\(\int \frac {(1+a x) (c-a c x)^p}{\sqrt {1-a^2 x^2}} \, dx\) [2]

Optimal result

Integrand size = 29, antiderivative size = 73 \[ \int \frac {(1+a x) (c-a c x)^p}{\sqrt {1-a^2 x^2}} \, dx=-\frac {2 \sqrt {2} (c-a c x)^p \sqrt {1-a^2 x^2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{2}+p,\frac {3}{2}+p,\frac {1}{2} (1-a x)\right )}{a (1+2 p) \sqrt {1+a x}} \] Output:

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

Mathematica [A] (verified)

Time = 1.11 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.40 \[ \int \frac {(1+a x) (c-a c x)^p}{\sqrt {1-a^2 x^2}} \, dx=\frac {2 (1-a x)^{-p} (1+a x) (c-a c x)^p \left (-(1-a x)^{1+p}+2^{\frac {1}{2}+p} \sqrt {1-a x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {1}{2}-p,\frac {3}{2},\frac {1}{2} (1+a x)\right )\right )}{(a+2 a p) \sqrt {1-a^2 x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.38, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {672, 474, 456, 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.

\(\displaystyle \int \frac {(a x+1) (c-a c x)^p}{\sqrt {1-a^2 x^2}} \, dx\)

\(\Big \downarrow \) 672

\(\displaystyle \frac {\int \frac {(c-a c x)^p}{\sqrt {1-a^2 x^2}}dx}{p+1}-\frac {\sqrt {1-a^2 x^2} (c-a c x)^p}{a (p+1)}\)

\(\Big \downarrow \) 474

\(\displaystyle \frac {(1-a x)^{-p} (c-a c x)^p \int \frac {(1-a x)^p}{\sqrt {1-a^2 x^2}}dx}{p+1}-\frac {\sqrt {1-a^2 x^2} (c-a c x)^p}{a (p+1)}\)

\(\Big \downarrow \) 456

\(\displaystyle \frac {(1-a x)^{-p} (c-a c x)^p \int \frac {(1-a x)^{p-\frac {1}{2}}}{\sqrt {a x+1}}dx}{p+1}-\frac {\sqrt {1-a^2 x^2} (c-a c x)^p}{a (p+1)}\)

\(\Big \downarrow \) 79

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

Input:

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

Output:

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

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 456

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ 
(c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x] /; FreeQ[{a, b, c, d, n, p}, x] && 
EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0] &&  !Integ 
erQ[n]))
 

rule 474

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
c^IntPart[n]*((c + d*x)^FracPart[n]/(1 + d*(x/c))^FracPart[n])   Int[(1 + d 
*(x/c))^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c^2 + 
 a*d^2, 0] &&  !(IntegerQ[n] || GtQ[c, 0])
 

rule 672

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_ 
), x_Symbol] :> Simp[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), 
 x] + Simp[(m*(d*g + e*f) + 2*e*f*(p + 1))/(e*(m + 2*p + 2))   Int[(d + e*x 
)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[c*d^ 
2 + a*e^2, 0] && NeQ[m + 2*p + 2, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral((-c*(a*x - 1))**p*(a*x + 1)/sqrt(-(a*x - 1)*(a*x + 1)), x)
 

Maxima [F]

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

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

Output:

integrate((a*x + 1)*(-a*c*x + c)^p/sqrt(-a^2*x^2 + 1), x)
 

Giac [F]

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

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

Output:

integrate((a*x + 1)*(-a*c*x + c)^p/sqrt(-a^2*x^2 + 1), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(( - a*c*x + c)**p/sqrt( - a**2*x**2 + 1),x) + int((( - a*c*x + c)**p*x 
)/sqrt( - a**2*x**2 + 1),x)*a