\(\int (A+B x) (c+d x)^n (c^2-d^2 x^2)^p \, dx\) [250]

Optimal result

Integrand size = 27, antiderivative size = 145 \[ \int (A+B x) (c+d x)^n \left (c^2-d^2 x^2\right )^p \, dx=-\frac {B (c+d x)^n \left (c^2-d^2 x^2\right )^{1+p}}{d^2 (2+n+2 p)}-\frac {2^{n+p} (B c n+A d (2+n+2 p)) (c+d x)^n \left (1+\frac {d x}{c}\right )^{-1-n-p} \left (c^2-d^2 x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (-n-p,1+p,2+p,\frac {c-d x}{2 c}\right )}{c d^2 (1+p) (2+n+2 p)} \] Output:

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.90 \[ \int (A+B x) (c+d x)^n \left (c^2-d^2 x^2\right )^p \, dx=\frac {(-c+d x) (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n-p} \left (c^2-d^2 x^2\right )^p \left (B (1+p) (c+d x) \left (1+\frac {d x}{c}\right )^{n+p}+2^{n+p} (B c n+A d (2+n+2 p)) \operatorname {Hypergeometric2F1}\left (-n-p,1+p,2+p,\frac {c-d x}{2 c}\right )\right )}{d^2 (1+p) (2+n+2 p)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {672, 474, 473, 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.

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

\(\Big \downarrow \) 672

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

\(\Big \downarrow \) 474

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

\(\Big \downarrow \) 473

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

\(\Big \downarrow \) 79

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

Input:

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

Output:

-((B*(c + d*x)^n*(c^2 - d^2*x^2)^(1 + p))/(d^2*(2 + n + 2*p))) - (2^(n + p 
)*(A + (B*c*n)/(d*(2 + n + 2*p)))*(c + d*x)^n*(1 + (d*x)/c)^(-1 - n - p)*( 
c^2 - d^2*x^2)^(1 + p)*Hypergeometric2F1[-n - p, 1 + p, 2 + p, (c - d*x)/( 
2*c)])/(c*d*(1 + 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 473

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
c^(n - 1)*((a + b*x^2)^(p + 1)/((1 + d*(x/c))^(p + 1)*(a/c + (b*x)/d)^(p + 
1)))   Int[(1 + d*(x/c))^(n + p)*(a/c + (b/d)*x)^p, x], x] /; FreeQ[{a, b, 
c, d, n}, x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[n] || GtQ[c, 0]) &&  !Gt 
Q[a, 0] &&  !(IntegerQ[n] && (IntegerQ[3*p] || IntegerQ[4*p]))
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (A+B x) (c+d x)^n \left (c^2-d^2 x^2\right )^p \, dx=\text {too large to display} \] Input:

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

Output:

((c + d*x)**n*(c**2 - d**2*x**2)**p*a*c*d*n**2 + 4*(c + d*x)**n*(c**2 - d* 
*2*x**2)**p*a*c*d*n*p + 2*(c + d*x)**n*(c**2 - d**2*x**2)**p*a*c*d*n + 4*( 
c + d*x)**n*(c**2 - d**2*x**2)**p*a*c*d*p**2 + 4*(c + d*x)**n*(c**2 - d**2 
*x**2)**p*a*c*d*p + (c + d*x)**n*(c**2 - d**2*x**2)**p*a*d**2*n**2*x + 2*( 
c + d*x)**n*(c**2 - d**2*x**2)**p*a*d**2*n*p*x + 2*(c + d*x)**n*(c**2 - d* 
*2*x**2)**p*a*d**2*n*x - (c + d*x)**n*(c**2 - d**2*x**2)**p*b*c**2*n + (c 
+ d*x)**n*(c**2 - d**2*x**2)**p*b*c*d*n**2*x + (c + d*x)**n*(c**2 - d**2*x 
**2)**p*b*d**2*n**2*x**2 + 2*(c + d*x)**n*(c**2 - d**2*x**2)**p*b*d**2*n*p 
*x**2 + (c + d*x)**n*(c**2 - d**2*x**2)**p*b*d**2*n*x**2 + 4*int(((c + d*x 
)**n*(c**2 - d**2*x**2)**p*x)/(c**2*n**2 + 4*c**2*n*p + 3*c**2*n + 4*c**2* 
p**2 + 6*c**2*p + 2*c**2 - d**2*n**2*x**2 - 4*d**2*n*p*x**2 - 3*d**2*n*x** 
2 - 4*d**2*p**2*x**2 - 6*d**2*p*x**2 - 2*d**2*x**2),x)*a*c*d**3*n**4*p + 2 
8*int(((c + d*x)**n*(c**2 - d**2*x**2)**p*x)/(c**2*n**2 + 4*c**2*n*p + 3*c 
**2*n + 4*c**2*p**2 + 6*c**2*p + 2*c**2 - d**2*n**2*x**2 - 4*d**2*n*p*x**2 
 - 3*d**2*n*x**2 - 4*d**2*p**2*x**2 - 6*d**2*p*x**2 - 2*d**2*x**2),x)*a*c* 
d**3*n**3*p**2 + 20*int(((c + d*x)**n*(c**2 - d**2*x**2)**p*x)/(c**2*n**2 
+ 4*c**2*n*p + 3*c**2*n + 4*c**2*p**2 + 6*c**2*p + 2*c**2 - d**2*n**2*x**2 
 - 4*d**2*n*p*x**2 - 3*d**2*n*x**2 - 4*d**2*p**2*x**2 - 6*d**2*p*x**2 - 2* 
d**2*x**2),x)*a*c*d**3*n**3*p + 72*int(((c + d*x)**n*(c**2 - d**2*x**2)**p 
*x)/(c**2*n**2 + 4*c**2*n*p + 3*c**2*n + 4*c**2*p**2 + 6*c**2*p + 2*c**...