\(\int \frac {(a+b x)^m (e+f x)^{-1-m}}{(c+d x)^2} \, dx\) [1802]

Optimal result

Integrand size = 26, antiderivative size = 149 \[ \int \frac {(a+b x)^m (e+f x)^{-1-m}}{(c+d x)^2} \, dx=\frac {d (a+b x)^{1+m} (e+f x)^{-m}}{(b c-a d) (d e-c f) (c+d x)}+\frac {(a d f (1+m)-b (c f+d e m)) (a+b x)^m (e+f x)^{-m} \operatorname {Hypergeometric2F1}\left (1,-m,1-m,-\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}\right )}{(b c-a d) (d e-c f)^2 m} \] Output:

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b x)^m (e+f x)^{-1-m}}{(c+d x)^2} \, dx=\frac {(a+b x)^{1+m} (e+f x)^{-m} \left (-\frac {d}{c+d x}+\frac {(-a d f (1+m)+b (c f+d e m)) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {(-d e+c f) (a+b x)}{(b c-a d) (e+f x)}\right )}{(b c-a d) (1+m) (e+f x)}\right )}{(b c-a d) (-d e+c f)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {107, 141}

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.

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

\(\Big \downarrow \) 107

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

\(\Big \downarrow \) 141

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 107

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[(a*d*f*(m + 1) + b*c*f*(n + 
 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 
 1)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x 
] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 1])
 

rule 141

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(b*c - a*d)^n*((a + b*x)^(m + 1)/((m + 1)*(b*e - a*f)^( 
n + 1)*(e + f*x)^(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2, (-(d*e - c*f 
))*((a + b*x)/((b*c - a*d)*(e + f*x)))], x] /; FreeQ[{a, b, c, d, e, f, m, 
p}, x] && EqQ[m + n + p + 2, 0] && ILtQ[n, 0] && (SumSimplerQ[m, 1] ||  !Su 
mSimplerQ[p, 1]) &&  !ILtQ[m, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(a+b x)^m (e+f x)^{-1-m}}{(c+d x)^2} \, dx=\int \frac {\left (b x +a \right )^{m}}{\left (f x +e \right )^{m} c^{2} e +\left (f x +e \right )^{m} c^{2} f x +2 \left (f x +e \right )^{m} c d e x +2 \left (f x +e \right )^{m} c d f \,x^{2}+\left (f x +e \right )^{m} d^{2} e \,x^{2}+\left (f x +e \right )^{m} d^{2} f \,x^{3}}d x \] Input:

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

Output:

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