^6 + c)*x^2),x, algorithm="giac")
[Out]