\(\int \frac {(a+b x)^n (c+d x)^{-n}}{x} \, dx\) [584]

Optimal result

Integrand size = 20, antiderivative size = 102 \[ \int \frac {(a+b x)^n (c+d x)^{-n}}{x} \, dx=\frac {b (a+b x)^n (c+d x)^{1-n} \operatorname {Hypergeometric2F1}\left (1,1,1+n,-\frac {d (a+b x)}{b c-a d}\right )}{(b c-a d) n}-\frac {(a+b x)^n (c+d x)^{-n} \operatorname {Hypergeometric2F1}\left (1,n,1+n,\frac {c (a+b x)}{a (c+d x)}\right )}{n} \] Output:

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.87 \[ \int \frac {(a+b x)^n (c+d x)^{-n}}{x} \, dx=\frac {(a+b x)^n (c+d x)^{-n} \left (-\operatorname {Hypergeometric2F1}\left (1,n,1+n,\frac {c (a+b x)}{a (c+d x)}\right )+\left (\frac {b (c+d x)}{b c-a d}\right )^n \operatorname {Hypergeometric2F1}\left (n,n,1+n,\frac {d (a+b x)}{-b c+a d}\right )\right )}{n} \] Input:

Integrate[(a + b*x)^n/(x*(c + d*x)^n),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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.

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

\(\Big \downarrow \) 140

\(\displaystyle b \int (a+b x)^{n-1} (c+d x)^{-n}dx+\int \frac {a (a+b x)^{n-1} (c+d x)^{-n}}{x}dx\)

\(\Big \downarrow \) 27

\(\displaystyle b \int (a+b x)^{n-1} (c+d x)^{-n}dx+a \int \frac {(a+b x)^{n-1} (c+d x)^{-n}}{x}dx\)

\(\Big \downarrow \) 80

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

\(\Big \downarrow \) 79

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

\(\Big \downarrow \) 141

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

Input:

Int[(a + b*x)^n/(x*(c + d*x)^n),x]
 

Output:

-(((a + b*x)^n*Hypergeometric2F1[1, n, 1 + n, (c*(a + b*x))/(a*(c + d*x))] 
)/(n*(c + d*x)^n)) + ((a + b*x)^n*((b*(c + d*x))/(b*c - a*d))^n*Hypergeome 
tric2F1[n, n, 1 + n, -((d*(a + b*x))/(b*c - a*d))])/(n*(c + d*x)^n)
 

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]

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

Input:

int((b*x+a)^n/x/((d*x+c)^n),x)
 

Output:

int((b*x+a)^n/x/((d*x+c)^n),x)
 

Fricas [F]

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

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

Output:

integral((b*x + a)^n/((d*x + c)^n*x), x)
 

Sympy [F]

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

integrate((b*x+a)**n/x/((d*x+c)**n),x)
 

Output:

Integral((a + b*x)**n/(x*(c + d*x)**n), x)
 

Maxima [F]

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

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

Output:

integrate((b*x + a)^n/((d*x + c)^n*x), x)
 

Giac [F]

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

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

Output:

integrate((b*x + a)^n/((d*x + c)^n*x), x)
 

Mupad [F(-1)]

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

int((a + b*x)^n/(x*(c + d*x)^n),x)
 

Output:

int((a + b*x)^n/(x*(c + d*x)^n), x)
 

Reduce [F]

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

int((b*x+a)^n/x/((d*x+c)^n),x)
 

Output:

int((a + b*x)**n/((c + d*x)**n*x),x)