Optimal. Leaf size=105 \[ \frac{3 b^2 \tan ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{64 a^{7/4}}+\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{64 a^{7/4}}+\frac{b \sqrt [4]{a-b x^4}}{32 a x^4}-\frac{\sqrt [4]{a-b x^4}}{8 x^8} \]
[Out]
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Rubi [A] time = 0.145679, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438 \[ \frac{3 b^2 \tan ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{64 a^{7/4}}+\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{64 a^{7/4}}+\frac{b \sqrt [4]{a-b x^4}}{32 a x^4}-\frac{\sqrt [4]{a-b x^4}}{8 x^8} \]
Antiderivative was successfully verified.
[In] Int[(a - b*x^4)^(1/4)/x^9,x]
[Out]
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Rubi in Sympy [A] time = 15.6796, size = 90, normalized size = 0.86 \[ - \frac{\sqrt [4]{a - b x^{4}}}{8 x^{8}} + \frac{b \sqrt [4]{a - b x^{4}}}{32 a x^{4}} + \frac{3 b^{2} \operatorname{atan}{\left (\frac{\sqrt [4]{a - b x^{4}}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{7}{4}}} + \frac{3 b^{2} \operatorname{atanh}{\left (\frac{\sqrt [4]{a - b x^{4}}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{7}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-b*x**4+a)**(1/4)/x**9,x)
[Out]
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Mathematica [C] time = 0.0485392, size = 83, normalized size = 0.79 \[ \frac{-4 a^2+b^2 x^8 \left (1-\frac{a}{b x^4}\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{a}{b x^4}\right )+5 a b x^4-b^2 x^8}{32 a x^8 \left (a-b x^4\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(a - b*x^4)^(1/4)/x^9,x]
[Out]
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Maple [F] time = 0.041, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{9}}\sqrt [4]{-b{x}^{4}+a}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-b*x^4+a)^(1/4)/x^9,x)
[Out]
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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)^(1/4)/x^9,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.269406, size = 267, normalized size = 2.54 \[ -\frac{12 \, a \left (\frac{b^{8}}{a^{7}}\right )^{\frac{1}{4}} x^{8} \arctan \left (\frac{a^{2} \left (\frac{b^{8}}{a^{7}}\right )^{\frac{1}{4}}}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} b^{2} + \sqrt{\sqrt{-b x^{4} + a} b^{4} + a^{4} \sqrt{\frac{b^{8}}{a^{7}}}}}\right ) - 3 \, a \left (\frac{b^{8}}{a^{7}}\right )^{\frac{1}{4}} x^{8} \log \left (3 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}} b^{2} + 3 \, a^{2} \left (\frac{b^{8}}{a^{7}}\right )^{\frac{1}{4}}\right ) + 3 \, a \left (\frac{b^{8}}{a^{7}}\right )^{\frac{1}{4}} x^{8} \log \left (3 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}} b^{2} - 3 \, a^{2} \left (\frac{b^{8}}{a^{7}}\right )^{\frac{1}{4}}\right ) - 4 \,{\left (b x^{4} - 4 \, a\right )}{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{128 \, a x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-b*x^4 + a)^(1/4)/x^9,x, algorithm="fricas")
[Out]
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Sympy [A] time = 8.56406, size = 44, normalized size = 0.42 \[ - \frac{\sqrt [4]{b} e^{- \frac{7 i \pi }{4}} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{a}{b x^{4}}} \right )}}{4 x^{7} \Gamma \left (\frac{11}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-b*x**4+a)**(1/4)/x**9,x)
[Out]
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GIAC/XCAS [A] time = 0.228905, size = 313, normalized size = 2.98 \[ \frac{1}{256} \, b^{2}{\left (\frac{6 \, \sqrt{2} \left (-a\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} + 2 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{a^{2}} + \frac{6 \, \sqrt{2} \left (-a\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} - 2 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{a^{2}} + \frac{3 \, \sqrt{2} \left (-a\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{-b x^{4} + a} + \sqrt{-a}\right )}{a^{2}} - \frac{3 \, \sqrt{2} \left (-a\right )^{\frac{1}{4}}{\rm ln}\left (-\sqrt{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{-b x^{4} + a} + \sqrt{-a}\right )}{a^{2}} - \frac{8 \,{\left ({\left (-b x^{4} + a\right )}^{\frac{5}{4}} + 3 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a\right )}}{a b^{2} x^{8}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-b*x^4 + a)^(1/4)/x^9,x, algorithm="giac")
[Out]