3.2.37 \(\int \frac {(a+b x)^m (A+B x) (c+d x)^{-m}}{e+f x} \, dx\) [137]

3.2.37.1 Optimal result

Integrand size = 29, antiderivative size = 233 \[ \int \frac {(a+b x)^m (A+B x) (c+d x)^{-m}}{e+f x} \, dx=-\frac {d (B e-A f) (a+b x)^{1+m} (c+d x)^{-m}}{(b c-a d) f^2 m}-\frac {(B e-A f) (a+b x)^m (c+d x)^{-m} \operatorname {Hypergeometric2F1}\left (1,-m,1-m,\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{f^2 m}-\frac {(a B d f m-b (B d e-A d f+B c f m)) (a+b x)^{1+m} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \operatorname {Hypergeometric2F1}\left (m,1+m,2+m,-\frac {d (a+b x)}{b c-a d}\right )}{b (b c-a d) f^2 m (1+m)} \]

output

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

3.2.37.2 Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.75 \[ \int \frac {(a+b x)^m (A+B x) (c+d x)^{-m}}{e+f x} \, dx=\frac {(a+b x)^m (c+d x)^{-m} \left (b (B e-A f) (1+m) \operatorname {Hypergeometric2F1}\left (1,m,1+m,\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )+\left (\frac {b (c+d x)}{b c-a d}\right )^m \left (-b (B e-A f) (1+m) \operatorname {Hypergeometric2F1}\left (m,m,1+m,\frac {d (a+b x)}{-b c+a d}\right )+B f m (a+b x) \operatorname {Hypergeometric2F1}\left (m,1+m,2+m,\frac {d (a+b x)}{-b c+a d}\right )\right )\right )}{b f^2 m (1+m)} \]

input

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

output

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

3.2.37.3 Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {173, 25, 88, 80, 79, 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.

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

\(\Big \downarrow \) 173

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 88

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

\(\Big \downarrow \) 80

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

\(\Big \downarrow \) 79

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

\(\Big \downarrow \) 141

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

input

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

output

-(((B*e - A*f)*(a + b*x)^m*Hypergeometric2F1[1, -m, 1 - m, ((b*e - a*f)*(c 
 + d*x))/((d*e - c*f)*(a + b*x))])/(f^2*m*(c + d*x)^m)) - ((d*(B*e - A*f)* 
(a + b*x)^(1 + m))/((b*c - a*d)*m*(c + d*x)^m) + ((a*B*d*f*m - b*(B*d*e - 
A*d*f + B*c*f*m))*(a + b*x)^(1 + m)*((b*(c + d*x))/(b*c - a*d))^m*Hypergeo 
metric2F1[m, 1 + m, 2 + m, -((d*(a + b*x))/(b*c - a*d))])/(b*(b*c - a*d)*m 
*(1 + m)*(c + d*x)^m))/f^2
 

3.2.37.3.1 Defintions of rubi rules used

rule 25

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

rule 79

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( 
a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 
, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] 
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] 
 ||  !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
 

rule 80

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c 
 + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) 
^FracPart[n])   Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) 
), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Integ 
erQ[n] && (RationalQ[m] ||  !SimplerQ[n + 1, m + 1])
 

rule 88

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^Simplify[p + 1], 
 x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] &&  !RationalQ[p] && SumSimpl 
erQ[p, 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]
 

rule 173

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((g_.) + (h_.)*(x_ 
)))/((e_.) + (f_.)*(x_)), x_] :> Simp[(f*g - e*h)*((c*f - d*e)^(m + n + 1)/ 
f^(m + n + 2))   Int[(a + b*x)^m/((c + d*x)^(m + 1)*(e + f*x)), x], x] + Si 
mp[1/f^(m + n + 2)   Int[((a + b*x)^m/(c + d*x)^(m + 1))*ExpandToSum[(f^(m 
+ n + 2)*(c + d*x)^(m + n + 1)*(g + h*x) - (f*g - e*h)*(c*f - d*e)^(m + n + 
 1))/(e + f*x), x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && IGtQ[m 
+ n + 1, 0] && (LtQ[m, 0] || SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])
 

3.2.37.4 Maple [F]

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

input

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

output

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

3.2.37.5 Fricas [F]

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

input

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

output

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

3.2.37.6 Sympy [F(-2)]

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

input

integrate((b*x+a)**m*(B*x+A)/((d*x+c)**m)/(f*x+e),x)
 

output

Exception raised: HeuristicGCDFailed >> no luck
 

3.2.37.7 Maxima [F]

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

input

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

output

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

3.2.37.8 Giac [F]

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

input

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

output

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

3.2.37.9 Mupad [F(-1)]

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

input

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

output

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