Optimal. Leaf size=55 \[ \frac{x^{m+1} \left (a+b x^n\right )^{3/2} \, _2F_1\left (1,\frac{m+1}{n}+\frac{3}{2};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a (m+1)} \]
[Out]
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Rubi [A] time = 0.0672495, 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 \[ \frac{x^{m+1} \sqrt{a+b x^n} \, _2F_1\left (-\frac{1}{2},\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{(m+1) \sqrt{\frac{b x^n}{a}+1}} \]
Antiderivative was successfully verified.
[In] Int[x^m*Sqrt[a + b*x^n],x]
[Out]
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Rubi in Sympy [A] time = 7.48744, size = 51, normalized size = 0.93 \[ \frac{x^{m + 1} \sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m + 1}{n} \\ \frac{m + n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{\sqrt{1 + \frac{b x^{n}}{a}} \left (m + 1\right )} \]
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 = 0.114909, size = 88, normalized size = 1.6 \[ \frac{x^{m+1} \left (a n \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )+2 (m+1) \left (a+b x^n\right )\right )}{(m+1) (2 m+n+2) \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
[In] Integrate[x^m*Sqrt[a + b*x^n],x]
[Out]
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Maple [F] time = 0.091, size = 0, normalized size = 0. \[ \int{x}^{m}\sqrt{a+b{x}^{n}}\, dx \]
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. \[ \int \sqrt{b x^{n} + a} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^n + a)*x^m,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(sqrt(b*x^n + a)*x^m,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. \[ \int \sqrt{b x^{n} + a} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^n + a)*x^m,x, algorithm="giac")
[Out]