\(\int (a+b x)^m (A+B x) (c+d x)^n \, dx\) [190]

Optimal result

Integrand size = 20, antiderivative size = 131 \[ \int (a+b x)^m (A+B x) (c+d x)^n \, dx=\frac {B (a+b x)^{1+m} (c+d x)^{1+n}}{b d (2+m+n)}-\frac {(b B c (1+m)+a B d (1+n)-A b d (2+m+n)) (a+b x)^{1+m} (c+d x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,2+m+n,2+m,-\frac {d (a+b x)}{b c-a d}\right )}{b d (b c-a d) (1+m) (2+m+n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.89 \[ \int (a+b x)^m (A+B x) (c+d x)^n \, dx=\frac {(a+b x)^{1+m} (c+d x)^n \left (b B (c+d x)-\frac {(b B c (1+m)+a B d (1+n)-A b d (2+m+n)) \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \operatorname {Hypergeometric2F1}\left (1+m,-n,2+m,\frac {d (a+b x)}{-b c+a d}\right )}{1+m}\right )}{b^2 d (2+m+n)} \] Input:

Integrate[(a + b*x)^m*(A + B*x)*(c + d*x)^n,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.05, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {90, 80, 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.

\(\displaystyle \int (A+B x) (a+b x)^m (c+d x)^n \, dx\)

\(\Big \downarrow \) 90

\(\displaystyle \left (A-\frac {B (a d (n+1)+b c (m+1))}{b d (m+n+2)}\right ) \int (a+b x)^m (c+d x)^ndx+\frac {B (a+b x)^{m+1} (c+d x)^{n+1}}{b d (m+n+2)}\)

\(\Big \downarrow \) 80

\(\displaystyle (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \left (A-\frac {B (a d (n+1)+b c (m+1))}{b d (m+n+2)}\right ) \int (a+b x)^m \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^ndx+\frac {B (a+b x)^{m+1} (c+d x)^{n+1}}{b d (m+n+2)}\)

\(\Big \downarrow \) 79

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

Input:

Int[(a + b*x)^m*(A + B*x)*(c + d*x)^n,x]
 

Output:

(B*(a + b*x)^(1 + m)*(c + d*x)^(1 + n))/(b*d*(2 + m + n)) + ((A - (B*(b*c* 
(1 + m) + a*d*(1 + n)))/(b*d*(2 + m + n)))*(a + b*x)^(1 + m)*(c + d*x)^n*H 
ypergeometric2F1[1 + m, -n, 2 + m, -((d*(a + b*x))/(b*c - a*d))])/(b*(1 + 
m)*((b*(c + d*x))/(b*c - a*d))^n)
 

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 80

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c 
 + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) 
^FracPart[n])   Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) 
), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Integ 
erQ[n] && (RationalQ[m] ||  !SimplerQ[n + 1, m + 1])
 

rule 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

Maple [F]

\[\int \left (b x +a \right )^{m} \left (B x +A \right ) \left (x d +c \right )^{n}d x\]

Input:

int((b*x+a)^m*(B*x+A)*(d*x+c)^n,x)
 

Output:

int((b*x+a)^m*(B*x+A)*(d*x+c)^n,x)
 

Fricas [F]

\[ \int (a+b x)^m (A+B x) (c+d x)^n \, dx=\int { {\left (B x + A\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n} \,d x } \] Input:

integrate((b*x+a)^m*(B*x+A)*(d*x+c)^n,x, algorithm="fricas")
 

Output:

integral((B*x + A)*(b*x + a)^m*(d*x + c)^n, x)
 

Sympy [F(-2)]

Exception generated. \[ \int (a+b x)^m (A+B x) (c+d x)^n \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

integrate((b*x+a)**m*(B*x+A)*(d*x+c)**n,x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

\[ \int (a+b x)^m (A+B x) (c+d x)^n \, dx=\int { {\left (B x + A\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n} \,d x } \] Input:

integrate((b*x+a)^m*(B*x+A)*(d*x+c)^n,x, algorithm="maxima")
 

Output:

integrate((B*x + A)*(b*x + a)^m*(d*x + c)^n, x)
 

Giac [F]

\[ \int (a+b x)^m (A+B x) (c+d x)^n \, dx=\int { {\left (B x + A\right )} {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n} \,d x } \] Input:

integrate((b*x+a)^m*(B*x+A)*(d*x+c)^n,x, algorithm="giac")
 

Output:

integrate((B*x + A)*(b*x + a)^m*(d*x + c)^n, x)
 

Mupad [F(-1)]

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

int((A + B*x)*(a + b*x)^m*(c + d*x)^n,x)
 

Output:

int((A + B*x)*(a + b*x)^m*(c + d*x)^n, x)
 

Reduce [F]

\[ \int (a+b x)^m (A+B x) (c+d x)^n \, dx=\text {too large to display} \] Input:

int((b*x+a)^m*(B*x+A)*(d*x+c)^n,x)
 

Output:

((c + d*x)**n*(a + b*x)**m*a**2*c*d*m**2 + 2*(c + d*x)**n*(a + b*x)**m*a** 
2*c*d*m*n + (c + d*x)**n*(a + b*x)**m*a**2*c*d*m + (c + d*x)**n*(a + b*x)* 
*m*a**2*c*d*n**2 + 2*(c + d*x)**n*(a + b*x)**m*a**2*c*d*n + 2*(c + d*x)**n 
*(a + b*x)**m*a**2*d**2*m*n*x + (c + d*x)**n*(a + b*x)**m*a**2*d**2*n**2*x 
 + 2*(c + d*x)**n*(a + b*x)**m*a**2*d**2*n*x - (c + d*x)**n*(a + b*x)**m*a 
*b*c**2*n + 2*(c + d*x)**n*(a + b*x)**m*a*b*c*d*m**2*x + (c + d*x)**n*(a + 
 b*x)**m*a*b*c*d*m*n*x + 2*(c + d*x)**n*(a + b*x)**m*a*b*c*d*m*x + (c + d* 
x)**n*(a + b*x)**m*a*b*c*d*n**2*x + (c + d*x)**n*(a + b*x)**m*a*b*d**2*m*n 
*x**2 + (c + d*x)**n*(a + b*x)**m*a*b*d**2*n**2*x**2 + (c + d*x)**n*(a + b 
*x)**m*a*b*d**2*n*x**2 + (c + d*x)**n*(a + b*x)**m*b**2*c**2*m*n*x + (c + 
d*x)**n*(a + b*x)**m*b**2*c*d*m**2*x**2 + (c + d*x)**n*(a + b*x)**m*b**2*c 
*d*m*n*x**2 + (c + d*x)**n*(a + b*x)**m*b**2*c*d*m*x**2 + int(((c + d*x)** 
n*(a + b*x)**m*x)/(a**2*c*d*m**2*n + 2*a**2*c*d*m*n**2 + 3*a**2*c*d*m*n + 
a**2*c*d*n**3 + 3*a**2*c*d*n**2 + 2*a**2*c*d*n + a**2*d**2*m**2*n*x + 2*a* 
*2*d**2*m*n**2*x + 3*a**2*d**2*m*n*x + a**2*d**2*n**3*x + 3*a**2*d**2*n**2 
*x + 2*a**2*d**2*n*x + a*b*c**2*m**3 + 2*a*b*c**2*m**2*n + 3*a*b*c**2*m**2 
 + a*b*c**2*m*n**2 + 3*a*b*c**2*m*n + 2*a*b*c**2*m + a*b*c*d*m**3*x + 3*a* 
b*c*d*m**2*n*x + 3*a*b*c*d*m**2*x + 3*a*b*c*d*m*n**2*x + 6*a*b*c*d*m*n*x + 
 2*a*b*c*d*m*x + a*b*c*d*n**3*x + 3*a*b*c*d*n**2*x + 2*a*b*c*d*n*x + a*b*d 
**2*m**2*n*x**2 + 2*a*b*d**2*m*n**2*x**2 + 3*a*b*d**2*m*n*x**2 + a*b*d*...