Optimal. Leaf size=65 \[ \frac{2 b x^{3 (m+1)}}{3 a^2 (m+1) \left (a+b x^{2 (m+1)}\right )^{3/2}}+\frac{x^{m+1}}{a (m+1) \left (a+b x^{2 (m+1)}\right )^{3/2}} \]
[Out]
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Rubi [A] time = 0.0801516, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{2 b x^{3 (m+1)}}{3 a^2 (m+1) \left (a+b x^{2 (m+1)}\right )^{3/2}}+\frac{x^{m+1}}{a (m+1) \left (a+b x^{2 (m+1)}\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[x^m/(a + b*x^(2 + 2*m))^(5/2),x]
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Rubi in Sympy [A] time = 7.42366, size = 58, normalized size = 0.89 \[ \frac{x^{m + 1} \sqrt{a + b x^{2 m + 2}}{{}_{2}F_{1}\left (\begin{matrix} \frac{5}{2}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{- \frac{b x^{2 m + 2}}{a}} \right )}}{a^{3} \sqrt{1 + \frac{b x^{2 m + 2}}{a}} \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**m/(a+b*x**(2+2*m))**(5/2),x)
[Out]
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Mathematica [A] time = 0.0641326, size = 46, normalized size = 0.71 \[ \frac{x^{m+1} \left (3 a+2 b x^{2 m+2}\right )}{3 a^2 (m+1) \left (a+b x^{2 m+2}\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^m/(a + b*x^(2 + 2*m))^(5/2),x]
[Out]
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Maple [F] time = 0.055, size = 0, normalized size = 0. \[ \int{{x}^{m} \left ( a+b{x}^{2+2\,m} \right ) ^{-{\frac{5}{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^m/(a+b*x^(2+2*m))^(5/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{{\left (b x^{2 \, m + 2} + a\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^m/(b*x^(2*m + 2) + a)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.228888, size = 126, normalized size = 1.94 \[ \frac{{\left (2 \, b x^{3} x^{3 \, m} + 3 \, a x x^{m}\right )} \sqrt{b x^{2} x^{2 \, m} + a}}{3 \,{\left ({\left (a^{2} b^{2} m + a^{2} b^{2}\right )} x^{4} x^{4 \, m} + a^{4} m + a^{4} + 2 \,{\left (a^{3} b m + a^{3} b\right )} x^{2} x^{2 \, m}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^m/(b*x^(2*m + 2) + a)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**m/(a+b*x**(2+2*m))**(5/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{{\left (b x^{2 \, m + 2} + a\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^m/(b*x^(2*m + 2) + a)^(5/2),x, algorithm="giac")
[Out]