3.31.93 \(\int \frac {(a+b x)^m (c+d x)^{-3-m}}{e+f x} \, dx\) [3093]

3.31.93.1 Optimal result

Integrand size = 26, antiderivative size = 196 \[ \int \frac {(a+b x)^m (c+d x)^{-3-m}}{e+f x} \, dx=\frac {d (a+b x)^{1+m} (c+d x)^{-2-m}}{(b c-a d) (d e-c f) (2+m)}+\frac {d (a d f (2+m)+b (d e-c f (3+m))) (a+b x)^{1+m} (c+d x)^{-1-m}}{(b c-a d)^2 (d e-c f)^2 (1+m) (2+m)}-\frac {f^2 (a+b x)^m (c+d x)^{-m} \operatorname {Hypergeometric2F1}\left (1,-m,1-m,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{(d e-c f)^3 m} \]

output

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

3.31.93.2 Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b x)^m (c+d x)^{-3-m}}{e+f x} \, dx=-\frac {(a+b x)^{1+m} (c+d x)^{-2-m} \left (d-\frac {d (b d e+a d f (2+m)-b c f (3+m)) (c+d x)}{(b c-a d) (-d e+c f) (1+m)}+\frac {(b c-a d) f^2 (2+m) (c+d x) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{(b e-a f) (d e-c f) (1+m)}\right )}{(b c-a d) (-d e+c f) (2+m)} \]

input

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

output

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

3.31.93.3 Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.16, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {144, 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.

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

\(\Big \downarrow \) 144

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

\(\Big \downarrow \) 172

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 141

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

input

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

output

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

3.31.93.3.1 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]
 

3.31.93.4 Maple [F]

\[\int \frac {\left (b x +a \right )^{m} \left (d x +c \right )^{-3-m}}{f x +e}d x\]

input

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

output

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

3.31.93.5 Fricas [F]

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

input

integrate((b*x+a)^m*(d*x+c)^(-3-m)/(f*x+e),x, algorithm="fricas")
 

output

integral((b*x + a)^m*(d*x + c)^(-m - 3)/(f*x + e), x)
 

3.31.93.6 Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^m (c+d x)^{-3-m}}{e+f x} \, dx=\text {Timed out} \]

input

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

output

Timed out
 

3.31.93.7 Maxima [F]

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

input

integrate((b*x+a)^m*(d*x+c)^(-3-m)/(f*x+e),x, algorithm="maxima")
 

output

integrate((b*x + a)^m*(d*x + c)^(-m - 3)/(f*x + e), x)
 

3.31.93.8 Giac [F]

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

input

integrate((b*x+a)^m*(d*x+c)^(-3-m)/(f*x+e),x, algorithm="giac")
 

output

integrate((b*x + a)^m*(d*x + c)^(-m - 3)/(f*x + e), x)
 

3.31.93.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^m (c+d x)^{-3-m}}{e+f x} \, dx=\int \frac {{\left (a+b\,x\right )}^m}{\left (e+f\,x\right )\,{\left (c+d\,x\right )}^{m+3}} \,d x \]

input

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

output

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