Optimal. Leaf size=33 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x^{m+1}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} (m+1)} \]
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Rubi [A] time = 0.0504952, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x^{m+1}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} (m+1)} \]
Antiderivative was successfully verified.
[In] Int[x^m/(a + b*x^(2 + 2*m)),x]
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Rubi in Sympy [A] time = 5.25988, size = 26, normalized size = 0.79 \[ \frac{x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{- \frac{b x^{2 m + 2}}{a}} \right )}}{a \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**m/(a+b*x**(2+2*m)),x)
[Out]
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Mathematica [A] time = 0.0178915, size = 33, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x^{m+1}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} (m+1)} \]
Antiderivative was successfully verified.
[In] Integrate[x^m/(a + b*x^(2 + 2*m)),x]
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Maple [B] time = 0.069, size = 61, normalized size = 1.9 \[ -{\frac{1}{2+2\,m}\ln \left ({x}^{m}-{\frac{a}{x}{\frac{1}{\sqrt{-ab}}}} \right ){\frac{1}{\sqrt{-ab}}}}+{\frac{1}{2+2\,m}\ln \left ({x}^{m}+{\frac{a}{x}{\frac{1}{\sqrt{-ab}}}} \right ){\frac{1}{\sqrt{-ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^m/(a+b*x^(2+2*m)),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{b x^{2 \, m + 2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^m/(b*x^(2*m + 2) + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.24023, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (\frac{\sqrt{-a b} b x^{2} x^{2 \, m} + 2 \, a b x x^{m} - \sqrt{-a b} a}{b x^{2} x^{2 \, m} + a}\right )}{2 \, \sqrt{-a b}{\left (m + 1\right )}}, -\frac{\arctan \left (\frac{a}{\sqrt{a b} x x^{m}}\right )}{\sqrt{a b}{\left (m + 1\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^m/(b*x^(2*m + 2) + a),x, algorithm="fricas")
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Sympy [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/(a+b*x**(2+2*m)),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{b x^{2 \, m + 2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^m/(b*x^(2*m + 2) + a),x, algorithm="giac")
[Out]