\(\int (d+e x) (a+b x+c x^2)^p \, dx\) [775]

Optimal result

Integrand size = 18, antiderivative size = 122 \[ \int (d+e x) \left (a+b x+c x^2\right )^p \, dx=\frac {e \left (a+b x+c x^2\right )^{1+p}}{2 c (1+p)}+\frac {2^{-2 (1+p)} (2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^p \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{c^2} \] Output:

1/2*e*(c*x^2+b*x+a)^(p+1)/c/(p+1)+(-b*e+2*c*d)*(2*c*x+b)*(c*x^2+b*x+a)^p*h 
ypergeom([1/2, -p],[3/2],(2*c*x+b)^2/(-4*a*c+b^2))/(2^(2*p+2))/c^2/((-c*(c 
*x^2+b*x+a)/(-4*a*c+b^2))^p)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 0.54 (sec) , antiderivative size = 268, normalized size of antiderivative = 2.20 \[ \int (d+e x) \left (a+b x+c x^2\right )^p \, dx=\frac {1}{2} (a+x (b+c x))^p \left (e x^2 \left (\frac {b-\sqrt {b^2-4 a c}+2 c x}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x}{b+\sqrt {b^2-4 a c}}\right )^{-p} \operatorname {AppellF1}\left (2,-p,-p,3,-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right )+\frac {2^p d \left (b-\sqrt {b^2-4 a c}+2 c x\right ) \left (\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,1+p,2+p,\frac {-b+\sqrt {b^2-4 a c}-2 c x}{2 \sqrt {b^2-4 a c}}\right )}{c (1+p)}\right ) \] Input:

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

Output:

((a + x*(b + c*x))^p*((e*x^2*AppellF1[2, -p, -p, 3, (-2*c*x)/(b + Sqrt[b^2 
 - 4*a*c]), (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])])/(((b - Sqrt[b^2 - 4*a*c] + 
2*c*x)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x)/(b + Sq 
rt[b^2 - 4*a*c]))^p) + (2^p*d*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)*Hypergeometr 
ic2F1[-p, 1 + p, 2 + p, (-b + Sqrt[b^2 - 4*a*c] - 2*c*x)/(2*Sqrt[b^2 - 4*a 
*c])])/(c*(1 + p)*((b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c])^p))) 
/2
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.31, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1160, 1096}

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.

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

\(\Big \downarrow \) 1160

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

\(\Big \downarrow \) 1096

\(\displaystyle \frac {e \left (a+b x+c x^2\right )^{p+1}}{2 c (p+1)}-\frac {2^p (2 c d-b e) \left (-\frac {-\sqrt {b^2-4 a c}+b+2 c x}{\sqrt {b^2-4 a c}}\right )^{-p-1} \left (a+b x+c x^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (-p,p+1,p+2,\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )}{c (p+1) \sqrt {b^2-4 a c}}\)

Input:

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

Output:

(e*(a + b*x + c*x^2)^(1 + p))/(2*c*(1 + p)) - (2^p*(2*c*d - b*e)*(-((b - S 
qrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]))^(-1 - p)*(a + b*x + c*x^2)^( 
1 + p)*Hypergeometric2F1[-p, 1 + p, 2 + p, (b + Sqrt[b^2 - 4*a*c] + 2*c*x) 
/(2*Sqrt[b^2 - 4*a*c])])/(c*Sqrt[b^2 - 4*a*c]*(1 + p))
 

Defintions of rubi rules used

rule 1096

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Rt[b^2 
 - 4*a*c, 2]}, Simp[(-(a + b*x + c*x^2)^(p + 1)/(q*(p + 1)*((q - b - 2*c*x) 
/(2*q))^(p + 1)))*Hypergeometric2F1[-p, p + 1, p + 2, (b + q + 2*c*x)/(2*q) 
], x]] /; FreeQ[{a, b, c, p}, x] &&  !IntegerQ[4*p] &&  !IntegerQ[3*p]
 

rule 1160

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol 
] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b 
*e)/(2*c)   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] 
 && NeQ[p, -1]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (d+e x) \left (a+b x+c x^2\right )^p \, dx=\frac {-\left (c \,x^{2}+b x +a \right )^{p} a b e +4 \left (c \,x^{2}+b x +a \right )^{p} a c d p +4 \left (c \,x^{2}+b x +a \right )^{p} a c d +\left (c \,x^{2}+b x +a \right )^{p} b^{2} e p x +2 \left (c \,x^{2}+b x +a \right )^{p} b c d p x +2 \left (c \,x^{2}+b x +a \right )^{p} b c d x +2 \left (c \,x^{2}+b x +a \right )^{p} b c e p \,x^{2}+\left (c \,x^{2}+b x +a \right )^{p} b c e \,x^{2}+8 \left (\int \frac {\left (c \,x^{2}+b x +a \right )^{p} x}{2 c p \,x^{2}+2 b p x +c \,x^{2}+2 a p +b x +a}d x \right ) a b c e \,p^{3}+12 \left (\int \frac {\left (c \,x^{2}+b x +a \right )^{p} x}{2 c p \,x^{2}+2 b p x +c \,x^{2}+2 a p +b x +a}d x \right ) a b c e \,p^{2}+4 \left (\int \frac {\left (c \,x^{2}+b x +a \right )^{p} x}{2 c p \,x^{2}+2 b p x +c \,x^{2}+2 a p +b x +a}d x \right ) a b c e p -16 \left (\int \frac {\left (c \,x^{2}+b x +a \right )^{p} x}{2 c p \,x^{2}+2 b p x +c \,x^{2}+2 a p +b x +a}d x \right ) a \,c^{2} d \,p^{3}-24 \left (\int \frac {\left (c \,x^{2}+b x +a \right )^{p} x}{2 c p \,x^{2}+2 b p x +c \,x^{2}+2 a p +b x +a}d x \right ) a \,c^{2} d \,p^{2}-8 \left (\int \frac {\left (c \,x^{2}+b x +a \right )^{p} x}{2 c p \,x^{2}+2 b p x +c \,x^{2}+2 a p +b x +a}d x \right ) a \,c^{2} d p -2 \left (\int \frac {\left (c \,x^{2}+b x +a \right )^{p} x}{2 c p \,x^{2}+2 b p x +c \,x^{2}+2 a p +b x +a}d x \right ) b^{3} e \,p^{3}-3 \left (\int \frac {\left (c \,x^{2}+b x +a \right )^{p} x}{2 c p \,x^{2}+2 b p x +c \,x^{2}+2 a p +b x +a}d x \right ) b^{3} e \,p^{2}-\left (\int \frac {\left (c \,x^{2}+b x +a \right )^{p} x}{2 c p \,x^{2}+2 b p x +c \,x^{2}+2 a p +b x +a}d x \right ) b^{3} e p +4 \left (\int \frac {\left (c \,x^{2}+b x +a \right )^{p} x}{2 c p \,x^{2}+2 b p x +c \,x^{2}+2 a p +b x +a}d x \right ) b^{2} c d \,p^{3}+6 \left (\int \frac {\left (c \,x^{2}+b x +a \right )^{p} x}{2 c p \,x^{2}+2 b p x +c \,x^{2}+2 a p +b x +a}d x \right ) b^{2} c d \,p^{2}+2 \left (\int \frac {\left (c \,x^{2}+b x +a \right )^{p} x}{2 c p \,x^{2}+2 b p x +c \,x^{2}+2 a p +b x +a}d x \right ) b^{2} c d p}{2 b c \left (2 p^{2}+3 p +1\right )} \] Input:

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

Output:

( - (a + b*x + c*x**2)**p*a*b*e + 4*(a + b*x + c*x**2)**p*a*c*d*p + 4*(a + 
 b*x + c*x**2)**p*a*c*d + (a + b*x + c*x**2)**p*b**2*e*p*x + 2*(a + b*x + 
c*x**2)**p*b*c*d*p*x + 2*(a + b*x + c*x**2)**p*b*c*d*x + 2*(a + b*x + c*x* 
*2)**p*b*c*e*p*x**2 + (a + b*x + c*x**2)**p*b*c*e*x**2 + 8*int(((a + b*x + 
 c*x**2)**p*x)/(2*a*p + a + 2*b*p*x + b*x + 2*c*p*x**2 + c*x**2),x)*a*b*c* 
e*p**3 + 12*int(((a + b*x + c*x**2)**p*x)/(2*a*p + a + 2*b*p*x + b*x + 2*c 
*p*x**2 + c*x**2),x)*a*b*c*e*p**2 + 4*int(((a + b*x + c*x**2)**p*x)/(2*a*p 
 + a + 2*b*p*x + b*x + 2*c*p*x**2 + c*x**2),x)*a*b*c*e*p - 16*int(((a + b* 
x + c*x**2)**p*x)/(2*a*p + a + 2*b*p*x + b*x + 2*c*p*x**2 + c*x**2),x)*a*c 
**2*d*p**3 - 24*int(((a + b*x + c*x**2)**p*x)/(2*a*p + a + 2*b*p*x + b*x + 
 2*c*p*x**2 + c*x**2),x)*a*c**2*d*p**2 - 8*int(((a + b*x + c*x**2)**p*x)/( 
2*a*p + a + 2*b*p*x + b*x + 2*c*p*x**2 + c*x**2),x)*a*c**2*d*p - 2*int(((a 
 + b*x + c*x**2)**p*x)/(2*a*p + a + 2*b*p*x + b*x + 2*c*p*x**2 + c*x**2),x 
)*b**3*e*p**3 - 3*int(((a + b*x + c*x**2)**p*x)/(2*a*p + a + 2*b*p*x + b*x 
 + 2*c*p*x**2 + c*x**2),x)*b**3*e*p**2 - int(((a + b*x + c*x**2)**p*x)/(2* 
a*p + a + 2*b*p*x + b*x + 2*c*p*x**2 + c*x**2),x)*b**3*e*p + 4*int(((a + b 
*x + c*x**2)**p*x)/(2*a*p + a + 2*b*p*x + b*x + 2*c*p*x**2 + c*x**2),x)*b* 
*2*c*d*p**3 + 6*int(((a + b*x + c*x**2)**p*x)/(2*a*p + a + 2*b*p*x + b*x + 
 2*c*p*x**2 + c*x**2),x)*b**2*c*d*p**2 + 2*int(((a + b*x + c*x**2)**p*x)/( 
2*a*p + a + 2*b*p*x + b*x + 2*c*p*x**2 + c*x**2),x)*b**2*c*d*p)/(2*b*c*...