Integrand size = 34, antiderivative size = 103 \[ \int (g x)^n (d+e x)^m \left (a d+(b d+a e) x+b e x^2\right )^p \, dx=\frac {(g x)^{1+n} \left (1+\frac {b x}{a}\right )^{-p} (d+e x)^m \left (1+\frac {e x}{d}\right )^{-m-p} \left (a d+(b d+a e) x+b e x^2\right )^p \operatorname {AppellF1}\left (1+n,-p,-m-p,2+n,-\frac {b x}{a},-\frac {e x}{d}\right )}{g (1+n)} \] Output:
(g*x)^(1+n)*(e*x+d)^m*(1+e*x/d)^(-m-p)*(a*d+(a*e+b*d)*x+b*e*x^2)^p*AppellF 1(1+n,-p,-m-p,2+n,-b*x/a,-e*x/d)/g/(1+n)/((1+b*x/a)^p)
Time = 0.36 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.90 \[ \int (g x)^n (d+e x)^m \left (a d+(b d+a e) x+b e x^2\right )^p \, dx=\frac {x (g x)^n \left (\frac {a+b x}{a}\right )^{-p} (d+e x)^m \left (\frac {d+e x}{d}\right )^{-m-p} ((a+b x) (d+e x))^p \operatorname {AppellF1}\left (1+n,-p,-m-p,2+n,-\frac {b x}{a},-\frac {e x}{d}\right )}{1+n} \] Input:
Integrate[(g*x)^n*(d + e*x)^m*(a*d + (b*d + a*e)*x + b*e*x^2)^p,x]
Output:
(x*(g*x)^n*(d + e*x)^m*((d + e*x)/d)^(-m - p)*((a + b*x)*(d + e*x))^p*Appe llF1[1 + n, -p, -m - p, 2 + n, -((b*x)/a), -((e*x)/d)])/((1 + n)*((a + b*x )/a)^p)
Time = 0.44 (sec) , antiderivative size = 103, 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.
| \(\displaystyle \int (g x)^n (d+e x)^m \left (x (a e+b d)+a d+b e x^2\right )^p \, dx\) |
| \(\Big \downarrow \) 1268 |
| \(\displaystyle (a+b x)^{-p} (d+e x)^{-p} \left (x (a e+b d)+a d+b e x^2\right )^p \int (g x)^n (a+b x)^p (d+e x)^{m+p}dx\) |
| \(\Big \downarrow \) 152 |
| \(\displaystyle \left (\frac {b x}{a}+1\right )^{-p} (d+e x)^{-p} \left (x (a e+b d)+a d+b e x^2\right )^p \int (g x)^n \left (\frac {b x}{a}+1\right )^p (d+e x)^{m+p}dx\) |
| \(\Big \downarrow \) 152 |
| \(\displaystyle \left (\frac {b x}{a}+1\right )^{-p} (d+e x)^m \left (\frac {e x}{d}+1\right )^{-m-p} \left (x (a e+b d)+a d+b e x^2\right )^p \int (g x)^n \left (\frac {b x}{a}+1\right )^p \left (\frac {e x}{d}+1\right )^{m+p}dx\) |
| \(\Big \downarrow \) 150 |
| \(\displaystyle \frac {(g x)^{n+1} \left (\frac {b x}{a}+1\right )^{-p} (d+e x)^m \left (\frac {e x}{d}+1\right )^{-m-p} \left (x (a e+b d)+a d+b e x^2\right )^p \operatorname {AppellF1}\left (n+1,-p,-m-p,n+2,-\frac {b x}{a},-\frac {e x}{d}\right )}{g (n+1)}\) |
Input:
Int[(g*x)^n*(d + e*x)^m*(a*d + (b*d + a*e)*x + b*e*x^2)^p,x]
Output:
((g*x)^(1 + n)*(d + e*x)^m*(1 + (e*x)/d)^(-m - p)*(a*d + (b*d + a*e)*x + b *e*x^2)^p*AppellF1[1 + n, -p, -m - p, 2 + n, -((b*x)/a), -((e*x)/d)])/(g*( 1 + n)*(1 + (b*x)/a)^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]
\[\int \left (g x \right )^{n} \left (e x +d \right )^{m} \left (a d +\left (a e +b d \right ) x +b e \,x^{2}\right )^{p}d x\]
Input:
int((g*x)^n*(e*x+d)^m*(a*d+(a*e+b*d)*x+b*e*x^2)^p,x)
Output:
int((g*x)^n*(e*x+d)^m*(a*d+(a*e+b*d)*x+b*e*x^2)^p,x)
\[ \int (g x)^n (d+e x)^m \left (a d+(b d+a e) x+b e x^2\right )^p \, dx=\int { {\left (b e x^{2} + a d + {\left (b d + a e\right )} x\right )}^{p} {\left (e x + d\right )}^{m} \left (g x\right )^{n} \,d x } \] Input:
integrate((g*x)^n*(e*x+d)^m*(a*d+(a*e+b*d)*x+b*e*x^2)^p,x, algorithm="fric as")
Output:
integral((b*e*x^2 + a*d + (b*d + a*e)*x)^p*(e*x + d)^m*(g*x)^n, x)
Timed out. \[ \int (g x)^n (d+e x)^m \left (a d+(b d+a e) x+b e x^2\right )^p \, dx=\text {Timed out} \] Input:
integrate((g*x)**n*(e*x+d)**m*(a*d+(a*e+b*d)*x+b*e*x**2)**p,x)
Output:
Timed out
\[ \int (g x)^n (d+e x)^m \left (a d+(b d+a e) x+b e x^2\right )^p \, dx=\int { {\left (b e x^{2} + a d + {\left (b d + a e\right )} x\right )}^{p} {\left (e x + d\right )}^{m} \left (g x\right )^{n} \,d x } \] Input:
integrate((g*x)^n*(e*x+d)^m*(a*d+(a*e+b*d)*x+b*e*x^2)^p,x, algorithm="maxi ma")
Output:
integrate((b*e*x^2 + a*d + (b*d + a*e)*x)^p*(e*x + d)^m*(g*x)^n, x)
\[ \int (g x)^n (d+e x)^m \left (a d+(b d+a e) x+b e x^2\right )^p \, dx=\int { {\left (b e x^{2} + a d + {\left (b d + a e\right )} x\right )}^{p} {\left (e x + d\right )}^{m} \left (g x\right )^{n} \,d x } \] Input:
integrate((g*x)^n*(e*x+d)^m*(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*(e*x + d)^m*(g*x)^n, x)
Timed out. \[ \int (g x)^n (d+e x)^m \left (a d+(b d+a e) x+b e x^2\right )^p \, dx=\int {\left (g\,x\right )}^n\,{\left (d+e\,x\right )}^m\,{\left (b\,e\,x^2+\left (a\,e+b\,d\right )\,x+a\,d\right )}^p \,d x \] Input:
int((g*x)^n*(d + e*x)^m*(a*d + x*(a*e + b*d) + b*e*x^2)^p,x)
Output:
int((g*x)^n*(d + e*x)^m*(a*d + x*(a*e + b*d) + b*e*x^2)^p, x)
\[ \int (g x)^n (d+e x)^m \left (a d+(b d+a e) x+b e x^2\right )^p \, dx=\text {too large to display} \] Input:
int((g*x)^n*(e*x+d)^m*(a*d+(a*e+b*d)*x+b*e*x^2)^p,x)
Output:
(g**n*(x**n*(d + e*x)**m*(a*d + a*e*x + b*d*x + b*e*x**2)**p*a*d*m + 2*x** n*(d + e*x)**m*(a*d + a*e*x + b*d*x + b*e*x**2)**p*a*d*p + x**n*(d + e*x)* *m*(a*d + a*e*x + b*d*x + b*e*x**2)**p*a*e*m*x + x**n*(d + e*x)**m*(a*d + a*e*x + b*d*x + b*e*x**2)**p*a*e*n*x + x**n*(d + e*x)**m*(a*d + a*e*x + b* d*x + b*e*x**2)**p*a*e*p*x + x**n*(d + e*x)**m*(a*d + a*e*x + b*d*x + b*e* x**2)**p*b*d*n*x + x**n*(d + e*x)**m*(a*d + a*e*x + b*d*x + b*e*x**2)**p*b *d*p*x + int((x**n*(d + e*x)**m*(a*d + a*e*x + b*d*x + b*e*x**2)**p*x)/(a* *2*d*e*m**2 + 2*a**2*d*e*m*n + 3*a**2*d*e*m*p + a**2*d*e*m + 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* m**2*x + 2*a**2*e**2*m*n*x + 3*a**2*e**2*m*p*x + a**2*e**2*m*x + 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*m*n + a*b*d**2*m*p + 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 + a*b*d*e*m**2*x + 3*a*b*d*e*m*n*x + 4*a*b*d*e*m*p*x + a*b*d*e*m*x + 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*m**2*x**2 + 2*a* b*e**2*m*n*x**2 + 3*a*b*e**2*m*p*x**2 + a*b*e**2*m*x**2 + 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*m*n*x + b**2*d**2*m*p*x + 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 *m*n*x**2 + b**2*d*e*m*p*x**2 + b**2*d*e*n**2*x**2 + 3*b**2*d*e*n*p*x**...