\(\int (d+e x)^m (f+g x)^n (a d+(b d+a e) x+b e x^2)^p \, dx\) [316]

Optimal result

Integrand size = 36, antiderivative size = 147 \[ \int (d+e x)^m (f+g x)^n \left (a d+(b d+a e) x+b e x^2\right )^p \, dx=\frac {(d+e x)^{-1+m} \left (\frac {b (d+e x)}{b d-a e}\right )^{-m-p} (f+g x)^n \left (\frac {b (f+g x)}{b f-a g}\right )^{-n} \left (a d+(b d+a e) x+b e x^2\right )^{1+p} \operatorname {AppellF1}\left (1+p,-m-p,-n,2+p,-\frac {e (a+b x)}{b d-a e},-\frac {g (a+b x)}{b f-a g}\right )}{b (1+p)} \] Output:

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

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.94 \[ \int (d+e x)^m (f+g x)^n \left (a d+(b d+a e) x+b e x^2\right )^p \, dx=\frac {(a+b x) (d+e x)^m \left (\frac {b (d+e x)}{b d-a e}\right )^{-m-p} ((a+b x) (d+e x))^p (f+g x)^n \left (\frac {b (f+g x)}{b f-a g}\right )^{-n} \operatorname {AppellF1}\left (1+p,-m-p,-n,2+p,\frac {e (a+b x)}{-b d+a e},\frac {g (a+b x)}{-b f+a g}\right )}{b (1+p)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1268, 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.

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

\(\Big \downarrow \) 1268

\(\displaystyle (a+b x)^{-p} (d+e x)^{-p} \left (x (a e+b d)+a d+b e x^2\right )^p \int (a+b x)^p (d+e x)^{m+p} (f+g x)^ndx\)

\(\Big \downarrow \) 157

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

\(\Big \downarrow \) 156

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

\(\Big \downarrow \) 155

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 155

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])
 

rule 156

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]
 

rule 157

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]
 

rule 1268

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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

integrate((e*x+d)^m*(g*x+f)^n*(a*d+(a*e+b*d)*x+b*e*x^2)^p,x, algorithm="fr 
icas")
 

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate((e*x+d)^m*(g*x+f)^n*(a*d+(a*e+b*d)*x+b*e*x^2)^p,x, algorithm="ma 
xima")
 

Output:

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

Giac [F]

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

integrate((e*x+d)^m*(g*x+f)^n*(a*d+(a*e+b*d)*x+b*e*x^2)^p,x, algorithm="gi 
ac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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