\(\int x^2 (a+b x+c x^2)^p \, dx\) [340]

Optimal result

Integrand size = 16, antiderivative size = 153 \[ \int x^2 \left (a+b x+c x^2\right )^p \, dx=-\frac {(b (2+p)-2 c (1+p) x) \left (a+b x+c x^2\right )^{1+p}}{2 c^2 (1+p) (3+2 p)}-\frac {2^{-2 (1+p)} \left (2 a c-b^2 (2+p)\right ) (b+2 c x) \left (a+b x+c x^2\right )^p \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{c^3 (3+2 p)} \] Output:

-1/2*(b*(2+p)-2*c*(p+1)*x)*(c*x^2+b*x+a)^(p+1)/c^2/(p+1)/(3+2*p)-(2*a*c-b^ 
2*(2+p))*(2*c*x+b)*(c*x^2+b*x+a)^p*hypergeom([1/2, -p],[3/2],(2*c*x+b)^2/( 
-4*a*c+b^2))/(2^(2*p+2))/c^3/(3+2*p)/((-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^p)
 

Mathematica [C] (verified)

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

Time = 0.36 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.99 \[ \int x^2 \left (a+b x+c x^2\right )^p \, dx=\frac {1}{3} x^3 \left (\frac {b-\sqrt {b^2-4 a c}+2 c x}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x}{b+\sqrt {b^2-4 a c}}\right )^{-p} (a+x (b+c x))^p \operatorname {AppellF1}\left (3,-p,-p,4,-\frac {2 c x}{b+\sqrt {b^2-4 a c}},\frac {2 c x}{-b+\sqrt {b^2-4 a c}}\right ) \] Input:

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

Output:

(x^3*(a + x*(b + c*x))^p*AppellF1[3, -p, -p, 4, (-2*c*x)/(b + Sqrt[b^2 - 4 
*a*c]), (2*c*x)/(-b + Sqrt[b^2 - 4*a*c])])/(3*((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 + Sqrt 
[b^2 - 4*a*c]))^p)
 

Rubi [A] (warning: unable to verify)

Time = 0.57 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.35, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1166, 25, 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.

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

\(\Big \downarrow \) 1166

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1160

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

\(\Big \downarrow \) 1096

\(\displaystyle \frac {x \left (a+b x+c x^2\right )^{p+1}}{c (2 p+3)}-\frac {\frac {b (p+2) \left (a+b x+c x^2\right )^{p+1}}{2 c (p+1)}-\frac {2^p \left (2 a c-b^2 (p+2)\right ) \left (-\frac {-\sqrt {b^2-4 a c}+b+2 c x}{\sqrt {b^2-4 a c}}\right )^{-p-1} \left (a+b x+c x^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (-p,p+1,p+2,\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )}{c (p+1) \sqrt {b^2-4 a c}}}{c (2 p+3)}\)

Input:

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

Output:

(x*(a + b*x + c*x^2)^(1 + p))/(c*(3 + 2*p)) - ((b*(2 + p)*(a + b*x + c*x^2 
)^(1 + p))/(2*c*(1 + p)) - (2^p*(2*a*c - b^2*(2 + p))*(-((b - Sqrt[b^2 - 4 
*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]))^(-1 - p)*(a + b*x + c*x^2)^(1 + p)*Hype 
rgeometric2F1[-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)))/(c*(3 + 2*p))
 

Defintions of rubi rules used

rule 25

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

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]
 

rule 1166

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 
 1))), x] + Simp[1/(c*(m + 2*p + 1))   Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m 
+ 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* 
(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration 
alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat 
icQ[a, b, c, d, e, m, p, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^2 \left (a+b x+c x^2\right )^p \, dx =\text {Too large to display} \] Input:

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

Output:

( - 4*(a + b*x + c*x**2)**p*a**2*c*p - 4*(a + b*x + c*x**2)**p*a**2*c + (a 
 + b*x + c*x**2)**p*a*b**2*p + 2*(a + b*x + c*x**2)**p*a*b**2 + 4*(a + b*x 
 + c*x**2)**p*a*b*c*p**2*x + 4*(a + b*x + c*x**2)**p*a*b*c*p*x - (a + b*x 
+ c*x**2)**p*b**3*p**2*x - 2*(a + b*x + c*x**2)**p*b**3*p*x + 2*(a + b*x + 
 c*x**2)**p*b**2*c*p**2*x**2 + (a + b*x + c*x**2)**p*b**2*c*p*x**2 + 4*(a 
+ b*x + c*x**2)**p*b*c**2*p**2*x**3 + 6*(a + b*x + c*x**2)**p*b*c**2*p*x** 
3 + 2*(a + b*x + c*x**2)**p*b*c**2*x**3 + 32*int(((a + b*x + c*x**2)**p*x) 
/(4*a*p**2 + 8*a*p + 3*a + 4*b*p**2*x + 8*b*p*x + 3*b*x + 4*c*p**2*x**2 + 
8*c*p*x**2 + 3*c*x**2),x)*a**2*c**2*p**4 + 96*int(((a + b*x + c*x**2)**p*x 
)/(4*a*p**2 + 8*a*p + 3*a + 4*b*p**2*x + 8*b*p*x + 3*b*x + 4*c*p**2*x**2 + 
 8*c*p*x**2 + 3*c*x**2),x)*a**2*c**2*p**3 + 88*int(((a + b*x + c*x**2)**p* 
x)/(4*a*p**2 + 8*a*p + 3*a + 4*b*p**2*x + 8*b*p*x + 3*b*x + 4*c*p**2*x**2 
+ 8*c*p*x**2 + 3*c*x**2),x)*a**2*c**2*p**2 + 24*int(((a + b*x + c*x**2)**p 
*x)/(4*a*p**2 + 8*a*p + 3*a + 4*b*p**2*x + 8*b*p*x + 3*b*x + 4*c*p**2*x**2 
 + 8*c*p*x**2 + 3*c*x**2),x)*a**2*c**2*p - 16*int(((a + b*x + c*x**2)**p*x 
)/(4*a*p**2 + 8*a*p + 3*a + 4*b*p**2*x + 8*b*p*x + 3*b*x + 4*c*p**2*x**2 + 
 8*c*p*x**2 + 3*c*x**2),x)*a*b**2*c*p**5 - 88*int(((a + b*x + c*x**2)**p*x 
)/(4*a*p**2 + 8*a*p + 3*a + 4*b*p**2*x + 8*b*p*x + 3*b*x + 4*c*p**2*x**2 + 
 8*c*p*x**2 + 3*c*x**2),x)*a*b**2*c*p**4 - 164*int(((a + b*x + c*x**2)**p* 
x)/(4*a*p**2 + 8*a*p + 3*a + 4*b*p**2*x + 8*b*p*x + 3*b*x + 4*c*p**2*x*...