Integrand size = 35, antiderivative size = 263 \[ \int (c+d x)^n (e+f x)^{1+n} (a (d e+c f)+2 a d f x)^m \, dx=\frac {(d e-c f) (c+d x)^n (e+f x)^n (a (d e+c f)+2 a d f x)^{1+m} \left (1-\frac {(d e+c f+2 d f x)^2}{(d e-c f)^2}\right )^{-n} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},-n,\frac {3+m}{2},\frac {(d e+c f+2 d f x)^2}{(d e-c f)^2}\right )}{4 a d^2 f (1+m)}+\frac {(c+d x)^n (e+f x)^n (a (d e+c f)+2 a d f x)^{2+m} \left (1-\frac {(d e+c f+2 d f x)^2}{(d e-c f)^2}\right )^{-n} \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},-n,\frac {4+m}{2},\frac {(d e+c f+2 d f x)^2}{(d e-c f)^2}\right )}{4 a^2 d^2 f (2+m)} \] Output:
1/4*(-c*f+d*e)*(d*x+c)^n*(f*x+e)^n*(a*(c*f+d*e)+2*a*d*f*x)^(1+m)*hypergeom ([-n, 1/2+1/2*m],[3/2+1/2*m],(2*d*f*x+c*f+d*e)^2/(-c*f+d*e)^2)/a/d^2/f/(1+ m)/((1-(2*d*f*x+c*f+d*e)^2/(-c*f+d*e)^2)^n)+1/4*(d*x+c)^n*(f*x+e)^n*(a*(c* f+d*e)+2*a*d*f*x)^(2+m)*hypergeom([-n, 1+1/2*m],[2+1/2*m],(2*d*f*x+c*f+d*e )^2/(-c*f+d*e)^2)/a^2/d^2/f/(2+m)/((1-(2*d*f*x+c*f+d*e)^2/(-c*f+d*e)^2)^n)
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 0.50 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.54 \[ \int (c+d x)^n (e+f x)^{1+n} (a (d e+c f)+2 a d f x)^m \, dx=\frac {(c+d x)^{1+n} (e+f x)^{1+n} \left (\frac {d (e+f x)}{d e-c f}\right )^{-1-n} \left (\frac {d e+c f+2 d f x}{d e-c f}\right )^{-m} (a (c f+d (e+2 f x)))^m \operatorname {AppellF1}\left (1+n,-1-n,-m,2+n,\frac {f (c+d x)}{-d e+c f},\frac {2 f (c+d x)}{-d e+c f}\right )}{d (1+n)} \] Input:
Integrate[(c + d*x)^n*(e + f*x)^(1 + n)*(a*(d*e + c*f) + 2*a*d*f*x)^m,x]
Output:
((c + d*x)^(1 + n)*(e + f*x)^(1 + n)*((d*(e + f*x))/(d*e - c*f))^(-1 - n)* (a*(c*f + d*(e + 2*f*x)))^m*AppellF1[1 + n, -1 - n, -m, 2 + n, (f*(c + d*x ))/(-(d*e) + c*f), (2*f*(c + d*x))/(-(d*e) + c*f)])/(d*(1 + n)*((d*e + c*f + 2*d*f*x)/(d*e - c*f))^m)
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 0.33 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.57, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {157, 156, 155}
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.
| \(\displaystyle \int (c+d x)^n (e+f x)^{n+1} (a (c f+d e)+2 a d f x)^m \, dx\) |
| \(\Big \downarrow \) 157 |
| \(\displaystyle \frac {(d e-c f) (e+f x)^n \left (\frac {d (e+f x)}{d e-c f}\right )^{-n} \int (c+d x)^n (a (d e+c f)+2 a d f x)^m \left (\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}\right )^{n+1}dx}{d}\) |
| \(\Big \downarrow \) 156 |
| \(\displaystyle \frac {(d e-c f) (e+f x)^n \left (\frac {c f+d e+2 d f x}{d e-c f}\right )^{-m} \left (\frac {d (e+f x)}{d e-c f}\right )^{-n} (a (c f+d e)+2 a d f x)^m \int (c+d x)^n \left (\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}\right )^{n+1} \left (\frac {d e+c f}{d e-c f}+\frac {2 d f x}{d e-c f}\right )^mdx}{d}\) |
| \(\Big \downarrow \) 155 |
| \(\displaystyle \frac {(d e-c f) (c+d x)^{n+1} (e+f x)^n \left (\frac {c f+d e+2 d f x}{d e-c f}\right )^{-m} \left (\frac {d (e+f x)}{d e-c f}\right )^{-n} (a (c f+d e)+2 a d f x)^m \operatorname {AppellF1}\left (n+1,-n-1,-m,n+2,-\frac {f (c+d x)}{d e-c f},-\frac {2 f (c+d x)}{d e-c f}\right )}{d^2 (n+1)}\) |
Input:
Int[(c + d*x)^n*(e + f*x)^(1 + n)*(a*(d*e + c*f) + 2*a*d*f*x)^m,x]
Output:
((d*e - c*f)*(c + d*x)^(1 + n)*(e + f*x)^n*(a*(d*e + c*f) + 2*a*d*f*x)^m*A ppellF1[1 + n, -1 - n, -m, 2 + n, -((f*(c + d*x))/(d*e - c*f)), (-2*f*(c + d*x))/(d*e - c*f)])/(d^2*(1 + n)*((d*(e + f*x))/(d*e - c*f))^n*((d*e + c* f + 2*d*f*x)/(d*e - c*f))^m)
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) ^(p_), x_] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*Simplify[b/(b*c - a*d)]^n* Simplify[b/(b*e - a*f)]^p))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/ (b*c - a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[m] && !IntegerQ[n] && !IntegerQ[p] && GtQ[Sim plify[b/(b*c - a*d)], 0] && GtQ[Simplify[b/(b*e - a*f)], 0] && !(GtQ[Simpl ify[d/(d*a - c*b)], 0] && GtQ[Simplify[d/(d*e - c*f)], 0] && SimplerQ[c + d *x, a + b*x]) && !(GtQ[Simplify[f/(f*a - e*b)], 0] && GtQ[Simplify[f/(f*c - e*d)], 0] && SimplerQ[e + f*x, a + b*x])
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) ^(p_), x_] :> Simp[(e + f*x)^FracPart[p]/(Simplify[b/(b*e - a*f)]^IntPart[p ]*(b*((e + f*x)/(b*e - a*f)))^FracPart[p]) Int[(a + b*x)^m*(c + d*x)^n*Si mp[b*(e/(b*e - a*f)) + b*f*(x/(b*e - a*f)), x]^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[m] && !IntegerQ[n] && !IntegerQ[p] & & GtQ[Simplify[b/(b*c - a*d)], 0] && !GtQ[Simplify[b/(b*e - a*f)], 0]
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) ^(p_), x_] :> Simp[(c + d*x)^FracPart[n]/(Simplify[b/(b*c - a*d)]^IntPart[n ]*(b*((c + d*x)/(b*c - a*d)))^FracPart[n]) Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d)), x]^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[m] && !IntegerQ[n] && !IntegerQ[p] & & !GtQ[Simplify[b/(b*c - a*d)], 0] && !SimplerQ[c + d*x, a + b*x] && !Si mplerQ[e + f*x, a + b*x]
\[\int \left (x d +c \right )^{n} \left (f x +e \right )^{1+n} \left (a \left (c f +d e \right )+2 a d f x \right )^{m}d x\]
Input:
int((d*x+c)^n*(f*x+e)^(1+n)*(a*(c*f+d*e)+2*a*d*f*x)^m,x)
Output:
int((d*x+c)^n*(f*x+e)^(1+n)*(a*(c*f+d*e)+2*a*d*f*x)^m,x)
\[ \int (c+d x)^n (e+f x)^{1+n} (a (d e+c f)+2 a d f x)^m \, dx=\int { {\left (2 \, a d f x + {\left (d e + c f\right )} a\right )}^{m} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{n + 1} \,d x } \] Input:
integrate((d*x+c)^n*(f*x+e)^(1+n)*(a*(c*f+d*e)+2*a*d*f*x)^m,x, algorithm=" fricas")
Output:
integral((2*a*d*f*x + a*d*e + a*c*f)^m*(d*x + c)^n*(f*x + e)^(n + 1), x)
Exception generated. \[ \int (c+d x)^n (e+f x)^{1+n} (a (d e+c f)+2 a d f x)^m \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:
integrate((d*x+c)**n*(f*x+e)**(1+n)*(a*(c*f+d*e)+2*a*d*f*x)**m,x)
Output:
Exception raised: HeuristicGCDFailed >> no luck
\[ \int (c+d x)^n (e+f x)^{1+n} (a (d e+c f)+2 a d f x)^m \, dx=\int { {\left (2 \, a d f x + {\left (d e + c f\right )} a\right )}^{m} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{n + 1} \,d x } \] Input:
integrate((d*x+c)^n*(f*x+e)^(1+n)*(a*(c*f+d*e)+2*a*d*f*x)^m,x, algorithm=" maxima")
Output:
integrate((2*a*d*f*x + (d*e + c*f)*a)^m*(d*x + c)^n*(f*x + e)^(n + 1), x)
\[ \int (c+d x)^n (e+f x)^{1+n} (a (d e+c f)+2 a d f x)^m \, dx=\int { {\left (2 \, a d f x + {\left (d e + c f\right )} a\right )}^{m} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{n + 1} \,d x } \] Input:
integrate((d*x+c)^n*(f*x+e)^(1+n)*(a*(c*f+d*e)+2*a*d*f*x)^m,x, algorithm=" giac")
Output:
integrate((2*a*d*f*x + (d*e + c*f)*a)^m*(d*x + c)^n*(f*x + e)^(n + 1), x)
Timed out. \[ \int (c+d x)^n (e+f x)^{1+n} (a (d e+c f)+2 a d f x)^m \, dx=\int {\left (e+f\,x\right )}^{n+1}\,{\left (a\,\left (c\,f+d\,e\right )+2\,a\,d\,f\,x\right )}^m\,{\left (c+d\,x\right )}^n \,d x \] Input:
int((e + f*x)^(n + 1)*(a*(c*f + d*e) + 2*a*d*f*x)^m*(c + d*x)^n,x)
Output:
int((e + f*x)^(n + 1)*(a*(c*f + d*e) + 2*a*d*f*x)^m*(c + d*x)^n, x)
\[ \int (c+d x)^n (e+f x)^{1+n} (a (d e+c f)+2 a d f x)^m \, dx=\text {too large to display} \] Input:
int((d*x+c)^n*(f*x+e)^(1+n)*(a*(c*f+d*e)+2*a*d*f*x)^m,x)
Output:
( - (e + f*x)**n*(c + d*x)**n*(a*c*f + a*d*e + 2*a*d*f*x)**m*c**3*f**3*m*n - (e + f*x)**n*(c + d*x)**n*(a*c*f + a*d*e + 2*a*d*f*x)**m*c**3*f**3*m - 2*(e + f*x)**n*(c + d*x)**n*(a*c*f + a*d*e + 2*a*d*f*x)**m*c**3*f**3*n**2 - 2*(e + f*x)**n*(c + d*x)**n*(a*c*f + a*d*e + 2*a*d*f*x)**m*c**3*f**3*n + 2*(e + f*x)**n*(c + d*x)**n*(a*c*f + a*d*e + 2*a*d*f*x)**m*c**2*d*e*f**2* m**2 + 7*(e + f*x)**n*(c + d*x)**n*(a*c*f + a*d*e + 2*a*d*f*x)**m*c**2*d*e *f**2*m*n + (e + f*x)**n*(c + d*x)**n*(a*c*f + a*d*e + 2*a*d*f*x)**m*c**2* d*e*f**2*m + 6*(e + f*x)**n*(c + d*x)**n*(a*c*f + a*d*e + 2*a*d*f*x)**m*c* *2*d*e*f**2*n**2 - 2*(e + f*x)**n*(c + d*x)**n*(a*c*f + a*d*e + 2*a*d*f*x) **m*c**2*d*e*f**2*n + 2*(e + f*x)**n*(c + d*x)**n*(a*c*f + a*d*e + 2*a*d*f *x)**m*c**2*d*f**3*m**2*x + 8*(e + f*x)**n*(c + d*x)**n*(a*c*f + a*d*e + 2 *a*d*f*x)**m*c**2*d*f**3*m*n*x + 8*(e + f*x)**n*(c + d*x)**n*(a*c*f + a*d* e + 2*a*d*f*x)**m*c**2*d*f**3*n**2*x + 4*(e + f*x)**n*(c + d*x)**n*(a*c*f + a*d*e + 2*a*d*f*x)**m*c*d**2*e**2*f*m**2 + 21*(e + f*x)**n*(c + d*x)**n* (a*c*f + a*d*e + 2*a*d*f*x)**m*c*d**2*e**2*f*m*n + 5*(e + f*x)**n*(c + d*x )**n*(a*c*f + a*d*e + 2*a*d*f*x)**m*c*d**2*e**2*f*m + 26*(e + f*x)**n*(c + d*x)**n*(a*c*f + a*d*e + 2*a*d*f*x)**m*c*d**2*e**2*f*n**2 + 18*(e + f*x)* *n*(c + d*x)**n*(a*c*f + a*d*e + 2*a*d*f*x)**m*c*d**2*e**2*f*n + 8*(e + f* x)**n*(c + d*x)**n*(a*c*f + a*d*e + 2*a*d*f*x)**m*c*d**2*e*f**2*m**2*x + 3 2*(e + f*x)**n*(c + d*x)**n*(a*c*f + a*d*e + 2*a*d*f*x)**m*c*d**2*e*f**...