Optimal. Leaf size=129 \[ \frac{8 b^{5/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{3 a^{7/2} \sqrt [4]{a+b x^4}}-\frac{4 b^2}{3 a^3 x \sqrt [4]{a+b x^4}}+\frac{2 b}{9 a^2 x^5 \sqrt [4]{a+b x^4}}-\frac{1}{9 a x^9 \sqrt [4]{a+b x^4}} \]
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Rubi [A] time = 0.178053, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{8 b^{5/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{3 a^{7/2} \sqrt [4]{a+b x^4}}-\frac{4 b^2}{3 a^3 x \sqrt [4]{a+b x^4}}+\frac{2 b}{9 a^2 x^5 \sqrt [4]{a+b x^4}}-\frac{1}{9 a x^9 \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^10*(a + b*x^4)^(5/4)),x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{1}{9 a x^{9} \sqrt [4]{a + b x^{4}}} + \frac{2 b}{9 a^{2} x^{5} \sqrt [4]{a + b x^{4}}} - \frac{4 b^{2} x \sqrt [4]{\frac{a}{b x^{4}} + 1} \int ^{\frac{1}{x^{2}}} \frac{1}{\sqrt [4]{\frac{a x^{2}}{b} + 1}}\, dx}{3 a^{3} \sqrt [4]{a + b x^{4}}} + \frac{4 b^{2}}{3 a^{3} x \sqrt [4]{a + b x^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**10/(b*x**4+a)**(5/4),x)
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Mathematica [C] time = 0.0756725, size = 94, normalized size = 0.73 \[ \frac{-a^3+2 a^2 b x^4+16 b^3 x^{12} \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )-12 a b^2 x^8-24 b^3 x^{12}}{9 a^4 x^9 \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^10*(a + b*x^4)^(5/4)),x]
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Maple [F] time = 0.087, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{10}} \left ( b{x}^{4}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^10/(b*x^4+a)^(5/4),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{5}{4}} x^{10}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)^(5/4)*x^10),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{14} + a x^{10}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)^(5/4)*x^10),x, algorithm="fricas")
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Sympy [A] time = 14.8666, size = 44, normalized size = 0.34 \[ \frac{\Gamma \left (- \frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{9}{4}, \frac{5}{4} \\ - \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{5}{4}} x^{9} \Gamma \left (- \frac{5}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**10/(b*x**4+a)**(5/4),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{5}{4}} x^{10}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)^(5/4)*x^10),x, algorithm="giac")
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