\(\int \frac {(b x)^m (c+d x)^n}{(e+f x)^2} \, dx\) [574]

Optimal result

Integrand size = 20, antiderivative size = 63 \[ \int \frac {(b x)^m (c+d x)^n}{(e+f x)^2} \, dx=\frac {(b x)^{1+m} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \operatorname {AppellF1}\left (1+m,-n,2,2+m,-\frac {d x}{c},-\frac {f x}{e}\right )}{b e^2 (1+m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.95 \[ \int \frac {(b x)^m (c+d x)^n}{(e+f x)^2} \, dx=\frac {x (b x)^m (c+d x)^n \left (\frac {c+d x}{c}\right )^{-n} \operatorname {AppellF1}\left (1+m,-n,2,2+m,-\frac {d x}{c},-\frac {f x}{e}\right )}{e^2 (1+m)} \] Input:

Integrate[((b*x)^m*(c + d*x)^n)/(e + f*x)^2,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {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.

\(\displaystyle \int \frac {(b x)^m (c+d x)^n}{(e+f x)^2} \, dx\)

\(\Big \downarrow \) 152

\(\displaystyle (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \int \frac {(b x)^m \left (\frac {d x}{c}+1\right )^n}{(e+f x)^2}dx\)

\(\Big \downarrow \) 150

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

Input:

Int[((b*x)^m*(c + d*x)^n)/(e + f*x)^2,x]
 

Output:

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

\[\int \frac {\left (b x \right )^{m} \left (x d +c \right )^{n}}{\left (f x +e \right )^{2}}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {(b x)^m (c+d x)^n}{(e+f x)^2} \, dx=\int { \frac {\left (b x\right )^{m} {\left (d x + c\right )}^{n}}{{\left (f x + e\right )}^{2}} \,d x } \] Input:

integrate((b*x)^m*(d*x+c)^n/(f*x+e)^2,x, algorithm="fricas")
 

Output:

integral((b*x)^m*(d*x + c)^n/(f^2*x^2 + 2*e*f*x + e^2), x)
 

Sympy [F]

\[ \int \frac {(b x)^m (c+d x)^n}{(e+f x)^2} \, dx=\int \frac {\left (b x\right )^{m} \left (c + d x\right )^{n}}{\left (e + f x\right )^{2}}\, dx \] Input:

integrate((b*x)**m*(d*x+c)**n/(f*x+e)**2,x)
 

Output:

Integral((b*x)**m*(c + d*x)**n/(e + f*x)**2, x)
 

Maxima [F]

\[ \int \frac {(b x)^m (c+d x)^n}{(e+f x)^2} \, dx=\int { \frac {\left (b x\right )^{m} {\left (d x + c\right )}^{n}}{{\left (f x + e\right )}^{2}} \,d x } \] Input:

integrate((b*x)^m*(d*x+c)^n/(f*x+e)^2,x, algorithm="maxima")
 

Output:

integrate((b*x)^m*(d*x + c)^n/(f*x + e)^2, x)
 

Giac [F]

\[ \int \frac {(b x)^m (c+d x)^n}{(e+f x)^2} \, dx=\int { \frac {\left (b x\right )^{m} {\left (d x + c\right )}^{n}}{{\left (f x + e\right )}^{2}} \,d x } \] Input:

integrate((b*x)^m*(d*x+c)^n/(f*x+e)^2,x, algorithm="giac")
 

Output:

integrate((b*x)^m*(d*x + c)^n/(f*x + e)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(b x)^m (c+d x)^n}{(e+f x)^2} \, dx=\int \frac {{\left (b\,x\right )}^m\,{\left (c+d\,x\right )}^n}{{\left (e+f\,x\right )}^2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {(b x)^m (c+d x)^n}{(e+f x)^2} \, dx=\text {too large to display} \] Input:

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

Output:

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