Optimal. Leaf size=97 \[ -\frac{\left (a+b x^4\right )^{3/4}}{x}+\frac{3 b x^3}{2 \sqrt [4]{a+b x^4}}+\frac{3 \sqrt{a} \sqrt{b} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt [4]{a+b x^4}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.152491, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{\left (a+b x^4\right )^{3/4}}{x}+\frac{3 b x^3}{2 \sqrt [4]{a+b x^4}}+\frac{3 \sqrt{a} \sqrt{b} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^4)^(3/4)/x^2,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{3 a x \sqrt [4]{\frac{a}{b x^{4}} + 1} \int ^{\frac{1}{x^{2}}} \frac{1}{\left (\frac{a x^{2}}{b} + 1\right )^{\frac{5}{4}}}\, dx}{4 \sqrt [4]{a + b x^{4}}} + \frac{3 b x^{3}}{2 \sqrt [4]{a + b x^{4}}} - \frac{\left (a + b x^{4}\right )^{\frac{3}{4}}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**4+a)**(3/4)/x**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.0380396, size = 63, normalized size = 0.65 \[ \frac{b x^4 \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )-a-b x^4}{x \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^4)^(3/4)/x^2,x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.043, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2}} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^4+a)^(3/4)/x^2,x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(3/4)/x^2,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(3/4)/x^2,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 3.20371, size = 41, normalized size = 0.42 \[ \frac{a^{\frac{3}{4}} \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 x \Gamma \left (\frac{3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**4+a)**(3/4)/x**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(3/4)/x^2,x, algorithm="giac")
[Out]