\(\int x (d+e x)^m (a+b x+c x^2)^p \, dx\) [135]

Optimal result

Integrand size = 21, antiderivative size = 377 \[ \int x (d+e x)^m \left (a+b x+c x^2\right )^p \, dx=-\frac {d (d+e x)^{1+m} \left (a+b x+c x^2\right )^p \left (1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )^{-p} \left (1-\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )^{-p} \operatorname {AppellF1}\left (1+m,-p,-p,2+m,\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e},\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{e^2 (1+m)}+\frac {(d+e x)^{2+m} \left (a+b x+c x^2\right )^p \left (1-\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}\right )^{-p} \left (1-\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )^{-p} \operatorname {AppellF1}\left (2+m,-p,-p,3+m,\frac {2 c (d+e x)}{2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e},\frac {2 c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{e^2 (2+m)} \] Output:

-d*(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)+(e*x+d)^(2+m)*(c*x^2+b*x+a)^p*App 
ellF1(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]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, 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.

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1179

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

\(\Big \downarrow \) 150

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

Input:

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

Output:

-((d*(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)) + ((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)
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

integrate(x*(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, x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate(x*(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, x)
 

Giac [F]

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

integrate(x*(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, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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