Optimal. Leaf size=79 \[ \frac{2 \sqrt{b} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{a^{3/2} \sqrt [4]{a+b x^4}}-\frac{1}{a x \sqrt [4]{a+b x^4}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.115236, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{2 \sqrt{b} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{a^{3/2} \sqrt [4]{a+b x^4}}-\frac{1}{a x \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^2*(a + b*x^4)^(5/4)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{x \sqrt [4]{\frac{a}{b x^{4}} + 1} \int ^{\frac{1}{x^{2}}} \frac{1}{\sqrt [4]{\frac{a x^{2}}{b} + 1}}\, dx}{a \sqrt [4]{a + b x^{4}}} + \frac{1}{a x \sqrt [4]{a + b x^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**2/(b*x**4+a)**(5/4),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.0502377, size = 71, normalized size = 0.9 \[ \frac{4 b x^4 \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )-3 \left (a+2 b x^4\right )}{3 a^2 x \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^2*(a + b*x^4)^(5/4)),x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.071, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2}} \left ( b{x}^{4}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^2/(b*x^4+a)^(5/4),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{5}{4}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)^(5/4)*x^2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{6} + a x^{2}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)^(5/4)*x^2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 3.53955, size = 39, normalized size = 0.49 \[ \frac{\Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{5}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{5}{4}} x \Gamma \left (\frac{3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**2/(b*x**4+a)**(5/4),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{5}{4}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)^(5/4)*x^2),x, algorithm="giac")
[Out]