Optimal. Leaf size=131 \[ \frac{40 b^{7/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{77 a^{7/2} \left (a+b x^4\right )^{3/4}}-\frac{20 b^2 \sqrt [4]{a+b x^4}}{77 a^3 x^3}+\frac{10 b \sqrt [4]{a+b x^4}}{77 a^2 x^7}-\frac{\sqrt [4]{a+b x^4}}{11 a x^{11}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.162316, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{40 b^{7/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{77 a^{7/2} \left (a+b x^4\right )^{3/4}}-\frac{20 b^2 \sqrt [4]{a+b x^4}}{77 a^3 x^3}+\frac{10 b \sqrt [4]{a+b x^4}}{77 a^2 x^7}-\frac{\sqrt [4]{a+b x^4}}{11 a x^{11}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^12*(a + b*x^4)^(3/4)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 18.8474, size = 119, normalized size = 0.91 \[ - \frac{\sqrt [4]{a + b x^{4}}}{11 a x^{11}} + \frac{10 b \sqrt [4]{a + b x^{4}}}{77 a^{2} x^{7}} - \frac{20 b^{2} \sqrt [4]{a + b x^{4}}}{77 a^{3} x^{3}} + \frac{40 b^{\frac{7}{2}} x^{3} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{77 a^{\frac{7}{2}} \left (a + b x^{4}\right )^{\frac{3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**12/(b*x**4+a)**(3/4),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.063467, size = 94, normalized size = 0.72 \[ \frac{-7 a^3+3 a^2 b x^4-40 b^3 x^{12} \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^4}{a}\right )-10 a b^2 x^8-20 b^3 x^{12}}{77 a^3 x^{11} \left (a+b x^4\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^12*(a + b*x^4)^(3/4)),x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.048, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{12}} \left ( b{x}^{4}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^12/(b*x^4+a)^(3/4),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{12}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)^(3/4)*x^12),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{12}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)^(3/4)*x^12),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 14.7161, size = 44, normalized size = 0.34 \[ \frac{\Gamma \left (- \frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{11}{4}, \frac{3}{4} \\ - \frac{7}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{4}} x^{11} \Gamma \left (- \frac{7}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**12/(b*x**4+a)**(3/4),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{12}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)^(3/4)*x^12),x, algorithm="giac")
[Out]