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]

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

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Rubi [A]  time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {66, 64} \[ \frac {2}{3} x^{3/2} (a+b x)^n \left (\frac {b x}{a}+1\right )^{-n} \, _2F_1\left (\frac {3}{2},-n;\frac {5}{2};-\frac {b x}{a}\right ) \]

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)

Rule 64

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c^n*(b*x)^(m + 1)*Hypergeometric2F1[-n, m +
1, m + 2, -((d*x)/c)])/(b*(m + 1)), x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[
c, 0] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-(d/(b*c)), 0])))

Rule 66

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[(c^IntPart[n]*(c + d*x)^FracPart[n])/(1 + (d
*x)/c)^FracPart[n], Int[(b*x)^m*(1 + (d*x)/c)^n, x], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Int
egerQ[n] &&  !GtQ[c, 0] &&  !GtQ[-(d/(b*c)), 0] && ((RationalQ[m] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0]))
||  !RationalQ[n])

Rubi steps

\begin {align*} \int \sqrt {x} (a+b x)^n \, dx &=\left ((a+b x)^n \left (1+\frac {b x}{a}\right )^{-n}\right ) \int \sqrt {x} \left (1+\frac {b x}{a}\right )^n \, dx\\ &=\frac {2}{3} x^{3/2} (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} \, _2F_1\left (\frac {3}{2},-n;\frac {5}{2};-\frac {b x}{a}\right )\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 45, normalized size = 1.00 \[ \frac {2}{3} x^{3/2} (a+b x)^n \left (\frac {b x}{a}+1\right )^{-n} \, _2F_1\left (\frac {3}{2},-n;\frac {5}{2};-\frac {b x}{a}\right ) \]

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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fricas [F]  time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x + a\right )}^{n} \sqrt {x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/2)*(b*x+a)^n,x, algorithm="fricas")

[Out]

integral((b*x + a)^n*sqrt(x), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x + a\right )}^{n} \sqrt {x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/2)*(b*x+a)^n,x, algorithm="giac")

[Out]

integrate((b*x + a)^n*sqrt(x), x)

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[ \int \sqrt {x}\, \left (b x +a \right )^{n}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x + a\right )}^{n} \sqrt {x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/2)*(b*x+a)^n,x, algorithm="maxima")

[Out]

integrate((b*x + a)^n*sqrt(x), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {x}\,{\left (a+b\,x\right )}^n \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)*(a + b*x)^n,x)

[Out]

int(x^(1/2)*(a + b*x)^n, x)

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sympy [C]  time = 8.38, size = 27, normalized size = 0.60 \[ \frac {2 a^{n} x^{\frac {3}{2}} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, - n \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )}}{3} \]

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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