Optimal. Leaf size=128 \[ \frac{b^{3/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{4 a^{3/2} \sqrt [4]{a+b x^4}}-\frac{b^2 x^2}{4 a^2 \sqrt [4]{a+b x^4}}+\frac{b \left (a+b x^4\right )^{3/4}}{4 a^2 x^2}-\frac{\left (a+b x^4\right )^{3/4}}{6 a x^6} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.172179, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{b^{3/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{4 a^{3/2} \sqrt [4]{a+b x^4}}-\frac{b^2 x^2}{4 a^2 \sqrt [4]{a+b x^4}}+\frac{b \left (a+b x^4\right )^{3/4}}{4 a^2 x^2}-\frac{\left (a+b x^4\right )^{3/4}}{6 a x^6} \]
Antiderivative was successfully verified.
[In] Int[1/(x^7*(a + b*x^4)^(1/4)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{\left (a + b x^{4}\right )^{\frac{3}{4}}}{6 a x^{6}} - \frac{b^{2} \int ^{x^{2}} \frac{1}{\sqrt [4]{a + b x^{2}}}\, dx}{8 a^{2}} + \frac{b \left (a + b x^{4}\right )^{\frac{3}{4}}}{4 a^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**7/(b*x**4+a)**(1/4),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.059074, size = 83, normalized size = 0.65 \[ \frac{-4 a^2-3 b^2 x^8 \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^4}{a}\right )+2 a b x^4+6 b^2 x^8}{24 a^2 x^6 \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^7*(a + b*x^4)^(1/4)),x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.043, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{7}}{\frac{1}{\sqrt [4]{b{x}^{4}+a}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^7/(b*x^4+a)^(1/4),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{7}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)^(1/4)*x^7),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{7}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)^(1/4)*x^7),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 4.494, size = 32, normalized size = 0.25 \[ - \frac{{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{1}{4} \\ - \frac{1}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{6 \sqrt [4]{a} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**7/(b*x**4+a)**(1/4),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{7}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)^(1/4)*x^7),x, algorithm="giac")
[Out]