Optimal. Leaf size=128 \[ -\frac{3 b^{5/2} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{8 a^{5/2} \left (a+b x^4\right )^{3/4}}-\frac{3 b^2 \sqrt [4]{a+b x^4}}{8 a^3 x^2}+\frac{3 b \sqrt [4]{a+b x^4}}{20 a^2 x^6}-\frac{\sqrt [4]{a+b x^4}}{10 a x^{10}} \]
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Rubi [A] time = 0.188425, 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 \[ -\frac{3 b^{5/2} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{8 a^{5/2} \left (a+b x^4\right )^{3/4}}-\frac{3 b^2 \sqrt [4]{a+b x^4}}{8 a^3 x^2}+\frac{3 b \sqrt [4]{a+b x^4}}{20 a^2 x^6}-\frac{\sqrt [4]{a+b x^4}}{10 a x^{10}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^11*(a + b*x^4)^(3/4)),x]
[Out]
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Rubi in Sympy [A] time = 19.1818, size = 116, normalized size = 0.91 \[ - \frac{\sqrt [4]{a + b x^{4}}}{10 a x^{10}} + \frac{3 b \sqrt [4]{a + b x^{4}}}{20 a^{2} x^{6}} - \frac{3 b^{2} \sqrt [4]{a + b x^{4}}}{8 a^{3} x^{2}} - \frac{3 b^{\frac{5}{2}} \left (1 + \frac{b x^{4}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{8 a^{\frac{5}{2}} \left (a + b x^{4}\right )^{\frac{3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**11/(b*x**4+a)**(3/4),x)
[Out]
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Mathematica [C] time = 0.061775, size = 94, normalized size = 0.73 \[ \frac{-8 a^3+4 a^2 b x^4-15 b^3 x^{12} \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^4}{a}\right )-18 a b^2 x^8-30 b^3 x^{12}}{80 a^3 x^{10} \left (a+b x^4\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^11*(a + b*x^4)^(3/4)),x]
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Maple [F] time = 0.048, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{11}} \left ( b{x}^{4}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^11/(b*x^4+a)^(3/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{11}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)^(3/4)*x^11),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{11}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)^(3/4)*x^11),x, algorithm="fricas")
[Out]
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Sympy [A] time = 12.1984, size = 32, normalized size = 0.25 \[ - \frac{{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, \frac{3}{4} \\ - \frac{3}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{10 a^{\frac{3}{4}} x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**11/(b*x**4+a)**(3/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{11}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)^(3/4)*x^11),x, algorithm="giac")
[Out]