\(\int x^m (a+b x)^{1+n} (c+d x)^n \, dx\) [600]

Optimal result

Integrand size = 20, antiderivative size = 79 \[ \int x^m (a+b x)^{1+n} (c+d x)^n \, dx=\frac {a x^{1+m} (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \operatorname {AppellF1}\left (1+m,-1-n,-n,2+m,-\frac {b x}{a},-\frac {d x}{c}\right )}{1+m} \] Output:

a*x^(1+m)*(b*x+a)^n*(d*x+c)^n*AppellF1(1+m,-1-n,-n,2+m,-b*x/a,-d*x/c)/(1+m 
)/((1+b*x/a)^n)/((1+d*x/c)^n)
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.52 \[ \int x^m (a+b x)^{1+n} (c+d x)^n \, dx=\frac {x^{1+m} (a+b x)^n \left (1+\frac {b x}{a}\right )^{-n} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \left (a (2+m) \operatorname {AppellF1}\left (1+m,-n,-n,2+m,-\frac {b x}{a},-\frac {d x}{c}\right )+b (1+m) x \operatorname {AppellF1}\left (2+m,-n,-n,3+m,-\frac {b x}{a},-\frac {d x}{c}\right )\right )}{(1+m) (2+m)} \] Input:

Integrate[x^m*(a + b*x)^(1 + n)*(c + d*x)^n,x]
 

Output:

(x^(1 + m)*(a + b*x)^n*(c + d*x)^n*(a*(2 + m)*AppellF1[1 + m, -n, -n, 2 + 
m, -((b*x)/a), -((d*x)/c)] + b*(1 + m)*x*AppellF1[2 + m, -n, -n, 3 + m, -( 
(b*x)/a), -((d*x)/c)]))/((1 + m)*(2 + m)*(1 + (b*x)/a)^n*(1 + (d*x)/c)^n)
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {152, 152, 150}

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.

\(\displaystyle \int x^m (a+b x)^{n+1} (c+d x)^n \, dx\)

\(\Big \downarrow \) 152

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

\(\Big \downarrow \) 152

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

\(\Big \downarrow \) 150

\(\displaystyle \frac {a x^{m+1} (a+b x)^n \left (\frac {b x}{a}+1\right )^{-n} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \operatorname {AppellF1}\left (m+1,-n-1,-n,m+2,-\frac {b x}{a},-\frac {d x}{c}\right )}{m+1}\)

Input:

Int[x^m*(a + b*x)^(1 + n)*(c + d*x)^n,x]
 

Output:

(a*x^(1 + m)*(a + b*x)^n*(c + d*x)^n*AppellF1[1 + m, -1 - n, -n, 2 + m, -( 
(b*x)/a), -((d*x)/c)])/((1 + m)*(1 + (b*x)/a)^n*(1 + (d*x)/c)^n)
 

Defintions of rubi rules used

rule 150

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 
, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] &&  !In 
tegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
 

rule 152

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^IntPart[n]*((c + d*x)^FracPart[n]/(1 + d*(x/c))^FracPart[n]) 
Int[(b*x)^m*(1 + d*(x/c))^n*(e + f*x)^p, x], x] /; FreeQ[{b, c, d, e, f, m, 
 n, p}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] &&  !GtQ[c, 0]
 

Maple [F]

\[\int x^{m} \left (b x +a \right )^{1+n} \left (x d +c \right )^{n}d x\]

Input:

int(x^m*(b*x+a)^(1+n)*(d*x+c)^n,x)
 

Output:

int(x^m*(b*x+a)^(1+n)*(d*x+c)^n,x)
 

Fricas [F]

\[ \int x^m (a+b x)^{1+n} (c+d x)^n \, dx=\int { {\left (b x + a\right )}^{n + 1} {\left (d x + c\right )}^{n} x^{m} \,d x } \] Input:

integrate(x^m*(b*x+a)^(1+n)*(d*x+c)^n,x, algorithm="fricas")
 

Output:

integral((b*x + a)^(n + 1)*(d*x + c)^n*x^m, x)
 

Sympy [F(-2)]

Exception generated. \[ \int x^m (a+b x)^{1+n} (c+d x)^n \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

integrate(x**m*(b*x+a)**(1+n)*(d*x+c)**n,x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

\[ \int x^m (a+b x)^{1+n} (c+d x)^n \, dx=\int { {\left (b x + a\right )}^{n + 1} {\left (d x + c\right )}^{n} x^{m} \,d x } \] Input:

integrate(x^m*(b*x+a)^(1+n)*(d*x+c)^n,x, algorithm="maxima")
 

Output:

integrate((b*x + a)^(n + 1)*(d*x + c)^n*x^m, x)
 

Giac [F]

\[ \int x^m (a+b x)^{1+n} (c+d x)^n \, dx=\int { {\left (b x + a\right )}^{n + 1} {\left (d x + c\right )}^{n} x^{m} \,d x } \] Input:

integrate(x^m*(b*x+a)^(1+n)*(d*x+c)^n,x, algorithm="giac")
 

Output:

integrate((b*x + a)^(n + 1)*(d*x + c)^n*x^m, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^m (a+b x)^{1+n} (c+d x)^n \, dx=\int x^m\,{\left (a+b\,x\right )}^{n+1}\,{\left (c+d\,x\right )}^n \,d x \] Input:

int(x^m*(a + b*x)^(n + 1)*(c + d*x)^n,x)
 

Output:

int(x^m*(a + b*x)^(n + 1)*(c + d*x)^n, x)
 

Reduce [F]

\[ \int x^m (a+b x)^{1+n} (c+d x)^n \, dx=\text {too large to display} \] Input:

int(x^m*(b*x+a)^(1+n)*(d*x+c)^n,x)
 

Output:

(x**m*(c + d*x)**n*(a + b*x)**n*a**2*c*d*m*n + 4*x**m*(c + d*x)**n*(a + b* 
x)**n*a**2*c*d*n**2 + 3*x**m*(c + d*x)**n*(a + b*x)**n*a**2*c*d*n + x**m*( 
c + d*x)**n*(a + b*x)**n*a**2*d**2*m**2*x + 4*x**m*(c + d*x)**n*(a + b*x)* 
*n*a**2*d**2*m*n*x + 2*x**m*(c + d*x)**n*(a + b*x)**n*a**2*d**2*m*x + 3*x* 
*m*(c + d*x)**n*(a + b*x)**n*a**2*d**2*n**2*x + 2*x**m*(c + d*x)**n*(a + b 
*x)**n*a**2*d**2*n*x - x**m*(c + d*x)**n*(a + b*x)**n*a*b*c**2*m*n - x**m* 
(c + d*x)**n*(a + b*x)**n*a*b*c**2*n + x**m*(c + d*x)**n*(a + b*x)**n*a*b* 
c*d*m**2*x + 5*x**m*(c + d*x)**n*(a + b*x)**n*a*b*c*d*m*n*x + 2*x**m*(c + 
d*x)**n*(a + b*x)**n*a*b*c*d*m*x + 4*x**m*(c + d*x)**n*(a + b*x)**n*a*b*c* 
d*n**2*x + 2*x**m*(c + d*x)**n*(a + b*x)**n*a*b*c*d*n*x + x**m*(c + d*x)** 
n*(a + b*x)**n*a*b*d**2*m**2*x**2 + 3*x**m*(c + d*x)**n*(a + b*x)**n*a*b*d 
**2*m*n*x**2 + x**m*(c + d*x)**n*(a + b*x)**n*a*b*d**2*m*x**2 + 2*x**m*(c 
+ d*x)**n*(a + b*x)**n*a*b*d**2*n**2*x**2 + x**m*(c + d*x)**n*(a + b*x)**n 
*a*b*d**2*n*x**2 + x**m*(c + d*x)**n*(a + b*x)**n*b**2*c**2*m*n*x + x**m*( 
c + d*x)**n*(a + b*x)**n*b**2*c**2*n**2*x + x**m*(c + d*x)**n*(a + b*x)**n 
*b**2*c*d*m**2*x**2 + 3*x**m*(c + d*x)**n*(a + b*x)**n*b**2*c*d*m*n*x**2 + 
 x**m*(c + d*x)**n*(a + b*x)**n*b**2*c*d*m*x**2 + 2*x**m*(c + d*x)**n*(a + 
 b*x)**n*b**2*c*d*n**2*x**2 + x**m*(c + d*x)**n*(a + b*x)**n*b**2*c*d*n*x* 
*2 + int((x**m*(c + d*x)**n*(a + b*x)**n*x)/(a**2*c*d*m**3 + 5*a**2*c*d*m* 
*2*n + 3*a**2*c*d*m**2 + 8*a**2*c*d*m*n**2 + 9*a**2*c*d*m*n + 2*a**2*c*...