Integrand size = 34, antiderivative size = 93 \[ \int (g x)^n (d+e x)^2 \left (a d+(b d+a e) x+b e x^2\right )^p \, dx=\frac {d^2 (g x)^{1+n} \left (1+\frac {b x}{a}\right )^{-p} \left (1+\frac {e x}{d}\right )^{-p} \left (a d+(b d+a e) x+b e x^2\right )^p \operatorname {AppellF1}\left (1+n,-p,-2-p,2+n,-\frac {b x}{a},-\frac {e x}{d}\right )}{g (1+n)} \] Output:
d^2*(g*x)^(1+n)*(a*d+(a*e+b*d)*x+b*e*x^2)^p*AppellF1(1+n,-p,-2-p,2+n,-b*x/ a,-e*x/d)/g/(1+n)/((1+b*x/a)^p)/((1+e*x/d)^p)
Time = 0.53 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.85 \[ \int (g x)^n (d+e x)^2 \left (a d+(b d+a e) x+b e x^2\right )^p \, dx=\frac {x (g x)^n \left (1+\frac {b x}{a}\right )^{-p} ((a+b x) (d+e x))^p \left (1+\frac {e x}{d}\right )^{-p} \left (d^2 \left (6+5 n+n^2\right ) \operatorname {AppellF1}\left (1+n,-p,-p,2+n,-\frac {b x}{a},-\frac {e x}{d}\right )+e (1+n) x \left (2 d (3+n) \operatorname {AppellF1}\left (2+n,-p,-p,3+n,-\frac {b x}{a},-\frac {e x}{d}\right )+e (2+n) x \operatorname {AppellF1}\left (3+n,-p,-p,4+n,-\frac {b x}{a},-\frac {e x}{d}\right )\right )\right )}{(1+n) (2+n) (3+n)} \] Input:
Integrate[(g*x)^n*(d + e*x)^2*(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*(6 + 5*n + n^2)*AppellF1[1 + n, -p , -p, 2 + n, -((b*x)/a), -((e*x)/d)] + e*(1 + n)*x*(2*d*(3 + n)*AppellF1[2 + n, -p, -p, 3 + n, -((b*x)/a), -((e*x)/d)] + e*(2 + n)*x*AppellF1[3 + n, -p, -p, 4 + n, -((b*x)/a), -((e*x)/d)])))/((1 + n)*(2 + n)*(3 + n)*(1 + ( b*x)/a)^p*(1 + (e*x)/d)^p)
Time = 0.44 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, 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.
| \(\displaystyle \int (d+e x)^2 (g x)^n \left (x (a e+b d)+a d+b e x^2\right )^p \, dx\) |
| \(\Big \downarrow \) 1268 |
| \(\displaystyle (a+b x)^{-p} (d+e x)^{-p} \left (x (a e+b d)+a d+b e x^2\right )^p \int (g x)^n (a+b x)^p (d+e x)^{p+2}dx\) |
| \(\Big \downarrow \) 152 |
| \(\displaystyle \left (\frac {b x}{a}+1\right )^{-p} (d+e x)^{-p} \left (x (a e+b d)+a d+b e x^2\right )^p \int (g x)^n \left (\frac {b x}{a}+1\right )^p (d+e x)^{p+2}dx\) |
| \(\Big \downarrow \) 152 |
| \(\displaystyle d^2 \left (\frac {b x}{a}+1\right )^{-p} \left (\frac {e x}{d}+1\right )^{-p} \left (x (a e+b d)+a d+b e x^2\right )^p \int (g x)^n \left (\frac {b x}{a}+1\right )^p \left (\frac {e x}{d}+1\right )^{p+2}dx\) |
| \(\Big \downarrow \) 150 |
| \(\displaystyle \frac {d^2 (g x)^{n+1} \left (\frac {b x}{a}+1\right )^{-p} \left (\frac {e x}{d}+1\right )^{-p} \left (x (a e+b d)+a d+b e x^2\right )^p \operatorname {AppellF1}\left (n+1,-p,-p-2,n+2,-\frac {b x}{a},-\frac {e x}{d}\right )}{g (n+1)}\) |
Input:
Int[(g*x)^n*(d + e*x)^2*(a*d + (b*d + a*e)*x + b*e*x^2)^p,x]
Output:
(d^2*(g*x)^(1 + n)*(a*d + (b*d + a*e)*x + b*e*x^2)^p*AppellF1[1 + n, -p, - 2 - p, 2 + n, -((b*x)/a), -((e*x)/d)])/(g*(1 + n)*(1 + (b*x)/a)^p*(1 + (e* x)/d)^p)
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])
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]
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]
\[\int \left (g x \right )^{n} \left (e x +d \right )^{2} \left (a d +\left (a e +b d \right ) x +b e \,x^{2}\right )^{p}d x\]
Input:
int((g*x)^n*(e*x+d)^2*(a*d+(a*e+b*d)*x+b*e*x^2)^p,x)
Output:
int((g*x)^n*(e*x+d)^2*(a*d+(a*e+b*d)*x+b*e*x^2)^p,x)
\[ \int (g x)^n (d+e x)^2 \left (a d+(b d+a e) x+b e x^2\right )^p \, dx=\int { {\left (e x + d\right )}^{2} {\left (b e x^{2} + a d + {\left (b d + a e\right )} x\right )}^{p} \left (g x\right )^{n} \,d x } \] Input:
integrate((g*x)^n*(e*x+d)^2*(a*d+(a*e+b*d)*x+b*e*x^2)^p,x, algorithm="fric as")
Output:
integral((e^2*x^2 + 2*d*e*x + d^2)*(b*e*x^2 + a*d + (b*d + a*e)*x)^p*(g*x) ^n, x)
Timed out. \[ \int (g x)^n (d+e x)^2 \left (a d+(b d+a e) x+b e x^2\right )^p \, dx=\text {Timed out} \] Input:
integrate((g*x)**n*(e*x+d)**2*(a*d+(a*e+b*d)*x+b*e*x**2)**p,x)
Output:
Timed out
\[ \int (g x)^n (d+e x)^2 \left (a d+(b d+a e) x+b e x^2\right )^p \, dx=\int { {\left (e x + d\right )}^{2} {\left (b e x^{2} + a d + {\left (b d + a e\right )} x\right )}^{p} \left (g x\right )^{n} \,d x } \] Input:
integrate((g*x)^n*(e*x+d)^2*(a*d+(a*e+b*d)*x+b*e*x^2)^p,x, algorithm="maxi ma")
Output:
integrate((e*x + d)^2*(b*e*x^2 + a*d + (b*d + a*e)*x)^p*(g*x)^n, x)
\[ \int (g x)^n (d+e x)^2 \left (a d+(b d+a e) x+b e x^2\right )^p \, dx=\int { {\left (e x + d\right )}^{2} {\left (b e x^{2} + a d + {\left (b d + a e\right )} x\right )}^{p} \left (g x\right )^{n} \,d x } \] Input:
integrate((g*x)^n*(e*x+d)^2*(a*d+(a*e+b*d)*x+b*e*x^2)^p,x, algorithm="giac ")
Output:
integrate((e*x + d)^2*(b*e*x^2 + a*d + (b*d + a*e)*x)^p*(g*x)^n, x)
Timed out. \[ \int (g x)^n (d+e x)^2 \left (a d+(b d+a e) x+b e x^2\right )^p \, dx=\int {\left (g\,x\right )}^n\,{\left (d+e\,x\right )}^2\,{\left (b\,e\,x^2+\left (a\,e+b\,d\right )\,x+a\,d\right )}^p \,d x \] Input:
int((g*x)^n*(d + e*x)^2*(a*d + x*(a*e + b*d) + b*e*x^2)^p,x)
Output:
int((g*x)^n*(d + e*x)^2*(a*d + x*(a*e + b*d) + b*e*x^2)^p, x)
\[ \int (g x)^n (d+e x)^2 \left (a d+(b d+a e) x+b e x^2\right )^p \, dx=\text {too large to display} \] Input:
int((g*x)^n*(e*x+d)^2*(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**3*d*e**2*n**2*p + x**n* (a*d + a*e*x + b*d*x + b*e*x**2)**p*a**3*d*e**2*n*p**2 + 3*x**n*(a*d + a*e *x + b*d*x + b*e*x**2)**p*a**3*d*e**2*n*p + x**n*(a*d + a*e*x + b*d*x + b* e*x**2)**p*a**3*d*e**2*p**2 + 2*x**n*(a*d + a*e*x + b*d*x + b*e*x**2)**p*a **3*d*e**2*p - x**n*(a*d + a*e*x + b*d*x + b*e*x**2)**p*a**3*e**3*n**2*p*x - 2*x**n*(a*d + a*e*x + b*d*x + b*e*x**2)**p*a**3*e**3*n*p**2*x - 2*x**n* (a*d + a*e*x + b*d*x + b*e*x**2)**p*a**3*e**3*n*p*x - x**n*(a*d + a*e*x + b*d*x + b*e*x**2)**p*a**3*e**3*p**3*x - 2*x**n*(a*d + a*e*x + b*d*x + b*e* x**2)**p*a**3*e**3*p**2*x - 2*x**n*(a*d + a*e*x + b*d*x + b*e*x**2)**p*a** 2*b*d**2*e*n**2*p - 6*x**n*(a*d + a*e*x + b*d*x + b*e*x**2)**p*a**2*b*d**2 *e*n*p**2 - 8*x**n*(a*d + a*e*x + b*d*x + b*e*x**2)**p*a**2*b*d**2*e*n*p - 6*x**n*(a*d + a*e*x + b*d*x + b*e*x**2)**p*a**2*b*d**2*e*p**2 - 6*x**n*(a *d + a*e*x + b*d*x + b*e*x**2)**p*a**2*b*d**2*e*p + x**n*(a*d + a*e*x + b* d*x + b*e*x**2)**p*a**2*b*d*e**2*n**2*p*x + 6*x**n*(a*d + a*e*x + b*d*x + b*e*x**2)**p*a**2*b*d*e**2*n*p**2*x + 4*x**n*(a*d + a*e*x + b*d*x + b*e*x* *2)**p*a**2*b*d*e**2*n*p*x + 5*x**n*(a*d + a*e*x + b*d*x + b*e*x**2)**p*a* *2*b*d*e**2*p**3*x + 4*x**n*(a*d + a*e*x + b*d*x + b*e*x**2)**p*a**2*b*d*e **2*p**2*x + x**n*(a*d + a*e*x + b*d*x + b*e*x**2)**p*a**2*b*e**3*n**2*p*x **2 + 3*x**n*(a*d + a*e*x + b*d*x + b*e*x**2)**p*a**2*b*e**3*n*p**2*x**2 + x**n*(a*d + a*e*x + b*d*x + b*e*x**2)**p*a**2*b*e**3*n*p*x**2 + 2*x**n...