\(\int (f+g x)^n (a d+(b d+a e) x+b e x^2)^p \, dx\) [313]

Optimal result

Integrand size = 29, antiderivative size = 134 \[ \int (f+g x)^n \left (a d+(b d+a e) x+b e x^2\right )^p \, dx=\frac {(f+g x)^{1+n} \left (a d+(b d+a e) x+b e x^2\right )^p \left (1-\frac {b (f+g x)}{b f-a g}\right )^{-p} \left (1-\frac {e (f+g x)}{e f-d g}\right )^{-p} \operatorname {AppellF1}\left (1+n,-p,-p,2+n,\frac {b (f+g x)}{b f-a g},\frac {e (f+g x)}{e f-d g}\right )}{g (1+n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.92 \[ \int (f+g x)^n \left (a d+(b d+a e) x+b e x^2\right )^p \, dx=\frac {(a+b x) \left (\frac {b (d+e x)}{b d-a e}\right )^{-p} ((a+b x) (d+e x))^p (f+g x)^n \left (\frac {b (f+g x)}{b f-a g}\right )^{-n} \operatorname {AppellF1}\left (1+p,-p,-n,2+p,\frac {e (a+b x)}{-b d+a e},\frac {g (a+b x)}{-b f+a g}\right )}{b (1+p)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, 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.

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

\(\Big \downarrow \) 1179

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

\(\Big \downarrow \) 150

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F]

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

integrate((g*x+f)^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 + f)^n, x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate((g*x+f)^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 + f)^n, x)
 

Giac [F]

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

integrate((g*x+f)^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 + f)^n, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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