3.32.1 \(\int \frac {(a+b x)^m (c+d x)^{-4-m}}{e+f x} \, dx\) [3101]

3.32.1.1 Optimal result

Integrand size = 26, antiderivative size = 330 \[ \int \frac {(a+b x)^m (c+d x)^{-4-m}}{e+f x} \, dx=\frac {d (a+b x)^{1+m} (c+d x)^{-3-m}}{(b c-a d) (d e-c f) (3+m)}+\frac {d (a d f (3+m)+b (2 d e-c f (5+m))) (a+b x)^{1+m} (c+d x)^{-2-m}}{(b c-a d)^2 (d e-c f)^2 (2+m) (3+m)}+\frac {d \left (a^2 d^2 f^2 \left (6+5 m+m^2\right )+a b d f (3+m) (d e-c f (5+2 m))+b^2 \left (2 d^2 e^2-c d e f (7+m)+c^2 f^2 \left (11+6 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{-1-m}}{(b c-a d)^3 (d e-c f)^3 (1+m) (2+m) (3+m)}+\frac {f^3 (a+b x)^m (c+d x)^{-m} \operatorname {Hypergeometric2F1}\left (1,-m,1-m,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{(d e-c f)^4 m} \]

output

d*(b*x+a)^(1+m)*(d*x+c)^(-3-m)/(-a*d+b*c)/(-c*f+d*e)/(3+m)+d*(a*d*f*(3+m)+ 
b*(2*d*e-c*f*(5+m)))*(b*x+a)^(1+m)*(d*x+c)^(-2-m)/(-a*d+b*c)^2/(-c*f+d*e)^ 
2/(2+m)/(3+m)+d*(a^2*d^2*f^2*(m^2+5*m+6)+a*b*d*f*(3+m)*(d*e-c*f*(5+2*m))+b 
^2*(2*d^2*e^2-c*d*e*f*(7+m)+c^2*f^2*(m^2+6*m+11)))*(b*x+a)^(1+m)*(d*x+c)^( 
-1-m)/(-a*d+b*c)^3/(-c*f+d*e)^3/(1+m)/(2+m)/(3+m)+f^3*(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)^4/m/((d*x+ 
c)^m)
 

3.32.1.2 Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 301, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x)^m (c+d x)^{-4-m}}{e+f x} \, dx=-\frac {(a+b x)^{1+m} (c+d x)^{-3-m} \left (d+\frac {d (-2 b d e-a d f (3+m)+b c f (5+m)) (c+d x)}{(b c-a d) (-d e+c f) (2+m)}-\frac {(c+d x)^2 \left (-d (b e-a f) \left (a^2 d^2 f^2 \left (6+5 m+m^2\right )+a b d f (3+m) (d e-c f (5+2 m))+b^2 \left (2 d^2 e^2-c d e f (7+m)+c^2 f^2 \left (11+6 m+m^2\right )\right )\right )+(b c-a d)^3 f^3 \left (6+5 m+m^2\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )\right )}{(b c-a d)^2 (b e-a f) (d e-c f)^2 (1+m) (2+m)}\right )}{(b c-a d) (-d e+c f) (3+m)} \]

input

Integrate[((a + b*x)^m*(c + d*x)^(-4 - m))/(e + f*x),x]
 

output

-(((a + b*x)^(1 + m)*(c + d*x)^(-3 - m)*(d + (d*(-2*b*d*e - a*d*f*(3 + m) 
+ b*c*f*(5 + m))*(c + d*x))/((b*c - a*d)*(-(d*e) + c*f)*(2 + m)) - ((c + d 
*x)^2*(-(d*(b*e - a*f)*(a^2*d^2*f^2*(6 + 5*m + m^2) + a*b*d*f*(3 + m)*(d*e 
 - c*f*(5 + 2*m)) + b^2*(2*d^2*e^2 - c*d*e*f*(7 + m) + c^2*f^2*(11 + 6*m + 
 m^2)))) + (b*c - a*d)^3*f^3*(6 + 5*m + m^2)*Hypergeometric2F1[1, 1 + m, 2 
 + m, ((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))]))/((b*c - a*d)^2*(b 
*e - a*f)*(d*e - c*f)^2*(1 + m)*(2 + m))))/((b*c - a*d)*(-(d*e) + c*f)*(3 
+ m)))
 

3.32.1.3 Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.16, 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, 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.

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

\(\Big \downarrow \) 144

\(\displaystyle \frac {\int \frac {(a+b x)^m (c+d x)^{-m-3} (2 b d e-b c f (m+3)+a d f (m+3)+2 b d f x)}{e+f x}dx}{(m+3) (b c-a d) (d e-c f)}+\frac {d (a+b x)^{m+1} (c+d x)^{-m-3}}{(m+3) (b c-a d) (d e-c f)}\)

\(\Big \downarrow \) 172

\(\displaystyle \frac {\frac {\int \frac {(a+b x)^m (c+d x)^{-m-2} \left (\left (2 d^2 e^2-c d f (m+5) e+c^2 f^2 \left (m^2+5 m+6\right )\right ) b^2+a d f (m+3) (d e-2 c f (m+2)) b+d f (2 b d e+a d f (m+3)-b c f (m+5)) x b+a^2 d^2 f^2 \left (m^2+5 m+6\right )\right )}{e+f x}dx}{(m+2) (b c-a d) (d e-c f)}+\frac {d (a+b x)^{m+1} (c+d x)^{-m-2} (a d f (m+3)-b c f (m+5)+2 b d e)}{(m+2) (b c-a d) (d e-c f)}}{(m+3) (b c-a d) (d e-c f)}+\frac {d (a+b x)^{m+1} (c+d x)^{-m-3}}{(m+3) (b c-a d) (d e-c f)}\)

\(\Big \downarrow \) 172

\(\displaystyle \frac {\frac {\frac {\int -\frac {(b c-a d)^3 f^3 (m+1) (m+2) (m+3) (a+b x)^m (c+d x)^{-m-1}}{e+f x}dx}{(m+1) (b c-a d) (d e-c f)}+\frac {d (a+b x)^{m+1} (c+d x)^{-m-1} \left (a^2 d^2 f^2 \left (m^2+5 m+6\right )+a b d f (m+3) (d e-c f (2 m+5))+b^2 \left (c^2 f^2 \left (m^2+6 m+11\right )-c d e f (m+7)+2 d^2 e^2\right )\right )}{(m+1) (b c-a d) (d e-c f)}}{(m+2) (b c-a d) (d e-c f)}+\frac {d (a+b x)^{m+1} (c+d x)^{-m-2} (a d f (m+3)-b c f (m+5)+2 b d e)}{(m+2) (b c-a d) (d e-c f)}}{(m+3) (b c-a d) (d e-c f)}+\frac {d (a+b x)^{m+1} (c+d x)^{-m-3}}{(m+3) (b c-a d) (d e-c f)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {d (a+b x)^{m+1} (c+d x)^{-m-1} \left (a^2 d^2 f^2 \left (m^2+5 m+6\right )+a b d f (m+3) (d e-c f (2 m+5))+b^2 \left (c^2 f^2 \left (m^2+6 m+11\right )-c d e f (m+7)+2 d^2 e^2\right )\right )}{(m+1) (b c-a d) (d e-c f)}-\frac {\int \frac {(b c-a d)^3 f^3 (m+1) (m+2) (m+3) (a+b x)^m (c+d x)^{-m-1}}{e+f x}dx}{(m+1) (b c-a d) (d e-c f)}}{(m+2) (b c-a d) (d e-c f)}+\frac {d (a+b x)^{m+1} (c+d x)^{-m-2} (a d f (m+3)-b c f (m+5)+2 b d e)}{(m+2) (b c-a d) (d e-c f)}}{(m+3) (b c-a d) (d e-c f)}+\frac {d (a+b x)^{m+1} (c+d x)^{-m-3}}{(m+3) (b c-a d) (d e-c f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {d (a+b x)^{m+1} (c+d x)^{-m-1} \left (a^2 d^2 f^2 \left (m^2+5 m+6\right )+a b d f (m+3) (d e-c f (2 m+5))+b^2 \left (c^2 f^2 \left (m^2+6 m+11\right )-c d e f (m+7)+2 d^2 e^2\right )\right )}{(m+1) (b c-a d) (d e-c f)}-\frac {f^3 (m+2) (m+3) (b c-a d)^2 \int \frac {(a+b x)^m (c+d x)^{-m-1}}{e+f x}dx}{d e-c f}}{(m+2) (b c-a d) (d e-c f)}+\frac {d (a+b x)^{m+1} (c+d x)^{-m-2} (a d f (m+3)-b c f (m+5)+2 b d e)}{(m+2) (b c-a d) (d e-c f)}}{(m+3) (b c-a d) (d e-c f)}+\frac {d (a+b x)^{m+1} (c+d x)^{-m-3}}{(m+3) (b c-a d) (d e-c f)}\)

\(\Big \downarrow \) 141

\(\displaystyle \frac {\frac {\frac {d (a+b x)^{m+1} (c+d x)^{-m-1} \left (a^2 d^2 f^2 \left (m^2+5 m+6\right )+a b d f (m+3) (d e-c f (2 m+5))+b^2 \left (c^2 f^2 \left (m^2+6 m+11\right )-c d e f (m+7)+2 d^2 e^2\right )\right )}{(m+1) (b c-a d) (d e-c f)}+\frac {f^3 (m+2) (m+3) (b c-a d)^2 (a+b x)^m (c+d x)^{-m} \operatorname {Hypergeometric2F1}\left (1,-m,1-m,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{m (d e-c f)^2}}{(m+2) (b c-a d) (d e-c f)}+\frac {d (a+b x)^{m+1} (c+d x)^{-m-2} (a d f (m+3)-b c f (m+5)+2 b d e)}{(m+2) (b c-a d) (d e-c f)}}{(m+3) (b c-a d) (d e-c f)}+\frac {d (a+b x)^{m+1} (c+d x)^{-m-3}}{(m+3) (b c-a d) (d e-c f)}\)

input

Int[((a + b*x)^m*(c + d*x)^(-4 - m))/(e + f*x),x]
 

output

(d*(a + b*x)^(1 + m)*(c + d*x)^(-3 - m))/((b*c - a*d)*(d*e - c*f)*(3 + m)) 
 + ((d*(2*b*d*e + a*d*f*(3 + m) - b*c*f*(5 + m))*(a + b*x)^(1 + m)*(c + d* 
x)^(-2 - m))/((b*c - a*d)*(d*e - c*f)*(2 + m)) + ((d*(a^2*d^2*f^2*(6 + 5*m 
 + m^2) + a*b*d*f*(3 + m)*(d*e - c*f*(5 + 2*m)) + b^2*(2*d^2*e^2 - c*d*e*f 
*(7 + m) + c^2*f^2*(11 + 6*m + m^2)))*(a + b*x)^(1 + m)*(c + d*x)^(-1 - m) 
)/((b*c - a*d)*(d*e - c*f)*(1 + m)) + ((b*c - a*d)^2*f^3*(2 + m)*(3 + 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))
 

3.32.1.3.1 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 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]
 

3.32.1.4 Maple [F]

\[\int \frac {\left (b x +a \right )^{m} \left (d x +c \right )^{-4-m}}{f x +e}d x\]

input

int((b*x+a)^m*(d*x+c)^(-4-m)/(f*x+e),x)
 

output

int((b*x+a)^m*(d*x+c)^(-4-m)/(f*x+e),x)
 

3.32.1.5 Fricas [F]

\[ \int \frac {(a+b x)^m (c+d x)^{-4-m}}{e+f x} \, dx=\int { \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 4}}{f x + e} \,d x } \]

input

integrate((b*x+a)^m*(d*x+c)^(-4-m)/(f*x+e),x, algorithm="fricas")
 

output

integral((b*x + a)^m*(d*x + c)^(-m - 4)/(f*x + e), x)
 

3.32.1.6 Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^m (c+d x)^{-4-m}}{e+f x} \, dx=\text {Timed out} \]

input

integrate((b*x+a)**m*(d*x+c)**(-4-m)/(f*x+e),x)
 

output

Timed out
 

3.32.1.7 Maxima [F]

\[ \int \frac {(a+b x)^m (c+d x)^{-4-m}}{e+f x} \, dx=\int { \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 4}}{f x + e} \,d x } \]

input

integrate((b*x+a)^m*(d*x+c)^(-4-m)/(f*x+e),x, algorithm="maxima")
 

output

integrate((b*x + a)^m*(d*x + c)^(-m - 4)/(f*x + e), x)
 

3.32.1.8 Giac [F]

\[ \int \frac {(a+b x)^m (c+d x)^{-4-m}}{e+f x} \, dx=\int { \frac {{\left (b x + a\right )}^{m} {\left (d x + c\right )}^{-m - 4}}{f x + e} \,d x } \]

input

integrate((b*x+a)^m*(d*x+c)^(-4-m)/(f*x+e),x, algorithm="giac")
 

output

integrate((b*x + a)^m*(d*x + c)^(-m - 4)/(f*x + e), x)
 

3.32.1.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^m (c+d x)^{-4-m}}{e+f x} \, dx=\int \frac {{\left (a+b\,x\right )}^m}{\left (e+f\,x\right )\,{\left (c+d\,x\right )}^{m+4}} \,d x \]

input

int((a + b*x)^m/((e + f*x)*(c + d*x)^(m + 4)),x)
 

output

int((a + b*x)^m/((e + f*x)*(c + d*x)^(m + 4)), x)