Optimal. Leaf size=55 \[ \frac{x^{m+1} \sqrt{a+b x^n} \, _2F_1\left (1,\frac{m+1}{n}+\frac{1}{2};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a (m+1)} \]
[Out]
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Rubi [A] time = 0.0670956, antiderivative size = 64, normalized size of antiderivative = 1.16, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{x^{m+1} \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{(m+1) \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
[In] Int[x^m/Sqrt[a + b*x^n],x]
[Out]
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Rubi in Sympy [A] time = 8.08685, size = 51, normalized size = 0.93 \[ \frac{x^{m + 1} \sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m + 1}{n} \\ \frac{m + n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a \sqrt{1 + \frac{b x^{n}}{a}} \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**m/(a+b*x**n)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0509487, size = 66, normalized size = 1.2 \[ \frac{x^{m+1} \sqrt{\frac{a+b x^n}{a}} \, _2F_1\left (\frac{1}{2},\frac{m+1}{n};\frac{m+1}{n}+1;-\frac{b x^n}{a}\right )}{(m+1) \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
[In] Integrate[x^m/Sqrt[a + b*x^n],x]
[Out]
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Maple [F] time = 0.051, size = 0, normalized size = 0. \[ \int{{x}^{m}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^m/(a+b*x^n)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{\sqrt{b x^{n} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^m/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/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/(a+b*x**n)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{\sqrt{b x^{n} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^m/sqrt(b*x^n + a),x, algorithm="giac")
[Out]