Optimal. Leaf size=76 \[ -\frac{\sqrt [4]{a+b x^2}}{a x}-\frac{\sqrt{b} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{a} \left (a+b x^2\right )^{3/4}} \]
[Out]
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Rubi [A] time = 0.065673, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{\sqrt [4]{a+b x^2}}{a x}-\frac{\sqrt{b} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{a} \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^2*(a + b*x^2)^(3/4)),x]
[Out]
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Rubi in Sympy [A] time = 7.93875, size = 63, normalized size = 0.83 \[ - \frac{\sqrt [4]{a + b x^{2}}}{a x} - \frac{\sqrt{b} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{\sqrt{a} \left (a + b x^{2}\right )^{\frac{3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**2/(b*x**2+a)**(3/4),x)
[Out]
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Mathematica [C] time = 0.0446168, size = 70, normalized size = 0.92 \[ \frac{-b x^2 \left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^2}{a}\right )-2 \left (a+b x^2\right )}{2 a x \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^2*(a + b*x^2)^(3/4)),x]
[Out]
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Maple [F] time = 0.037, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^2/(b*x^2+a)^(3/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^(3/4)*x^2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^(3/4)*x^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.12821, size = 27, normalized size = 0.36 \[ - \frac{{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{1}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{a^{\frac{3}{4}} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**2/(b*x**2+a)**(3/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^(3/4)*x^2),x, algorithm="giac")
[Out]