\(\int \frac {(a+b x)^m (c+d x)^{-m}}{e+f x} \, dx\) [1716]

Optimal result

Integrand size = 24, antiderivative size = 122 \[ \int \frac {(a+b x)^m (c+d x)^{-m}}{e+f x} \, dx=\frac {b (a+b x)^m (c+d x)^{1-m} \operatorname {Hypergeometric2F1}\left (1,1,1+m,-\frac {d (a+b x)}{b c-a d}\right )}{(b c-a d) f m}-\frac {(a+b x)^m (c+d x)^{-m} \operatorname {Hypergeometric2F1}\left (1,m,1+m,\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{f m} \] Output:

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b x)^m (c+d x)^{-m}}{e+f x} \, dx=\frac {(a+b x)^m (c+d x)^{-m} \left (-\operatorname {Hypergeometric2F1}\left (1,m,1+m,\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )+\left (\frac {b (c+d x)}{b c-a d}\right )^m \operatorname {Hypergeometric2F1}\left (m,m,1+m,\frac {d (a+b x)}{-b c+a d}\right )\right )}{f m} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.17, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {140, 27, 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.

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

\(\Big \downarrow \) 140

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 80

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

\(\Big \downarrow \) 79

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

\(\Big \downarrow \) 141

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

Input:

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

Output:

((a - (b*e)/f)*(a + b*x)^m*Hypergeometric2F1[1, m, 1 + m, ((d*e - c*f)*(a 
+ b*x))/((b*e - a*f)*(c + d*x))])/((b*e - a*f)*m*(c + d*x)^m) + ((a + b*x) 
^m*((b*(c + d*x))/(b*c - a*d))^m*Hypergeometric2F1[m, m, 1 + m, -((d*(a + 
b*x))/(b*c - a*d))])/(f*m*(c + d*x)^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 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 140

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-2)]

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

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

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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