\(\int (d+e x)^m (f+g x) (a+b x+c x^2)^p \, dx\) [1125]

Optimal result

Integrand size = 25, antiderivative size = 384 \[ \int (d+e x)^m (f+g x) \left (a+b x+c x^2\right )^p \, dx=\frac {(e f-d g) (d+e x)^{1+m} \left (a+b x+c x^2\right )^p \left (1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )^{-p} \left (1-\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )^{-p} \operatorname {AppellF1}\left (1+m,-p,-p,2+m,\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e},\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{e^2 (1+m)}+\frac {g (d+e x)^{2+m} \left (a+b x+c x^2\right )^p \left (1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )^{-p} \left (1-\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )^{-p} \operatorname {AppellF1}\left (2+m,-p,-p,3+m,\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e},\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{e^2 (2+m)} \] Output:

(-d*g+e*f)*(e*x+d)^(1+m)*(c*x^2+b*x+a)^p*AppellF1(1+m,-p,-p,2+m,2*c*(e*x+d 
)/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e),2*c*(e*x+d)/(2*c*d-(b+(-4*a*c+b^2)^(1/2 
))*e))/e^2/(1+m)/((1-2*c*(e*x+d)/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e))^p)/((1- 
2*c*(e*x+d)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e))^p)+g*(e*x+d)^(2+m)*(c*x^2+b* 
x+a)^p*AppellF1(2+m,-p,-p,3+m,2*c*(e*x+d)/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e) 
,2*c*(e*x+d)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e))/e^2/(2+m)/((1-2*c*(e*x+d)/( 
2*c*d-(b-(-4*a*c+b^2)^(1/2))*e))^p)/((1-2*c*(e*x+d)/(2*c*d-(b+(-4*a*c+b^2) 
^(1/2))*e))^p)
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {1269, 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.

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

\(\Big \downarrow \) 1269

\(\displaystyle \frac {(e f-d g) \int (d+e x)^m \left (c x^2+b x+a\right )^pdx}{e}+\frac {g \int (d+e x)^{m+1} \left (c x^2+b x+a\right )^pdx}{e}\)

\(\Big \downarrow \) 1179

\(\displaystyle \frac {(e f-d g) \left (a+b x+c x^2\right )^p \left (1-\frac {2 c (d+e x)}{2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}\right )^{-p} \left (1-\frac {2 c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}\right )^{-p} \int (d+e x)^m \left (1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )^p \left (1-\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )^pd(d+e x)}{e^2}+\frac {g \left (a+b x+c x^2\right )^p \left (1-\frac {2 c (d+e x)}{2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}\right )^{-p} \left (1-\frac {2 c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}\right )^{-p} \int (d+e x)^{m+1} \left (1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )^p \left (1-\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )^pd(d+e x)}{e^2}\)

\(\Big \downarrow \) 150

\(\displaystyle \frac {(e f-d g) (d+e x)^{m+1} \left (a+b x+c x^2\right )^p \left (1-\frac {2 c (d+e x)}{2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}\right )^{-p} \left (1-\frac {2 c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}\right )^{-p} \operatorname {AppellF1}\left (m+1,-p,-p,m+2,\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e},\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{e^2 (m+1)}+\frac {g (d+e x)^{m+2} \left (a+b x+c x^2\right )^p \left (1-\frac {2 c (d+e x)}{2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}\right )^{-p} \left (1-\frac {2 c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}\right )^{-p} \operatorname {AppellF1}\left (m+2,-p,-p,m+3,\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e},\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{e^2 (m+2)}\)

Input:

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

Output:

((e*f - d*g)*(d + e*x)^(1 + m)*(a + b*x + c*x^2)^p*AppellF1[1 + m, -p, -p, 
 2 + m, (2*c*(d + e*x))/(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e), (2*c*(d + e*x 
))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(e^2*(1 + m)*(1 - (2*c*(d + e*x)) 
/(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e))^p*(1 - (2*c*(d + e*x))/(2*c*d - (b + 
 Sqrt[b^2 - 4*a*c])*e))^p) + (g*(d + e*x)^(2 + m)*(a + b*x + c*x^2)^p*Appe 
llF1[2 + m, -p, -p, 3 + m, (2*c*(d + e*x))/(2*c*d - (b - Sqrt[b^2 - 4*a*c] 
)*e), (2*c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(e^2*(2 + m)*( 
1 - (2*c*(d + e*x))/(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e))^p*(1 - (2*c*(d + 
e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e))^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]
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (d+e x)^m (f+g x) \left (a+b x+c x^2\right )^p \, dx=\text {too large to display} \] Input:

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

Output:

( - (d + e*x)**m*(a + b*x + c*x**2)**p*a*b*e**2*g*m*p - (d + e*x)**m*(a + 
b*x + c*x**2)**p*a*b*e**2*g*p + 4*(d + e*x)**m*(a + b*x + c*x**2)**p*a*c*d 
*e*g*m*p + 4*(d + e*x)**m*(a + b*x + c*x**2)**p*a*c*d*e*g*p**2 + 2*(d + e* 
x)**m*(a + b*x + c*x**2)**p*a*c*d*e*g*p + 2*(d + e*x)**m*(a + b*x + c*x**2 
)**p*a*c*e**2*f*m*p + 4*(d + e*x)**m*(a + b*x + c*x**2)**p*a*c*e**2*f*p**2 
 + 4*(d + e*x)**m*(a + b*x + c*x**2)**p*a*c*e**2*f*p - (d + e*x)**m*(a + b 
*x + c*x**2)**p*b**2*d*e*g*p**2 - (d + e*x)**m*(a + b*x + c*x**2)**p*b**2* 
d*e*g*p + (d + e*x)**m*(a + b*x + c*x**2)**p*b**2*e**2*g*m*p*x + (d + e*x) 
**m*(a + b*x + c*x**2)**p*b**2*e**2*g*p**2*x - (d + e*x)**m*(a + b*x + c*x 
**2)**p*b*c*d**2*g*m*p - (d + e*x)**m*(a + b*x + c*x**2)**p*b*c*d**2*g*m + 
 (d + e*x)**m*(a + b*x + c*x**2)**p*b*c*d*e*f*m**2 + 3*(d + e*x)**m*(a + b 
*x + c*x**2)**p*b*c*d*e*f*m*p + 2*(d + e*x)**m*(a + b*x + c*x**2)**p*b*c*d 
*e*f*m + 2*(d + e*x)**m*(a + b*x + c*x**2)**p*b*c*d*e*f*p**2 + 2*(d + e*x) 
**m*(a + b*x + c*x**2)**p*b*c*d*e*f*p + (d + e*x)**m*(a + b*x + c*x**2)**p 
*b*c*d*e*g*m**2*x + (d + e*x)**m*(a + b*x + c*x**2)**p*b*c*d*e*g*m*p*x + 2 
*(d + e*x)**m*(a + b*x + c*x**2)**p*b*c*d*e*g*p**2*x + (d + e*x)**m*(a + b 
*x + c*x**2)**p*b*c*e**2*f*m**2*x + 3*(d + e*x)**m*(a + b*x + c*x**2)**p*b 
*c*e**2*f*m*p*x + 2*(d + e*x)**m*(a + b*x + c*x**2)**p*b*c*e**2*f*m*x + 2* 
(d + e*x)**m*(a + b*x + c*x**2)**p*b*c*e**2*f*p**2*x + 2*(d + e*x)**m*(a + 
 b*x + c*x**2)**p*b*c*e**2*f*p*x + (d + e*x)**m*(a + b*x + c*x**2)**p*b...