\(\int \frac {(a c-b c x)^n}{(a+b x)^{3/2}} \, dx\) [294]

Optimal result

Integrand size = 21, antiderivative size = 77 \[ \int \frac {(a c-b c x)^n}{(a+b x)^{3/2}} \, dx=-\frac {\left (1+\frac {b x}{a}\right )^{3/2} (a c-b c x)^{1+n} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},1+n,2+n,\frac {a-b x}{2 a}\right )}{2 \sqrt {2} b c (1+n) (a+b x)^{3/2}} \] Output:

-1/4*(1+b*x/a)^(3/2)*(-b*c*x+a*c)^(1+n)*hypergeom([3/2, 1+n],[2+n],1/2*(-b 
*x+a)/a)*2^(1/2)/b/c/(1+n)/(b*x+a)^(3/2)
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.84 \[ \int \frac {(a c-b c x)^n}{(a+b x)^{3/2}} \, dx=-\frac {2^{1+n} \left (\frac {a-b x}{a}\right )^{-n} (c (a-b x))^n \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-n,\frac {1}{2},\frac {a+b x}{2 a}\right )}{b \sqrt {a+b x}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.86, 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.

\(\displaystyle \int \frac {(a c-b c x)^n}{(a+b x)^{3/2}} \, dx\)

\(\Big \downarrow \) 80

\(\displaystyle 2^n \left (\frac {a-b x}{a}\right )^{-n} (a c-b c x)^n \int \frac {\left (\frac {1}{2}-\frac {b x}{2 a}\right )^n}{(a+b x)^{3/2}}dx\)

\(\Big \downarrow \) 79

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

Input:

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

Output:

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

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]

\[\int \frac {\left (-b c x +a c \right )^{n}}{\left (b x +a \right )^{\frac {3}{2}}}d x\]

Input:

int((-b*c*x+a*c)^n/(b*x+a)^(3/2),x)
 

Output:

int((-b*c*x+a*c)^n/(b*x+a)^(3/2),x)
 

Fricas [F]

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

integrate((-b*c*x+a*c)^n/(b*x+a)^(3/2),x, algorithm="fricas")
 

Output:

integral(sqrt(b*x + a)*(-b*c*x + a*c)^n/(b^2*x^2 + 2*a*b*x + a^2), x)
 

Sympy [F]

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

integrate((-b*c*x+a*c)**n/(b*x+a)**(3/2),x)
 

Output:

Integral((-c*(-a + b*x))**n/(a + b*x)**(3/2), x)
 

Maxima [F]

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

integrate((-b*c*x+a*c)^n/(b*x+a)^(3/2),x, algorithm="maxima")
 

Output:

integrate((-b*c*x + a*c)^n/(b*x + a)^(3/2), x)
 

Giac [F]

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

integrate((-b*c*x+a*c)^n/(b*x+a)^(3/2),x, algorithm="giac")
 

Output:

integrate((-b*c*x + a*c)^n/(b*x + a)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a c-b c x)^n}{(a+b x)^{3/2}} \, dx=\int \frac {{\left (a\,c-b\,c\,x\right )}^n}{{\left (a+b\,x\right )}^{3/2}} \,d x \] Input:

int((a*c - b*c*x)^n/(a + b*x)^(3/2),x)
 

Output:

int((a*c - b*c*x)^n/(a + b*x)^(3/2), x)
 

Reduce [F]

\[ \int \frac {(a c-b c x)^n}{(a+b x)^{3/2}} \, dx=\frac {-2 \sqrt {b x +a}\, \left (-b c x +a c \right )^{n}-8 \left (\int \frac {\sqrt {b x +a}\, \left (-b c x +a c \right )^{n} x}{-2 b^{3} n \,x^{3}-2 a \,b^{2} n \,x^{2}-b^{3} x^{3}+2 a^{2} b n x -a \,b^{2} x^{2}+2 a^{3} n +a^{2} b x +a^{3}}d x \right ) a \,b^{2} n^{2}-4 \left (\int \frac {\sqrt {b x +a}\, \left (-b c x +a c \right )^{n} x}{-2 b^{3} n \,x^{3}-2 a \,b^{2} n \,x^{2}-b^{3} x^{3}+2 a^{2} b n x -a \,b^{2} x^{2}+2 a^{3} n +a^{2} b x +a^{3}}d x \right ) a \,b^{2} n -8 \left (\int \frac {\sqrt {b x +a}\, \left (-b c x +a c \right )^{n} x}{-2 b^{3} n \,x^{3}-2 a \,b^{2} n \,x^{2}-b^{3} x^{3}+2 a^{2} b n x -a \,b^{2} x^{2}+2 a^{3} n +a^{2} b x +a^{3}}d x \right ) b^{3} n^{2} x -4 \left (\int \frac {\sqrt {b x +a}\, \left (-b c x +a c \right )^{n} x}{-2 b^{3} n \,x^{3}-2 a \,b^{2} n \,x^{2}-b^{3} x^{3}+2 a^{2} b n x -a \,b^{2} x^{2}+2 a^{3} n +a^{2} b x +a^{3}}d x \right ) b^{3} n x}{b \left (2 b n x +2 a n +b x +a \right )} \] Input:

int((-b*c*x+a*c)^n/(b*x+a)^(3/2),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**3*n + a**3 + 2*a**2*b*n*x + a**2*b*x - 2*a*b**2*n*x**2 - a*b* 
*2*x**2 - 2*b**3*n*x**3 - b**3*x**3),x)*a*b**2*n**2 - 2*int((sqrt(a + b*x) 
*(a*c - b*c*x)**n*x)/(2*a**3*n + a**3 + 2*a**2*b*n*x + a**2*b*x - 2*a*b**2 
*n*x**2 - a*b**2*x**2 - 2*b**3*n*x**3 - b**3*x**3),x)*a*b**2*n - 4*int((sq 
rt(a + b*x)*(a*c - b*c*x)**n*x)/(2*a**3*n + a**3 + 2*a**2*b*n*x + a**2*b*x 
 - 2*a*b**2*n*x**2 - a*b**2*x**2 - 2*b**3*n*x**3 - b**3*x**3),x)*b**3*n**2 
*x - 2*int((sqrt(a + b*x)*(a*c - b*c*x)**n*x)/(2*a**3*n + a**3 + 2*a**2*b* 
n*x + a**2*b*x - 2*a*b**2*n*x**2 - a*b**2*x**2 - 2*b**3*n*x**3 - b**3*x**3 
),x)*b**3*n*x))/(b*(2*a*n + a + 2*b*n*x + b*x))