\(\int \frac {(a+2 b x)^q (a x+b x^2)^p}{x} \, dx\) [221]

Optimal result

Integrand size = 23, antiderivative size = 165 \[ \int \frac {(a+2 b x)^q \left (a x+b x^2\right )^p}{x} \, dx=-\frac {(a+2 b x)^{1+q} \left (a x+b x^2\right )^p \left (1-\frac {(a+2 b x)^2}{a^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (1-p,\frac {1+q}{2},\frac {3+q}{2},\frac {(a+2 b x)^2}{a^2}\right )}{a (1+q)}-\frac {(a+2 b x)^{2+q} \left (a x+b x^2\right )^p \left (1-\frac {(a+2 b x)^2}{a^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (1-p,\frac {2+q}{2},\frac {4+q}{2},\frac {(a+2 b x)^2}{a^2}\right )}{a^2 (2+q)} \] Output:

-(2*b*x+a)^(1+q)*(b*x^2+a*x)^p*hypergeom([1-p, 1/2+1/2*q],[3/2+1/2*q],(2*b 
*x+a)^2/a^2)/a/(1+q)/((1-(2*b*x+a)^2/a^2)^p)-(2*b*x+a)^(2+q)*(b*x^2+a*x)^p 
*hypergeom([1-p, 1+1/2*q],[2+1/2*q],(2*b*x+a)^2/a^2)/a^2/(2+q)/((1-(2*b*x+ 
a)^2/a^2)^p)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 0.15 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.44 \[ \int \frac {(a+2 b x)^q \left (a x+b x^2\right )^p}{x} \, dx=\frac {\left (\frac {a+b x}{a}\right )^{-p} (x (a+b x))^p (a+2 b x)^q \left (\frac {a+2 b x}{a}\right )^{-q} \operatorname {AppellF1}\left (p,-p,-q,1+p,-\frac {b x}{a},-\frac {2 b x}{a}\right )}{p} \] Input:

Integrate[((a + 2*b*x)^q*(a*x + b*x^2)^p)/x,x]
 

Output:

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

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 0.37 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.44, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {1261, 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.

\(\displaystyle \int \frac {\left (a x+b x^2\right )^p (a+2 b x)^q}{x} \, dx\)

\(\Big \downarrow \) 1261

\(\displaystyle x^{-p} (a+b x)^{-p} \left (a x+b x^2\right )^p \int x^{p-1} (a+b x)^p (a+2 b x)^qdx\)

\(\Big \downarrow \) 152

\(\displaystyle x^{-p} \left (\frac {b x}{a}+1\right )^{-p} \left (a x+b x^2\right )^p \int x^{p-1} (a+2 b x)^q \left (\frac {b x}{a}+1\right )^pdx\)

\(\Big \downarrow \) 152

\(\displaystyle x^{-p} \left (\frac {b x}{a}+1\right )^{-p} \left (a x+b x^2\right )^p (a+2 b x)^q \left (\frac {2 b x}{a}+1\right )^{-q} \int x^{p-1} \left (\frac {b x}{a}+1\right )^p \left (\frac {2 b x}{a}+1\right )^qdx\)

\(\Big \downarrow \) 150

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

Input:

Int[((a + 2*b*x)^q*(a*x + b*x^2)^p)/x,x]
 

Output:

((a + 2*b*x)^q*(a*x + b*x^2)^p*AppellF1[p, -p, -q, 1 + p, -((b*x)/a), (-2* 
b*x)/a])/(p*(1 + (b*x)/a)^p*(1 + (2*b*x)/a)^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 1261

Int[((e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((b_.)*(x_) + (c_.)*(x_)^2) 
^(p_), x_Symbol] :> Simp[(e*x)^m*((b*x + c*x^2)^p/(x^(m + p)*(b + c*x)^p)) 
  Int[x^(m + p)*(f + g*x)^n*(b + c*x)^p, x], x] /; FreeQ[{b, c, e, f, g, m, 
 n}, x] &&  !IGtQ[n, 0]
 

Maple [F]

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

Input:

int((2*b*x+a)^q*(b*x^2+a*x)^p/x,x)
 

Output:

int((2*b*x+a)^q*(b*x^2+a*x)^p/x,x)
 

Fricas [F]

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

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

Output:

integral((b*x^2 + a*x)^p*(2*b*x + a)^q/x, x)
 

Sympy [F]

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

integrate((2*b*x+a)**q*(b*x**2+a*x)**p/x,x)
 

Output:

Integral((x*(a + b*x))**p*(a + 2*b*x)**q/x, x)
 

Maxima [F]

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

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

Output:

integrate((b*x^2 + a*x)^p*(2*b*x + a)^q/x, x)
 

Giac [F]

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

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

Output:

integrate((b*x^2 + a*x)^p*(2*b*x + a)^q/x, x)
 

Mupad [F(-1)]

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

int(((a*x + b*x^2)^p*(a + 2*b*x)^q)/x,x)
 

Output:

int(((a*x + b*x^2)^p*(a + 2*b*x)^q)/x, x)
 

Reduce [F]

\[ \int \frac {(a+2 b x)^q \left (a x+b x^2\right )^p}{x} \, dx=\frac {3 \left (2 b x +a \right )^{q} \left (b \,x^{2}+a x \right )^{p}-8 \left (\int \frac {\left (2 b x +a \right )^{q} \left (b \,x^{2}+a x \right )^{p} x}{4 b^{2} p \,x^{2}+2 b^{2} q \,x^{2}+6 a b p x +3 a b q x +2 a^{2} p +a^{2} q}d x \right ) b^{2} p^{2}-8 \left (\int \frac {\left (2 b x +a \right )^{q} \left (b \,x^{2}+a x \right )^{p} x}{4 b^{2} p \,x^{2}+2 b^{2} q \,x^{2}+6 a b p x +3 a b q x +2 a^{2} p +a^{2} q}d x \right ) b^{2} p q -2 \left (\int \frac {\left (2 b x +a \right )^{q} \left (b \,x^{2}+a x \right )^{p} x}{4 b^{2} p \,x^{2}+2 b^{2} q \,x^{2}+6 a b p x +3 a b q x +2 a^{2} p +a^{2} q}d x \right ) b^{2} q^{2}+2 \left (\int \frac {\left (2 b x +a \right )^{q} \left (b \,x^{2}+a x \right )^{p}}{4 b^{2} p \,x^{3}+2 b^{2} q \,x^{3}+6 a b p \,x^{2}+3 a b q \,x^{2}+2 a^{2} p x +a^{2} q x}d x \right ) a^{2} p^{2}+5 \left (\int \frac {\left (2 b x +a \right )^{q} \left (b \,x^{2}+a x \right )^{p}}{4 b^{2} p \,x^{3}+2 b^{2} q \,x^{3}+6 a b p \,x^{2}+3 a b q \,x^{2}+2 a^{2} p x +a^{2} q x}d x \right ) a^{2} p q +2 \left (\int \frac {\left (2 b x +a \right )^{q} \left (b \,x^{2}+a x \right )^{p}}{4 b^{2} p \,x^{3}+2 b^{2} q \,x^{3}+6 a b p \,x^{2}+3 a b q \,x^{2}+2 a^{2} p x +a^{2} q x}d x \right ) a^{2} q^{2}}{4 p +2 q} \] Input:

int((2*b*x+a)^q*(b*x^2+a*x)^p/x,x)
 

Output:

(3*(a + 2*b*x)**q*(a*x + b*x**2)**p - 8*int(((a + 2*b*x)**q*(a*x + b*x**2) 
**p*x)/(2*a**2*p + a**2*q + 6*a*b*p*x + 3*a*b*q*x + 4*b**2*p*x**2 + 2*b**2 
*q*x**2),x)*b**2*p**2 - 8*int(((a + 2*b*x)**q*(a*x + b*x**2)**p*x)/(2*a**2 
*p + a**2*q + 6*a*b*p*x + 3*a*b*q*x + 4*b**2*p*x**2 + 2*b**2*q*x**2),x)*b* 
*2*p*q - 2*int(((a + 2*b*x)**q*(a*x + b*x**2)**p*x)/(2*a**2*p + a**2*q + 6 
*a*b*p*x + 3*a*b*q*x + 4*b**2*p*x**2 + 2*b**2*q*x**2),x)*b**2*q**2 + 2*int 
(((a + 2*b*x)**q*(a*x + b*x**2)**p)/(2*a**2*p*x + a**2*q*x + 6*a*b*p*x**2 
+ 3*a*b*q*x**2 + 4*b**2*p*x**3 + 2*b**2*q*x**3),x)*a**2*p**2 + 5*int(((a + 
 2*b*x)**q*(a*x + b*x**2)**p)/(2*a**2*p*x + a**2*q*x + 6*a*b*p*x**2 + 3*a* 
b*q*x**2 + 4*b**2*p*x**3 + 2*b**2*q*x**3),x)*a**2*p*q + 2*int(((a + 2*b*x) 
**q*(a*x + b*x**2)**p)/(2*a**2*p*x + a**2*q*x + 6*a*b*p*x**2 + 3*a*b*q*x** 
2 + 4*b**2*p*x**3 + 2*b**2*q*x**3),x)*a**2*q**2)/(2*(2*p + q))