Integrand size = 12, antiderivative size = 85 \[ \int \left (a+b x+c x^2\right )^p \, dx=\frac {2^{-1-2 p} (b+2 c x) \left (a+b x+c x^2\right )^p \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{c} \] Output:
2^(-1-2*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))/c/((-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^p)
Time = 0.10 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.48 \[ \int \left (a+b x+c x^2\right )^p \, dx=\frac {2^{-1+p} \left (b-\sqrt {b^2-4 a c}+2 c x\right ) \left (\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}\right )^{-p} (a+x (b+c x))^p \operatorname {Hypergeometric2F1}\left (-p,1+p,2+p,\frac {-b+\sqrt {b^2-4 a c}-2 c x}{2 \sqrt {b^2-4 a c}}\right )}{c (1+p)} \] Input:
Integrate[(a + b*x + c*x^2)^p,x]
Output:
(2^(-1 + p)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)*(a + x*(b + c*x))^p*Hypergeome tric2F1[-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)
Time = 0.38 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.44, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {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.
\(\displaystyle \int \left (a+b x+c x^2\right )^p \, dx\) |
\(\Big \downarrow \) 1096 |
\(\displaystyle -\frac {2^{p+1} \left (-\frac {-\sqrt {b^2-4 a c}+b+2 c x}{\sqrt {b^2-4 a c}}\right )^{-p-1} \left (a+b x+c x^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (-p,p+1,p+2,\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )}{(p+1) \sqrt {b^2-4 a c}}\) |
Input:
Int[(a + b*x + c*x^2)^p,x]
Output:
-((2^(1 + 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)*Hypergeometric2F1[-p, 1 + p, 2 + p, (b + Sqrt [b^2 - 4*a*c] + 2*c*x)/(2*Sqrt[b^2 - 4*a*c])])/(Sqrt[b^2 - 4*a*c]*(1 + p)) )
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]
\[\int \left (c \,x^{2}+b x +a \right )^{p}d x\]
Input:
int((c*x^2+b*x+a)^p,x)
Output:
int((c*x^2+b*x+a)^p,x)
\[ \int \left (a+b x+c x^2\right )^p \, dx=\int { {\left (c x^{2} + b x + a\right )}^{p} \,d x } \] Input:
integrate((c*x^2+b*x+a)^p,x, algorithm="fricas")
Output:
integral((c*x^2 + b*x + a)^p, x)
\[ \int \left (a+b x+c x^2\right )^p \, dx=\int \left (a + b x + c x^{2}\right )^{p}\, dx \] Input:
integrate((c*x**2+b*x+a)**p,x)
Output:
Integral((a + b*x + c*x**2)**p, x)
\[ \int \left (a+b x+c x^2\right )^p \, dx=\int { {\left (c x^{2} + b x + a\right )}^{p} \,d x } \] Input:
integrate((c*x^2+b*x+a)^p,x, algorithm="maxima")
Output:
integrate((c*x^2 + b*x + a)^p, x)
\[ \int \left (a+b x+c x^2\right )^p \, dx=\int { {\left (c x^{2} + b x + a\right )}^{p} \,d x } \] Input:
integrate((c*x^2+b*x+a)^p,x, algorithm="giac")
Output:
integrate((c*x^2 + b*x + a)^p, x)
Timed out. \[ \int \left (a+b x+c x^2\right )^p \, dx=\int {\left (c\,x^2+b\,x+a\right )}^p \,d x \] Input:
int((a + b*x + c*x^2)^p,x)
Output:
int((a + b*x + c*x^2)^p, x)
\[ \int \left (a+b x+c x^2\right )^p \, dx=\frac {2 \left (c \,x^{2}+b x +a \right )^{p} a +\left (c \,x^{2}+b x +a \right )^{p} b x -8 \left (\int \frac {\left (c \,x^{2}+b x +a \right )^{p} x}{2 c p \,x^{2}+2 b p x +c \,x^{2}+2 a p +b x +a}d x \right ) a c \,p^{2}-4 \left (\int \frac {\left (c \,x^{2}+b x +a \right )^{p} x}{2 c p \,x^{2}+2 b p x +c \,x^{2}+2 a p +b x +a}d x \right ) a c p +2 \left (\int \frac {\left (c \,x^{2}+b x +a \right )^{p} x}{2 c p \,x^{2}+2 b p x +c \,x^{2}+2 a p +b x +a}d x \right ) b^{2} p^{2}+\left (\int \frac {\left (c \,x^{2}+b x +a \right )^{p} x}{2 c p \,x^{2}+2 b p x +c \,x^{2}+2 a p +b x +a}d x \right ) b^{2} p}{b \left (2 p +1\right )} \] Input:
int((c*x^2+b*x+a)^p,x)
Output:
(2*(a + b*x + c*x**2)**p*a + (a + b*x + c*x**2)**p*b*x - 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*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*c*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**2*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**2*p)/(b*(2*p + 1))