Optimal. Leaf size=82 \[ -\frac{\sqrt [4]{a+b x^4}}{2 a x^2}-\frac{\sqrt{b} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{a} \left (a+b x^4\right )^{3/4}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.107731, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{\sqrt [4]{a+b x^4}}{2 a x^2}-\frac{\sqrt{b} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{a} \left (a+b x^4\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^3*(a + b*x^4)^(3/4)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 10.7267, size = 70, normalized size = 0.85 \[ - \frac{\sqrt [4]{a + b x^{4}}}{2 a x^{2}} - \frac{\sqrt{b} \left (1 + \frac{b x^{4}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{2 \sqrt{a} \left (a + b x^{4}\right )^{\frac{3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**3/(b*x**4+a)**(3/4),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.0453499, size = 70, normalized size = 0.85 \[ \frac{-b x^4 \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^4}{a}\right )-2 \left (a+b x^4\right )}{4 a x^2 \left (a+b x^4\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^3*(a + b*x^4)^(3/4)),x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.04, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3}} \left ( b{x}^{4}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^3/(b*x^4+a)^(3/4),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)^(3/4)*x^3),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)^(3/4)*x^3),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 3.06132, size = 31, normalized size = 0.38 \[ - \frac{{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{1}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{2 a^{\frac{3}{4}} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**3/(b*x**4+a)**(3/4),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)^(3/4)*x^3),x, algorithm="giac")
[Out]