Optimal. Leaf size=60 \[ \frac{x^{m+n+1} \sqrt{a+b x^n} \, _2F_1\left (1,\frac{m+n+1}{n}+\frac{1}{2};\frac{m+2 n+1}{n};-\frac{b x^n}{a}\right )}{a (m+n+1)} \]
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Rubi [A] time = 0.0771706, antiderivative size = 69, normalized size of antiderivative = 1.15, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{x^{m+n+1} \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+n+1}{n};\frac{m+2 n+1}{n};-\frac{b x^n}{a}\right )}{(m+n+1) \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
[In] Int[x^(m + n)/Sqrt[a + b*x^n],x]
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Rubi in Sympy [A] time = 8.29935, size = 58, normalized size = 0.97 \[ \frac{x^{m + n + 1} \sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m + n + 1}{n} \\ \frac{m + 2 n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a \sqrt{1 + \frac{b x^{n}}{a}} \left (m + n + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(m+n)/(a+b*x**n)**(1/2),x)
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Mathematica [A] time = 0.0575739, size = 70, normalized size = 1.17 \[ \frac{x^{m+n+1} \sqrt{\frac{a+b x^n}{a}} \, _2F_1\left (\frac{1}{2},\frac{m+n+1}{n};\frac{m+n+1}{n}+1;-\frac{b x^n}{a}\right )}{(m+n+1) \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
[In] Integrate[x^(m + n)/Sqrt[a + b*x^n],x]
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Maple [F] time = 0.075, size = 0, normalized size = 0. \[ \int{{x}^{m+n}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(m+n)/(a+b*x^n)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m + n}}{\sqrt{b x^{n} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(m + n)/sqrt(b*x^n + a),x, algorithm="maxima")
[Out]
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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(x^(m + n)/sqrt(b*x^n + a),x, algorithm="fricas")
[Out]
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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+n)/(a+b*x**n)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m + n}}{\sqrt{b x^{n} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(m + n)/sqrt(b*x^n + a),x, algorithm="giac")
[Out]