\(\int \frac {(1-a x)^{3/2} (c-a c x)^m}{(1+a x)^{3/2}} \, dx\) [1710]

Optimal result

Integrand size = 29, antiderivative size = 60 \[ \int \frac {(1-a x)^{3/2} (c-a c x)^m}{(1+a x)^{3/2}} \, dx=-\frac {(1-a x)^{5/2} (c-a c x)^m \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {5}{2}+m,\frac {7}{2}+m,\frac {1}{2} (1-a x)\right )}{\sqrt {2} a (5+2 m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.05 \[ \int \frac {(1-a x)^{3/2} (c-a c x)^m}{(1+a x)^{3/2}} \, dx=-\frac {2^{\frac {5}{2}+m} (1-a x)^{-m} (c-a c x)^m \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {3}{2}-m,\frac {1}{2},\frac {1}{2} (1+a x)\right )}{a \sqrt {1+a x}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.05, 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.

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

\(\Big \downarrow \) 37

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

\(\Big \downarrow \) 79

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

Input:

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

Output:

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

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

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

Output:

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

Reduce [F]

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