\(\int (g x)^n (d+e x) (a d+(b d+a e) x+b e x^2)^p \, dx\) [147]

Optimal result

Integrand size = 32, antiderivative size = 91 \[ \int (g x)^n (d+e x) \left (a d+(b d+a e) x+b e x^2\right )^p \, dx=\frac {d (g x)^{1+n} \left (1+\frac {b x}{a}\right )^{-p} \left (1+\frac {e x}{d}\right )^{-p} \left (a d+(b d+a e) x+b e x^2\right )^p \operatorname {AppellF1}\left (1+n,-p,-1-p,2+n,-\frac {b x}{a},-\frac {e x}{d}\right )}{g (1+n)} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 0.39 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.32 \[ \int (g x)^n (d+e x) \left (a d+(b d+a e) x+b e x^2\right )^p \, dx=\frac {x (g x)^n \left (1+\frac {b x}{a}\right )^{-p} ((a+b x) (d+e x))^p \left (1+\frac {e x}{d}\right )^{-p} \left (d (2+n) \operatorname {AppellF1}\left (1+n,-p,-p,2+n,-\frac {b x}{a},-\frac {e x}{d}\right )+e (1+n) x \operatorname {AppellF1}\left (2+n,-p,-p,3+n,-\frac {b x}{a},-\frac {e x}{d}\right )\right )}{(1+n) (2+n)} \] Input:

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

Output:

(x*(g*x)^n*((a + b*x)*(d + e*x))^p*(d*(2 + n)*AppellF1[1 + n, -p, -p, 2 + 
n, -((b*x)/a), -((e*x)/d)] + e*(1 + n)*x*AppellF1[2 + n, -p, -p, 3 + n, -( 
(b*x)/a), -((e*x)/d)]))/((1 + n)*(2 + n)*(1 + (b*x)/a)^p*(1 + (e*x)/d)^p)
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1268, 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.

\(\displaystyle \int (d+e x) (g x)^n \left (x (a e+b d)+a d+b e x^2\right )^p \, dx\)

\(\Big \downarrow \) 1268

\(\displaystyle (a+b x)^{-p} (d+e x)^{-p} \left (x (a e+b d)+a d+b e x^2\right )^p \int (g x)^n (a+b x)^p (d+e x)^{p+1}dx\)

\(\Big \downarrow \) 152

\(\displaystyle \left (\frac {b x}{a}+1\right )^{-p} (d+e x)^{-p} \left (x (a e+b d)+a d+b e x^2\right )^p \int (g x)^n \left (\frac {b x}{a}+1\right )^p (d+e x)^{p+1}dx\)

\(\Big \downarrow \) 152

\(\displaystyle d \left (\frac {b x}{a}+1\right )^{-p} \left (\frac {e x}{d}+1\right )^{-p} \left (x (a e+b d)+a d+b e x^2\right )^p \int (g x)^n \left (\frac {b x}{a}+1\right )^p \left (\frac {e x}{d}+1\right )^{p+1}dx\)

\(\Big \downarrow \) 150

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

Input:

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

Output:

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

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 1268

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

Maple [F]

\[\int \left (g x \right )^{n} \left (e x +d \right ) \left (a d +\left (a e +b d \right ) x +b e \,x^{2}\right )^{p}d x\]

Input:

int((g*x)^n*(e*x+d)*(a*d+(a*e+b*d)*x+b*e*x^2)^p,x)
 

Output:

int((g*x)^n*(e*x+d)*(a*d+(a*e+b*d)*x+b*e*x^2)^p,x)
 

Fricas [F]

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

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

Output:

integral((e*x + d)*(b*e*x^2 + a*d + (b*d + a*e)*x)^p*(g*x)^n, x)
 

Sympy [F(-1)]

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

integrate((g*x)**n*(e*x+d)*(a*d+(a*e+b*d)*x+b*e*x**2)**p,x)
 

Output:

Timed out
 

Maxima [F]

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

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

Output:

integrate((e*x + d)*(b*e*x^2 + a*d + (b*d + a*e)*x)^p*(g*x)^n, x)
 

Giac [F]

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

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

Output:

integrate((e*x + d)*(b*e*x^2 + a*d + (b*d + a*e)*x)^p*(g*x)^n, x)
 

Mupad [F(-1)]

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

int((g*x)^n*(d + e*x)*(a*d + x*(a*e + b*d) + b*e*x^2)^p,x)
 

Output:

int((g*x)^n*(d + e*x)*(a*d + x*(a*e + b*d) + b*e*x^2)^p, x)
 

Reduce [F]

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

int((g*x)^n*(e*x+d)*(a*d+(a*e+b*d)*x+b*e*x^2)^p,x)
 

Output:

(g**n*( - x**n*(a*d + a*e*x + b*d*x + b*e*x**2)**p*a**2*d*e*n*p - x**n*(a* 
d + a*e*x + b*d*x + b*e*x**2)**p*a**2*d*e*p + x**n*(a*d + a*e*x + b*d*x + 
b*e*x**2)**p*a**2*e**2*n*p*x + x**n*(a*d + a*e*x + b*d*x + b*e*x**2)**p*a* 
*2*e**2*p**2*x + x**n*(a*d + a*e*x + b*d*x + b*e*x**2)**p*a*b*d**2*n*p + 4 
*x**n*(a*d + a*e*x + b*d*x + b*e*x**2)**p*a*b*d**2*p**2 + 3*x**n*(a*d + a* 
e*x + b*d*x + b*e*x**2)**p*a*b*d**2*p + x**n*(a*d + a*e*x + b*d*x + b*e*x* 
*2)**p*a*b*d*e*n**2*x + 5*x**n*(a*d + a*e*x + b*d*x + b*e*x**2)**p*a*b*d*e 
*n*p*x + 2*x**n*(a*d + a*e*x + b*d*x + b*e*x**2)**p*a*b*d*e*n*x + 4*x**n*( 
a*d + a*e*x + b*d*x + b*e*x**2)**p*a*b*d*e*p**2*x + 2*x**n*(a*d + a*e*x + 
b*d*x + b*e*x**2)**p*a*b*d*e*p*x + x**n*(a*d + a*e*x + b*d*x + b*e*x**2)** 
p*a*b*e**2*n**2*x**2 + 3*x**n*(a*d + a*e*x + b*d*x + b*e*x**2)**p*a*b*e**2 
*n*p*x**2 + x**n*(a*d + a*e*x + b*d*x + b*e*x**2)**p*a*b*e**2*n*x**2 + 2*x 
**n*(a*d + a*e*x + b*d*x + b*e*x**2)**p*a*b*e**2*p**2*x**2 + x**n*(a*d + a 
*e*x + b*d*x + b*e*x**2)**p*a*b*e**2*p*x**2 + x**n*(a*d + a*e*x + b*d*x + 
b*e*x**2)**p*b**2*d**2*n**2*x + 4*x**n*(a*d + a*e*x + b*d*x + b*e*x**2)**p 
*b**2*d**2*n*p*x + 2*x**n*(a*d + a*e*x + b*d*x + b*e*x**2)**p*b**2*d**2*n* 
x + 3*x**n*(a*d + a*e*x + b*d*x + b*e*x**2)**p*b**2*d**2*p**2*x + 2*x**n*( 
a*d + a*e*x + b*d*x + b*e*x**2)**p*b**2*d**2*p*x + x**n*(a*d + a*e*x + b*d 
*x + b*e*x**2)**p*b**2*d*e*n**2*x**2 + 3*x**n*(a*d + a*e*x + b*d*x + b*e*x 
**2)**p*b**2*d*e*n*p*x**2 + x**n*(a*d + a*e*x + b*d*x + b*e*x**2)**p*b*...