\(\int \frac {(a^2+2 a c \sqrt {x}+c^2 x)^p}{x^2} \, dx\) [91]

Optimal result

Integrand size = 24, antiderivative size = 68 \[ \int \frac {\left (a^2+2 a c \sqrt {x}+c^2 x\right )^p}{x^2} \, dx=-\frac {2 c^2 \left (a+c \sqrt {x}\right ) \left (a^2+2 a c \sqrt {x}+c^2 x\right )^p \operatorname {Hypergeometric2F1}\left (3,1+2 p,2 (1+p),1+\frac {c \sqrt {x}}{a}\right )}{a^3 (1+2 p)} \] Output:

-2*c^2*(a+c*x^(1/2))*(a^2+2*a*c*x^(1/2)+c^2*x)^p*hypergeom([3, 1+2*p],[2*p 
+2],1+c*x^(1/2)/a)/a^3/(1+2*p)
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a^2+2 a c \sqrt {x}+c^2 x\right )^p}{x^2} \, dx=-\frac {2 c^2 \left (a+c \sqrt {x}\right ) \left (\left (a+c \sqrt {x}\right )^2\right )^p \operatorname {Hypergeometric2F1}\left (3,1+2 p,2+2 p,1+\frac {c \sqrt {x}}{a}\right )}{a^3 (1+2 p)} \] Input:

Integrate[(a^2 + 2*a*c*Sqrt[x] + c^2*x)^p/x^2,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1385, 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.

\(\displaystyle \int \frac {\left (a^2+2 a c \sqrt {x}+c^2 x\right )^p}{x^2} \, dx\)

\(\Big \downarrow \) 1385

\(\displaystyle \left (\frac {c \sqrt {x}}{a}+1\right )^{-2 p} \left (a^2+2 a c \sqrt {x}+c^2 x\right )^p \int \frac {\left (\frac {\sqrt {x} c}{a}+1\right )^{2 p}}{x^2}dx\)

\(\Big \downarrow \) 798

\(\displaystyle 2 \left (\frac {c \sqrt {x}}{a}+1\right )^{-2 p} \left (a^2+2 a c \sqrt {x}+c^2 x\right )^p \int \frac {\left (\frac {\sqrt {x} c}{a}+1\right )^{2 p}}{x^{3/2}}d\sqrt {x}\)

\(\Big \downarrow \) 75

\(\displaystyle -\frac {2 c^2 \left (\frac {c \sqrt {x}}{a}+1\right ) \left (a^2+2 a c \sqrt {x}+c^2 x\right )^p \operatorname {Hypergeometric2F1}\left (3,2 p+1,2 (p+1),\frac {\sqrt {x} c}{a}+1\right )}{a^2 (2 p+1)}\)

Input:

Int[(a^2 + 2*a*c*Sqrt[x] + c^2*x)^p/x^2,x]
 

Output:

(-2*c^2*(1 + (c*Sqrt[x])/a)*(a^2 + 2*a*c*Sqrt[x] + c^2*x)^p*Hypergeometric 
2F1[3, 1 + 2*p, 2*(1 + p), 1 + (c*Sqrt[x])/a])/(a^2*(1 + 2*p))
 

Defintions of rubi rules used

rule 75

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])
 

rule 798

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]]
 

rule 1385

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> S 
imp[a^IntPart[p]*((a + b*x^n + c*x^(2*n))^FracPart[p]/(1 + 2*c*(x^n/b))^(2* 
FracPart[p]))   Int[u*(1 + 2*c*(x^n/b))^(2*p), x], x] /; FreeQ[{a, b, c, n, 
 p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[2*p] && NeQ[u, 
 x^(n - 1)] && NeQ[u, x^(2*n - 1)]
 

Maple [F]

\[\int \frac {\left (a^{2}+2 a c \sqrt {x}+c^{2} x \right )^{p}}{x^{2}}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {\left (a^2+2 a c \sqrt {x}+c^2 x\right )^p}{x^2} \, dx=\int { \frac {{\left (c^{2} x + 2 \, a c \sqrt {x} + a^{2}\right )}^{p}}{x^{2}} \,d x } \] Input:

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

Output:

integral((c^2*x + 2*a*c*sqrt(x) + a^2)^p/x^2, x)
 

Sympy [F]

\[ \int \frac {\left (a^2+2 a c \sqrt {x}+c^2 x\right )^p}{x^2} \, dx=\int \frac {\left (a^{2} + 2 a c \sqrt {x} + c^{2} x\right )^{p}}{x^{2}}\, dx \] Input:

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

Output:

Integral((a**2 + 2*a*c*sqrt(x) + c**2*x)**p/x**2, x)
 

Maxima [F]

\[ \int \frac {\left (a^2+2 a c \sqrt {x}+c^2 x\right )^p}{x^2} \, dx=\int { \frac {{\left (c^{2} x + 2 \, a c \sqrt {x} + a^{2}\right )}^{p}}{x^{2}} \,d x } \] Input:

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

Output:

integrate((c^2*x + 2*a*c*sqrt(x) + a^2)^p/x^2, x)
 

Giac [F]

\[ \int \frac {\left (a^2+2 a c \sqrt {x}+c^2 x\right )^p}{x^2} \, dx=\int { \frac {{\left (c^{2} x + 2 \, a c \sqrt {x} + a^{2}\right )}^{p}}{x^{2}} \,d x } \] Input:

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

Output:

integrate((c^2*x + 2*a*c*sqrt(x) + a^2)^p/x^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a^2+2 a c \sqrt {x}+c^2 x\right )^p}{x^2} \, dx=\int \frac {{\left (c^2\,x+a^2+2\,a\,c\,\sqrt {x}\right )}^p}{x^2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\left (a^2+2 a c \sqrt {x}+c^2 x\right )^p}{x^2} \, dx=\frac {-2 \sqrt {x}\, \left (2 \sqrt {x}\, a c +a^{2}+c^{2} x \right )^{p} c p -\left (2 \sqrt {x}\, a c +a^{2}+c^{2} x \right )^{p} a +2 \left (\int \frac {\left (2 \sqrt {x}\, a c +a^{2}+c^{2} x \right )^{p}}{-c^{2} x^{2}+a^{2} x}d x \right ) a \,c^{2} p^{2} x -\left (\int \frac {\left (2 \sqrt {x}\, a c +a^{2}+c^{2} x \right )^{p}}{-c^{2} x^{2}+a^{2} x}d x \right ) a \,c^{2} p x -2 \left (\int \frac {\sqrt {x}\, \left (2 \sqrt {x}\, a c +a^{2}+c^{2} x \right )^{p}}{-c^{2} x^{2}+a^{2} x}d x \right ) c^{3} p^{2} x +\left (\int \frac {\sqrt {x}\, \left (2 \sqrt {x}\, a c +a^{2}+c^{2} x \right )^{p}}{-c^{2} x^{2}+a^{2} x}d x \right ) c^{3} p x}{a x} \] Input:

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

Output:

( - 2*sqrt(x)*(2*sqrt(x)*a*c + a**2 + c**2*x)**p*c*p - (2*sqrt(x)*a*c + a* 
*2 + c**2*x)**p*a + 2*int((2*sqrt(x)*a*c + a**2 + c**2*x)**p/(a**2*x - c** 
2*x**2),x)*a*c**2*p**2*x - int((2*sqrt(x)*a*c + a**2 + c**2*x)**p/(a**2*x 
- c**2*x**2),x)*a*c**2*p*x - 2*int((sqrt(x)*(2*sqrt(x)*a*c + a**2 + c**2*x 
)**p)/(a**2*x - c**2*x**2),x)*c**3*p**2*x + int((sqrt(x)*(2*sqrt(x)*a*c + 
a**2 + c**2*x)**p)/(a**2*x - c**2*x**2),x)*c**3*p*x)/(a*x)