\(\int \frac {(c+d x)^q (a x^n+b x^{1+n})^p}{x} \, dx\) [274]

Optimal result

Integrand size = 26, antiderivative size = 82 \[ \int \frac {(c+d x)^q \left (a x^n+b x^{1+n}\right )^p}{x} \, dx=\frac {\left (1+\frac {b x}{a}\right )^{-p} (c+d x)^q \left (1+\frac {d x}{c}\right )^{-q} \left (a x^n+b x^{1+n}\right )^p \operatorname {AppellF1}\left (n p,-p,-q,1+n p,-\frac {b x}{a},-\frac {d x}{c}\right )}{n p} \] Output:

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

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.98 \[ \int \frac {(c+d x)^q \left (a x^n+b x^{1+n}\right )^p}{x} \, dx=\frac {\left (\frac {a+b x}{a}\right )^{-p} \left (x^n (a+b x)\right )^p (c+d x)^q \left (\frac {c+d x}{c}\right )^{-q} \operatorname {AppellF1}\left (n p,-p,-q,1+n p,-\frac {b x}{a},-\frac {d x}{c}\right )}{n p} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1948, 152, 152, 150}

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.

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

\(\Big \downarrow \) 1948

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

\(\Big \downarrow \) 152

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

\(\Big \downarrow \) 152

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

\(\Big \downarrow \) 150

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 150

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

rule 152

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

rule 1948

Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + 
(d_.)*(x_)^(n_.))^(q_.), x_Symbol] :> Simp[e^IntPart[m]*(e*x)^FracPart[m]*( 
(a*x^j + b*x^(j + n))^FracPart[p]/(x^(FracPart[m] + j*FracPart[p])*(a + b*x 
^n)^FracPart[p]))   Int[x^(m + j*p)*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; 
FreeQ[{a, b, c, d, e, j, m, n, p, q}, x] && EqQ[jn, j + n] &&  !IntegerQ[p] 
 && NeQ[b*c - a*d, 0] &&  !(EqQ[n, 1] && EqQ[j, 1])
 

Maple [F]

\[\int \frac {\left (d x +c \right )^{q} \left (a \,x^{n}+b \,x^{1+n}\right )^{p}}{x}d x\]

Input:

int((d*x+c)^q*(a*x^n+b*x^(1+n))^p/x,x)
 

Output:

int((d*x+c)^q*(a*x^n+b*x^(1+n))^p/x,x)
 

Fricas [F]

\[ \int \frac {(c+d x)^q \left (a x^n+b x^{1+n}\right )^p}{x} \, dx=\int { \frac {{\left (b x^{n + 1} + a x^{n}\right )}^{p} {\left (d x + c\right )}^{q}}{x} \,d x } \] Input:

integrate((d*x+c)^q*(a*x^n+b*x^(1+n))^p/x,x, algorithm="fricas")
 

Output:

integral((b*x^(n + 1) + a*x^n)^p*(d*x + c)^q/x, x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x)^q \left (a x^n+b x^{1+n}\right )^p}{x} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**q*(a*x**n+b*x**(1+n))**p/x,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(c+d x)^q \left (a x^n+b x^{1+n}\right )^p}{x} \, dx=\int { \frac {{\left (b x^{n + 1} + a x^{n}\right )}^{p} {\left (d x + c\right )}^{q}}{x} \,d x } \] Input:

integrate((d*x+c)^q*(a*x^n+b*x^(1+n))^p/x,x, algorithm="maxima")
 

Output:

integrate((b*x^(n + 1) + a*x^n)^p*(d*x + c)^q/x, x)
 

Giac [F]

\[ \int \frac {(c+d x)^q \left (a x^n+b x^{1+n}\right )^p}{x} \, dx=\int { \frac {{\left (b x^{n + 1} + a x^{n}\right )}^{p} {\left (d x + c\right )}^{q}}{x} \,d x } \] Input:

integrate((d*x+c)^q*(a*x^n+b*x^(1+n))^p/x,x, algorithm="giac")
 

Output:

integrate((b*x^(n + 1) + a*x^n)^p*(d*x + c)^q/x, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^q \left (a x^n+b x^{1+n}\right )^p}{x} \, dx=\int \frac {{\left (a\,x^n+b\,x^{n+1}\right )}^p\,{\left (c+d\,x\right )}^q}{x} \,d x \] Input:

int(((a*x^n + b*x^(n + 1))^p*(c + d*x)^q)/x,x)
 

Output:

int(((a*x^n + b*x^(n + 1))^p*(c + d*x)^q)/x, x)
 

Reduce [F]

\[ \int \frac {(c+d x)^q \left (a x^n+b x^{1+n}\right )^p}{x} \, dx=\text {too large to display} \] Input:

int((d*x+c)^q*(a*x^n+b*x^(1+n))^p/x,x)
 

Output:

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