3.31.85 \(\int \frac {(a+b x)^m (c+d x)^{-2-m}}{(e+f x)^2} \, dx\) [3085]

3.31.85.1 Optimal result

Integrand size = 26, antiderivative size = 233 \[ \int \frac {(a+b x)^m (c+d x)^{-2-m}}{(e+f x)^2} \, dx=-\frac {d (a d f (2+m)-b (d e+c f (1+m))) (a+b x)^{1+m} (c+d x)^{-1-m}}{(b c-a d) (b e-a f) (d e-c f)^2 (1+m)}-\frac {f (a+b x)^{1+m} (c+d x)^{-1-m}}{(b e-a f) (d e-c f) (e+f x)}-\frac {f (a d f (2+m)-b (2 d e+c f m)) (a+b x)^m (c+d x)^{-m} \operatorname {Hypergeometric2F1}\left (1,-m,1-m,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{(b e-a f) (d e-c f)^3 m} \]

output

-d*(a*d*f*(2+m)-b*(d*e+c*f*(1+m)))*(b*x+a)^(1+m)*(d*x+c)^(-1-m)/(-a*d+b*c) 
/(-a*f+b*e)/(-c*f+d*e)^2/(1+m)-f*(b*x+a)^(1+m)*(d*x+c)^(-1-m)/(-a*f+b*e)/( 
-c*f+d*e)/(f*x+e)-f*(a*d*f*(2+m)-b*(c*f*m+2*d*e))*(b*x+a)^m*hypergeom([1, 
-m],[1-m],(-a*f+b*e)*(d*x+c)/(-c*f+d*e)/(b*x+a))/(-a*f+b*e)/(-c*f+d*e)^3/m 
/((d*x+c)^m)
 

3.31.85.2 Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b x)^m (c+d x)^{-2-m}}{(e+f x)^2} \, dx=\frac {(a+b x)^{1+m} (c+d x)^{-1-m} \left (d (b e-a f)^2 (d e-c f)+f (b e-a f) (b d e+b c f (1+m)-a d f (2+m)) (c+d x)+(b c-a d) f (a d f (2+m)-b (2 d e+c f m)) (e+f x) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )\right )}{(b c-a d) (b e-a f)^2 (d e-c f)^2 (1+m) (e+f x)} \]

input

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

output

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

3.31.85.3 Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {114, 172, 25, 27, 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.

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 172

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 141

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

input

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

output

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

3.31.85.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 114

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[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*Simp[a*d*f*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
 

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]
 

rule 172

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> With[{mnp = Simplify[m + n + p]}, Simp[ 
(b*g - a*h)*(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[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*Simp[(a*d*f*g - b*(d*e + c*f 
)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g 
 - a*h)*(mnp + 3)*x, x], x], x] /; ILtQ[mnp + 2, 0] && (SumSimplerQ[m, 1] | 
| ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) &&  !(NeQ[p, -1] && SumSimplerQ[p, 1 
])))] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && NeQ[m, -1]
 

3.31.85.4 Maple [F]

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

input

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

output

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

3.31.85.5 Fricas [F]

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

input

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

output

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

3.31.85.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.31.85.7 Maxima [F]

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

input

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

output

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

3.31.85.8 Giac [F]

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

input

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

output

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

3.31.85.9 Mupad [F(-1)]

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

input

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

output

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