Integrand size = 28, antiderivative size = 56 \[ \int \frac {\sqrt {1+a x} (c+a c x)^m}{\sqrt {1-a x}} \, dx=\frac {\sqrt {2} (1+a x)^{3/2} (c+a c x)^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{2}+m,\frac {5}{2}+m,\frac {1}{2} (1+a x)\right )}{a (3+2 m)} \] Output:
2^(1/2)*(a*x+1)^(3/2)*(a*c*x+c)^m*hypergeom([1/2, 3/2+m],[5/2+m],1/2*a*x+1 /2)/a/(3+2*m)
Time = 0.09 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.12 \[ \int \frac {\sqrt {1+a x} (c+a c x)^m}{\sqrt {1-a x}} \, dx=-\frac {2^{\frac {3}{2}+m} \sqrt {1-a x} (1+a x)^{-m} (c+a c x)^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {1}{2}-m,\frac {3}{2},\frac {1}{2} (1-a x)\right )}{a} \] Input:
Integrate[(Sqrt[1 + a*x]*(c + a*c*x)^m)/Sqrt[1 - a*x],x]
Output:
-((2^(3/2 + m)*Sqrt[1 - a*x]*(c + a*c*x)^m*Hypergeometric2F1[1/2, -1/2 - m , 3/2, (1 - a*x)/2])/(a*(1 + a*x)^m))
Time = 0.16 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.12, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, 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.
\(\displaystyle \int \frac {\sqrt {a x+1} (a c x+c)^m}{\sqrt {1-a x}} \, dx\) |
\(\Big \downarrow \) 37 |
\(\displaystyle (a x+1)^{-m} (a c x+c)^m \int \frac {(a x+1)^{m+\frac {1}{2}}}{\sqrt {1-a x}}dx\) |
\(\Big \downarrow \) 79 |
\(\displaystyle -\frac {2^{m+\frac {3}{2}} \sqrt {1-a x} (a x+1)^{-m} (a c x+c)^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m-\frac {1}{2},\frac {3}{2},\frac {1}{2} (1-a x)\right )}{a}\) |
Input:
Int[(Sqrt[1 + a*x]*(c + a*c*x)^m)/Sqrt[1 - a*x],x]
Output:
-((2^(3/2 + m)*Sqrt[1 - a*x]*(c + a*c*x)^m*Hypergeometric2F1[1/2, -1/2 - m , 3/2, (1 - a*x)/2])/(a*(1 + a*x)^m))
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]
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]))
\[\int \frac {\sqrt {a x +1}\, \left (a c x +c \right )^{m}}{\sqrt {-a x +1}}d x\]
Input:
int((a*x+1)^(1/2)*(a*c*x+c)^m/(-a*x+1)^(1/2),x)
Output:
int((a*x+1)^(1/2)*(a*c*x+c)^m/(-a*x+1)^(1/2),x)
\[ \int \frac {\sqrt {1+a x} (c+a c x)^m}{\sqrt {1-a x}} \, dx=\int { \frac {\sqrt {a x + 1} {\left (a c x + c\right )}^{m}}{\sqrt {-a x + 1}} \,d x } \] Input:
integrate((a*x+1)^(1/2)*(a*c*x+c)^m/(-a*x+1)^(1/2),x, algorithm="fricas")
Output:
integral(-sqrt(a*x + 1)*sqrt(-a*x + 1)*(a*c*x + c)^m/(a*x - 1), x)
\[ \int \frac {\sqrt {1+a x} (c+a c x)^m}{\sqrt {1-a x}} \, dx=\int \frac {\left (c \left (a x + 1\right )\right )^{m} \sqrt {a x + 1}}{\sqrt {- a x + 1}}\, dx \] Input:
integrate((a*x+1)**(1/2)*(a*c*x+c)**m/(-a*x+1)**(1/2),x)
Output:
Integral((c*(a*x + 1))**m*sqrt(a*x + 1)/sqrt(-a*x + 1), x)
\[ \int \frac {\sqrt {1+a x} (c+a c x)^m}{\sqrt {1-a x}} \, dx=\int { \frac {\sqrt {a x + 1} {\left (a c x + c\right )}^{m}}{\sqrt {-a x + 1}} \,d x } \] Input:
integrate((a*x+1)^(1/2)*(a*c*x+c)^m/(-a*x+1)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(a*x + 1)*(a*c*x + c)^m/sqrt(-a*x + 1), x)
\[ \int \frac {\sqrt {1+a x} (c+a c x)^m}{\sqrt {1-a x}} \, dx=\int { \frac {\sqrt {a x + 1} {\left (a c x + c\right )}^{m}}{\sqrt {-a x + 1}} \,d x } \] Input:
integrate((a*x+1)^(1/2)*(a*c*x+c)^m/(-a*x+1)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(a*x + 1)*(a*c*x + c)^m/sqrt(-a*x + 1), x)
Timed out. \[ \int \frac {\sqrt {1+a x} (c+a c x)^m}{\sqrt {1-a x}} \, dx=\int \frac {{\left (c+a\,c\,x\right )}^m\,\sqrt {a\,x+1}}{\sqrt {1-a\,x}} \,d x \] Input:
int(((c + a*c*x)^m*(a*x + 1)^(1/2))/(1 - a*x)^(1/2),x)
Output:
int(((c + a*c*x)^m*(a*x + 1)^(1/2))/(1 - a*x)^(1/2), x)
\[ \int \frac {\sqrt {1+a x} (c+a c x)^m}{\sqrt {1-a x}} \, dx=\int \frac {\sqrt {a x +1}\, \left (a c x +c \right )^{m}}{\sqrt {-a x +1}}d x \] Input:
int((a*x+1)^(1/2)*(a*c*x+c)^m/(-a*x+1)^(1/2),x)
Output:
int((sqrt(a*x + 1)*(a*c*x + c)**m)/sqrt( - a*x + 1),x)