\(\int \frac {(1+a x)^{3/2} (c-a c x)^m}{(1-a x)^{3/2}} \, dx\) [1707]

Optimal result

Integrand size = 29, antiderivative size = 60 \[ \int \frac {(1+a x)^{3/2} (c-a c x)^m}{(1-a x)^{3/2}} \, dx=\frac {4 \sqrt {2} (c-a c x)^m \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {1}{2}+m,\frac {1}{2}+m,\frac {1}{2} (1-a x)\right )}{a (1-2 m) \sqrt {1-a x}} \] Output:

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.08 \[ \int \frac {(1+a x)^{3/2} (c-a c x)^m}{(1-a x)^{3/2}} \, dx=\frac {2^{-\frac {1}{2}+m} (1-a x)^{-m} (1+a x)^{5/2} (c-a c x)^m \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {3}{2}-m,\frac {7}{2},\frac {1}{2} (1+a x)\right )}{5 a} \] Input:

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

Output:

(2^(-1/2 + m)*(1 + a*x)^(5/2)*(c - a*c*x)^m*Hypergeometric2F1[5/2, 3/2 - m 
, 7/2, (1 + a*x)/2])/(5*a*(1 - a*x)^m)
 

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.08, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {37, 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.

\(\displaystyle \int \frac {(a x+1)^{3/2} (c-a c x)^m}{(1-a x)^{3/2}} \, dx\)

\(\Big \downarrow \) 37

\(\displaystyle (1-a x)^{-m} (c-a c x)^m \int (1-a x)^{m-\frac {3}{2}} (a x+1)^{3/2}dx\)

\(\Big \downarrow \) 79

\(\displaystyle \frac {2^{m-\frac {1}{2}} (a x+1)^{5/2} (1-a x)^{-m} (c-a c x)^m \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {3}{2}-m,\frac {7}{2},\frac {1}{2} (a x+1)\right )}{5 a}\)

Input:

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

Output:

(2^(-1/2 + m)*(1 + a*x)^(5/2)*(c - a*c*x)^m*Hypergeometric2F1[5/2, 3/2 - m 
, 7/2, (1 + a*x)/2])/(5*a*(1 - a*x)^m)
 

Defintions of rubi rules used

rule 37

Int[(u_.)*((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> S 
imp[(a + b*x)^m/(c + d*x)^m   Int[u*(c + d*x)^(m + n), x], x] /; FreeQ[{a, 
b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] &&  !SimplerQ[a + b*x, c + d*x]
 

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]))
 

Maple [F]

\[\int \frac {\left (a x +1\right )^{\frac {3}{2}} \left (-a c x +c \right )^{m}}{\left (-a x +1\right )^{\frac {3}{2}}}d x\]

Input:

int((a*x+1)^(3/2)*(-a*c*x+c)^m/(-a*x+1)^(3/2),x)
 

Output:

int((a*x+1)^(3/2)*(-a*c*x+c)^m/(-a*x+1)^(3/2),x)
 

Fricas [F]

\[ \int \frac {(1+a x)^{3/2} (c-a c x)^m}{(1-a x)^{3/2}} \, dx=\int { \frac {{\left (a x + 1\right )}^{\frac {3}{2}} {\left (-a c x + c\right )}^{m}}{{\left (-a x + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((a*x+1)^(3/2)*(-a*c*x+c)^m/(-a*x+1)^(3/2),x, algorithm="fricas")
 

Output:

integral((a*x + 1)^(3/2)*sqrt(-a*x + 1)*(-a*c*x + c)^m/(a^2*x^2 - 2*a*x + 
1), x)
 

Sympy [F]

\[ \int \frac {(1+a x)^{3/2} (c-a c x)^m}{(1-a x)^{3/2}} \, dx=\int \frac {\left (- c \left (a x - 1\right )\right )^{m} \left (a x + 1\right )^{\frac {3}{2}}}{\left (- a x + 1\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((a*x+1)**(3/2)*(-a*c*x+c)**m/(-a*x+1)**(3/2),x)
 

Output:

Integral((-c*(a*x - 1))**m*(a*x + 1)**(3/2)/(-a*x + 1)**(3/2), x)
 

Maxima [F]

\[ \int \frac {(1+a x)^{3/2} (c-a c x)^m}{(1-a x)^{3/2}} \, dx=\int { \frac {{\left (a x + 1\right )}^{\frac {3}{2}} {\left (-a c x + c\right )}^{m}}{{\left (-a x + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((a*x+1)^(3/2)*(-a*c*x+c)^m/(-a*x+1)^(3/2),x, algorithm="maxima")
 

Output:

integrate((a*x + 1)^(3/2)*(-a*c*x + c)^m/(-a*x + 1)^(3/2), x)
 

Giac [F]

\[ \int \frac {(1+a x)^{3/2} (c-a c x)^m}{(1-a x)^{3/2}} \, dx=\int { \frac {{\left (a x + 1\right )}^{\frac {3}{2}} {\left (-a c x + c\right )}^{m}}{{\left (-a x + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((a*x+1)^(3/2)*(-a*c*x+c)^m/(-a*x+1)^(3/2),x, algorithm="giac")
 

Output:

integrate((a*x + 1)^(3/2)*(-a*c*x + c)^m/(-a*x + 1)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(1+a x)^{3/2} (c-a c x)^m}{(1-a x)^{3/2}} \, dx=\int \frac {{\left (c-a\,c\,x\right )}^m\,{\left (a\,x+1\right )}^{3/2}}{{\left (1-a\,x\right )}^{3/2}} \,d x \] Input:

int(((c - a*c*x)^m*(a*x + 1)^(3/2))/(1 - a*x)^(3/2),x)
 

Output:

int(((c - a*c*x)^m*(a*x + 1)^(3/2))/(1 - a*x)^(3/2), x)
 

Reduce [F]

\[ \int \frac {(1+a x)^{3/2} (c-a c x)^m}{(1-a x)^{3/2}} \, dx=-\left (\int \frac {\sqrt {a x +1}\, \left (-a c x +c \right )^{m} x}{\sqrt {-a x +1}\, a x -\sqrt {-a x +1}}d x \right ) a -\left (\int \frac {\sqrt {a x +1}\, \left (-a c x +c \right )^{m}}{\sqrt {-a x +1}\, a x -\sqrt {-a x +1}}d x \right ) \] Input:

int((a*x+1)^(3/2)*(-a*c*x+c)^m/(-a*x+1)^(3/2),x)
 

Output:

 - (int((sqrt(a*x + 1)*( - a*c*x + c)**m*x)/(sqrt( - a*x + 1)*a*x - sqrt( 
- a*x + 1)),x)*a + int((sqrt(a*x + 1)*( - a*c*x + c)**m)/(sqrt( - a*x + 1) 
*a*x - sqrt( - a*x + 1)),x))