Integrand size = 11, antiderivative size = 46 \[ \int \left (a+\frac {b}{\sqrt {x}}\right )^p \, dx=\frac {2 b^2 \left (a+\frac {b}{\sqrt {x}}\right )^{1+p} \operatorname {Hypergeometric2F1}\left (3,1+p,2+p,1+\frac {b}{a \sqrt {x}}\right )}{a^3 (1+p)} \] Output:
2*b^2*(a+b/x^(1/2))^(p+1)*hypergeom([3, p+1],[2+p],1+b/a/x^(1/2))/a^3/(p+1 )
Time = 0.11 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00 \[ \int \left (a+\frac {b}{\sqrt {x}}\right )^p \, dx=\frac {2 b^2 \left (a+\frac {b}{\sqrt {x}}\right )^{1+p} \operatorname {Hypergeometric2F1}\left (3,1+p,2+p,1+\frac {b}{a \sqrt {x}}\right )}{a^3 (1+p)} \] Input:
Integrate[(a + b/Sqrt[x])^p,x]
Output:
(2*b^2*(a + b/Sqrt[x])^(1 + p)*Hypergeometric2F1[3, 1 + p, 2 + p, 1 + b/(a *Sqrt[x])])/(a^3*(1 + p))
Time = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {774, 798, 75}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \left (a+\frac {b}{\sqrt {x}}\right )^p \, dx\) |
\(\Big \downarrow \) 774 |
\(\displaystyle 2 \int \left (a+\frac {b}{\sqrt {x}}\right )^p \sqrt {x}d\sqrt {x}\) |
\(\Big \downarrow \) 798 |
\(\displaystyle -2 \int \frac {\left (a+\frac {b}{\sqrt {x}}\right )^p}{x^{3/2}}d\frac {1}{\sqrt {x}}\) |
\(\Big \downarrow \) 75 |
\(\displaystyle \frac {2 b^2 \left (a+\frac {b}{\sqrt {x}}\right )^{p+1} \operatorname {Hypergeometric2F1}\left (3,p+1,p+2,\frac {b}{a \sqrt {x}}+1\right )}{a^3 (p+1)}\) |
Input:
Int[(a + b/Sqrt[x])^p,x]
Output:
(2*b^2*(a + b/Sqrt[x])^(1 + p)*Hypergeometric2F1[3, 1 + p, 2 + p, 1 + b/(a *Sqrt[x])])/(a^3*(1 + p))
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x )^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))*Hypergeometric2F1[-m, n + 1, n + 2, 1 + d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[n] && (IntegerQ[m] || GtQ[-d/(b*c), 0])
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Simp[k Subst[Int[x^(k - 1)*(a + b*x^(k*n))^p, x], x, x^(1/k)], x]] /; Fre eQ[{a, b, p}, x] && FractionQ[n]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[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]]
\[\int \left (a +\frac {b}{\sqrt {x}}\right )^{p}d x\]
Input:
int((a+b/x^(1/2))^p,x)
Output:
int((a+b/x^(1/2))^p,x)
\[ \int \left (a+\frac {b}{\sqrt {x}}\right )^p \, dx=\int { {\left (a + \frac {b}{\sqrt {x}}\right )}^{p} \,d x } \] Input:
integrate((a+b/x^(1/2))^p,x, algorithm="fricas")
Output:
integral(((a*x + b*sqrt(x))/x)^p, x)
Result contains complex when optimal does not.
Time = 2.95 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.89 \[ \int \left (a+\frac {b}{\sqrt {x}}\right )^p \, dx=\frac {2 b^{p} x^{1 - \frac {p}{2}} \Gamma \left (2 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 2 - p \\ 3 - p \end {matrix}\middle | {\frac {a \sqrt {x} e^{i \pi }}{b}} \right )}}{\Gamma \left (3 - p\right )} \] Input:
integrate((a+b/x**(1/2))**p,x)
Output:
2*b**p*x**(1 - p/2)*gamma(2 - p)*hyper((-p, 2 - p), (3 - p,), a*sqrt(x)*ex p_polar(I*pi)/b)/gamma(3 - p)
\[ \int \left (a+\frac {b}{\sqrt {x}}\right )^p \, dx=\int { {\left (a + \frac {b}{\sqrt {x}}\right )}^{p} \,d x } \] Input:
integrate((a+b/x^(1/2))^p,x, algorithm="maxima")
Output:
integrate((a + b/sqrt(x))^p, x)
\[ \int \left (a+\frac {b}{\sqrt {x}}\right )^p \, dx=\int { {\left (a + \frac {b}{\sqrt {x}}\right )}^{p} \,d x } \] Input:
integrate((a+b/x^(1/2))^p,x, algorithm="giac")
Output:
integrate((a + b/sqrt(x))^p, x)
Time = 0.20 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.24 \[ \int \left (a+\frac {b}{\sqrt {x}}\right )^p \, dx=-\frac {x\,{\left (a+\frac {b}{\sqrt {x}}\right )}^p\,{{}}_2{\mathrm {F}}_1\left (2-p,-p;\ 3-p;\ -\frac {a\,\sqrt {x}}{b}\right )}{\left (\frac {p}{2}-1\right )\,{\left (\frac {a\,\sqrt {x}}{b}+1\right )}^p} \] Input:
int((a + b/x^(1/2))^p,x)
Output:
-(x*(a + b/x^(1/2))^p*hypergeom([2 - p, -p], 3 - p, -(a*x^(1/2))/b))/((p/2 - 1)*((a*x^(1/2))/b + 1)^p)
\[ \int \left (a+\frac {b}{\sqrt {x}}\right )^p \, dx=\frac {2 \sqrt {x}\, \left (\sqrt {x}\, a +b \right )^{p} a b p +2 \left (\sqrt {x}\, a +b \right )^{p} a^{2} x +2 \left (\sqrt {x}\, a +b \right )^{p} b^{2} p -2 \left (\sqrt {x}\, a +b \right )^{p} b^{2}-x^{\frac {p}{2}} \left (\int \frac {\left (\sqrt {x}\, a +b \right )^{p}}{x^{\frac {p}{2}} a^{2} x^{2}-x^{\frac {p}{2}} b^{2} x}d x \right ) b^{4} p^{2}+x^{\frac {p}{2}} \left (\int \frac {\left (\sqrt {x}\, a +b \right )^{p}}{x^{\frac {p}{2}} a^{2} x^{2}-x^{\frac {p}{2}} b^{2} x}d x \right ) b^{4} p +x^{\frac {p}{2}} \left (\int \frac {\left (\sqrt {x}\, a +b \right )^{p}}{x^{\frac {p}{2}} a^{2} x -x^{\frac {p}{2}} b^{2}}d x \right ) a^{2} b^{2} p^{2}-x^{\frac {p}{2}} \left (\int \frac {\left (\sqrt {x}\, a +b \right )^{p}}{x^{\frac {p}{2}} a^{2} x -x^{\frac {p}{2}} b^{2}}d x \right ) a^{2} b^{2} p}{2 x^{\frac {p}{2}} a^{2}} \] Input:
int((a+b/x^(1/2))^p,x)
Output:
(2*sqrt(x)*(sqrt(x)*a + b)**p*a*b*p + 2*(sqrt(x)*a + b)**p*a**2*x + 2*(sqr t(x)*a + b)**p*b**2*p - 2*(sqrt(x)*a + b)**p*b**2 - x**(p/2)*int((sqrt(x)* a + b)**p/(x**(p/2)*a**2*x**2 - x**(p/2)*b**2*x),x)*b**4*p**2 + x**(p/2)*i nt((sqrt(x)*a + b)**p/(x**(p/2)*a**2*x**2 - x**(p/2)*b**2*x),x)*b**4*p + x **(p/2)*int((sqrt(x)*a + b)**p/(x**(p/2)*a**2*x - x**(p/2)*b**2),x)*a**2*b **2*p**2 - x**(p/2)*int((sqrt(x)*a + b)**p/(x**(p/2)*a**2*x - x**(p/2)*b** 2),x)*a**2*b**2*p)/(2*x**(p/2)*a**2)