\(\int \frac {(a+b x)^n (c+d x)^p}{x^2} \, dx\) [579]

Optimal result

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

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.11 \[ \int \frac {(a+b x)^n (c+d x)^p}{x^2} \, dx=\frac {\left (1+\frac {a}{b x}\right )^{-n} \left (1+\frac {c}{d x}\right )^{-p} (a+b x)^n (c+d x)^p \operatorname {AppellF1}\left (1-n-p,-n,-p,2-n-p,-\frac {a}{b x},-\frac {c}{d x}\right )}{(-1+n+p) x} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.22 (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.

\(\displaystyle \int \frac {(a+b x)^n (c+d x)^p}{x^2} \, dx\)

\(\Big \downarrow \) 154

\(\displaystyle (c+d x)^p \left (\frac {b (c+d x)}{b c-a d}\right )^{-p} \int \frac {(a+b x)^n \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^p}{x^2}dx\)

\(\Big \downarrow \) 153

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

Input:

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

Output:

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

\[\int \frac {\left (b x +a \right )^{n} \left (x d +c \right )^{p}}{x^{2}}d x\]

Input:

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

Output:

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

Fricas [F]

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

integrate((b*x+a)^n*(d*x+c)^p/x^2,x, algorithm="fricas")
 

Output:

integral((b*x + a)^n*(d*x + c)^p/x^2, x)
 

Sympy [F]

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

integrate((b*x+a)**n*(d*x+c)**p/x**2,x)
 

Output:

Integral((a + b*x)**n*(c + d*x)**p/x**2, x)
 

Maxima [F]

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

integrate((b*x+a)^n*(d*x+c)^p/x^2,x, algorithm="maxima")
 

Output:

integrate((b*x + a)^n*(d*x + c)^p/x^2, x)
 

Giac [F]

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

integrate((b*x+a)^n*(d*x+c)^p/x^2,x, algorithm="giac")
 

Output:

integrate((b*x + a)^n*(d*x + c)^p/x^2, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(a+b x)^n (c+d x)^p}{x^2} \, dx=\text {too large to display} \] Input:

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

Output:

( - (c + d*x)**p*(a + b*x)**n*a**2*d**2*p**2 + (c + d*x)**p*(a + b*x)**n*a 
**2*d**2*p - 2*(c + d*x)**p*(a + b*x)**n*a*b*c*d*n*p + (c + d*x)**p*(a + b 
*x)**n*a*b*c*d*n + (c + d*x)**p*(a + b*x)**n*a*b*c*d*p - (c + d*x)**p*(a + 
 b*x)**n*a*b*d**2*n*x - (c + d*x)**p*(a + b*x)**n*b**2*c**2*n**2 + (c + d* 
x)**p*(a + b*x)**n*b**2*c**2*n - (c + d*x)**p*(a + b*x)**n*b**2*c*d*p*x + 
int(((c + d*x)**p*(a + b*x)**n*x)/(a**3*c*d**2*p**2 - a**3*c*d**2*p + a**3 
*d**3*p**2*x - a**3*d**3*p*x + 2*a**2*b*c**2*d*n*p - a**2*b*c**2*d*n - a** 
2*b*c**2*d*p + 2*a**2*b*c*d**2*n*p*x - a**2*b*c*d**2*n*x + a**2*b*c*d**2*p 
**2*x - 2*a**2*b*c*d**2*p*x + a**2*b*d**3*p**2*x**2 - a**2*b*d**3*p*x**2 + 
 a*b**2*c**3*n**2 - a*b**2*c**3*n + a*b**2*c**2*d*n**2*x + 2*a*b**2*c**2*d 
*n*p*x - 2*a*b**2*c**2*d*n*x - a*b**2*c**2*d*p*x + 2*a*b**2*c*d**2*n*p*x** 
2 - a*b**2*c*d**2*n*x**2 - a*b**2*c*d**2*p*x**2 + b**3*c**3*n**2*x - b**3* 
c**3*n*x + b**3*c**2*d*n**2*x**2 - b**3*c**2*d*n*x**2),x)*a**3*b**2*d**5*n 
**2*p**2*x - int(((c + d*x)**p*(a + b*x)**n*x)/(a**3*c*d**2*p**2 - a**3*c* 
d**2*p + a**3*d**3*p**2*x - a**3*d**3*p*x + 2*a**2*b*c**2*d*n*p - a**2*b*c 
**2*d*n - a**2*b*c**2*d*p + 2*a**2*b*c*d**2*n*p*x - a**2*b*c*d**2*n*x + a* 
*2*b*c*d**2*p**2*x - 2*a**2*b*c*d**2*p*x + a**2*b*d**3*p**2*x**2 - a**2*b* 
d**3*p*x**2 + a*b**2*c**3*n**2 - a*b**2*c**3*n + a*b**2*c**2*d*n**2*x + 2* 
a*b**2*c**2*d*n*p*x - 2*a*b**2*c**2*d*n*x - a*b**2*c**2*d*p*x + 2*a*b**2*c 
*d**2*n*p*x**2 - a*b**2*c*d**2*n*x**2 - a*b**2*c*d**2*p*x**2 + b**3*c**...