\(\int (b x)^m (c+d x)^n (e+f x)^p \, dx\) [591]

Optimal result

Integrand size = 20, antiderivative size = 81 \[ \int (b x)^m (c+d x)^n (e+f x)^p \, dx=\frac {(b x)^{1+m} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} (e+f x)^p \left (1+\frac {f x}{e}\right )^{-p} \operatorname {AppellF1}\left (1+m,-n,-p,2+m,-\frac {d x}{c},-\frac {f x}{e}\right )}{b (1+m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.98 \[ \int (b x)^m (c+d x)^n (e+f x)^p \, dx=\frac {x (b x)^m (c+d x)^n \left (\frac {c+d x}{c}\right )^{-n} (e+f x)^p \left (\frac {e+f x}{e}\right )^{-p} \operatorname {AppellF1}\left (1+m,-n,-p,2+m,-\frac {d x}{c},-\frac {f x}{e}\right )}{1+m} \] Input:

Integrate[(b*x)^m*(c + d*x)^n*(e + f*x)^p,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 81, 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.

\(\displaystyle \int (b x)^m (c+d x)^n (e+f x)^p \, dx\)

\(\Big \downarrow \) 152

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

\(\Big \downarrow \) 152

\(\displaystyle (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} (e+f x)^p \left (\frac {f x}{e}+1\right )^{-p} \int (b x)^m \left (\frac {d x}{c}+1\right )^n \left (\frac {f x}{e}+1\right )^pdx\)

\(\Big \downarrow \) 150

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

Input:

Int[(b*x)^m*(c + d*x)^n*(e + f*x)^p,x]
 

Output:

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

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]

\[\int \left (b x \right )^{m} \left (x d +c \right )^{n} \left (f x +e \right )^{p}d x\]

Input:

int((b*x)^m*(d*x+c)^n*(f*x+e)^p,x)
 

Output:

int((b*x)^m*(d*x+c)^n*(f*x+e)^p,x)
 

Fricas [F]

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

integrate((b*x)^m*(d*x+c)^n*(f*x+e)^p,x, algorithm="fricas")
 

Output:

integral((b*x)^m*(d*x + c)^n*(f*x + e)^p, x)
 

Sympy [F(-1)]

Timed out. \[ \int (b x)^m (c+d x)^n (e+f x)^p \, dx=\text {Timed out} \] Input:

integrate((b*x)**m*(d*x+c)**n*(f*x+e)**p,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((b*x)^m*(d*x+c)^n*(f*x+e)^p,x, algorithm="maxima")
 

Output:

integrate((b*x)^m*(d*x + c)^n*(f*x + e)^p, x)
 

Giac [F]

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

integrate((b*x)^m*(d*x+c)^n*(f*x+e)^p,x, algorithm="giac")
 

Output:

integrate((b*x)^m*(d*x + c)^n*(f*x + e)^p, x)
 

Mupad [F(-1)]

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

int((e + f*x)^p*(b*x)^m*(c + d*x)^n,x)
 

Output:

int((e + f*x)^p*(b*x)^m*(c + d*x)^n, x)
 

Reduce [F]

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

int((b*x)^m*(d*x+c)^n*(f*x+e)^p,x)
 

Output:

(b**m*(x**m*(e + f*x)**p*(c + d*x)**n*c*e*n + x**m*(e + f*x)**p*(c + d*x)* 
*n*c*e*p + x**m*(e + f*x)**p*(c + d*x)**n*c*f*m*x + x**m*(e + f*x)**p*(c + 
 d*x)**n*c*f*p*x + x**m*(e + f*x)**p*(c + d*x)**n*d*e*m*x + x**m*(e + f*x) 
**p*(c + d*x)**n*d*e*n*x + int((x**m*(e + f*x)**p*(c + d*x)**n*x)/(c**2*e* 
f*m**2 + c**2*e*f*m*n + 2*c**2*e*f*m*p + c**2*e*f*m + c**2*e*f*n*p + c**2* 
e*f*p**2 + c**2*e*f*p + c**2*f**2*m**2*x + c**2*f**2*m*n*x + 2*c**2*f**2*m 
*p*x + c**2*f**2*m*x + c**2*f**2*n*p*x + c**2*f**2*p**2*x + c**2*f**2*p*x 
+ c*d*e**2*m**2 + 2*c*d*e**2*m*n + c*d*e**2*m*p + c*d*e**2*m + c*d*e**2*n* 
*2 + c*d*e**2*n*p + c*d*e**2*n + 2*c*d*e*f*m**2*x + 3*c*d*e*f*m*n*x + 3*c* 
d*e*f*m*p*x + 2*c*d*e*f*m*x + c*d*e*f*n**2*x + 2*c*d*e*f*n*p*x + c*d*e*f*n 
*x + c*d*e*f*p**2*x + c*d*e*f*p*x + c*d*f**2*m**2*x**2 + c*d*f**2*m*n*x**2 
 + 2*c*d*f**2*m*p*x**2 + c*d*f**2*m*x**2 + c*d*f**2*n*p*x**2 + c*d*f**2*p* 
*2*x**2 + c*d*f**2*p*x**2 + d**2*e**2*m**2*x + 2*d**2*e**2*m*n*x + d**2*e* 
*2*m*p*x + d**2*e**2*m*x + d**2*e**2*n**2*x + d**2*e**2*n*p*x + d**2*e**2* 
n*x + d**2*e*f*m**2*x**2 + 2*d**2*e*f*m*n*x**2 + d**2*e*f*m*p*x**2 + d**2* 
e*f*m*x**2 + d**2*e*f*n**2*x**2 + d**2*e*f*n*p*x**2 + d**2*e*f*n*x**2),x)* 
c**3*f**3*m**3*n + int((x**m*(e + f*x)**p*(c + d*x)**n*x)/(c**2*e*f*m**2 + 
 c**2*e*f*m*n + 2*c**2*e*f*m*p + c**2*e*f*m + c**2*e*f*n*p + c**2*e*f*p**2 
 + c**2*e*f*p + c**2*f**2*m**2*x + c**2*f**2*m*n*x + 2*c**2*f**2*m*p*x + c 
**2*f**2*m*x + c**2*f**2*n*p*x + c**2*f**2*p**2*x + c**2*f**2*p*x + c*d...