\(\int (d+e x)^{-6-2 p} (a+b x+c x^2)^p \, dx\) [2580]

Optimal result

Integrand size = 24, antiderivative size = 809 \[ \int (d+e x)^{-6-2 p} \left (a+b x+c x^2\right )^p \, dx=-\frac {e (d+e x)^{-5-2 p} \left (a+b x+c x^2\right )^{1+p}}{\left (c d^2-b d e+a e^2\right ) (5+2 p)}-\frac {e \left (b^2 e^2 \left (12+7 p+p^2\right )+2 c^2 d^2 \left (18+11 p+2 p^2\right )-2 c e \left (3 a e (2+p)+b d \left (18+11 p+2 p^2\right )\right )\right ) (d+e x)^{-3-2 p} \left (a+b x+c x^2\right )^{1+p}}{2 \left (c d^2-b d e+a e^2\right )^3 (2+p) (3+2 p) (5+2 p)}-\frac {e (2 c d-b e) (3+p) \left (b^2 e^2 \left (8+6 p+p^2\right )+2 c^2 d^2 \left (8+7 p+2 p^2\right )-2 c e \left (a e (8+5 p)+b d \left (8+7 p+2 p^2\right )\right )\right ) (d+e x)^{-2 (1+p)} \left (a+b x+c x^2\right )^{1+p}}{4 \left (c d^2-b d e+a e^2\right )^4 (1+p) (2+p) (3+2 p) (5+2 p)}-\frac {e (2 c d-b e) (4+p) (d+e x)^{-2 (2+p)} \left (a+b x+c x^2\right )^{1+p}}{2 \left (c d^2-b d e+a e^2\right )^2 (2+p) (5+2 p)}+\frac {\left (b^4 e^4 \left (12+7 p+p^2\right )+4 c^4 d^4 \left (15+16 p+4 p^2\right )-8 c^3 d^2 e (5+2 p) (3 a e+b d (3+2 p))-4 b^2 c e^3 (3+p) (3 a e+b d (5+2 p))+12 c^2 e^2 \left (a^2 e^2+2 a b d e (5+2 p)+b^2 d^2 \left (10+9 p+2 p^2\right )\right )\right ) \left (b-\sqrt {b^2-4 a c}+2 c x\right ) \left (\frac {\left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b-\sqrt {b^2-4 a c}+2 c x\right )}\right )^{-p} (d+e x)^{-1-2 p} \left (a+b x+c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-1-2 p,-p,-2 p,-\frac {4 c \sqrt {b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b-\sqrt {b^2-4 a c}+2 c x\right )}\right )}{4 \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \left (c d^2-b d e+a e^2\right )^4 (1+2 p) (3+2 p) (5+2 p)} \]

[Out]

-e*(e*x+d)^(-5-2*p)*(c*x^2+b*x+a)^(p+1)/(a*e^2-b*d*e+c*d^2)/(5+2*p)-1/2*e*(b^2*e^2*(p^2+7*p+12)+2*c^2*d^2*(2*p
^2+11*p+18)-2*c*e*(3*a*e*(2+p)+b*d*(2*p^2+11*p+18)))*(e*x+d)^(-3-2*p)*(c*x^2+b*x+a)^(p+1)/(a*e^2-b*d*e+c*d^2)^
3/(2+p)/(3+2*p)/(5+2*p)-1/4*e*(-b*e+2*c*d)*(3+p)*(b^2*e^2*(p^2+6*p+8)+2*c^2*d^2*(2*p^2+7*p+8)-2*c*e*(a*e*(8+5*
p)+b*d*(2*p^2+7*p+8)))*(c*x^2+b*x+a)^(p+1)/(a*e^2-b*d*e+c*d^2)^4/(p+1)/(2+p)/(3+2*p)/(5+2*p)/((e*x+d)^(2+2*p))
-1/2*e*(-b*e+2*c*d)*(4+p)*(c*x^2+b*x+a)^(p+1)/(a*e^2-b*d*e+c*d^2)^2/(2+p)/(5+2*p)/((e*x+d)^(4+2*p))+1/4*(b^4*e
^4*(p^2+7*p+12)+4*c^4*d^4*(4*p^2+16*p+15)-8*c^3*d^2*e*(5+2*p)*(3*a*e+b*d*(3+2*p))-4*b^2*c*e^3*(3+p)*(3*a*e+b*d
*(5+2*p))+12*c^2*e^2*(a^2*e^2+2*a*b*d*e*(5+2*p)+b^2*d^2*(2*p^2+9*p+10)))*(e*x+d)^(-1-2*p)*(c*x^2+b*x+a)^p*hype
rgeom([-p, -1-2*p],[-2*p],-4*c*(e*x+d)*(-4*a*c+b^2)^(1/2)/(b+2*c*x-(-4*a*c+b^2)^(1/2))/(2*c*d-e*(b+(-4*a*c+b^2
)^(1/2))))*(b+2*c*x-(-4*a*c+b^2)^(1/2))/(a*e^2-b*d*e+c*d^2)^4/(1+2*p)/(3+2*p)/(5+2*p)/(2*c*d-e*(b-(-4*a*c+b^2)
^(1/2)))/(((2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+2*c*x-(-4*a*c+b^2)^(1/2))/(2*c*d-e
*(b+(-4*a*c+b^2)^(1/2))))^p)

Rubi [A] (verified)

Time = 1.11 (sec) , antiderivative size = 809, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {758, 850, 820, 740} \[ \int (d+e x)^{-6-2 p} \left (a+b x+c x^2\right )^p \, dx=-\frac {e \left (c x^2+b x+a\right )^{p+1} (d+e x)^{-2 p-5}}{\left (c d^2-b e d+a e^2\right ) (2 p+5)}-\frac {e \left (2 c^2 \left (2 p^2+11 p+18\right ) d^2+b^2 e^2 \left (p^2+7 p+12\right )-2 c e \left (3 a e (p+2)+b d \left (2 p^2+11 p+18\right )\right )\right ) \left (c x^2+b x+a\right )^{p+1} (d+e x)^{-2 p-3}}{2 \left (c d^2-b e d+a e^2\right )^3 (p+2) (2 p+3) (2 p+5)}+\frac {\left (4 c^4 \left (4 p^2+16 p+15\right ) d^4-8 c^3 e (2 p+5) (3 a e+b d (2 p+3)) d^2+b^4 e^4 \left (p^2+7 p+12\right )-4 b^2 c e^3 (p+3) (3 a e+b d (2 p+5))+12 c^2 e^2 \left (b^2 \left (2 p^2+9 p+10\right ) d^2+2 a b e (2 p+5) d+a^2 e^2\right )\right ) \left (b+2 c x-\sqrt {b^2-4 a c}\right ) \left (\frac {\left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt {b^2-4 a c}\right )}\right )^{-p} \left (c x^2+b x+a\right )^p \operatorname {Hypergeometric2F1}\left (-2 p-1,-p,-2 p,-\frac {4 c \sqrt {b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt {b^2-4 a c}\right )}\right ) (d+e x)^{-2 p-1}}{4 \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \left (c d^2-b e d+a e^2\right )^4 (2 p+1) (2 p+3) (2 p+5)}-\frac {e (2 c d-b e) (p+3) \left (2 c^2 \left (2 p^2+7 p+8\right ) d^2+b^2 e^2 \left (p^2+6 p+8\right )-2 c e \left (a e (5 p+8)+b d \left (2 p^2+7 p+8\right )\right )\right ) \left (c x^2+b x+a\right )^{p+1} (d+e x)^{-2 (p+1)}}{4 \left (c d^2-b e d+a e^2\right )^4 (p+1) (p+2) (2 p+3) (2 p+5)}-\frac {e (2 c d-b e) (p+4) \left (c x^2+b x+a\right )^{p+1} (d+e x)^{-2 (p+2)}}{2 \left (c d^2-b e d+a e^2\right )^2 (p+2) (2 p+5)} \]

[In]

Int[(d + e*x)^(-6 - 2*p)*(a + b*x + c*x^2)^p,x]

[Out]

-((e*(d + e*x)^(-5 - 2*p)*(a + b*x + c*x^2)^(1 + p))/((c*d^2 - b*d*e + a*e^2)*(5 + 2*p))) - (e*(b^2*e^2*(12 +
7*p + p^2) + 2*c^2*d^2*(18 + 11*p + 2*p^2) - 2*c*e*(3*a*e*(2 + p) + b*d*(18 + 11*p + 2*p^2)))*(d + e*x)^(-3 -
2*p)*(a + b*x + c*x^2)^(1 + p))/(2*(c*d^2 - b*d*e + a*e^2)^3*(2 + p)*(3 + 2*p)*(5 + 2*p)) - (e*(2*c*d - b*e)*(
3 + p)*(b^2*e^2*(8 + 6*p + p^2) + 2*c^2*d^2*(8 + 7*p + 2*p^2) - 2*c*e*(a*e*(8 + 5*p) + b*d*(8 + 7*p + 2*p^2)))
*(a + b*x + c*x^2)^(1 + p))/(4*(c*d^2 - b*d*e + a*e^2)^4*(1 + p)*(2 + p)*(3 + 2*p)*(5 + 2*p)*(d + e*x)^(2*(1 +
p))) - (e*(2*c*d - b*e)*(4 + p)*(a + b*x + c*x^2)^(1 + p))/(2*(c*d^2 - b*d*e + a*e^2)^2*(2 + p)*(5 + 2*p)*(d
+ e*x)^(2*(2 + p))) + ((b^4*e^4*(12 + 7*p + p^2) + 4*c^4*d^4*(15 + 16*p + 4*p^2) - 8*c^3*d^2*e*(5 + 2*p)*(3*a*
e + b*d*(3 + 2*p)) - 4*b^2*c*e^3*(3 + p)*(3*a*e + b*d*(5 + 2*p)) + 12*c^2*e^2*(a^2*e^2 + 2*a*b*d*e*(5 + 2*p) +
b^2*d^2*(10 + 9*p + 2*p^2)))*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)*(d + e*x)^(-1 - 2*p)*(a + b*x + c*x^2)^p*Hyperge
ometric2F1[-1 - 2*p, -p, -2*p, (-4*c*Sqrt[b^2 - 4*a*c]*(d + e*x))/((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*(b - Sq
rt[b^2 - 4*a*c] + 2*c*x))])/(4*(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)*(c*d^2 - b*d*e + a*e^2)^4*(1 + 2*p)*(3 + 2*
p)*(5 + 2*p)*(((2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/((2*c*d - (b + Sqrt[b^2 -
4*a*c])*e)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)))^p)

Rule 740

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(b - Rt[b^2 - 4*a*
c, 2] + 2*c*x))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/((m + 1)*(2*c*d - b*e + e*Rt[b^2 - 4*a*c, 2])*((2*c*d -
b*e + e*Rt[b^2 - 4*a*c, 2])*((b + Rt[b^2 - 4*a*c, 2] + 2*c*x)/((2*c*d - b*e - e*Rt[b^2 - 4*a*c, 2])*(b - Rt[b
^2 - 4*a*c, 2] + 2*c*x))))^p))*Hypergeometric2F1[m + 1, -p, m + 2, -4*c*Rt[b^2 - 4*a*c, 2]*((d + e*x)/((2*c*d
- b*e - e*Rt[b^2 - 4*a*c, 2])*(b - Rt[b^2 - 4*a*c, 2] + 2*c*x)))], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && Ne
Q[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] &&  !IntegerQ[p] && EqQ[m + 2*p + 2,
0]

Rule 758

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m + 1)*
((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)),
Int[(d + e*x)^(m + 1)*Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*x, x]*(a + b*x + c*x^2)^p, x], x]
/; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e
, 0] && NeQ[m, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimplerQ[m, 1] && IntegerQ
[p]) || ILtQ[Simplify[m + 2*p + 3], 0])

Rule 820

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Dist[
(b*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x]
, x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[S
implify[m + 2*p + 3], 0]

Rule 850

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x]
&& NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[Simplify[m + 2*p + 3], 0] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {e (d+e x)^{-5-2 p} \left (a+b x+c x^2\right )^{1+p}}{\left (c d^2-b d e+a e^2\right ) (5+2 p)}-\frac {\int (d+e x)^{-5-2 p} (b e (4+p)-c d (5+2 p)+3 c e x) \left (a+b x+c x^2\right )^p \, dx}{\left (c d^2-b d e+a e^2\right ) (5+2 p)} \\ & = -\frac {e (d+e x)^{-5-2 p} \left (a+b x+c x^2\right )^{1+p}}{\left (c d^2-b d e+a e^2\right ) (5+2 p)}-\frac {e (2 c d-b e) (4+p) (d+e x)^{-2 (2+p)} \left (a+b x+c x^2\right )^{1+p}}{2 \left (c d^2-b d e+a e^2\right )^2 (2+p) (5+2 p)}+\frac {\int (d+e x)^{-4-2 p} \left (b^2 e^2 \left (12+7 p+p^2\right )+2 c^2 d^2 \left (10+9 p+2 p^2\right )-2 c e \left (3 a e (2+p)+2 b d \left (7+5 p+p^2\right )\right )-2 c e (2 c d-b e) (4+p) x\right ) \left (a+b x+c x^2\right )^p \, dx}{2 \left (c d^2-b d e+a e^2\right )^2 (2+p) (5+2 p)} \\ & = -\frac {e (d+e x)^{-5-2 p} \left (a+b x+c x^2\right )^{1+p}}{\left (c d^2-b d e+a e^2\right ) (5+2 p)}-\frac {e \left (b^2 e^2 \left (12+7 p+p^2\right )+2 c^2 d^2 \left (18+11 p+2 p^2\right )-2 c e \left (3 a e (2+p)+b d \left (18+11 p+2 p^2\right )\right )\right ) (d+e x)^{-3-2 p} \left (a+b x+c x^2\right )^{1+p}}{2 \left (c d^2-b d e+a e^2\right )^3 (2+p) (3+2 p) (5+2 p)}-\frac {e (2 c d-b e) (4+p) (d+e x)^{-2 (2+p)} \left (a+b x+c x^2\right )^{1+p}}{2 \left (c d^2-b d e+a e^2\right )^2 (2+p) (5+2 p)}-\frac {\int (d+e x)^{-3-2 p} \left (b^3 e^3 \left (24+26 p+9 p^2+p^3\right )-2 c^3 d^3 \left (30+47 p+24 p^2+4 p^3\right )-b c e^2 (3+p) \left (2 a e (8+5 p)+b d \left (28+25 p+6 p^2\right )\right )+2 c^2 d e \left (a e \left (42+43 p+10 p^2\right )+b d \left (54+76 p+37 p^2+6 p^3\right )\right )+c e \left (b^2 e^2 \left (12+7 p+p^2\right )+2 c^2 d^2 \left (18+11 p+2 p^2\right )-2 c e \left (3 a e (2+p)+b d \left (18+11 p+2 p^2\right )\right )\right ) x\right ) \left (a+b x+c x^2\right )^p \, dx}{2 \left (c d^2-b d e+a e^2\right )^3 (2+p) (3+2 p) (5+2 p)} \\ & = -\frac {e (d+e x)^{-5-2 p} \left (a+b x+c x^2\right )^{1+p}}{\left (c d^2-b d e+a e^2\right ) (5+2 p)}-\frac {e \left (b^2 e^2 \left (12+7 p+p^2\right )+2 c^2 d^2 \left (18+11 p+2 p^2\right )-2 c e \left (3 a e (2+p)+b d \left (18+11 p+2 p^2\right )\right )\right ) (d+e x)^{-3-2 p} \left (a+b x+c x^2\right )^{1+p}}{2 \left (c d^2-b d e+a e^2\right )^3 (2+p) (3+2 p) (5+2 p)}-\frac {e (2 c d-b e) (3+p) \left (b^2 e^2 \left (8+6 p+p^2\right )+2 c^2 d^2 \left (8+7 p+2 p^2\right )-2 c e \left (a e (8+5 p)+b d \left (8+7 p+2 p^2\right )\right )\right ) (d+e x)^{-2 (1+p)} \left (a+b x+c x^2\right )^{1+p}}{4 \left (c d^2-b d e+a e^2\right )^4 (1+p) (2+p) (3+2 p) (5+2 p)}-\frac {e (2 c d-b e) (4+p) (d+e x)^{-2 (2+p)} \left (a+b x+c x^2\right )^{1+p}}{2 \left (c d^2-b d e+a e^2\right )^2 (2+p) (5+2 p)}+\frac {\left (b^4 e^4 \left (12+7 p+p^2\right )+4 c^4 d^4 \left (15+16 p+4 p^2\right )-8 c^3 d^2 e (5+2 p) (3 a e+b d (3+2 p))-4 b^2 c e^3 (3+p) (3 a e+b d (5+2 p))+12 c^2 e^2 \left (a^2 e^2+2 a b d e (5+2 p)+b^2 d^2 \left (10+9 p+2 p^2\right )\right )\right ) \int (d+e x)^{-2-2 p} \left (a+b x+c x^2\right )^p \, dx}{4 \left (c d^2-b d e+a e^2\right )^4 (3+2 p) (5+2 p)} \\ & = -\frac {e (d+e x)^{-5-2 p} \left (a+b x+c x^2\right )^{1+p}}{\left (c d^2-b d e+a e^2\right ) (5+2 p)}-\frac {e \left (b^2 e^2 \left (12+7 p+p^2\right )+2 c^2 d^2 \left (18+11 p+2 p^2\right )-2 c e \left (3 a e (2+p)+b d \left (18+11 p+2 p^2\right )\right )\right ) (d+e x)^{-3-2 p} \left (a+b x+c x^2\right )^{1+p}}{2 \left (c d^2-b d e+a e^2\right )^3 (2+p) (3+2 p) (5+2 p)}-\frac {e (2 c d-b e) (3+p) \left (b^2 e^2 \left (8+6 p+p^2\right )+2 c^2 d^2 \left (8+7 p+2 p^2\right )-2 c e \left (a e (8+5 p)+b d \left (8+7 p+2 p^2\right )\right )\right ) (d+e x)^{-2 (1+p)} \left (a+b x+c x^2\right )^{1+p}}{4 \left (c d^2-b d e+a e^2\right )^4 (1+p) (2+p) (3+2 p) (5+2 p)}-\frac {e (2 c d-b e) (4+p) (d+e x)^{-2 (2+p)} \left (a+b x+c x^2\right )^{1+p}}{2 \left (c d^2-b d e+a e^2\right )^2 (2+p) (5+2 p)}+\frac {\left (b^4 e^4 \left (12+7 p+p^2\right )+4 c^4 d^4 \left (15+16 p+4 p^2\right )-8 c^3 d^2 e (5+2 p) (3 a e+b d (3+2 p))-4 b^2 c e^3 (3+p) (3 a e+b d (5+2 p))+12 c^2 e^2 \left (a^2 e^2+2 a b d e (5+2 p)+b^2 d^2 \left (10+9 p+2 p^2\right )\right )\right ) \left (b-\sqrt {b^2-4 a c}+2 c x\right ) \left (\frac {\left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b-\sqrt {b^2-4 a c}+2 c x\right )}\right )^{-p} (d+e x)^{-1-2 p} \left (a+b x+c x^2\right )^p \, _2F_1\left (-1-2 p,-p;-2 p;-\frac {4 c \sqrt {b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b-\sqrt {b^2-4 a c}+2 c x\right )}\right )}{4 \left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \left (c d^2-b d e+a e^2\right )^4 (1+2 p) (3+2 p) (5+2 p)} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.57 (sec) , antiderivative size = 1577, normalized size of antiderivative = 1.95 \[ \int (d+e x)^{-6-2 p} \left (a+b x+c x^2\right )^p \, dx=\frac {e (d+e x)^{1-2 (3+p)} \left (a+b x+c x^2\right )^{1+p}}{\left (c d^2-b d e+a e^2\right ) (1-2 (3+p))}+\frac {3 c \left (\frac {e (d+e x)^{3-2 (3+p)} \left (a+b x+c x^2\right )^{1+p}}{\left (c d^2-b d e+a e^2\right ) (3-2 (3+p))}+\frac {\frac {(c d e-e (b e (2+p)-c d (3+2 p))) (d+e x)^{4-2 (3+p)} \left (a+b x+c x^2\right )^{1+p}}{2 \left (c d^2-b d e+a e^2\right ) (1+p)}-\frac {\left (-2 \left (a c e^2+c d (b e (2+p)-c d (3+2 p))\right )+b (c d e+e (b e (2+p)-c d (3+2 p)))\right ) \left (-b+\sqrt {b^2-4 a c}-2 c x\right ) \left (\frac {\left (2 c d-b e+\sqrt {b^2-4 a c} e\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c d-b e-\sqrt {b^2-4 a c} e\right ) \left (b-\sqrt {b^2-4 a c}+2 c x\right )}\right )^{-p} (d+e x)^{5-2 (3+p)} \left (a+b x+c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-p,5-2 (3+p),6-2 (3+p),-\frac {4 c \sqrt {b^2-4 a c} (d+e x)}{\left (2 c d-b e-\sqrt {b^2-4 a c} e\right ) \left (b-\sqrt {b^2-4 a c}+2 c x\right )}\right )}{2 \left (2 c d-b e+\sqrt {b^2-4 a c} e\right ) \left (c d^2-b d e+a e^2\right ) (5-2 (3+p))}}{\left (c d^2-b d e+a e^2\right ) (3-2 (3+p))}\right )+\frac {(-3 c d e+e (b e (4+p)-c d (5+2 p))) \left (\frac {e (d+e x)^{2-2 (3+p)} \left (a+b x+c x^2\right )^{1+p}}{\left (c d^2-b d e+a e^2\right ) (2-2 (3+p))}+\frac {2 c \left (\frac {e (d+e x)^{4-2 (3+p)} \left (a+b x+c x^2\right )^{1+p}}{\left (c d^2-b d e+a e^2\right ) (4-2 (3+p))}+\frac {(2 c d-b e) \left (-b+\sqrt {b^2-4 a c}-2 c x\right ) \left (\frac {\left (2 c d-b e+\sqrt {b^2-4 a c} e\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c d-b e-\sqrt {b^2-4 a c} e\right ) \left (b-\sqrt {b^2-4 a c}+2 c x\right )}\right )^{-p} (d+e x)^{5-2 (3+p)} \left (a+b x+c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-p,5-2 (3+p),6-2 (3+p),-\frac {4 c \sqrt {b^2-4 a c} (d+e x)}{\left (2 c d-b e-\sqrt {b^2-4 a c} e\right ) \left (b-\sqrt {b^2-4 a c}+2 c x\right )}\right )}{2 \left (2 c d-b e+\sqrt {b^2-4 a c} e\right ) \left (c d^2-b d e+a e^2\right ) (5-2 (3+p))}\right )+\frac {(-2 c d e+e (-2 c d (2+p)+b e (3+p))) \left (\frac {e (d+e x)^{3-2 (3+p)} \left (a+b x+c x^2\right )^{1+p}}{\left (c d^2-b d e+a e^2\right ) (3-2 (3+p))}+\frac {\frac {(c d e-e (b e (2+p)-c d (3+2 p))) (d+e x)^{4-2 (3+p)} \left (a+b x+c x^2\right )^{1+p}}{2 \left (c d^2-b d e+a e^2\right ) (1+p)}-\frac {\left (-2 \left (a c e^2+c d (b e (2+p)-c d (3+2 p))\right )+b (c d e+e (b e (2+p)-c d (3+2 p)))\right ) \left (-b+\sqrt {b^2-4 a c}-2 c x\right ) \left (\frac {\left (2 c d-b e+\sqrt {b^2-4 a c} e\right ) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c d-b e-\sqrt {b^2-4 a c} e\right ) \left (b-\sqrt {b^2-4 a c}+2 c x\right )}\right )^{-p} (d+e x)^{5-2 (3+p)} \left (a+b x+c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-p,5-2 (3+p),6-2 (3+p),-\frac {4 c \sqrt {b^2-4 a c} (d+e x)}{\left (2 c d-b e-\sqrt {b^2-4 a c} e\right ) \left (b-\sqrt {b^2-4 a c}+2 c x\right )}\right )}{2 \left (2 c d-b e+\sqrt {b^2-4 a c} e\right ) \left (c d^2-b d e+a e^2\right ) (5-2 (3+p))}}{\left (c d^2-b d e+a e^2\right ) (3-2 (3+p))}\right )}{e}}{\left (c d^2-b d e+a e^2\right ) (2-2 (3+p))}\right )}{e}}{\left (c d^2-b d e+a e^2\right ) (1-2 (3+p))} \]

[In]

Integrate[(d + e*x)^(-6 - 2*p)*(a + b*x + c*x^2)^p,x]

[Out]

(e*(d + e*x)^(1 - 2*(3 + p))*(a + b*x + c*x^2)^(1 + p))/((c*d^2 - b*d*e + a*e^2)*(1 - 2*(3 + p))) + (3*c*((e*(
d + e*x)^(3 - 2*(3 + p))*(a + b*x + c*x^2)^(1 + p))/((c*d^2 - b*d*e + a*e^2)*(3 - 2*(3 + p))) + (((c*d*e - e*(
b*e*(2 + p) - c*d*(3 + 2*p)))*(d + e*x)^(4 - 2*(3 + p))*(a + b*x + c*x^2)^(1 + p))/(2*(c*d^2 - b*d*e + a*e^2)*
(1 + p)) - ((-2*(a*c*e^2 + c*d*(b*e*(2 + p) - c*d*(3 + 2*p))) + b*(c*d*e + e*(b*e*(2 + p) - c*d*(3 + 2*p))))*(
-b + Sqrt[b^2 - 4*a*c] - 2*c*x)*(d + e*x)^(5 - 2*(3 + p))*(a + b*x + c*x^2)^p*Hypergeometric2F1[-p, 5 - 2*(3 +
p), 6 - 2*(3 + p), (-4*c*Sqrt[b^2 - 4*a*c]*(d + e*x))/((2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e)*(b - Sqrt[b^2 - 4*
a*c] + 2*c*x))])/(2*(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)*(c*d^2 - b*d*e + a*e^2)*(5 - 2*(3 + p))*(((2*c*d - b*e
+ Sqrt[b^2 - 4*a*c]*e)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/((2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e)*(b - Sqrt[b^2 -
4*a*c] + 2*c*x)))^p))/((c*d^2 - b*d*e + a*e^2)*(3 - 2*(3 + p)))) + ((-3*c*d*e + e*(b*e*(4 + p) - c*d*(5 + 2*p)
))*((e*(d + e*x)^(2 - 2*(3 + p))*(a + b*x + c*x^2)^(1 + p))/((c*d^2 - b*d*e + a*e^2)*(2 - 2*(3 + p))) + (2*c*(
(e*(d + e*x)^(4 - 2*(3 + p))*(a + b*x + c*x^2)^(1 + p))/((c*d^2 - b*d*e + a*e^2)*(4 - 2*(3 + p))) + ((2*c*d -
b*e)*(-b + Sqrt[b^2 - 4*a*c] - 2*c*x)*(d + e*x)^(5 - 2*(3 + p))*(a + b*x + c*x^2)^p*Hypergeometric2F1[-p, 5 -
2*(3 + p), 6 - 2*(3 + p), (-4*c*Sqrt[b^2 - 4*a*c]*(d + e*x))/((2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e)*(b - Sqrt[b^
2 - 4*a*c] + 2*c*x))])/(2*(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)*(c*d^2 - b*d*e + a*e^2)*(5 - 2*(3 + p))*(((2*c*d
- b*e + Sqrt[b^2 - 4*a*c]*e)*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/((2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e)*(b - Sqrt[
b^2 - 4*a*c] + 2*c*x)))^p)) + ((-2*c*d*e + e*(-2*c*d*(2 + p) + b*e*(3 + p)))*((e*(d + e*x)^(3 - 2*(3 + p))*(a
+ b*x + c*x^2)^(1 + p))/((c*d^2 - b*d*e + a*e^2)*(3 - 2*(3 + p))) + (((c*d*e - e*(b*e*(2 + p) - c*d*(3 + 2*p))
)*(d + e*x)^(4 - 2*(3 + p))*(a + b*x + c*x^2)^(1 + p))/(2*(c*d^2 - b*d*e + a*e^2)*(1 + p)) - ((-2*(a*c*e^2 + c
*d*(b*e*(2 + p) - c*d*(3 + 2*p))) + b*(c*d*e + e*(b*e*(2 + p) - c*d*(3 + 2*p))))*(-b + Sqrt[b^2 - 4*a*c] - 2*c
*x)*(d + e*x)^(5 - 2*(3 + p))*(a + b*x + c*x^2)^p*Hypergeometric2F1[-p, 5 - 2*(3 + p), 6 - 2*(3 + p), (-4*c*Sq
rt[b^2 - 4*a*c]*(d + e*x))/((2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))])/(2*(2*c*d -
b*e + Sqrt[b^2 - 4*a*c]*e)*(c*d^2 - b*d*e + a*e^2)*(5 - 2*(3 + p))*(((2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)*(b +
Sqrt[b^2 - 4*a*c] + 2*c*x))/((2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)))^p))/((c*d^
2 - b*d*e + a*e^2)*(3 - 2*(3 + p)))))/e)/((c*d^2 - b*d*e + a*e^2)*(2 - 2*(3 + p)))))/e)/((c*d^2 - b*d*e + a*e^
2)*(1 - 2*(3 + p)))

Maple [F]

\[\int \left (e x +d \right )^{-6-2 p} \left (c \,x^{2}+b x +a \right )^{p}d x\]

[In]

int((e*x+d)^(-6-2*p)*(c*x^2+b*x+a)^p,x)

[Out]

int((e*x+d)^(-6-2*p)*(c*x^2+b*x+a)^p,x)

Fricas [F]

\[ \int (d+e x)^{-6-2 p} \left (a+b x+c x^2\right )^p \, dx=\int { {\left (c x^{2} + b x + a\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 6} \,d x } \]

[In]

integrate((e*x+d)^(-6-2*p)*(c*x^2+b*x+a)^p,x, algorithm="fricas")

[Out]

integral((c*x^2 + b*x + a)^p*(e*x + d)^(-2*p - 6), x)

Sympy [F(-1)]

Timed out. \[ \int (d+e x)^{-6-2 p} \left (a+b x+c x^2\right )^p \, dx=\text {Timed out} \]

[In]

integrate((e*x+d)**(-6-2*p)*(c*x**2+b*x+a)**p,x)

[Out]

Timed out

Maxima [F]

\[ \int (d+e x)^{-6-2 p} \left (a+b x+c x^2\right )^p \, dx=\int { {\left (c x^{2} + b x + a\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 6} \,d x } \]

[In]

integrate((e*x+d)^(-6-2*p)*(c*x^2+b*x+a)^p,x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^p*(e*x + d)^(-2*p - 6), x)

Giac [F]

\[ \int (d+e x)^{-6-2 p} \left (a+b x+c x^2\right )^p \, dx=\int { {\left (c x^{2} + b x + a\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 6} \,d x } \]

[In]

integrate((e*x+d)^(-6-2*p)*(c*x^2+b*x+a)^p,x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^p*(e*x + d)^(-2*p - 6), x)

Mupad [F(-1)]

Timed out. \[ \int (d+e x)^{-6-2 p} \left (a+b x+c x^2\right )^p \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^p}{{\left (d+e\,x\right )}^{2\,p+6}} \,d x \]

[In]

int((a + b*x + c*x^2)^p/(d + e*x)^(2*p + 6),x)

[Out]

int((a + b*x + c*x^2)^p/(d + e*x)^(2*p + 6), x)