Optimal. Leaf size=126 \[ -\frac{3 b^{5/2} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{4 a^{5/2} \left (a+b x^2\right )^{3/4}}-\frac{3 b^2 \sqrt [4]{a+b x^2}}{4 a^3 x}+\frac{3 b \sqrt [4]{a+b x^2}}{10 a^2 x^3}-\frac{\sqrt [4]{a+b x^2}}{5 a x^5} \]
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Rubi [A] time = 0.133766, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{3 b^{5/2} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{4 a^{5/2} \left (a+b x^2\right )^{3/4}}-\frac{3 b^2 \sqrt [4]{a+b x^2}}{4 a^3 x}+\frac{3 b \sqrt [4]{a+b x^2}}{10 a^2 x^3}-\frac{\sqrt [4]{a+b x^2}}{5 a x^5} \]
Antiderivative was successfully verified.
[In] Int[1/(x^6*(a + b*x^2)^(3/4)),x]
[Out]
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Rubi in Sympy [A] time = 15.9738, size = 112, normalized size = 0.89 \[ - \frac{\sqrt [4]{a + b x^{2}}}{5 a x^{5}} + \frac{3 b \sqrt [4]{a + b x^{2}}}{10 a^{2} x^{3}} - \frac{3 b^{2} \sqrt [4]{a + b x^{2}}}{4 a^{3} x} - \frac{3 b^{\frac{5}{2}} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{4 a^{\frac{5}{2}} \left (a + b x^{2}\right )^{\frac{3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**6/(b*x**2+a)**(3/4),x)
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Mathematica [C] time = 0.0631611, size = 94, normalized size = 0.75 \[ \frac{-8 a^3+4 a^2 b x^2-15 b^3 x^6 \left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^2}{a}\right )-18 a b^2 x^4-30 b^3 x^6}{40 a^3 x^5 \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^6*(a + b*x^2)^(3/4)),x]
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Maple [F] time = 0.04, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{6}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^6/(b*x^2+a)^(3/4),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^(3/4)*x^6),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} x^{6}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^(3/4)*x^6),x, algorithm="fricas")
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Sympy [A] time = 5.07331, size = 32, normalized size = 0.25 \[ - \frac{{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, \frac{3}{4} \\ - \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{5 a^{\frac{3}{4}} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**6/(b*x**2+a)**(3/4),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^(3/4)*x^6),x, algorithm="giac")
[Out]