\(\int \frac {(a+\frac {b}{c+d x})^p}{x} \, dx\) [57]

Optimal result

Integrand size = 17, antiderivative size = 105 \[ \int \frac {\left (a+\frac {b}{c+d x}\right )^p}{x} \, dx=-\frac {c \left (a+\frac {b}{c+d x}\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,\frac {c \left (a+\frac {b}{c+d x}\right )}{b+a c}\right )}{(b+a c) (1+p)}+\frac {\left (a+\frac {b}{c+d x}\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b}{a (c+d x)}\right )}{a (1+p)} \] Output:

-c*(a+b/(d*x+c))^(p+1)*hypergeom([1, p+1],[2+p],c*(a+b/(d*x+c))/(a*c+b))/( 
a*c+b)/(p+1)+(a+b/(d*x+c))^(p+1)*hypergeom([1, p+1],[2+p],1+b/a/(d*x+c))/a 
/(p+1)
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {896, 25, 941, 948, 25, 97, 75, 78}

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.

\(\displaystyle \int \frac {\left (a+\frac {b}{c+d x}\right )^p}{x} \, dx\)

\(\Big \downarrow \) 896

\(\displaystyle \int \frac {\left (a+\frac {b}{c+d x}\right )^p}{d x}d(c+d x)\)

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 941

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

\(\Big \downarrow \) 948

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 97

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

\(\Big \downarrow \) 75

\(\displaystyle \frac {\left (a+\frac {b}{c+d x}\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,\frac {b}{a (c+d x)}+1\right )}{a (p+1)}-c \int \frac {\left (a+\frac {b}{c+d x}\right )^p}{1-\frac {c}{c+d x}}d\frac {1}{c+d x}\)

\(\Big \downarrow \) 78

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

Input:

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

Output:

-((c*(a + b/(c + d*x))^(1 + p)*Hypergeometric2F1[1, 1 + p, 2 + p, (c*(a + 
b/(c + d*x)))/(b + a*c)])/((b + a*c)*(1 + p))) + ((a + b/(c + d*x))^(1 + p 
)*Hypergeometric2F1[1, 1 + p, 2 + p, 1 + b/(a*(c + d*x))])/(a*(1 + p))
 

Defintions of rubi rules used

rule 25

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

rule 75

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

rule 78

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

rule 97

Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), 
x_] :> Simp[b/(b*c - a*d)   Int[(e + f*x)^p/(a + b*x), x], x] - Simp[d/(b*c 
 - a*d)   Int[(e + f*x)^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, p}, 
 x] &&  !IntegerQ[p]
 

rule 896

Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coeff 
icient[v, x, 0], d = Coefficient[v, x, 1]}, Simp[1/d^(m + 1)   Subst[Int[Si 
mplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]] /; 
 FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]
 

rule 941

Int[((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Sym 
bol] :> Int[(a + b*x^n)^p*((d + c*x^n)^q/x^(n*q)), x] /; FreeQ[{a, b, c, d, 
 n, p}, x] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] ||  !IntegerQ[p])
 

rule 948

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. 
), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ 
p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ 
[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (a+\frac {b}{c+d x}\right )^p}{x} \, dx=\text {Exception raised: RuntimeError} \] Input:

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

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:Unable to divide, perhaps due to rounding error%%%{-1,[ 
0,1,1,0]%%%} / %%%{1,[0,0,0,1]%%%} Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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