\(\int x^2 (d+e x)^m (a+b x+c x^2)^p \, dx\) [134]

Optimal result

Integrand size = 23, antiderivative size = 471 \[ \int x^2 (d+e x)^m \left (a+b x+c x^2\right )^p \, dx=\frac {(d+e x)^{1+m} \left (a+b x+c x^2\right )^{1+p}}{c e (3+m+2 p)}+\frac {\left (b d-a e+\frac {2 c d^2 (1+p)}{e (1+m)}\right ) (d+e x)^{1+m} \left (a+b x+c x^2\right )^p \left (1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )^{-p} \left (1-\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )^{-p} \operatorname {AppellF1}\left (1+m,-p,-p,2+m,\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e},\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e^2 (3+m+2 p)}-\frac {(2 c d (1+p)+b e (2+m+p)) (d+e x)^{2+m} \left (a+b x+c x^2\right )^p \left (1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )^{-p} \left (1-\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )^{-p} \operatorname {AppellF1}\left (2+m,-p,-p,3+m,\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e},\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e^3 (2+m) (3+m+2 p)} \] Output:

(e*x+d)^(1+m)*(c*x^2+b*x+a)^(p+1)/c/e/(3+m+2*p)+(b*d-a*e+2*c*d^2*(p+1)/e/( 
1+m))*(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) 
)/c/e^2/(3+m+2*p)/((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)-(2*c*d*(p+1)+b*e*(2+m+p) 
)*(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))/c/ 
e^3/(2+m)/(3+m+2*p)/((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]

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

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

Output:

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

Rubi [A] (verified)

Time = 1.04 (sec) , antiderivative size = 469, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {1291, 25, 27, 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.

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

\(\Big \downarrow \) 1291

\(\displaystyle \frac {\int -e (d+e x)^m (a e (m+1)+b d (p+1)+(2 c d (p+1)+b e (m+p+2)) x) \left (c x^2+b x+a\right )^pdx}{c e^2 (m+2 p+3)}+\frac {(d+e x)^{m+1} \left (a+b x+c x^2\right )^{p+1}}{c e (m+2 p+3)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {(d+e x)^{m+1} \left (a+b x+c x^2\right )^{p+1}}{c e (m+2 p+3)}-\frac {\int e (d+e x)^m (a e (m+1)+b d (p+1)+(2 c d (p+1)+b e (m+p+2)) x) \left (c x^2+b x+a\right )^pdx}{c e^2 (m+2 p+3)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(d+e x)^{m+1} \left (a+b x+c x^2\right )^{p+1}}{c e (m+2 p+3)}-\frac {\int (d+e x)^m (a e (m+1)+b d (p+1)+(2 c d (p+1)+b e (m+p+2)) x) \left (c x^2+b x+a\right )^pdx}{c e (m+2 p+3)}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {(d+e x)^{m+1} \left (a+b x+c x^2\right )^{p+1}}{c e (m+2 p+3)}-\frac {\frac {(b e (m+p+2)+2 c d (p+1)) \int (d+e x)^{m+1} \left (c x^2+b x+a\right )^pdx}{e}-\frac {\left (e (m+1) (b d-a e)+2 c d^2 (p+1)\right ) \int (d+e x)^m \left (c x^2+b x+a\right )^pdx}{e}}{c e (m+2 p+3)}\)

\(\Big \downarrow \) 1179

\(\displaystyle \frac {(d+e x)^{m+1} \left (a+b x+c x^2\right )^{p+1}}{c e (m+2 p+3)}-\frac {\frac {\left (a+b x+c x^2\right )^p (b e (m+p+2)+2 c d (p+1)) \left (1-\frac {2 c (d+e x)}{2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}\right )^{-p} \left (1-\frac {2 c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}\right )^{-p} \int (d+e x)^{m+1} \left (1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )^p \left (1-\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )^pd(d+e x)}{e^2}-\frac {\left (a+b x+c x^2\right )^p \left (e (m+1) (b d-a e)+2 c d^2 (p+1)\right ) \left (1-\frac {2 c (d+e x)}{2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}\right )^{-p} \left (1-\frac {2 c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}\right )^{-p} \int (d+e x)^m \left (1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )^p \left (1-\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )^pd(d+e x)}{e^2}}{c e (m+2 p+3)}\)

\(\Big \downarrow \) 150

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

Input:

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

Output:

((d + e*x)^(1 + m)*(a + b*x + c*x^2)^(1 + p))/(c*e*(3 + m + 2*p)) - (-(((e 
*(b*d - a*e)*(1 + m) + 2*c*d^2*(1 + p))*(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)) + ((2*c*d*(1 + p) + 
 b*e*(2 + m + p))*(d + e*x)^(2 + m)*(a + b*x + c*x^2)^p*AppellF1[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))/(c*e*(3 + m + 2*p))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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]
 

rule 1291

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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