Optimal. Leaf size=15 \[ \frac{x^m}{\sqrt{a+b x^n}} \]
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Rubi [C] time = 0.190587, antiderivative size = 126, normalized size of antiderivative = 8.4, number of steps used = 5, number of rules used = 2, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ \frac{x^m \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m}{n};\frac{m+n}{n};-\frac{b x^n}{a}\right )}{\sqrt{a+b x^n}}-\frac{b n x^{m+n} \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{3}{2},\frac{m+n}{n};\frac{m}{n}+2;-\frac{b x^n}{a}\right )}{2 a (m+n) \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
[In] Int[-(b*n*x^(-1 + m + n))/(2*(a + b*x^n)^(3/2)) + (m*x^(-1 + m))/Sqrt[a + b*x^n],x]
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Rubi in Sympy [A] time = 19.0928, size = 100, normalized size = 6.67 \[ \frac{x^{m} \sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{n} \\ \frac{m + n}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a \sqrt{1 + \frac{b x^{n}}{a}}} - \frac{b n x^{m + n} \sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{m + n}{n} \\ \frac{m}{n} + 2 \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{2 a^{2} \sqrt{1 + \frac{b x^{n}}{a}} \left (m + n\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(-1/2*b*n*x**(-1+m+n)/(a+b*x**n)**(3/2)+m*x**(-1+m)/(a+b*x**n)**(1/2),x)
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Mathematica [A] time = 0.0663734, size = 15, normalized size = 1. \[ \frac{x^m}{\sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
[In] Integrate[-(b*n*x^(-1 + m + n))/(2*(a + b*x^n)^(3/2)) + (m*x^(-1 + m))/Sqrt[a + b*x^n],x]
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Maple [F] time = 0., size = 0, normalized size = 0. \[ \int -{\frac{bn{x}^{-1+m+n}}{2} \left ( a+b{x}^{n} \right ) ^{-{\frac{3}{2}}}}+{m{x}^{-1+m}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(-1/2*b*n*x^(-1+m+n)/(a+b*x^n)^(3/2)+m*x^(-1+m)/(a+b*x^n)^(1/2),x)
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Maxima [A] time = 1.64235, size = 18, normalized size = 1.2 \[ \frac{x^{m}}{\sqrt{b x^{n} + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/2*b*n*x^(m + n - 1)/(b*x^n + a)^(3/2) + m*x^(m - 1)/sqrt(b*x^n + a),x, algorithm="maxima")
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Fricas [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(-1/2*b*n*x^(m + n - 1)/(b*x^n + a)^(3/2) + m*x^(m - 1)/sqrt(b*x^n + 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(-1/2*b*n*x**(-1+m+n)/(a+b*x**n)**(3/2)+m*x**(-1+m)/(a+b*x**n)**(1/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{b n x^{m + n - 1}}{2 \,{\left (b x^{n} + a\right )}^{\frac{3}{2}}} + \frac{m x^{m - 1}}{\sqrt{b x^{n} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/2*b*n*x^(m + n - 1)/(b*x^n + a)^(3/2) + m*x^(m - 1)/sqrt(b*x^n + a),x, algorithm="giac")
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