\(\int \frac {(a+b x)^m (c+d x)^{-5-m}}{e+f x} \, dx\) [1756]

Optimal result

Integrand size = 26, antiderivative size = 556 \[ \int \frac {(a+b x)^m (c+d x)^{-5-m}}{e+f x} \, dx=\frac {d (a+b x)^{1+m} (c+d x)^{-4-m}}{(b c-a d) (d e-c f) (4+m)}+\frac {d (3 b d e+a d f (4+m)-b c f (7+m)) (a+b x)^{1+m} (c+d x)^{-3-m}}{(b c-a d)^2 (d e-c f)^2 (3+m) (4+m)}+\frac {d \left (a^2 d^2 f^2 \left (12+7 m+m^2\right )+2 a b d f (4+m) (d e-c f (4+m))+b^2 \left (6 d^2 e^2-2 c d e f (10+m)+c^2 f^2 \left (26+9 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-2-m}}{(b c-a d)^3 (d e-c f)^3 (2+m) (3+m) (4+m)}+\frac {d \left (a^3 d^3 f^3 \left (24+26 m+9 m^2+m^3\right )+a^2 b d^2 f^2 \left (12+7 m+m^2\right ) (d e-c f (7+3 m))+a b^2 d f (4+m) \left (2 d^2 e^2-2 c d e f (5+m)+c^2 f^2 \left (26+17 m+3 m^2\right )\right )+b^3 \left (6 d^3 e^3-2 c d^2 e^2 f (13+m)+c^2 d e f^2 \left (46+11 m+m^2\right )-c^3 f^3 \left (50+35 m+10 m^2+m^3\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-1-m}}{(b c-a d)^4 (d e-c f)^4 (1+m) (2+m) (3+m) (4+m)}-\frac {f^4 (a+b x)^m (c+d x)^{-m} \operatorname {Hypergeometric2F1}\left (1,-m,1-m,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{(d e-c f)^5 m} \] Output:

d*(b*x+a)^(1+m)*(d*x+c)^(-4-m)/(-a*d+b*c)/(-c*f+d*e)/(4+m)+d*(3*b*d*e+a*d* 
f*(4+m)-b*c*f*(7+m))*(b*x+a)^(1+m)*(d*x+c)^(-3-m)/(-a*d+b*c)^2/(-c*f+d*e)^ 
2/(3+m)/(4+m)+d*(a^2*d^2*f^2*(m^2+7*m+12)+2*a*b*d*f*(4+m)*(d*e-c*f*(4+m))+ 
b^2*(6*d^2*e^2-2*c*d*e*f*(10+m)+c^2*f^2*(m^2+9*m+26)))*(b*x+a)^(1+m)*(d*x+ 
c)^(-2-m)/(-a*d+b*c)^3/(-c*f+d*e)^3/(2+m)/(3+m)/(4+m)+d*(a^3*d^3*f^3*(m^3+ 
9*m^2+26*m+24)+a^2*b*d^2*f^2*(m^2+7*m+12)*(d*e-c*f*(7+3*m))+a*b^2*d*f*(4+m 
)*(2*d^2*e^2-2*c*d*e*f*(5+m)+c^2*f^2*(3*m^2+17*m+26))+b^3*(6*d^3*e^3-2*c*d 
^2*e^2*f*(13+m)+c^2*d*e*f^2*(m^2+11*m+46)-c^3*f^3*(m^3+10*m^2+35*m+50)))*( 
b*x+a)^(1+m)*(d*x+c)^(-1-m)/(-a*d+b*c)^4/(-c*f+d*e)^4/(1+m)/(2+m)/(3+m)/(4 
+m)-f^4*(b*x+a)^m*hypergeom([1, -m],[1-m],(-a*f+b*e)*(d*x+c)/(-c*f+d*e)/(b 
*x+a))/(-c*f+d*e)^5/m/((d*x+c)^m)
 

Mathematica [A] (verified)

Time = 1.24 (sec) , antiderivative size = 525, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b x)^m (c+d x)^{-5-m}}{e+f x} \, dx=-\frac {(a+b x)^{1+m} (c+d x)^{-4-m} \left (d-\frac {d (3 b d e+a d f (4+m)-b c f (7+m)) (c+d x)}{(b c-a d) (-d e+c f) (3+m)}-\frac {(c+d x)^2 \left (d (b c-a d) (b e-a f) (d e-c f) (1+m)^2 \left (-a^2 d^2 f^2 \left (12+7 m+m^2\right )+2 a b d f (4+m) (-d e+c f (4+m))-b^2 \left (6 d^2 e^2-2 c d e f (10+m)+c^2 f^2 \left (26+9 m+m^2\right )\right )\right )+(c+d x) \left (-d (b e-a f) (1+m) \left (a^3 d^3 f^3 \left (24+26 m+9 m^2+m^3\right )+a^2 b d^2 f^2 \left (12+7 m+m^2\right ) (d e-c f (7+3 m))+a b^2 d f (4+m) \left (2 d^2 e^2-2 c d e f (5+m)+c^2 f^2 \left (26+17 m+3 m^2\right )\right )+b^3 \left (6 d^3 e^3-2 c d^2 e^2 f (13+m)+c^2 d e f^2 \left (46+11 m+m^2\right )-c^3 f^3 \left (50+35 m+10 m^2+m^3\right )\right )\right )-(b c-a d)^4 f^4 \left (24+50 m+35 m^2+10 m^3+m^4\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )\right )\right )}{(b c-a d)^3 (b e-a f) (d e-c f)^3 (1+m)^2 (2+m) (3+m)}\right )}{(b c-a d) (-d e+c f) (4+m)} \] Input:

Integrate[((a + b*x)^m*(c + d*x)^(-5 - m))/(e + f*x),x]
 

Output:

-(((a + b*x)^(1 + m)*(c + d*x)^(-4 - m)*(d - (d*(3*b*d*e + a*d*f*(4 + m) - 
 b*c*f*(7 + m))*(c + d*x))/((b*c - a*d)*(-(d*e) + c*f)*(3 + m)) - ((c + d* 
x)^2*(d*(b*c - a*d)*(b*e - a*f)*(d*e - c*f)*(1 + m)^2*(-(a^2*d^2*f^2*(12 + 
 7*m + m^2)) + 2*a*b*d*f*(4 + m)*(-(d*e) + c*f*(4 + m)) - b^2*(6*d^2*e^2 - 
 2*c*d*e*f*(10 + m) + c^2*f^2*(26 + 9*m + m^2))) + (c + d*x)*(-(d*(b*e - a 
*f)*(1 + m)*(a^3*d^3*f^3*(24 + 26*m + 9*m^2 + m^3) + a^2*b*d^2*f^2*(12 + 7 
*m + m^2)*(d*e - c*f*(7 + 3*m)) + a*b^2*d*f*(4 + m)*(2*d^2*e^2 - 2*c*d*e*f 
*(5 + m) + c^2*f^2*(26 + 17*m + 3*m^2)) + b^3*(6*d^3*e^3 - 2*c*d^2*e^2*f*( 
13 + m) + c^2*d*e*f^2*(46 + 11*m + m^2) - c^3*f^3*(50 + 35*m + 10*m^2 + m^ 
3)))) - (b*c - a*d)^4*f^4*(24 + 50*m + 35*m^2 + 10*m^3 + m^4)*Hypergeometr 
ic2F1[1, 1 + m, 2 + m, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))]))) 
/((b*c - a*d)^3*(b*e - a*f)*(d*e - c*f)^3*(1 + m)^2*(2 + m)*(3 + m))))/((b 
*c - a*d)*(-(d*e) + c*f)*(4 + m)))
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 626, normalized size of antiderivative = 1.13, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {144, 172, 172, 172, 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.

\(\displaystyle \int \frac {(a+b x)^m (c+d x)^{-m-5}}{e+f x} \, dx\)

\(\Big \downarrow \) 144

\(\displaystyle \frac {\int \frac {(a+b x)^m (c+d x)^{-m-4} (3 b d e-b c f (m+4)+a d f (m+4)+3 b d f x)}{e+f x}dx}{(m+4) (b c-a d) (d e-c f)}+\frac {d (a+b x)^{m+1} (c+d x)^{-m-4}}{(m+4) (b c-a d) (d e-c f)}\)

\(\Big \downarrow \) 172

\(\displaystyle \frac {\frac {\int \frac {(a+b x)^m (c+d x)^{-m-3} \left (\left (6 d^2 e^2-2 c d f (m+7) e+c^2 f^2 \left (m^2+7 m+12\right )\right ) b^2+2 a d f (m+4) (d e-c f (m+3)) b+2 d f (3 b d e+a d f (m+4)-b c f (m+7)) x b+a^2 d^2 f^2 \left (m^2+7 m+12\right )\right )}{e+f x}dx}{(m+3) (b c-a d) (d e-c f)}+\frac {d (a+b x)^{m+1} (c+d x)^{-m-3} (a d f (m+4)-b c f (m+7)+3 b d e)}{(m+3) (b c-a d) (d e-c f)}}{(m+4) (b c-a d) (d e-c f)}+\frac {d (a+b x)^{m+1} (c+d x)^{-m-4}}{(m+4) (b c-a d) (d e-c f)}\)

\(\Big \downarrow \) 172

\(\displaystyle \frac {\frac {\frac {\int \frac {(a+b x)^m (c+d x)^{-m-2} \left (\left (6 d^3 e^3-2 c d^2 f (m+10) e^2+c^2 d f^2 \left (m^2+9 m+26\right ) e-c^3 f^3 \left (m^3+9 m^2+26 m+24\right )\right ) b^3+a d f (m+4) \left (2 d^2 e^2-2 c d f (m+4) e+3 c^2 f^2 \left (m^2+5 m+6\right )\right ) b^2+a^2 d^2 f^2 \left (m^2+7 m+12\right ) (d e-3 c f (m+2)) b+d f \left (\left (6 d^2 e^2-2 c d f (m+10) e+c^2 f^2 \left (m^2+9 m+26\right )\right ) b^2+2 a d f (m+4) (d e-c f (m+4)) b+a^2 d^2 f^2 \left (m^2+7 m+12\right )\right ) x b+a^3 d^3 f^3 \left (m^3+9 m^2+26 m+24\right )\right )}{e+f x}dx}{(m+2) (b c-a d) (d e-c f)}+\frac {d (a+b x)^{m+1} (c+d x)^{-m-2} \left (a^2 d^2 f^2 \left (m^2+7 m+12\right )+2 a b d f (m+4) (d e-c f (m+4))+b^2 \left (c^2 f^2 \left (m^2+9 m+26\right )-2 c d e f (m+10)+6 d^2 e^2\right )\right )}{(m+2) (b c-a d) (d e-c f)}}{(m+3) (b c-a d) (d e-c f)}+\frac {d (a+b x)^{m+1} (c+d x)^{-m-3} (a d f (m+4)-b c f (m+7)+3 b d e)}{(m+3) (b c-a d) (d e-c f)}}{(m+4) (b c-a d) (d e-c f)}+\frac {d (a+b x)^{m+1} (c+d x)^{-m-4}}{(m+4) (b c-a d) (d e-c f)}\)

\(\Big \downarrow \) 172

\(\displaystyle \frac {\frac {\frac {\frac {\int \frac {(b c-a d)^4 f^4 (m+1) (m+2) (m+3) (m+4) (a+b x)^m (c+d x)^{-m-1}}{e+f x}dx}{(m+1) (b c-a d) (d e-c f)}+\frac {d (a+b x)^{m+1} (c+d x)^{-m-1} \left (a^3 d^3 f^3 \left (m^3+9 m^2+26 m+24\right )+a^2 b d^2 f^2 \left (m^2+7 m+12\right ) (d e-c f (3 m+7))+a b^2 d f (m+4) \left (c^2 f^2 \left (3 m^2+17 m+26\right )-2 c d e f (m+5)+2 d^2 e^2\right )+b^3 \left (-c^3 f^3 \left (m^3+10 m^2+35 m+50\right )+c^2 d e f^2 \left (m^2+11 m+46\right )-2 c d^2 e^2 f (m+13)+6 d^3 e^3\right )\right )}{(m+1) (b c-a d) (d e-c f)}}{(m+2) (b c-a d) (d e-c f)}+\frac {d (a+b x)^{m+1} (c+d x)^{-m-2} \left (a^2 d^2 f^2 \left (m^2+7 m+12\right )+2 a b d f (m+4) (d e-c f (m+4))+b^2 \left (c^2 f^2 \left (m^2+9 m+26\right )-2 c d e f (m+10)+6 d^2 e^2\right )\right )}{(m+2) (b c-a d) (d e-c f)}}{(m+3) (b c-a d) (d e-c f)}+\frac {d (a+b x)^{m+1} (c+d x)^{-m-3} (a d f (m+4)-b c f (m+7)+3 b d e)}{(m+3) (b c-a d) (d e-c f)}}{(m+4) (b c-a d) (d e-c f)}+\frac {d (a+b x)^{m+1} (c+d x)^{-m-4}}{(m+4) (b c-a d) (d e-c f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\frac {f^4 (m+2) (m+3) (m+4) (b c-a d)^3 \int \frac {(a+b x)^m (c+d x)^{-m-1}}{e+f x}dx}{d e-c f}+\frac {d (a+b x)^{m+1} (c+d x)^{-m-1} \left (a^3 d^3 f^3 \left (m^3+9 m^2+26 m+24\right )+a^2 b d^2 f^2 \left (m^2+7 m+12\right ) (d e-c f (3 m+7))+a b^2 d f (m+4) \left (c^2 f^2 \left (3 m^2+17 m+26\right )-2 c d e f (m+5)+2 d^2 e^2\right )+b^3 \left (-c^3 f^3 \left (m^3+10 m^2+35 m+50\right )+c^2 d e f^2 \left (m^2+11 m+46\right )-2 c d^2 e^2 f (m+13)+6 d^3 e^3\right )\right )}{(m+1) (b c-a d) (d e-c f)}}{(m+2) (b c-a d) (d e-c f)}+\frac {d (a+b x)^{m+1} (c+d x)^{-m-2} \left (a^2 d^2 f^2 \left (m^2+7 m+12\right )+2 a b d f (m+4) (d e-c f (m+4))+b^2 \left (c^2 f^2 \left (m^2+9 m+26\right )-2 c d e f (m+10)+6 d^2 e^2\right )\right )}{(m+2) (b c-a d) (d e-c f)}}{(m+3) (b c-a d) (d e-c f)}+\frac {d (a+b x)^{m+1} (c+d x)^{-m-3} (a d f (m+4)-b c f (m+7)+3 b d e)}{(m+3) (b c-a d) (d e-c f)}}{(m+4) (b c-a d) (d e-c f)}+\frac {d (a+b x)^{m+1} (c+d x)^{-m-4}}{(m+4) (b c-a d) (d e-c f)}\)

\(\Big \downarrow \) 141

\(\displaystyle \frac {\frac {\frac {d (a+b x)^{m+1} (c+d x)^{-m-2} \left (a^2 d^2 f^2 \left (m^2+7 m+12\right )+2 a b d f (m+4) (d e-c f (m+4))+b^2 \left (c^2 f^2 \left (m^2+9 m+26\right )-2 c d e f (m+10)+6 d^2 e^2\right )\right )}{(m+2) (b c-a d) (d e-c f)}+\frac {\frac {d (a+b x)^{m+1} (c+d x)^{-m-1} \left (a^3 d^3 f^3 \left (m^3+9 m^2+26 m+24\right )+a^2 b d^2 f^2 \left (m^2+7 m+12\right ) (d e-c f (3 m+7))+a b^2 d f (m+4) \left (c^2 f^2 \left (3 m^2+17 m+26\right )-2 c d e f (m+5)+2 d^2 e^2\right )+b^3 \left (-c^3 f^3 \left (m^3+10 m^2+35 m+50\right )+c^2 d e f^2 \left (m^2+11 m+46\right )-2 c d^2 e^2 f (m+13)+6 d^3 e^3\right )\right )}{(m+1) (b c-a d) (d e-c f)}-\frac {f^4 (m+2) (m+3) (m+4) (b c-a d)^3 (a+b x)^m (c+d x)^{-m} \operatorname {Hypergeometric2F1}\left (1,-m,1-m,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{m (d e-c f)^2}}{(m+2) (b c-a d) (d e-c f)}}{(m+3) (b c-a d) (d e-c f)}+\frac {d (a+b x)^{m+1} (c+d x)^{-m-3} (a d f (m+4)-b c f (m+7)+3 b d e)}{(m+3) (b c-a d) (d e-c f)}}{(m+4) (b c-a d) (d e-c f)}+\frac {d (a+b x)^{m+1} (c+d x)^{-m-4}}{(m+4) (b c-a d) (d e-c f)}\)

Input:

Int[((a + b*x)^m*(c + d*x)^(-5 - m))/(e + f*x),x]
 

Output:

(d*(a + b*x)^(1 + m)*(c + d*x)^(-4 - m))/((b*c - a*d)*(d*e - c*f)*(4 + m)) 
 + ((d*(3*b*d*e + a*d*f*(4 + m) - b*c*f*(7 + m))*(a + b*x)^(1 + m)*(c + d* 
x)^(-3 - m))/((b*c - a*d)*(d*e - c*f)*(3 + m)) + ((d*(a^2*d^2*f^2*(12 + 7* 
m + m^2) + 2*a*b*d*f*(4 + m)*(d*e - c*f*(4 + m)) + b^2*(6*d^2*e^2 - 2*c*d* 
e*f*(10 + m) + c^2*f^2*(26 + 9*m + m^2)))*(a + b*x)^(1 + m)*(c + d*x)^(-2 
- m))/((b*c - a*d)*(d*e - c*f)*(2 + m)) + ((d*(a^3*d^3*f^3*(24 + 26*m + 9* 
m^2 + m^3) + a^2*b*d^2*f^2*(12 + 7*m + m^2)*(d*e - c*f*(7 + 3*m)) + a*b^2* 
d*f*(4 + m)*(2*d^2*e^2 - 2*c*d*e*f*(5 + m) + c^2*f^2*(26 + 17*m + 3*m^2)) 
+ b^3*(6*d^3*e^3 - 2*c*d^2*e^2*f*(13 + m) + c^2*d*e*f^2*(46 + 11*m + m^2) 
- c^3*f^3*(50 + 35*m + 10*m^2 + m^3)))*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m 
))/((b*c - a*d)*(d*e - c*f)*(1 + m)) - ((b*c - a*d)^3*f^4*(2 + m)*(3 + m)* 
(4 + 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*c - a*d)*(d 
*e - c*f)*(2 + m)))/((b*c - a*d)*(d*e - c*f)*(3 + m)))/((b*c - a*d)*(d*e - 
 c*f)*(4 + m))
 

Defintions of rubi rules used

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 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 144

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

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]
 

Maple [F]

\[\int \frac {\left (b x +a \right )^{m} \left (x d +c \right )^{-5-m}}{f x +e}d x\]

Input:

int((b*x+a)^m*(d*x+c)^(-5-m)/(f*x+e),x)
 

Output:

int((b*x+a)^m*(d*x+c)^(-5-m)/(f*x+e),x)
 

Fricas [F]

\[ \int \frac {(a+b x)^m (c+d x)^{-5-m}}{e+f x} \, dx=\int { \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 5}}{f x + e} \,d x } \] Input:

integrate((b*x+a)^m*(d*x+c)^(-5-m)/(f*x+e),x, algorithm="fricas")
 

Output:

integral((b*x + a)^m*(d*x + c)^(-m - 5)/(f*x + e), x)
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {(a+b x)^m (c+d x)^{-5-m}}{e+f x} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

integrate((b*x+a)**m*(d*x+c)**(-5-m)/(f*x+e),x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

\[ \int \frac {(a+b x)^m (c+d x)^{-5-m}}{e+f x} \, dx=\int { \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 5}}{f x + e} \,d x } \] Input:

integrate((b*x+a)^m*(d*x+c)^(-5-m)/(f*x+e),x, algorithm="maxima")
 

Output:

integrate((b*x + a)^m*(d*x + c)^(-m - 5)/(f*x + e), x)
 

Giac [F]

\[ \int \frac {(a+b x)^m (c+d x)^{-5-m}}{e+f x} \, dx=\int { \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 5}}{f x + e} \,d x } \] Input:

integrate((b*x+a)^m*(d*x+c)^(-5-m)/(f*x+e),x, algorithm="giac")
 

Output:

integrate((b*x + a)^m*(d*x + c)^(-m - 5)/(f*x + e), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^m (c+d x)^{-5-m}}{e+f x} \, dx=\int \frac {{\left (a+b\,x\right )}^m}{\left (e+f\,x\right )\,{\left (c+d\,x\right )}^{m+5}} \,d x \] Input:

int((a + b*x)^m/((e + f*x)*(c + d*x)^(m + 5)),x)
 

Output:

int((a + b*x)^m/((e + f*x)*(c + d*x)^(m + 5)), x)
 

Reduce [F]

\[ \int \frac {(a+b x)^m (c+d x)^{-5-m}}{e+f x} \, dx=\int \frac {\left (b x +a \right )^{m}}{\left (d x +c \right )^{m} c^{5} e +\left (d x +c \right )^{m} c^{5} f x +5 \left (d x +c \right )^{m} c^{4} d e x +5 \left (d x +c \right )^{m} c^{4} d f \,x^{2}+10 \left (d x +c \right )^{m} c^{3} d^{2} e \,x^{2}+10 \left (d x +c \right )^{m} c^{3} d^{2} f \,x^{3}+10 \left (d x +c \right )^{m} c^{2} d^{3} e \,x^{3}+10 \left (d x +c \right )^{m} c^{2} d^{3} f \,x^{4}+5 \left (d x +c \right )^{m} c \,d^{4} e \,x^{4}+5 \left (d x +c \right )^{m} c \,d^{4} f \,x^{5}+\left (d x +c \right )^{m} d^{5} e \,x^{5}+\left (d x +c \right )^{m} d^{5} f \,x^{6}}d x \] Input:

int((b*x+a)^m*(d*x+c)^(-5-m)/(f*x+e),x)
 

Output:

int((a + b*x)**m/((c + d*x)**m*c**5*e + (c + d*x)**m*c**5*f*x + 5*(c + d*x 
)**m*c**4*d*e*x + 5*(c + d*x)**m*c**4*d*f*x**2 + 10*(c + d*x)**m*c**3*d**2 
*e*x**2 + 10*(c + d*x)**m*c**3*d**2*f*x**3 + 10*(c + d*x)**m*c**2*d**3*e*x 
**3 + 10*(c + d*x)**m*c**2*d**3*f*x**4 + 5*(c + d*x)**m*c*d**4*e*x**4 + 5* 
(c + d*x)**m*c*d**4*f*x**5 + (c + d*x)**m*d**5*e*x**5 + (c + d*x)**m*d**5* 
f*x**6),x)