\(\int \sqrt {a+b x} (a c-b c x)^n \, dx\) [292]

Optimal result

Integrand size = 21, antiderivative size = 75 \[ \int \sqrt {a+b x} (a c-b c x)^n \, dx=-\frac {\sqrt {2} \sqrt {a+b x} (a c-b c x)^{1+n} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1+n,2+n,\frac {a-b x}{2 a}\right )}{b c (1+n) \sqrt {1+\frac {b x}{a}}} \] Output:

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.89 \[ \int \sqrt {a+b x} (a c-b c x)^n \, dx=\frac {2^{1+n} \left (\frac {a-b x}{a}\right )^{-n} (c (a-b x))^n (a+b x)^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-n,\frac {5}{2},\frac {a+b x}{2 a}\right )}{3 b} \] Input:

Integrate[Sqrt[a + b*x]*(a*c - b*c*x)^n,x]
 

Output:

(2^(1 + n)*(c*(a - b*x))^n*(a + b*x)^(3/2)*Hypergeometric2F1[3/2, -n, 5/2, 
 (a + b*x)/(2*a)])/(3*b*((a - b*x)/a)^n)
 

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.91, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {80, 79}

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.

\(\displaystyle \int \sqrt {a+b x} (a c-b c x)^n \, dx\)

\(\Big \downarrow \) 80

\(\displaystyle 2^n \left (\frac {a-b x}{a}\right )^{-n} (a c-b c x)^n \int \sqrt {a+b x} \left (\frac {1}{2}-\frac {b x}{2 a}\right )^ndx\)

\(\Big \downarrow \) 79

\(\displaystyle \frac {2^{n+1} (a+b x)^{3/2} \left (\frac {a-b x}{a}\right )^{-n} (a c-b c x)^n \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-n,\frac {5}{2},\frac {a+b x}{2 a}\right )}{3 b}\)

Input:

Int[Sqrt[a + b*x]*(a*c - b*c*x)^n,x]
 

Output:

(2^(1 + n)*(a + b*x)^(3/2)*(a*c - b*c*x)^n*Hypergeometric2F1[3/2, -n, 5/2, 
 (a + b*x)/(2*a)])/(3*b*((a - b*x)/a)^n)
 

Defintions of rubi rules used

rule 79

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

rule 80

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c 
 + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) 
^FracPart[n])   Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) 
), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Integ 
erQ[n] && (RationalQ[m] ||  !SimplerQ[n + 1, m + 1])
 

Maple [F]

\[\int \sqrt {b x +a}\, \left (-b c x +a c \right )^{n}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \sqrt {a+b x} (a c-b c x)^n \, dx=\int { \sqrt {b x + a} {\left (-b c x + a c\right )}^{n} \,d x } \] Input:

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

Output:

integral(sqrt(b*x + a)*(-b*c*x + a*c)^n, x)
 

Sympy [F]

\[ \int \sqrt {a+b x} (a c-b c x)^n \, dx=\int \left (- c \left (- a + b x\right )\right )^{n} \sqrt {a + b x}\, dx \] Input:

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

Output:

Integral((-c*(-a + b*x))**n*sqrt(a + b*x), x)
 

Maxima [F]

\[ \int \sqrt {a+b x} (a c-b c x)^n \, dx=\int { \sqrt {b x + a} {\left (-b c x + a c\right )}^{n} \,d x } \] Input:

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

Output:

integrate(sqrt(b*x + a)*(-b*c*x + a*c)^n, x)
 

Giac [F]

\[ \int \sqrt {a+b x} (a c-b c x)^n \, dx=\int { \sqrt {b x + a} {\left (-b c x + a c\right )}^{n} \,d x } \] Input:

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

Output:

integrate(sqrt(b*x + a)*(-b*c*x + a*c)^n, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \sqrt {a+b x} (a c-b c x)^n \, dx=\frac {-4 \sqrt {b x +a}\, \left (-b c x +a c \right )^{n} a n -2 \sqrt {b x +a}\, \left (-b c x +a c \right )^{n} a +4 \sqrt {b x +a}\, \left (-b c x +a c \right )^{n} b n x -2 \sqrt {b x +a}\, \left (-b c x +a c \right )^{n} b x -32 \left (\int \frac {\sqrt {b x +a}\, \left (-b c x +a c \right )^{n} x}{-4 b^{2} n^{2} x^{2}-4 b^{2} n \,x^{2}+4 a^{2} n^{2}+3 b^{2} x^{2}+4 a^{2} n -3 a^{2}}d x \right ) a \,b^{2} n^{3}-32 \left (\int \frac {\sqrt {b x +a}\, \left (-b c x +a c \right )^{n} x}{-4 b^{2} n^{2} x^{2}-4 b^{2} n \,x^{2}+4 a^{2} n^{2}+3 b^{2} x^{2}+4 a^{2} n -3 a^{2}}d x \right ) a \,b^{2} n^{2}+24 \left (\int \frac {\sqrt {b x +a}\, \left (-b c x +a c \right )^{n} x}{-4 b^{2} n^{2} x^{2}-4 b^{2} n \,x^{2}+4 a^{2} n^{2}+3 b^{2} x^{2}+4 a^{2} n -3 a^{2}}d x \right ) a \,b^{2} n}{b \left (4 n^{2}+4 n -3\right )} \] Input:

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

Output:

(2*( - 2*sqrt(a + b*x)*(a*c - b*c*x)**n*a*n - sqrt(a + b*x)*(a*c - b*c*x)* 
*n*a + 2*sqrt(a + b*x)*(a*c - b*c*x)**n*b*n*x - sqrt(a + b*x)*(a*c - b*c*x 
)**n*b*x - 16*int((sqrt(a + b*x)*(a*c - b*c*x)**n*x)/(4*a**2*n**2 + 4*a**2 
*n - 3*a**2 - 4*b**2*n**2*x**2 - 4*b**2*n*x**2 + 3*b**2*x**2),x)*a*b**2*n* 
*3 - 16*int((sqrt(a + b*x)*(a*c - b*c*x)**n*x)/(4*a**2*n**2 + 4*a**2*n - 3 
*a**2 - 4*b**2*n**2*x**2 - 4*b**2*n*x**2 + 3*b**2*x**2),x)*a*b**2*n**2 + 1 
2*int((sqrt(a + b*x)*(a*c - b*c*x)**n*x)/(4*a**2*n**2 + 4*a**2*n - 3*a**2 
- 4*b**2*n**2*x**2 - 4*b**2*n*x**2 + 3*b**2*x**2),x)*a*b**2*n))/(b*(4*n**2 
 + 4*n - 3))