\(\int \frac {(a+b x)^n (c+d x)^p}{x} \, dx\) [578]

Optimal result

Integrand size = 18, antiderivative size = 84 \[ \int \frac {(a+b x)^n (c+d x)^p}{x} \, dx=-\frac {(a+b x)^{1+n} (c+d x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-p} \operatorname {AppellF1}\left (1+n,1,-p,2+n,1+\frac {b x}{a},-\frac {d (a+b x)}{b c-a d}\right )}{a (1+n)} \] Output:

-(b*x+a)^(1+n)*(d*x+c)^p*AppellF1(1+n,-p,1,2+n,-d*(b*x+a)/(-a*d+b*c),1+b*x 
/a)/a/(1+n)/((b*(d*x+c)/(-a*d+b*c))^p)
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.05 \[ \int \frac {(a+b x)^n (c+d x)^p}{x} \, dx=\frac {\left (1+\frac {a}{b x}\right )^{-n} \left (1+\frac {c}{d x}\right )^{-p} (a+b x)^n (c+d x)^p \operatorname {AppellF1}\left (-n-p,-n,-p,1-n-p,-\frac {a}{b x},-\frac {c}{d x}\right )}{n+p} \] Input:

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

Output:

((a + b*x)^n*(c + d*x)^p*AppellF1[-n - p, -n, -p, 1 - n - p, -(a/(b*x)), - 
(c/(d*x))])/((n + p)*(1 + a/(b*x))^n*(1 + c/(d*x))^p)
 

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.01, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {154, 153}

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.

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

\(\Big \downarrow \) 154

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

\(\Big \downarrow \) 153

\(\displaystyle -\frac {(a+b x)^{n+1} (c+d x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-p} \operatorname {AppellF1}\left (n+1,-p,1,n+2,-\frac {d (a+b x)}{b c-a d},\frac {a+b x}{a}\right )}{a (n+1)}\)

Input:

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

Output:

-(((a + b*x)^(1 + n)*(c + d*x)^p*AppellF1[1 + n, -p, 1, 2 + n, -((d*(a + b 
*x))/(b*c - a*d)), (a + b*x)/a])/(a*(1 + n)*((b*(c + d*x))/(b*c - a*d))^p) 
)
 

Defintions of rubi rules used

rule 153

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) 
^(p_), x_] :> Simp[(b*e - a*f)^p*((a + b*x)^(m + 1)/(b^(p + 1)*(m + 1)*Simp 
lify[b/(b*c - a*d)]^n))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/(b*c 
 - a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, m, 
n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] && IntegerQ[p] && GtQ[Simplify[b/( 
b*c - a*d)], 0] &&  !(GtQ[Simplify[d/(d*a - c*b)], 0] && SimplerQ[c + d*x, 
a + b*x])
 

rule 154

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) 
^(p_), x_] :> Simp[(c + d*x)^FracPart[n]/(Simplify[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*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, 
 d, e, f, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] && IntegerQ[p] &&  !G 
tQ[Simplify[b/(b*c - a*d)], 0] &&  !SimplerQ[c + d*x, a + b*x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(a+b x)^n (c+d x)^p}{x} \, dx=\frac {\left (d x +c \right )^{p} \left (b x +a \right )^{n} a d +\left (d x +c \right )^{p} \left (b x +a \right )^{n} b c -\left (\int \frac {\left (d x +c \right )^{p} \left (b x +a \right )^{n} x}{a b \,d^{2} p \,x^{2}+b^{2} c d n \,x^{2}+a^{2} d^{2} p x +a b c d n x +a b c d p x +b^{2} c^{2} n x +a^{2} c d p +a b \,c^{2} n}d x \right ) a^{2} b \,d^{3} n p -\left (\int \frac {\left (d x +c \right )^{p} \left (b x +a \right )^{n} x}{a b \,d^{2} p \,x^{2}+b^{2} c d n \,x^{2}+a^{2} d^{2} p x +a b c d n x +a b c d p x +b^{2} c^{2} n x +a^{2} c d p +a b \,c^{2} n}d x \right ) a \,b^{2} c \,d^{2} n^{2}-\left (\int \frac {\left (d x +c \right )^{p} \left (b x +a \right )^{n} x}{a b \,d^{2} p \,x^{2}+b^{2} c d n \,x^{2}+a^{2} d^{2} p x +a b c d n x +a b c d p x +b^{2} c^{2} n x +a^{2} c d p +a b \,c^{2} n}d x \right ) a \,b^{2} c \,d^{2} p^{2}-\left (\int \frac {\left (d x +c \right )^{p} \left (b x +a \right )^{n} x}{a b \,d^{2} p \,x^{2}+b^{2} c d n \,x^{2}+a^{2} d^{2} p x +a b c d n x +a b c d p x +b^{2} c^{2} n x +a^{2} c d p +a b \,c^{2} n}d x \right ) b^{3} c^{2} d n p +\left (\int \frac {\left (d x +c \right )^{p} \left (b x +a \right )^{n}}{a b \,d^{2} p \,x^{3}+b^{2} c d n \,x^{3}+a^{2} d^{2} p \,x^{2}+a b c d n \,x^{2}+a b c d p \,x^{2}+b^{2} c^{2} n \,x^{2}+a^{2} c d p x +a b \,c^{2} n x}d x \right ) a^{3} c \,d^{2} p^{2}+2 \left (\int \frac {\left (d x +c \right )^{p} \left (b x +a \right )^{n}}{a b \,d^{2} p \,x^{3}+b^{2} c d n \,x^{3}+a^{2} d^{2} p \,x^{2}+a b c d n \,x^{2}+a b c d p \,x^{2}+b^{2} c^{2} n \,x^{2}+a^{2} c d p x +a b \,c^{2} n x}d x \right ) a^{2} b \,c^{2} d n p +\left (\int \frac {\left (d x +c \right )^{p} \left (b x +a \right )^{n}}{a b \,d^{2} p \,x^{3}+b^{2} c d n \,x^{3}+a^{2} d^{2} p \,x^{2}+a b c d n \,x^{2}+a b c d p \,x^{2}+b^{2} c^{2} n \,x^{2}+a^{2} c d p x +a b \,c^{2} n x}d x \right ) a \,b^{2} c^{3} n^{2}}{a d p +b c n} \] Input:

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

Output:

((c + d*x)**p*(a + b*x)**n*a*d + (c + d*x)**p*(a + b*x)**n*b*c - int(((c + 
 d*x)**p*(a + b*x)**n*x)/(a**2*c*d*p + a**2*d**2*p*x + a*b*c**2*n + a*b*c* 
d*n*x + a*b*c*d*p*x + a*b*d**2*p*x**2 + b**2*c**2*n*x + b**2*c*d*n*x**2),x 
)*a**2*b*d**3*n*p - int(((c + d*x)**p*(a + b*x)**n*x)/(a**2*c*d*p + a**2*d 
**2*p*x + a*b*c**2*n + a*b*c*d*n*x + a*b*c*d*p*x + a*b*d**2*p*x**2 + b**2* 
c**2*n*x + b**2*c*d*n*x**2),x)*a*b**2*c*d**2*n**2 - int(((c + d*x)**p*(a + 
 b*x)**n*x)/(a**2*c*d*p + a**2*d**2*p*x + a*b*c**2*n + a*b*c*d*n*x + a*b*c 
*d*p*x + a*b*d**2*p*x**2 + b**2*c**2*n*x + b**2*c*d*n*x**2),x)*a*b**2*c*d* 
*2*p**2 - int(((c + d*x)**p*(a + b*x)**n*x)/(a**2*c*d*p + a**2*d**2*p*x + 
a*b*c**2*n + a*b*c*d*n*x + a*b*c*d*p*x + a*b*d**2*p*x**2 + b**2*c**2*n*x + 
 b**2*c*d*n*x**2),x)*b**3*c**2*d*n*p + int(((c + d*x)**p*(a + b*x)**n)/(a* 
*2*c*d*p*x + a**2*d**2*p*x**2 + a*b*c**2*n*x + a*b*c*d*n*x**2 + a*b*c*d*p* 
x**2 + a*b*d**2*p*x**3 + b**2*c**2*n*x**2 + b**2*c*d*n*x**3),x)*a**3*c*d** 
2*p**2 + 2*int(((c + d*x)**p*(a + b*x)**n)/(a**2*c*d*p*x + a**2*d**2*p*x** 
2 + a*b*c**2*n*x + a*b*c*d*n*x**2 + a*b*c*d*p*x**2 + a*b*d**2*p*x**3 + b** 
2*c**2*n*x**2 + b**2*c*d*n*x**3),x)*a**2*b*c**2*d*n*p + int(((c + d*x)**p* 
(a + b*x)**n)/(a**2*c*d*p*x + a**2*d**2*p*x**2 + a*b*c**2*n*x + a*b*c*d*n* 
x**2 + a*b*c*d*p*x**2 + a*b*d**2*p*x**3 + b**2*c**2*n*x**2 + b**2*c*d*n*x* 
*3),x)*a*b**2*c**3*n**2)/(a*d*p + b*c*n)