Optimal. Leaf size=45 \[ \frac{2}{3} x^{3/2} (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (\frac{3}{2},-n;\frac{5}{2};-\frac{b x}{a}\right ) \]
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Rubi [A] time = 0.0269656, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{2}{3} x^{3/2} (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (\frac{3}{2},-n;\frac{5}{2};-\frac{b x}{a}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[x]*(a + b*x)^n,x]
[Out]
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Rubi in Sympy [A] time = 5.27297, size = 36, normalized size = 0.8 \[ \frac{2 x^{\frac{3}{2}} \left (1 + \frac{b x}{a}\right )^{- n} \left (a + b x\right )^{n}{{}_{2}F_{1}\left (\begin{matrix} - n, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{- \frac{b x}{a}} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(1/2)*(b*x+a)**n,x)
[Out]
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Mathematica [A] time = 0.0170199, size = 45, normalized size = 1. \[ \frac{2}{3} x^{3/2} (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (\frac{3}{2},-n;\frac{5}{2};-\frac{b x}{a}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[x]*(a + b*x)^n,x]
[Out]
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Maple [F] time = 0.027, size = 0, normalized size = 0. \[ \int \sqrt{x} \left ( bx+a \right ) ^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(1/2)*(b*x+a)^n,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{n} \sqrt{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^n*sqrt(x),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x + a\right )}^{n} \sqrt{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^n*sqrt(x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.44883, size = 27, normalized size = 0.6 \[ \frac{2 a^{n} x^{\frac{3}{2}}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, - n \\ \frac{5}{2} \end{matrix}\middle |{\frac{b x e^{i \pi }}{a}} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(1/2)*(b*x+a)**n,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{n} \sqrt{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^n*sqrt(x),x, algorithm="giac")
[Out]