\(\int (a+b x)^{3/2} (a c-b c x)^n \, dx\) [291]

Optimal result

Integrand size = 21, antiderivative size = 75 \[ \int (a+b x)^{3/2} (a c-b c x)^n \, dx=-\frac {2 \sqrt {2} (a+b x)^{3/2} (a c-b c x)^{1+n} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1+n,2+n,\frac {a-b x}{2 a}\right )}{b c (1+n) \left (1+\frac {b x}{a}\right )^{3/2}} \] Output:

-2*2^(1/2)*(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)/b/c/(1+n)/(1+b*x/a)^(3/2)
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.89 \[ \int (a+b x)^{3/2} (a c-b c x)^n \, dx=\frac {2^{1+n} \left (\frac {a-b x}{a}\right )^{-n} (c (a-b x))^n (a+b x)^{5/2} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-n,\frac {7}{2},\frac {a+b x}{2 a}\right )}{5 b} \] Input:

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

Output:

(2^(1 + n)*(c*(a - b*x))^n*(a + b*x)^(5/2)*Hypergeometric2F1[5/2, -n, 7/2, 
 (a + b*x)/(2*a)])/(5*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.

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

\(\Big \downarrow \) 80

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

\(\Big \downarrow \) 79

\(\displaystyle \frac {2^{n+1} (a+b x)^{5/2} \left (\frac {a-b x}{a}\right )^{-n} (a c-b c x)^n \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-n,\frac {7}{2},\frac {a+b x}{2 a}\right )}{5 b}\)

Input:

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

Output:

(2^(1 + n)*(a + b*x)^(5/2)*(a*c - b*c*x)^n*Hypergeometric2F1[5/2, -n, 7/2, 
 (a + b*x)/(2*a)])/(5*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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

\[ \int (a+b x)^{3/2} (a c-b c x)^n \, dx=\frac {-8 \sqrt {b x +a}\, \left (-b c x +a c \right )^{n} a^{2} n^{2}-32 \sqrt {b x +a}\, \left (-b c x +a c \right )^{n} a^{2} n -6 \sqrt {b x +a}\, \left (-b c x +a c \right )^{n} a^{2}+24 \sqrt {b x +a}\, \left (-b c x +a c \right )^{n} a b n x -12 \sqrt {b x +a}\, \left (-b c x +a c \right )^{n} a b x +8 \sqrt {b x +a}\, \left (-b c x +a c \right )^{n} b^{2} n^{2} x^{2}+8 \sqrt {b x +a}\, \left (-b c x +a c \right )^{n} b^{2} n \,x^{2}-6 \sqrt {b x +a}\, \left (-b c x +a c \right )^{n} b^{2} x^{2}-384 \left (\int \frac {\sqrt {b x +a}\, \left (-b c x +a c \right )^{n} x}{-8 b^{2} n^{3} x^{2}-28 b^{2} n^{2} x^{2}+8 a^{2} n^{3}-14 b^{2} n \,x^{2}+28 a^{2} n^{2}+15 b^{2} x^{2}+14 a^{2} n -15 a^{2}}d x \right ) a^{2} b^{2} n^{4}-1344 \left (\int \frac {\sqrt {b x +a}\, \left (-b c x +a c \right )^{n} x}{-8 b^{2} n^{3} x^{2}-28 b^{2} n^{2} x^{2}+8 a^{2} n^{3}-14 b^{2} n \,x^{2}+28 a^{2} n^{2}+15 b^{2} x^{2}+14 a^{2} n -15 a^{2}}d x \right ) a^{2} b^{2} n^{3}-672 \left (\int \frac {\sqrt {b x +a}\, \left (-b c x +a c \right )^{n} x}{-8 b^{2} n^{3} x^{2}-28 b^{2} n^{2} x^{2}+8 a^{2} n^{3}-14 b^{2} n \,x^{2}+28 a^{2} n^{2}+15 b^{2} x^{2}+14 a^{2} n -15 a^{2}}d x \right ) a^{2} b^{2} n^{2}+720 \left (\int \frac {\sqrt {b x +a}\, \left (-b c x +a c \right )^{n} x}{-8 b^{2} n^{3} x^{2}-28 b^{2} n^{2} x^{2}+8 a^{2} n^{3}-14 b^{2} n \,x^{2}+28 a^{2} n^{2}+15 b^{2} x^{2}+14 a^{2} n -15 a^{2}}d x \right ) a^{2} b^{2} n}{b \left (8 n^{3}+28 n^{2}+14 n -15\right )} \] Input:

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

Output:

(2*( - 4*sqrt(a + b*x)*(a*c - b*c*x)**n*a**2*n**2 - 16*sqrt(a + b*x)*(a*c 
- b*c*x)**n*a**2*n - 3*sqrt(a + b*x)*(a*c - b*c*x)**n*a**2 + 12*sqrt(a + b 
*x)*(a*c - b*c*x)**n*a*b*n*x - 6*sqrt(a + b*x)*(a*c - b*c*x)**n*a*b*x + 4* 
sqrt(a + b*x)*(a*c - b*c*x)**n*b**2*n**2*x**2 + 4*sqrt(a + b*x)*(a*c - b*c 
*x)**n*b**2*n*x**2 - 3*sqrt(a + b*x)*(a*c - b*c*x)**n*b**2*x**2 - 192*int( 
(sqrt(a + b*x)*(a*c - b*c*x)**n*x)/(8*a**2*n**3 + 28*a**2*n**2 + 14*a**2*n 
 - 15*a**2 - 8*b**2*n**3*x**2 - 28*b**2*n**2*x**2 - 14*b**2*n*x**2 + 15*b* 
*2*x**2),x)*a**2*b**2*n**4 - 672*int((sqrt(a + b*x)*(a*c - b*c*x)**n*x)/(8 
*a**2*n**3 + 28*a**2*n**2 + 14*a**2*n - 15*a**2 - 8*b**2*n**3*x**2 - 28*b* 
*2*n**2*x**2 - 14*b**2*n*x**2 + 15*b**2*x**2),x)*a**2*b**2*n**3 - 336*int( 
(sqrt(a + b*x)*(a*c - b*c*x)**n*x)/(8*a**2*n**3 + 28*a**2*n**2 + 14*a**2*n 
 - 15*a**2 - 8*b**2*n**3*x**2 - 28*b**2*n**2*x**2 - 14*b**2*n*x**2 + 15*b* 
*2*x**2),x)*a**2*b**2*n**2 + 360*int((sqrt(a + b*x)*(a*c - b*c*x)**n*x)/(8 
*a**2*n**3 + 28*a**2*n**2 + 14*a**2*n - 15*a**2 - 8*b**2*n**3*x**2 - 28*b* 
*2*n**2*x**2 - 14*b**2*n*x**2 + 15*b**2*x**2),x)*a**2*b**2*n))/(b*(8*n**3 
+ 28*n**2 + 14*n - 15))