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

Optimal. Leaf size=43 \[ 2 \sqrt{x} (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};-\frac{b x}{a}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0086755, antiderivative size = 43, 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, Rules used = {66, 64} \[ 2 \sqrt{x} (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};-\frac{b x}{a}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^n/Sqrt[x],x]

[Out]

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

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])

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])))

Rubi steps

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

Mathematica [A]  time = 0.0060289, size = 43, normalized size = 1. \[ 2 \sqrt{x} (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};-\frac{b x}{a}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^n/Sqrt[x],x]

[Out]

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

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Maple [F]  time = 0.021, size = 0, normalized size = 0. \begin{align*} \int{ \left ( bx+a \right ) ^{n}{\frac{1}{\sqrt{x}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{n}}{\sqrt{x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^n/x^(1/2),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. \begin{align*}{\rm integral}\left (\frac{{\left (b x + a\right )}^{n}}{\sqrt{x}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x + a)^n/sqrt(x), x)

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Sympy [C]  time = 7.76647, size = 26, normalized size = 0.6 \begin{align*} 2 a^{n} \sqrt{x}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, - n \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x e^{i \pi }}{a}} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a**n*sqrt(x)*hyper((1/2, -n), (3/2,), b*x*exp_polar(I*pi)/a)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{n}}{\sqrt{x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x + a)^n/sqrt(x), x)