\(\int (g x)^n (a d+(b d+a e) x+b e x^2)^p \, dx\) [148]

Optimal result

Integrand size = 27, antiderivative size = 88 \[ \int (g x)^n \left (a d+(b d+a e) x+b e x^2\right )^p \, dx=\frac {(g x)^{1+n} \left (1+\frac {b x}{a}\right )^{-p} \left (1+\frac {e x}{d}\right )^{-p} \left (a d+(b d+a e) x+b e x^2\right )^p \operatorname {AppellF1}\left (1+n,-p,-p,2+n,-\frac {b x}{a},-\frac {e x}{d}\right )}{g (1+n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.89 \[ \int (g x)^n \left (a d+(b d+a e) x+b e x^2\right )^p \, dx=\frac {x (g x)^n \left (\frac {a+b x}{a}\right )^{-p} \left (\frac {d+e x}{d}\right )^{-p} ((a+b x) (d+e x))^p \operatorname {AppellF1}\left (1+n,-p,-p,2+n,-\frac {b x}{a},-\frac {e x}{d}\right )}{1+n} \] Input:

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

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1179, 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.

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

\(\Big \downarrow \) 1179

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

\(\Big \downarrow \) 150

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

Input:

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

Output:

((g*x)^(1 + n)*(a*d + (b*d + a*e)*x + b*e*x^2)^p*AppellF1[1 + n, -p, -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 1179

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(a + b*x + c*x^2)^p/(e*(1 - ( 
d + e*x)/(d - e*((b - q)/(2*c))))^p*(1 - (d + e*x)/(d - e*((b + q)/(2*c)))) 
^p)   Subst[Int[x^m*Simp[1 - x/(d - e*((b - q)/(2*c))), x]^p*Simp[1 - x/(d 
- e*((b + q)/(2*c))), x]^p, x], x, d + e*x], x]] /; FreeQ[{a, b, c, d, e, m 
, p}, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(g**n*(2*x**n*(a*d + a*e*x + b*d*x + b*e*x**2)**p*a*d*p + x**n*(a*d + a*e* 
x + b*d*x + b*e*x**2)**p*a*e*n*x + x**n*(a*d + a*e*x + b*d*x + b*e*x**2)** 
p*a*e*p*x + x**n*(a*d + a*e*x + b*d*x + b*e*x**2)**p*b*d*n*x + x**n*(a*d + 
 a*e*x + b*d*x + b*e*x**2)**p*b*d*p*x + int((x**n*(a*d + a*e*x + b*d*x + b 
*e*x**2)**p*x)/(a**2*d*e*n**2 + 3*a**2*d*e*n*p + a**2*d*e*n + 2*a**2*d*e*p 
**2 + a**2*d*e*p + a**2*e**2*n**2*x + 3*a**2*e**2*n*p*x + a**2*e**2*n*x + 
2*a**2*e**2*p**2*x + a**2*e**2*p*x + a*b*d**2*n**2 + 3*a*b*d**2*n*p + a*b* 
d**2*n + 2*a*b*d**2*p**2 + a*b*d**2*p + 2*a*b*d*e*n**2*x + 6*a*b*d*e*n*p*x 
 + 2*a*b*d*e*n*x + 4*a*b*d*e*p**2*x + 2*a*b*d*e*p*x + a*b*e**2*n**2*x**2 + 
 3*a*b*e**2*n*p*x**2 + a*b*e**2*n*x**2 + 2*a*b*e**2*p**2*x**2 + a*b*e**2*p 
*x**2 + b**2*d**2*n**2*x + 3*b**2*d**2*n*p*x + b**2*d**2*n*x + 2*b**2*d**2 
*p**2*x + b**2*d**2*p*x + b**2*d*e*n**2*x**2 + 3*b**2*d*e*n*p*x**2 + b**2* 
d*e*n*x**2 + 2*b**2*d*e*p**2*x**2 + b**2*d*e*p*x**2),x)*a**3*e**3*n**3*p + 
 4*int((x**n*(a*d + a*e*x + b*d*x + b*e*x**2)**p*x)/(a**2*d*e*n**2 + 3*a** 
2*d*e*n*p + a**2*d*e*n + 2*a**2*d*e*p**2 + a**2*d*e*p + a**2*e**2*n**2*x + 
 3*a**2*e**2*n*p*x + a**2*e**2*n*x + 2*a**2*e**2*p**2*x + a**2*e**2*p*x + 
a*b*d**2*n**2 + 3*a*b*d**2*n*p + a*b*d**2*n + 2*a*b*d**2*p**2 + a*b*d**2*p 
 + 2*a*b*d*e*n**2*x + 6*a*b*d*e*n*p*x + 2*a*b*d*e*n*x + 4*a*b*d*e*p**2*x + 
 2*a*b*d*e*p*x + a*b*e**2*n**2*x**2 + 3*a*b*e**2*n*p*x**2 + a*b*e**2*n*x** 
2 + 2*a*b*e**2*p**2*x**2 + a*b*e**2*p*x**2 + b**2*d**2*n**2*x + 3*b**2*...