Integrand size = 31, antiderivative size = 108 \[ \int \left (a+\frac {e x}{2}+c x^2\right )^p \left (2 a+e x+2 c x^2\right )^q \, dx=\frac {2^{-2-4 p-3 q} (e+4 c x) \left (2 a+e x+2 c x^2\right )^{p+q} \left (\frac {c \left (2 a+e x+2 c x^2\right )}{16 a c-e^2}\right )^{-p-q} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p-q,\frac {3}{2},-\frac {(e+4 c x)^2}{16 a c-e^2}\right )}{c} \] Output:
2^(-2-4*p-3*q)*(4*c*x+e)*(2*c*x^2+e*x+2*a)^(p+q)*(c*(2*c*x^2+e*x+2*a)/(16* a*c-e^2))^(-p-q)*hypergeom([1/2, -p-q],[3/2],-(4*c*x+e)^2/(16*a*c-e^2))/c
Time = 0.24 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.31 \[ \int \left (a+\frac {e x}{2}+c x^2\right )^p \left (2 a+e x+2 c x^2\right )^q \, dx=\frac {2^{-2+q} \left (e-\sqrt {-16 a c+e^2}+4 c x\right ) \left (\frac {e+\sqrt {-16 a c+e^2}+4 c x}{\sqrt {-16 a c+e^2}}\right )^{-p-q} (2 a+x (e+2 c x))^{p+q} \operatorname {Hypergeometric2F1}\left (-p-q,1+p+q,2+p+q,\frac {-e+\sqrt {-16 a c+e^2}-4 c x}{2 \sqrt {-16 a c+e^2}}\right )}{c (1+p+q)} \] Input:
Integrate[(a + (e*x)/2 + c*x^2)^p*(2*a + e*x + 2*c*x^2)^q,x]
Output:
(2^(-2 + q)*(e - Sqrt[-16*a*c + e^2] + 4*c*x)*((e + Sqrt[-16*a*c + e^2] + 4*c*x)/Sqrt[-16*a*c + e^2])^(-p - q)*(2*a + x*(e + 2*c*x))^(p + q)*Hyperge ometric2F1[-p - q, 1 + p + q, 2 + p + q, (-e + Sqrt[-16*a*c + e^2] - 4*c*x )/(2*Sqrt[-16*a*c + e^2])])/(c*(1 + p + q))
Time = 0.27 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.26, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {1295, 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.
| \(\displaystyle \int \left (a+c x^2+\frac {e x}{2}\right )^p \left (2 a+2 c x^2+e x\right )^q \, dx\) |
| \(\Big \downarrow \) 1295 |
| \(\displaystyle 2^{-p} \int \left (2 c x^2+e x+2 a\right )^{p+q}dx\) |
| \(\Big \downarrow \) 1096 |
| \(\displaystyle -\frac {2^{q+1} \left (-\frac {-\sqrt {e^2-16 a c}+4 c x+e}{\sqrt {e^2-16 a c}}\right )^{-p-q-1} \left (2 a+2 c x^2+e x\right )^{p+q+1} \operatorname {Hypergeometric2F1}\left (-p-q,p+q+1,p+q+2,\frac {e+4 c x+\sqrt {e^2-16 a c}}{2 \sqrt {e^2-16 a c}}\right )}{(p+q+1) \sqrt {e^2-16 a c}}\) |
Input:
Int[(a + (e*x)/2 + c*x^2)^p*(2*a + e*x + 2*c*x^2)^q,x]
Output:
-((2^(1 + q)*(-((e - Sqrt[-16*a*c + e^2] + 4*c*x)/Sqrt[-16*a*c + e^2]))^(- 1 - p - q)*(2*a + e*x + 2*c*x^2)^(1 + p + q)*Hypergeometric2F1[-p - q, 1 + p + q, 2 + p + q, (e + Sqrt[-16*a*c + e^2] + 4*c*x)/(2*Sqrt[-16*a*c + e^2 ])])/(Sqrt[-16*a*c + e^2]*(1 + p + 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[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.)*((d_) + (e_.)*(x_) + (f_.)*(x_ )^2)^(q_.), x_Symbol] :> Simp[(c/f)^p Int[(d + e*x + f*x^2)^(p + q), x], x] /; FreeQ[{a, b, c, d, e, f, p, q}, x] && EqQ[c*d - a*f, 0] && EqQ[b*d - a*e, 0] && (IntegerQ[p] || GtQ[c/f, 0]) && ( !IntegerQ[q] || LeafCount[d + e*x + f*x^2] <= LeafCount[a + b*x + c*x^2])
\[\int \left (a +\frac {1}{2} e x +c \,x^{2}\right )^{p} \left (2 c \,x^{2}+e x +2 a \right )^{q}d x\]
Input:
int((a+1/2*e*x+c*x^2)^p*(2*c*x^2+e*x+2*a)^q,x)
Output:
int((a+1/2*e*x+c*x^2)^p*(2*c*x^2+e*x+2*a)^q,x)
\[ \int \left (a+\frac {e x}{2}+c x^2\right )^p \left (2 a+e x+2 c x^2\right )^q \, dx=\int { {\left (2 \, c x^{2} + e x + 2 \, a\right )}^{q} {\left (c x^{2} + \frac {1}{2} \, e x + a\right )}^{p} \,d x } \] Input:
integrate((a+1/2*e*x+c*x^2)^p*(2*c*x^2+e*x+2*a)^q,x, algorithm="fricas")
Output:
integral((2*c*x^2 + e*x + 2*a)^q*(c*x^2 + 1/2*e*x + a)^p, x)
Timed out. \[ \int \left (a+\frac {e x}{2}+c x^2\right )^p \left (2 a+e x+2 c x^2\right )^q \, dx=\text {Timed out} \] Input:
integrate((a+1/2*e*x+c*x**2)**p*(2*c*x**2+e*x+2*a)**q,x)
Output:
Timed out
\[ \int \left (a+\frac {e x}{2}+c x^2\right )^p \left (2 a+e x+2 c x^2\right )^q \, dx=\int { {\left (2 \, c x^{2} + e x + 2 \, a\right )}^{q} {\left (c x^{2} + \frac {1}{2} \, e x + a\right )}^{p} \,d x } \] Input:
integrate((a+1/2*e*x+c*x^2)^p*(2*c*x^2+e*x+2*a)^q,x, algorithm="maxima")
Output:
integrate((2*c*x^2 + e*x + 2*a)^q*(c*x^2 + 1/2*e*x + a)^p, x)
\[ \int \left (a+\frac {e x}{2}+c x^2\right )^p \left (2 a+e x+2 c x^2\right )^q \, dx=\int { {\left (2 \, c x^{2} + e x + 2 \, a\right )}^{q} {\left (c x^{2} + \frac {1}{2} \, e x + a\right )}^{p} \,d x } \] Input:
integrate((a+1/2*e*x+c*x^2)^p*(2*c*x^2+e*x+2*a)^q,x, algorithm="giac")
Output:
integrate((2*c*x^2 + e*x + 2*a)^q*(c*x^2 + 1/2*e*x + a)^p, x)
Timed out. \[ \int \left (a+\frac {e x}{2}+c x^2\right )^p \left (2 a+e x+2 c x^2\right )^q \, dx=\int {\left (c\,x^2+\frac {e\,x}{2}+a\right )}^p\,{\left (2\,c\,x^2+e\,x+2\,a\right )}^q \,d x \] Input:
int((a + (e*x)/2 + c*x^2)^p*(2*a + e*x + 2*c*x^2)^q,x)
Output:
int((a + (e*x)/2 + c*x^2)^p*(2*a + e*x + 2*c*x^2)^q, x)
\[ \int \left (a+\frac {e x}{2}+c x^2\right )^p \left (2 a+e x+2 c x^2\right )^q \, dx=\frac {4 \left (2 c \,x^{2}+e x +2 a \right )^{p +q} a +\left (2 c \,x^{2}+e x +2 a \right )^{p +q} e x -32 \left (\int \frac {\left (2 c \,x^{2}+e x +2 a \right )^{p +q} x}{4 c p \,x^{2}+4 c q \,x^{2}+2 c \,x^{2}+2 e p x +2 e q x +4 a p +4 a q +e x +2 a}d x \right ) a c \,p^{2}-64 \left (\int \frac {\left (2 c \,x^{2}+e x +2 a \right )^{p +q} x}{4 c p \,x^{2}+4 c q \,x^{2}+2 c \,x^{2}+2 e p x +2 e q x +4 a p +4 a q +e x +2 a}d x \right ) a c p q -16 \left (\int \frac {\left (2 c \,x^{2}+e x +2 a \right )^{p +q} x}{4 c p \,x^{2}+4 c q \,x^{2}+2 c \,x^{2}+2 e p x +2 e q x +4 a p +4 a q +e x +2 a}d x \right ) a c p -32 \left (\int \frac {\left (2 c \,x^{2}+e x +2 a \right )^{p +q} x}{4 c p \,x^{2}+4 c q \,x^{2}+2 c \,x^{2}+2 e p x +2 e q x +4 a p +4 a q +e x +2 a}d x \right ) a c \,q^{2}-16 \left (\int \frac {\left (2 c \,x^{2}+e x +2 a \right )^{p +q} x}{4 c p \,x^{2}+4 c q \,x^{2}+2 c \,x^{2}+2 e p x +2 e q x +4 a p +4 a q +e x +2 a}d x \right ) a c q +2 \left (\int \frac {\left (2 c \,x^{2}+e x +2 a \right )^{p +q} x}{4 c p \,x^{2}+4 c q \,x^{2}+2 c \,x^{2}+2 e p x +2 e q x +4 a p +4 a q +e x +2 a}d x \right ) e^{2} p^{2}+4 \left (\int \frac {\left (2 c \,x^{2}+e x +2 a \right )^{p +q} x}{4 c p \,x^{2}+4 c q \,x^{2}+2 c \,x^{2}+2 e p x +2 e q x +4 a p +4 a q +e x +2 a}d x \right ) e^{2} p q +\left (\int \frac {\left (2 c \,x^{2}+e x +2 a \right )^{p +q} x}{4 c p \,x^{2}+4 c q \,x^{2}+2 c \,x^{2}+2 e p x +2 e q x +4 a p +4 a q +e x +2 a}d x \right ) e^{2} p +2 \left (\int \frac {\left (2 c \,x^{2}+e x +2 a \right )^{p +q} x}{4 c p \,x^{2}+4 c q \,x^{2}+2 c \,x^{2}+2 e p x +2 e q x +4 a p +4 a q +e x +2 a}d x \right ) e^{2} q^{2}+\left (\int \frac {\left (2 c \,x^{2}+e x +2 a \right )^{p +q} x}{4 c p \,x^{2}+4 c q \,x^{2}+2 c \,x^{2}+2 e p x +2 e q x +4 a p +4 a q +e x +2 a}d x \right ) e^{2} q}{2^{p} e \left (2 p +2 q +1\right )} \] Input:
int((a+1/2*e*x+c*x^2)^p*(2*c*x^2+e*x+2*a)^q,x)
Output:
(4*(2*a + 2*c*x**2 + e*x)**(p + q)*a + (2*a + 2*c*x**2 + e*x)**(p + q)*e*x - 32*int(((2*a + 2*c*x**2 + e*x)**(p + q)*x)/(4*a*p + 4*a*q + 2*a + 4*c*p *x**2 + 4*c*q*x**2 + 2*c*x**2 + 2*e*p*x + 2*e*q*x + e*x),x)*a*c*p**2 - 64* int(((2*a + 2*c*x**2 + e*x)**(p + q)*x)/(4*a*p + 4*a*q + 2*a + 4*c*p*x**2 + 4*c*q*x**2 + 2*c*x**2 + 2*e*p*x + 2*e*q*x + e*x),x)*a*c*p*q - 16*int(((2 *a + 2*c*x**2 + e*x)**(p + q)*x)/(4*a*p + 4*a*q + 2*a + 4*c*p*x**2 + 4*c*q *x**2 + 2*c*x**2 + 2*e*p*x + 2*e*q*x + e*x),x)*a*c*p - 32*int(((2*a + 2*c* x**2 + e*x)**(p + q)*x)/(4*a*p + 4*a*q + 2*a + 4*c*p*x**2 + 4*c*q*x**2 + 2 *c*x**2 + 2*e*p*x + 2*e*q*x + e*x),x)*a*c*q**2 - 16*int(((2*a + 2*c*x**2 + e*x)**(p + q)*x)/(4*a*p + 4*a*q + 2*a + 4*c*p*x**2 + 4*c*q*x**2 + 2*c*x** 2 + 2*e*p*x + 2*e*q*x + e*x),x)*a*c*q + 2*int(((2*a + 2*c*x**2 + e*x)**(p + q)*x)/(4*a*p + 4*a*q + 2*a + 4*c*p*x**2 + 4*c*q*x**2 + 2*c*x**2 + 2*e*p* x + 2*e*q*x + e*x),x)*e**2*p**2 + 4*int(((2*a + 2*c*x**2 + e*x)**(p + q)*x )/(4*a*p + 4*a*q + 2*a + 4*c*p*x**2 + 4*c*q*x**2 + 2*c*x**2 + 2*e*p*x + 2* e*q*x + e*x),x)*e**2*p*q + int(((2*a + 2*c*x**2 + e*x)**(p + q)*x)/(4*a*p + 4*a*q + 2*a + 4*c*p*x**2 + 4*c*q*x**2 + 2*c*x**2 + 2*e*p*x + 2*e*q*x + e *x),x)*e**2*p + 2*int(((2*a + 2*c*x**2 + e*x)**(p + q)*x)/(4*a*p + 4*a*q + 2*a + 4*c*p*x**2 + 4*c*q*x**2 + 2*c*x**2 + 2*e*p*x + 2*e*q*x + e*x),x)*e* *2*q**2 + int(((2*a + 2*c*x**2 + e*x)**(p + q)*x)/(4*a*p + 4*a*q + 2*a + 4 *c*p*x**2 + 4*c*q*x**2 + 2*c*x**2 + 2*e*p*x + 2*e*q*x + e*x),x)*e**2*q)...