\(\int \frac {(a c-b c x)^n}{\sqrt {a+b x}} \, dx\) [293]

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.85 \[ \int \frac {(a c-b c x)^n}{\sqrt {a+b x}} \, dx=\frac {2^{1+n} \left (\frac {a-b x}{a}\right )^{-n} (c (a-b x))^n \sqrt {a+b x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-n,\frac {3}{2},\frac {a+b x}{2 a}\right )}{b} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.87, 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.

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

\(\Big \downarrow \) 80

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

\(\Big \downarrow \) 79

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

\[ \int \frac {(a c-b c x)^n}{\sqrt {a+b x}} \, dx=\frac {-2 \sqrt {b x +a}\, \left (-b c x +a c \right )^{n}-8 \left (\int \frac {\sqrt {b x +a}\, \left (-b c x +a c \right )^{n} x}{-2 b^{2} n \,x^{2}+b^{2} x^{2}+2 a^{2} n -a^{2}}d x \right ) b^{2} n^{2}+4 \left (\int \frac {\sqrt {b x +a}\, \left (-b c x +a c \right )^{n} x}{-2 b^{2} n \,x^{2}+b^{2} x^{2}+2 a^{2} n -a^{2}}d x \right ) b^{2} n}{b \left (2 n -1\right )} \] Input:

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

Output:

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