3.746 \(\int \sqrt{x} (a+b x)^n \, dx\)

Optimal. Leaf size=45 \[ \frac{2}{3} x^{3/2} (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (\frac{3}{2},-n;\frac{5}{2};-\frac{b x}{a}\right ) \]

[Out]

(2*x^(3/2)*(a + b*x)^n*Hypergeometric2F1[3/2, -n, 5/2, -((b*x)/a)])/(3*(1 + (b*x
)/a)^n)

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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]

(2*x^(3/2)*(a + b*x)^n*Hypergeometric2F1[3/2, -n, 5/2, -((b*x)/a)])/(3*(1 + (b*x
)/a)^n)

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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]

2*x**(3/2)*(1 + b*x/a)**(-n)*(a + b*x)**n*hyper((-n, 3/2), (5/2,), -b*x/a)/3

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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]

(2*x^(3/2)*(a + b*x)^n*Hypergeometric2F1[3/2, -n, 5/2, -((b*x)/a)])/(3*(1 + (b*x
)/a)^n)

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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]

int(x^(1/2)*(b*x+a)^n,x)

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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]

integrate((b*x + a)^n*sqrt(x), x)

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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]

integral((b*x + a)^n*sqrt(x), x)

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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]

2*a**n*x**(3/2)*hyper((3/2, -n), (5/2,), b*x*exp_polar(I*pi)/a)/3

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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]

integrate((b*x + a)^n*sqrt(x), x)