3.2269 \(\int \frac{(a+b \sqrt{x})^p}{x^2} \, dx\)

Optimal. Leaf size=46 \[ -\frac{2 b^2 \left (a+b \sqrt{x}\right )^{p+1} \, _2F_1\left (3,p+1;p+2;\frac{\sqrt{x} b}{a}+1\right )}{a^3 (p+1)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0188881, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 65} \[ -\frac{2 b^2 \left (a+b \sqrt{x}\right )^{p+1} \, _2F_1\left (3,p+1;p+2;\frac{\sqrt{x} b}{a}+1\right )}{a^3 (p+1)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^p/x^2,x]

[Out]

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 65

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

Rubi steps

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

Mathematica [A]  time = 0.0098022, size = 46, normalized size = 1. \[ -\frac{2 b^2 \left (a+b \sqrt{x}\right )^{p+1} \, _2F_1\left (3,p+1;p+2;\frac{\sqrt{x} b}{a}+1\right )}{a^3 (p+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^p/x^2,x]

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.019, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( a+b\sqrt{x} \right ) ^{p}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \sqrt{x} + a\right )}^{p}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*sqrt(x) + a)^p/x^2, x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b \sqrt{x} + a\right )}^{p}}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*sqrt(x) + a)^p/x^2, x)

________________________________________________________________________________________

Sympy [C]  time = 7.46967, size = 42, normalized size = 0.91 \begin{align*} - \frac{2 b^{p} x^{\frac{p}{2}} \Gamma \left (2 - p\right ){{}_{2}F_{1}\left (\begin{matrix} - p, 2 - p \\ 3 - p \end{matrix}\middle |{\frac{a e^{i \pi }}{b \sqrt{x}}} \right )}}{x \Gamma \left (3 - p\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \sqrt{x} + a\right )}^{p}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*sqrt(x) + a)^p/x^2, x)