Integrand size = 12, antiderivative size = 85 \[ \int \left (d+e x+f x^2\right )^q \, dx=\frac {2^{-1-2 q} (e+2 f x) \left (d+e x+f x^2\right )^q \left (-\frac {f \left (d+e x+f x^2\right )}{e^2-4 d f}\right )^{-q} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-q,\frac {3}{2},\frac {(e+2 f x)^2}{e^2-4 d f}\right )}{f} \] Output:
2^(-1-2*q)*(2*f*x+e)*(f*x^2+e*x+d)^q*hypergeom([1/2, -q],[3/2],(2*f*x+e)^2 /(-4*d*f+e^2))/f/((-f*(f*x^2+e*x+d)/(-4*d*f+e^2))^q)
Time = 0.11 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.48 \[ \int \left (d+e x+f x^2\right )^q \, dx=\frac {2^{-1+q} \left (e-\sqrt {e^2-4 d f}+2 f x\right ) \left (\frac {e+\sqrt {e^2-4 d f}+2 f x}{\sqrt {e^2-4 d f}}\right )^{-q} (d+x (e+f x))^q \operatorname {Hypergeometric2F1}\left (-q,1+q,2+q,\frac {-e+\sqrt {e^2-4 d f}-2 f x}{2 \sqrt {e^2-4 d f}}\right )}{f (1+q)} \] Input:
Integrate[(d + e*x + f*x^2)^q,x]
Output:
(2^(-1 + q)*(e - Sqrt[e^2 - 4*d*f] + 2*f*x)*(d + x*(e + f*x))^q*Hypergeome tric2F1[-q, 1 + q, 2 + q, (-e + Sqrt[e^2 - 4*d*f] - 2*f*x)/(2*Sqrt[e^2 - 4 *d*f])])/(f*(1 + q)*((e + Sqrt[e^2 - 4*d*f] + 2*f*x)/Sqrt[e^2 - 4*d*f])^q)
Time = 0.22 (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.
\(\displaystyle \int \left (d+e x+f x^2\right )^q \, dx\) |
\(\Big \downarrow \) 1096 |
\(\displaystyle -\frac {2^{q+1} \left (-\frac {-\sqrt {e^2-4 d f}+e+2 f x}{\sqrt {e^2-4 d f}}\right )^{-q-1} \left (d+e x+f x^2\right )^{q+1} \operatorname {Hypergeometric2F1}\left (-q,q+1,q+2,\frac {e+2 f x+\sqrt {e^2-4 d f}}{2 \sqrt {e^2-4 d f}}\right )}{(q+1) \sqrt {e^2-4 d f}}\) |
Input:
Int[(d + e*x + f*x^2)^q,x]
Output:
-((2^(1 + q)*(-((e - Sqrt[e^2 - 4*d*f] + 2*f*x)/Sqrt[e^2 - 4*d*f]))^(-1 - q)*(d + e*x + f*x^2)^(1 + q)*Hypergeometric2F1[-q, 1 + q, 2 + q, (e + Sqrt [e^2 - 4*d*f] + 2*f*x)/(2*Sqrt[e^2 - 4*d*f])])/(Sqrt[e^2 - 4*d*f]*(1 + q)) )
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]
\[\int \left (f \,x^{2}+e x +d \right )^{q}d x\]
Input:
int((f*x^2+e*x+d)^q,x)
Output:
int((f*x^2+e*x+d)^q,x)
\[ \int \left (d+e x+f x^2\right )^q \, dx=\int { {\left (f x^{2} + e x + d\right )}^{q} \,d x } \] Input:
integrate((f*x^2+e*x+d)^q,x, algorithm="fricas")
Output:
integral((f*x^2 + e*x + d)^q, x)
\[ \int \left (d+e x+f x^2\right )^q \, dx=\int \left (d + e x + f x^{2}\right )^{q}\, dx \] Input:
integrate((f*x**2+e*x+d)**q,x)
Output:
Integral((d + e*x + f*x**2)**q, x)
\[ \int \left (d+e x+f x^2\right )^q \, dx=\int { {\left (f x^{2} + e x + d\right )}^{q} \,d x } \] Input:
integrate((f*x^2+e*x+d)^q,x, algorithm="maxima")
Output:
integrate((f*x^2 + e*x + d)^q, x)
\[ \int \left (d+e x+f x^2\right )^q \, dx=\int { {\left (f x^{2} + e x + d\right )}^{q} \,d x } \] Input:
integrate((f*x^2+e*x+d)^q,x, algorithm="giac")
Output:
integrate((f*x^2 + e*x + d)^q, x)
Timed out. \[ \int \left (d+e x+f x^2\right )^q \, dx=\int {\left (f\,x^2+e\,x+d\right )}^q \,d x \] Input:
int((d + e*x + f*x^2)^q,x)
Output:
int((d + e*x + f*x^2)^q, x)
\[ \int \left (d+e x+f x^2\right )^q \, dx=\frac {2 \left (f \,x^{2}+e x +d \right )^{q} d +\left (f \,x^{2}+e x +d \right )^{q} e x -8 \left (\int \frac {\left (f \,x^{2}+e x +d \right )^{q} x}{2 f q \,x^{2}+2 e q x +f \,x^{2}+2 d q +e x +d}d x \right ) d f \,q^{2}-4 \left (\int \frac {\left (f \,x^{2}+e x +d \right )^{q} x}{2 f q \,x^{2}+2 e q x +f \,x^{2}+2 d q +e x +d}d x \right ) d f q +2 \left (\int \frac {\left (f \,x^{2}+e x +d \right )^{q} x}{2 f q \,x^{2}+2 e q x +f \,x^{2}+2 d q +e x +d}d x \right ) e^{2} q^{2}+\left (\int \frac {\left (f \,x^{2}+e x +d \right )^{q} x}{2 f q \,x^{2}+2 e q x +f \,x^{2}+2 d q +e x +d}d x \right ) e^{2} q}{e \left (2 q +1\right )} \] Input:
int((f*x^2+e*x+d)^q,x)
Output:
(2*(d + e*x + f*x**2)**q*d + (d + e*x + f*x**2)**q*e*x - 8*int(((d + e*x + f*x**2)**q*x)/(2*d*q + d + 2*e*q*x + e*x + 2*f*q*x**2 + f*x**2),x)*d*f*q* *2 - 4*int(((d + e*x + f*x**2)**q*x)/(2*d*q + d + 2*e*q*x + e*x + 2*f*q*x* *2 + f*x**2),x)*d*f*q + 2*int(((d + e*x + f*x**2)**q*x)/(2*d*q + d + 2*e*q *x + e*x + 2*f*q*x**2 + f*x**2),x)*e**2*q**2 + int(((d + e*x + f*x**2)**q* x)/(2*d*q + d + 2*e*q*x + e*x + 2*f*q*x**2 + f*x**2),x)*e**2*q)/(e*(2*q + 1))