Optimal. Leaf size=101 \[ -\frac{3 b^2 \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{5/4}}+\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{5/4}}-\frac{3 b \left (a+b x^4\right )^{3/4}}{32 a x^4}-\frac{\left (a+b x^4\right )^{3/4}}{8 x^8} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.147771, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467 \[ -\frac{3 b^2 \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{5/4}}+\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{5/4}}-\frac{3 b \left (a+b x^4\right )^{3/4}}{32 a x^4}-\frac{\left (a+b x^4\right )^{3/4}}{8 x^8} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^4)^(3/4)/x^9,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 15.6954, size = 92, normalized size = 0.91 \[ - \frac{\left (a + b x^{4}\right )^{\frac{3}{4}}}{8 x^{8}} - \frac{3 b \left (a + b x^{4}\right )^{\frac{3}{4}}}{32 a x^{4}} - \frac{3 b^{2} \operatorname{atan}{\left (\frac{\sqrt [4]{a + b x^{4}}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{5}{4}}} + \frac{3 b^{2} \operatorname{atanh}{\left (\frac{\sqrt [4]{a + b x^{4}}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{5}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**4+a)**(3/4)/x**9,x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 0.0561714, size = 83, normalized size = 0.82 \[ \frac{-4 a^2+3 b^2 x^8 \sqrt [4]{\frac{a}{b x^4}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};-\frac{a}{b x^4}\right )-7 a b x^4-3 b^2 x^8}{32 a x^8 \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^4)^(3/4)/x^9,x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.052, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{9}} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^4+a)^(3/4)/x^9,x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(3/4)/x^9,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.421864, size = 266, normalized size = 2.63 \[ \frac{12 \, \left (\frac{b^{8}}{a^{5}}\right )^{\frac{1}{4}} a x^{8} \arctan \left (\frac{\left (\frac{b^{8}}{a^{5}}\right )^{\frac{3}{4}} a^{4}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{6} + \sqrt{\sqrt{b x^{4} + a} b^{12} + \sqrt{\frac{b^{8}}{a^{5}}} a^{3} b^{8}}}\right ) + 3 \, \left (\frac{b^{8}}{a^{5}}\right )^{\frac{1}{4}} a x^{8} \log \left (27 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{6} + 27 \, \left (\frac{b^{8}}{a^{5}}\right )^{\frac{3}{4}} a^{4}\right ) - 3 \, \left (\frac{b^{8}}{a^{5}}\right )^{\frac{1}{4}} a x^{8} \log \left (27 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{6} - 27 \, \left (\frac{b^{8}}{a^{5}}\right )^{\frac{3}{4}} a^{4}\right ) - 4 \,{\left (3 \, b x^{4} + 4 \, a\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{128 \, a x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(3/4)/x^9,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 10.8896, size = 41, normalized size = 0.41 \[ - \frac{b^{\frac{3}{4}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{4}}} \right )}}{4 x^{5} \Gamma \left (\frac{9}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**4+a)**(3/4)/x**9,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.228808, size = 304, normalized size = 3.01 \[ \frac{1}{256} \, b^{2}{\left (\frac{6 \, \sqrt{2} \left (-a\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} + 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{a^{2}} + \frac{6 \, \sqrt{2} \left (-a\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} - 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{a^{2}} - \frac{3 \, \sqrt{2} \left (-a\right )^{\frac{3}{4}}{\rm ln}\left (\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right )}{a^{2}} + \frac{3 \, \sqrt{2} \left (-a\right )^{\frac{3}{4}}{\rm ln}\left (-\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right )}{a^{2}} - \frac{8 \,{\left (3 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} +{\left (b x^{4} + a\right )}^{\frac{3}{4}} a\right )}}{a b^{2} x^{8}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(3/4)/x^9,x, algorithm="giac")
[Out]