Optimal. Leaf size=36 \[ \frac{x^{m+1} \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};-\frac{a x^2}{b}\right )}{m+1} \]
[Out]
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Rubi [A] time = 0.060137, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.049 \[ \frac{x^{m+1} \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};-\frac{a x^2}{b}\right )}{m+1} \]
Antiderivative was successfully verified.
[In] Int[x^m/((1 - (Sqrt[a]*x)/Sqrt[-b])^2*(1 + (Sqrt[a]*x)/Sqrt[-b])^2),x]
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Rubi in Sympy [A] time = 8.94023, size = 27, normalized size = 0.75 \[ \frac{x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 2, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{a x^{2}}{b}} \right )}}{m + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**m/(1-x*a**(1/2)/(-b)**(1/2))**2/(1+x*a**(1/2)/(-b)**(1/2))**2,x)
[Out]
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Mathematica [A] time = 0.0188131, size = 38, normalized size = 1.06 \[ \frac{x^{m+1} \, _2F_1\left (2,\frac{m+1}{2};\frac{m+1}{2}+1;-\frac{a x^2}{b}\right )}{m+1} \]
Antiderivative was successfully verified.
[In] Integrate[x^m/((1 - (Sqrt[a]*x)/Sqrt[-b])^2*(1 + (Sqrt[a]*x)/Sqrt[-b])^2),x]
[Out]
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Maple [F] time = 0.185, size = 0, normalized size = 0. \[ \int{{x}^{m} \left ( 1-{x\sqrt{a}{\frac{1}{\sqrt{-b}}}} \right ) ^{-2} \left ( 1+{x\sqrt{a}{\frac{1}{\sqrt{-b}}}} \right ) ^{-2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^m/(1-x*a^(1/2)/(-b)^(1/2))^2/(1+x*a^(1/2)/(-b)^(1/2))^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{{\left (\frac{\sqrt{a} x}{\sqrt{-b}} + 1\right )}^{2}{\left (\frac{\sqrt{a} x}{\sqrt{-b}} - 1\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^m/((sqrt(a)*x/sqrt(-b) + 1)^2*(sqrt(a)*x/sqrt(-b) - 1)^2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{b^{2} x^{m}}{a^{2} x^{4} + 2 \, a b x^{2} + b^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^m/((sqrt(a)*x/sqrt(-b) + 1)^2*(sqrt(a)*x/sqrt(-b) - 1)^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 33.8287, size = 541, normalized size = 15.03 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**m/(1-x*a**(1/2)/(-b)**(1/2))**2/(1+x*a**(1/2)/(-b)**(1/2))**2,x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^m/((sqrt(a)*x/sqrt(-b) + 1)^2*(sqrt(a)*x/sqrt(-b) - 1)^2),x, algorithm="giac")
[Out]