Optimal. Leaf size=128 \[ -\frac{77 b^{5/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{40 a^{7/2} \sqrt [4]{a+b x^4}}-\frac{77 b^2}{120 a^3 x^2 \sqrt [4]{a+b x^4}}+\frac{11 b}{60 a^2 x^6 \sqrt [4]{a+b x^4}}-\frac{1}{10 a x^{10} \sqrt [4]{a+b x^4}} \]
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Rubi [A] time = 0.194603, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{77 b^{5/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{40 a^{7/2} \sqrt [4]{a+b x^4}}-\frac{77 b^2}{120 a^3 x^2 \sqrt [4]{a+b x^4}}+\frac{11 b}{60 a^2 x^6 \sqrt [4]{a+b x^4}}-\frac{1}{10 a x^{10} \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^11*(a + b*x^4)^(5/4)),x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{1}{10 a x^{10} \sqrt [4]{a + b x^{4}}} + \frac{11 b}{60 a^{2} x^{6} \sqrt [4]{a + b x^{4}}} - \frac{77 b^{2}}{120 a^{3} x^{2} \sqrt [4]{a + b x^{4}}} - \frac{77 b^{3} x^{2}}{40 a^{4} \sqrt [4]{a + b x^{4}}} + \frac{77 b^{3} \int ^{x^{2}} \frac{1}{\sqrt [4]{a + b x^{2}}}\, dx}{80 a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**11/(b*x**4+a)**(5/4),x)
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Mathematica [C] time = 0.078535, size = 94, normalized size = 0.73 \[ \frac{-24 a^3+44 a^2 b x^4+231 b^3 x^{12} \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^4}{a}\right )-154 a b^2 x^8-462 b^3 x^{12}}{240 a^4 x^{10} \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^11*(a + b*x^4)^(5/4)),x]
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Maple [F] time = 0.084, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{11}} \left ( b{x}^{4}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^11/(b*x^4+a)^(5/4),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{5}{4}} x^{11}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)^(5/4)*x^11),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{15} + a x^{11}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)^(5/4)*x^11),x, algorithm="fricas")
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Sympy [A] time = 17.7205, size = 32, normalized size = 0.25 \[ - \frac{{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, \frac{5}{4} \\ - \frac{3}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{10 a^{\frac{5}{4}} x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**11/(b*x**4+a)**(5/4),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{5}{4}} x^{11}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)^(5/4)*x^11),x, algorithm="giac")
[Out]