\(\int \frac {(c+d x)^{-1+m} (e+f x)}{(a+b x)^2} \, dx\) [245]

Optimal result

Integrand size = 22, antiderivative size = 107 \[ \int \frac {(c+d x)^{-1+m} (e+f x)}{(a+b x)^2} \, dx=-\frac {f (c+d x)^m}{b d (1-m) (a+b x)}-\frac {(b c f-b d e (1-m)-a d f m) (c+d x)^m \operatorname {Hypergeometric2F1}\left (2,m,1+m,\frac {b (c+d x)}{b c-a d}\right )}{b (b c-a d)^2 (1-m) m} \] Output:

-f*(d*x+c)^m/b/d/(1-m)/(b*x+a)-(b*c*f-b*d*e*(1-m)-a*d*f*m)*(d*x+c)^m*hyper 
geom([2, m],[1+m],b*(d*x+c)/(-a*d+b*c))/b/(-a*d+b*c)^2/(1-m)/m
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.87 \[ \int \frac {(c+d x)^{-1+m} (e+f x)}{(a+b x)^2} \, dx=\frac {(c+d x)^m \left (-\frac {(b c-a d) (b e-a f)}{a+b x}-\frac {(b c f+b d e (-1+m)-a d f m) \operatorname {Hypergeometric2F1}\left (1,m,1+m,\frac {b (c+d x)}{b c-a d}\right )}{m}\right )}{b (b c-a d)^2} \] Input:

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

Output:

((c + d*x)^m*(-(((b*c - a*d)*(b*e - a*f))/(a + b*x)) - ((b*c*f + b*d*e*(-1 
 + m) - a*d*f*m)*Hypergeometric2F1[1, m, 1 + m, (b*(c + d*x))/(b*c - a*d)] 
)/m))/(b*(b*c - a*d)^2)
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {87, 78}

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.

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

\(\Big \downarrow \) 87

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

\(\Big \downarrow \) 78

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

Input:

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

Output:

-(((b*e - a*f)*(c + d*x)^m)/(b*(b*c - a*d)*(a + b*x))) - ((b*c*f - b*d*e*( 
1 - m) - a*d*f*m)*(c + d*x)^m*Hypergeometric2F1[1, m, 1 + m, (b*(c + d*x)) 
/(b*c - a*d)])/(b*(b*c - a*d)^2*m)
 

Defintions of rubi rules used

rule 78

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b 
*c - a*d)^n*((a + b*x)^(m + 1)/(b^(n + 1)*(m + 1)))*Hypergeometric2F1[-n, m 
 + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m}, x] 
 &&  !IntegerQ[m] && IntegerQ[n]
 

rule 87

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

Maple [F]

\[\int \frac {\left (x d +c \right )^{m -1} \left (f x +e \right )}{\left (b x +a \right )^{2}}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {(c+d x)^{-1+m} (e+f x)}{(a+b x)^2} \, dx=\int { \frac {{\left (f x + e\right )} {\left (d x + c\right )}^{m - 1}}{{\left (b x + a\right )}^{2}} \,d x } \] Input:

integrate((d*x+c)^(-1+m)*(f*x+e)/(b*x+a)^2,x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {(c+d x)^{-1+m} (e+f x)}{(a+b x)^2} \, dx=\int \frac {\left (c + d x\right )^{m - 1} \left (e + f x\right )}{\left (a + b x\right )^{2}}\, dx \] Input:

integrate((d*x+c)**(-1+m)*(f*x+e)/(b*x+a)**2,x)
 

Output:

Integral((c + d*x)**(m - 1)*(e + f*x)/(a + b*x)**2, x)
 

Maxima [F]

\[ \int \frac {(c+d x)^{-1+m} (e+f x)}{(a+b x)^2} \, dx=\int { \frac {{\left (f x + e\right )} {\left (d x + c\right )}^{m - 1}}{{\left (b x + a\right )}^{2}} \,d x } \] Input:

integrate((d*x+c)^(-1+m)*(f*x+e)/(b*x+a)^2,x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {(c+d x)^{-1+m} (e+f x)}{(a+b x)^2} \, dx=\int { \frac {{\left (f x + e\right )} {\left (d x + c\right )}^{m - 1}}{{\left (b x + a\right )}^{2}} \,d x } \] Input:

integrate((d*x+c)^(-1+m)*(f*x+e)/(b*x+a)^2,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{-1+m} (e+f x)}{(a+b x)^2} \, dx=\int \frac {\left (e+f\,x\right )\,{\left (c+d\,x\right )}^{m-1}}{{\left (a+b\,x\right )}^2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {(c+d x)^{-1+m} (e+f x)}{(a+b x)^2} \, dx=\text {too large to display} \] Input:

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

Output:

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