\(\int (A+B x) (c+d x)^{-3-2 p} (A^2-B^2 x^2)^p \, dx\) [5]

Optimal result

Integrand size = 31, antiderivative size = 123 \[ \int (A+B x) (c+d x)^{-3-2 p} \left (A^2-B^2 x^2\right )^p \, dx=-\frac {2^{1+p} \left (\frac {(B c+A d) (A+B x)}{A B (c+d x)}\right )^{-1-p} (c+d x)^{-2 (1+p)} \left (A^2-B^2 x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (-1-p,1+p,2+p,\frac {(B c-A d) (A-B x)}{2 A B (c+d x)}\right )}{(B c+A d) (1+p)} \] Output:

-2^(p+1)*((A*d+B*c)*(B*x+A)/A/B/(d*x+c))^(-1-p)*(-B^2*x^2+A^2)^(p+1)*hyper 
geom([p+1, -1-p],[2+p],1/2*(-A*d+B*c)*(-B*x+A)/A/B/(d*x+c))/(A*d+B*c)/(p+1 
)/((d*x+c)^(2*p+2))
 

Mathematica [F]

\[ \int (A+B x) (c+d x)^{-3-2 p} \left (A^2-B^2 x^2\right )^p \, dx=\int (A+B x) (c+d x)^{-3-2 p} \left (A^2-B^2 x^2\right )^p \, dx \] Input:

Integrate[(A + B*x)*(c + d*x)^(-3 - 2*p)*(A^2 - B^2*x^2)^p,x]
 

Output:

Integrate[(A + B*x)*(c + d*x)^(-3 - 2*p)*(A^2 - B^2*x^2)^p, x]
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.45, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {679, 489}

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.

\(\displaystyle \int (A+B x) \left (A^2-B^2 x^2\right )^p (c+d x)^{-2 p-3} \, dx\)

\(\Big \downarrow \) 679

\(\displaystyle \frac {A B \int (c+d x)^{-2 (p+1)} \left (A^2-B^2 x^2\right )^pdx}{A d+B c}-\frac {\left (A^2-B^2 x^2\right )^{p+1} (c+d x)^{-2 (p+1)}}{2 (p+1) (A d+B c)}\)

\(\Big \downarrow \) 489

\(\displaystyle \frac {A B (A+B x) \left (A^2-B^2 x^2\right )^p (c+d x)^{-2 p-1} \left (-\frac {(A-B x) (B c-A d)}{(A+B x) (A d+B c)}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-2 p-1,-p,-2 p,\frac {2 A B (c+d x)}{(B c+A d) (A+B x)}\right )}{(2 p+1) (B c-A d) (A d+B c)}-\frac {\left (A^2-B^2 x^2\right )^{p+1} (c+d x)^{-2 (p+1)}}{2 (p+1) (A d+B c)}\)

Input:

Int[(A + B*x)*(c + d*x)^(-3 - 2*p)*(A^2 - B^2*x^2)^p,x]
 

Output:

-1/2*(A^2 - B^2*x^2)^(1 + p)/((B*c + A*d)*(1 + p)*(c + d*x)^(2*(1 + p))) + 
 (A*B*(A + B*x)*(c + d*x)^(-1 - 2*p)*(A^2 - B^2*x^2)^p*Hypergeometric2F1[- 
1 - 2*p, -p, -2*p, (2*A*B*(c + d*x))/((B*c + A*d)*(A + B*x))])/((B*c - A*d 
)*(B*c + A*d)*(1 + 2*p)*(-(((B*c - A*d)*(A - B*x))/((B*c + A*d)*(A + B*x)) 
))^p)
 

Defintions of rubi rules used

rule 489

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{q = Rt[(-a)*b, 2]}, Simp[(q - b*x)*(c + d*x)^(n + 1)*((a + b*x^2)^p/((n + 
1)*(b*c + d*q)*((b*c + d*q)*((q + b*x)/((b*c - d*q)*(-q + b*x))))^p))*Hyper 
geometric2F1[n + 1, -p, n + 2, 2*b*q*((c + d*x)/((b*c - d*q)*(q - b*x)))], 
x]] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[n + 2*p + 2, 0]
 

rule 679

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1 
)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[(c*d*f + a*e*g)/(c*d^2 + a*e^2) 
 Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, 
 p}, x] && EqQ[Simplify[m + 2*p + 3], 0]
 

Maple [F]

\[\int \left (B x +A \right ) \left (d x +c \right )^{-3-2 p} \left (-B^{2} x^{2}+A^{2}\right )^{p}d x\]

Input:

int((B*x+A)*(d*x+c)^(-3-2*p)*(-B^2*x^2+A^2)^p,x)
 

Output:

int((B*x+A)*(d*x+c)^(-3-2*p)*(-B^2*x^2+A^2)^p,x)
 

Fricas [F]

\[ \int (A+B x) (c+d x)^{-3-2 p} \left (A^2-B^2 x^2\right )^p \, dx=\int { {\left (B x + A\right )} {\left (-B^{2} x^{2} + A^{2}\right )}^{p} {\left (d x + c\right )}^{-2 \, p - 3} \,d x } \] Input:

integrate((B*x+A)*(d*x+c)^(-3-2*p)*(-B^2*x^2+A^2)^p,x, algorithm="fricas")
 

Output:

integral((B*x + A)*(-B^2*x^2 + A^2)^p*(d*x + c)^(-2*p - 3), x)
 

Sympy [F]

\[ \int (A+B x) (c+d x)^{-3-2 p} \left (A^2-B^2 x^2\right )^p \, dx=\int \left (- \left (- A + B x\right ) \left (A + B x\right )\right )^{p} \left (A + B x\right ) \left (c + d x\right )^{- 2 p - 3}\, dx \] Input:

integrate((B*x+A)*(d*x+c)**(-3-2*p)*(-B**2*x**2+A**2)**p,x)
 

Output:

Integral((-(-A + B*x)*(A + B*x))**p*(A + B*x)*(c + d*x)**(-2*p - 3), x)
 

Maxima [F]

\[ \int (A+B x) (c+d x)^{-3-2 p} \left (A^2-B^2 x^2\right )^p \, dx=\int { {\left (B x + A\right )} {\left (-B^{2} x^{2} + A^{2}\right )}^{p} {\left (d x + c\right )}^{-2 \, p - 3} \,d x } \] Input:

integrate((B*x+A)*(d*x+c)^(-3-2*p)*(-B^2*x^2+A^2)^p,x, algorithm="maxima")
 

Output:

integrate((B*x + A)*(-B^2*x^2 + A^2)^p*(d*x + c)^(-2*p - 3), x)
 

Giac [F]

\[ \int (A+B x) (c+d x)^{-3-2 p} \left (A^2-B^2 x^2\right )^p \, dx=\int { {\left (B x + A\right )} {\left (-B^{2} x^{2} + A^{2}\right )}^{p} {\left (d x + c\right )}^{-2 \, p - 3} \,d x } \] Input:

integrate((B*x+A)*(d*x+c)^(-3-2*p)*(-B^2*x^2+A^2)^p,x, algorithm="giac")
 

Output:

integrate((B*x + A)*(-B^2*x^2 + A^2)^p*(d*x + c)^(-2*p - 3), x)
 

Mupad [F(-1)]

Timed out. \[ \int (A+B x) (c+d x)^{-3-2 p} \left (A^2-B^2 x^2\right )^p \, dx=\int \frac {\left (A+B\,x\right )\,{\left (A^2-B^2\,x^2\right )}^p}{{\left (c+d\,x\right )}^{2\,p+3}} \,d x \] Input:

int(((A + B*x)*(A^2 - B^2*x^2)^p)/(c + d*x)^(2*p + 3),x)
 

Output:

int(((A + B*x)*(A^2 - B^2*x^2)^p)/(c + d*x)^(2*p + 3), x)
 

Reduce [F]

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

int((B*x+A)*(d*x+c)^(-3-2*p)*(-B^2*x^2+A^2)^p,x)
 

Output:

int((a**2 - b**2*x**2)**p/((c + d*x)**(2*p)*c**3 + 3*(c + d*x)**(2*p)*c**2 
*d*x + 3*(c + d*x)**(2*p)*c*d**2*x**2 + (c + d*x)**(2*p)*d**3*x**3),x)*a + 
 int(((a**2 - b**2*x**2)**p*x)/((c + d*x)**(2*p)*c**3 + 3*(c + d*x)**(2*p) 
*c**2*d*x + 3*(c + d*x)**(2*p)*c*d**2*x**2 + (c + d*x)**(2*p)*d**3*x**3),x 
)*b