3.2196 \(\int (d+e x)^2 (f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2} \, dx\)

Optimal. Leaf size=487 \[ \frac{5 (b+2 c x) (2 c d-b e)^6 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-11 b e g+4 c d g+18 c e f)}{32768 c^6 e}+\frac{5 (b+2 c x) (2 c d-b e)^4 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-11 b e g+4 c d g+18 c e f)}{12288 c^5 e}+\frac{(b+2 c x) (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-11 b e g+4 c d g+18 c e f)}{768 c^4 e}-\frac{(2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-11 b e g+4 c d g+18 c e f)}{224 c^3 e^2}-\frac{(d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-11 b e g+4 c d g+18 c e f)}{144 c^2 e^2}+\frac{5 (2 c d-b e)^8 (-11 b e g+4 c d g+18 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{65536 c^{13/2} e^2}-\frac{g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e^2} \]

[Out]

(5*(2*c*d - b*e)^6*(18*c*e*f + 4*c*d*g - 11*b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(327
68*c^6*e) + (5*(2*c*d - b*e)^4*(18*c*e*f + 4*c*d*g - 11*b*e*g)*(b + 2*c*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^
2)^(3/2))/(12288*c^5*e) + ((2*c*d - b*e)^2*(18*c*e*f + 4*c*d*g - 11*b*e*g)*(b + 2*c*x)*(d*(c*d - b*e) - b*e^2*
x - c*e^2*x^2)^(5/2))/(768*c^4*e) - ((2*c*d - b*e)*(18*c*e*f + 4*c*d*g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x -
c*e^2*x^2)^(7/2))/(224*c^3*e^2) - ((18*c*e*f + 4*c*d*g - 11*b*e*g)*(d + e*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*
x^2)^(7/2))/(144*c^2*e^2) - (g*(d + e*x)^2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(9*c*e^2) + (5*(2*c*d
- b*e)^8*(18*c*e*f + 4*c*d*g - 11*b*e*g)*ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^
2*x^2])])/(65536*c^(13/2)*e^2)

________________________________________________________________________________________

Rubi [A]  time = 0.811349, antiderivative size = 487, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.159, Rules used = {1638, 12, 670, 640, 612, 621, 204} \[ \frac{5 (b+2 c x) (2 c d-b e)^6 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-11 b e g+4 c d g+18 c e f)}{32768 c^6 e}+\frac{5 (b+2 c x) (2 c d-b e)^4 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-11 b e g+4 c d g+18 c e f)}{12288 c^5 e}+\frac{(b+2 c x) (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-11 b e g+4 c d g+18 c e f)}{768 c^4 e}-\frac{(2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-11 b e g+4 c d g+18 c e f)}{224 c^3 e^2}-\frac{(d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-11 b e g+4 c d g+18 c e f)}{144 c^2 e^2}+\frac{5 (2 c d-b e)^8 (-11 b e g+4 c d g+18 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{65536 c^{13/2} e^2}-\frac{g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e^2} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^2*(f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]

[Out]

(5*(2*c*d - b*e)^6*(18*c*e*f + 4*c*d*g - 11*b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(327
68*c^6*e) + (5*(2*c*d - b*e)^4*(18*c*e*f + 4*c*d*g - 11*b*e*g)*(b + 2*c*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^
2)^(3/2))/(12288*c^5*e) + ((2*c*d - b*e)^2*(18*c*e*f + 4*c*d*g - 11*b*e*g)*(b + 2*c*x)*(d*(c*d - b*e) - b*e^2*
x - c*e^2*x^2)^(5/2))/(768*c^4*e) - ((2*c*d - b*e)*(18*c*e*f + 4*c*d*g - 11*b*e*g)*(d*(c*d - b*e) - b*e^2*x -
c*e^2*x^2)^(7/2))/(224*c^3*e^2) - ((18*c*e*f + 4*c*d*g - 11*b*e*g)*(d + e*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*
x^2)^(7/2))/(144*c^2*e^2) - (g*(d + e*x)^2*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(9*c*e^2) + (5*(2*c*d
- b*e)^8*(18*c*e*f + 4*c*d*g - 11*b*e*g)*ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^
2*x^2])])/(65536*c^(13/2)*e^2)

Rule 1638

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq
, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + b*x + c*x^2)^(p + 1))/(c*e^(q - 1)*(
m + q + 2*p + 1)), x] + Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^
q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q + e*f*(m + p + q)*(d + e*x)^(q - 2)*(b*d - 2*a*e +
(2*c*d - b*e)*x), x], x], x] /; NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] &&
NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 670

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))/(c*(m + 2*p + 1)), x] + Dist[((m + p)*(2*c*d - b*e))/(c*(m + 2*p + 1)), Int[(d + e
*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p]

Rule 640

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] + Dist[(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[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int (d+e x)^2 (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx &=-\frac{g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e^2}-\frac{\int -\frac{1}{2} e^2 (18 c e f+4 c d g-11 b e g) (d+e x)^2 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx}{9 c e^3}\\ &=-\frac{g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e^2}+\frac{(18 c e f+4 c d g-11 b e g) \int (d+e x)^2 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx}{18 c e}\\ &=-\frac{(18 c e f+4 c d g-11 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{144 c^2 e^2}-\frac{g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e^2}+\frac{((2 c d-b e) (18 c e f+4 c d g-11 b e g)) \int (d+e x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx}{32 c^2 e}\\ &=-\frac{(2 c d-b e) (18 c e f+4 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{224 c^3 e^2}-\frac{(18 c e f+4 c d g-11 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{144 c^2 e^2}-\frac{g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e^2}+\frac{\left ((2 c d-b e)^2 (18 c e f+4 c d g-11 b e g)\right ) \int \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx}{64 c^3 e}\\ &=\frac{(2 c d-b e)^2 (18 c e f+4 c d g-11 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{768 c^4 e}-\frac{(2 c d-b e) (18 c e f+4 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{224 c^3 e^2}-\frac{(18 c e f+4 c d g-11 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{144 c^2 e^2}-\frac{g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e^2}+\frac{\left (5 (2 c d-b e)^4 (18 c e f+4 c d g-11 b e g)\right ) \int \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2} \, dx}{1536 c^4 e}\\ &=\frac{5 (2 c d-b e)^4 (18 c e f+4 c d g-11 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{12288 c^5 e}+\frac{(2 c d-b e)^2 (18 c e f+4 c d g-11 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{768 c^4 e}-\frac{(2 c d-b e) (18 c e f+4 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{224 c^3 e^2}-\frac{(18 c e f+4 c d g-11 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{144 c^2 e^2}-\frac{g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e^2}+\frac{\left (5 (2 c d-b e)^6 (18 c e f+4 c d g-11 b e g)\right ) \int \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{8192 c^5 e}\\ &=\frac{5 (2 c d-b e)^6 (18 c e f+4 c d g-11 b e g) (b+2 c x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{32768 c^6 e}+\frac{5 (2 c d-b e)^4 (18 c e f+4 c d g-11 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{12288 c^5 e}+\frac{(2 c d-b e)^2 (18 c e f+4 c d g-11 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{768 c^4 e}-\frac{(2 c d-b e) (18 c e f+4 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{224 c^3 e^2}-\frac{(18 c e f+4 c d g-11 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{144 c^2 e^2}-\frac{g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e^2}+\frac{\left (5 (2 c d-b e)^8 (18 c e f+4 c d g-11 b e g)\right ) \int \frac{1}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{65536 c^6 e}\\ &=\frac{5 (2 c d-b e)^6 (18 c e f+4 c d g-11 b e g) (b+2 c x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{32768 c^6 e}+\frac{5 (2 c d-b e)^4 (18 c e f+4 c d g-11 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{12288 c^5 e}+\frac{(2 c d-b e)^2 (18 c e f+4 c d g-11 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{768 c^4 e}-\frac{(2 c d-b e) (18 c e f+4 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{224 c^3 e^2}-\frac{(18 c e f+4 c d g-11 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{144 c^2 e^2}-\frac{g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e^2}+\frac{\left (5 (2 c d-b e)^8 (18 c e f+4 c d g-11 b e g)\right ) \operatorname{Subst}\left (\int \frac{1}{-4 c e^2-x^2} \, dx,x,\frac{-b e^2-2 c e^2 x}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}}\right )}{32768 c^6 e}\\ &=\frac{5 (2 c d-b e)^6 (18 c e f+4 c d g-11 b e g) (b+2 c x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{32768 c^6 e}+\frac{5 (2 c d-b e)^4 (18 c e f+4 c d g-11 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{12288 c^5 e}+\frac{(2 c d-b e)^2 (18 c e f+4 c d g-11 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{768 c^4 e}-\frac{(2 c d-b e) (18 c e f+4 c d g-11 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{224 c^3 e^2}-\frac{(18 c e f+4 c d g-11 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{144 c^2 e^2}-\frac{g (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{9 c e^2}+\frac{5 (2 c d-b e)^8 (18 c e f+4 c d g-11 b e g) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{65536 c^{13/2} e^2}\\ \end{align*}

Mathematica [B]  time = 6.63542, size = 1511, normalized size = 3.1 \[ -\frac{2 (c d e+(c d-b e) e)^2 \left (-9 c f e^2-\left (\frac{11}{2} e (c d-b e)-\frac{7 c d e}{2}\right ) g\right ) (d+e x)^3 ((d+e x) (c (d-e x)-b e))^{5/2} \left (\frac{495 (c d e+(c d-b e) e)^6 \left (-\frac{256 c^5 (d+e x)^5 e^{10}}{315 (c d e+(c d-b e) e)^5 \left (\frac{c d e^2}{c d e+(c d-b e) e}+\frac{(c d-b e) e^2}{c d e+(c d-b e) e}\right )^5}-\frac{32 c^4 (d+e x)^4 e^8}{35 (c d e+(c d-b e) e)^4 \left (\frac{c d e^2}{c d e+(c d-b e) e}+\frac{(c d-b e) e^2}{c d e+(c d-b e) e}\right )^4}-\frac{16 c^3 (d+e x)^3 e^6}{15 (c d e+(c d-b e) e)^3 \left (\frac{c d e^2}{c d e+(c d-b e) e}+\frac{(c d-b e) e^2}{c d e+(c d-b e) e}\right )^3}-\frac{4 c^2 (d+e x)^2 e^4}{3 (c d e+(c d-b e) e)^2 \left (\frac{c d e^2}{c d e+(c d-b e) e}+\frac{(c d-b e) e^2}{c d e+(c d-b e) e}\right )^2}-\frac{2 c (d+e x) e^2}{(c d e+(c d-b e) e) \left (\frac{c d e^2}{c d e+(c d-b e) e}+\frac{(c d-b e) e^2}{c d e+(c d-b e) e}\right )}+\frac{2 \sqrt{c} \sqrt{d+e x} \sin ^{-1}\left (\frac{\sqrt{c} e \sqrt{d+e x}}{\sqrt{c d e+(c d-b e) e} \sqrt{\frac{c d e^2}{c d e+(c d-b e) e}+\frac{(c d-b e) e^2}{c d e+(c d-b e) e}}}\right ) e}{\sqrt{c d e+(c d-b e) e} \sqrt{\frac{c d e^2}{c d e+(c d-b e) e}+\frac{(c d-b e) e^2}{c d e+(c d-b e) e}} \sqrt{1-\frac{c e^2 (d+e x)}{(c d e+(c d-b e) e) \left (\frac{c d e^2}{c d e+(c d-b e) e}+\frac{(c d-b e) e^2}{c d e+(c d-b e) e}\right )}}}\right ) \left (\frac{c d e^2}{c d e+(c d-b e) e}+\frac{(c d-b e) e^2}{c d e+(c d-b e) e}\right )^6}{65536 c^6 e^{12} (d+e x)^6 \left (1-\frac{c e^2 (d+e x)}{(c d e+(c d-b e) e) \left (\frac{c d e^2}{c d e+(c d-b e) e}+\frac{(c d-b e) e^2}{c d e+(c d-b e) e}\right )}\right )^3}+\frac{11}{16} \left (\frac{1}{1-\frac{c e^2 (d+e x)}{(c d e+(c d-b e) e) \left (\frac{c d e^2}{c d e+(c d-b e) e}+\frac{(c d-b e) e^2}{c d e+(c d-b e) e}\right )}}+\frac{5}{14 \left (1-\frac{c e^2 (d+e x)}{(c d e+(c d-b e) e) \left (\frac{c d e^2}{c d e+(c d-b e) e}+\frac{(c d-b e) e^2}{c d e+(c d-b e) e}\right )}\right )^2}+\frac{5}{56 \left (1-\frac{c e^2 (d+e x)}{(c d e+(c d-b e) e) \left (\frac{c d e^2}{c d e+(c d-b e) e}+\frac{(c d-b e) e^2}{c d e+(c d-b e) e}\right )}\right )^3}\right )\right ) \left (1-\frac{c e^2 (d+e x)}{(c d e+(c d-b e) e) \left (\frac{c d e^2}{c d e+(c d-b e) e}+\frac{(c d-b e) e^2}{c d e+(c d-b e) e}\right )}\right )^{7/2}}{99 c e^5 \left (\frac{e}{\frac{c d e^2}{c d e+(c d-b e) e}+\frac{(c d-b e) e^2}{c d e+(c d-b e) e}}\right )^{5/2} (c d-b e-c e x)^2 \sqrt{\frac{e (c d-b e-c e x)}{c d e+(c d-b e) e}}}-\frac{g (d+e x)^3 (c d-b e-c e x) ((d+e x) (c (d-e x)-b e))^{5/2}}{9 c e^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^2*(f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]

[Out]

-(g*(d + e*x)^3*(c*d - b*e - c*e*x)*((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2))/(9*c*e^2) - (2*(c*d*e + e*(c*d -
b*e))^2*(-9*c*e^2*f - ((-7*c*d*e)/2 + (11*e*(c*d - b*e))/2)*g)*(d + e*x)^3*((d + e*x)*(-(b*e) + c*(d - e*x)))
^(5/2)*(1 - (c*e^2*(d + e*x))/((c*d*e + e*(c*d - b*e))*((c*d*e^2)/(c*d*e + e*(c*d - b*e)) + (e^2*(c*d - b*e))/
(c*d*e + e*(c*d - b*e)))))^(7/2)*((11*(5/(56*(1 - (c*e^2*(d + e*x))/((c*d*e + e*(c*d - b*e))*((c*d*e^2)/(c*d*e
+ e*(c*d - b*e)) + (e^2*(c*d - b*e))/(c*d*e + e*(c*d - b*e)))))^3) + 5/(14*(1 - (c*e^2*(d + e*x))/((c*d*e + e
*(c*d - b*e))*((c*d*e^2)/(c*d*e + e*(c*d - b*e)) + (e^2*(c*d - b*e))/(c*d*e + e*(c*d - b*e)))))^2) + (1 - (c*e
^2*(d + e*x))/((c*d*e + e*(c*d - b*e))*((c*d*e^2)/(c*d*e + e*(c*d - b*e)) + (e^2*(c*d - b*e))/(c*d*e + e*(c*d
- b*e)))))^(-1)))/16 + (495*(c*d*e + e*(c*d - b*e))^6*((c*d*e^2)/(c*d*e + e*(c*d - b*e)) + (e^2*(c*d - b*e))/(
c*d*e + e*(c*d - b*e)))^6*((-2*c*e^2*(d + e*x))/((c*d*e + e*(c*d - b*e))*((c*d*e^2)/(c*d*e + e*(c*d - b*e)) +
(e^2*(c*d - b*e))/(c*d*e + e*(c*d - b*e)))) - (4*c^2*e^4*(d + e*x)^2)/(3*(c*d*e + e*(c*d - b*e))^2*((c*d*e^2)/
(c*d*e + e*(c*d - b*e)) + (e^2*(c*d - b*e))/(c*d*e + e*(c*d - b*e)))^2) - (16*c^3*e^6*(d + e*x)^3)/(15*(c*d*e
+ e*(c*d - b*e))^3*((c*d*e^2)/(c*d*e + e*(c*d - b*e)) + (e^2*(c*d - b*e))/(c*d*e + e*(c*d - b*e)))^3) - (32*c^
4*e^8*(d + e*x)^4)/(35*(c*d*e + e*(c*d - b*e))^4*((c*d*e^2)/(c*d*e + e*(c*d - b*e)) + (e^2*(c*d - b*e))/(c*d*e
+ e*(c*d - b*e)))^4) - (256*c^5*e^10*(d + e*x)^5)/(315*(c*d*e + e*(c*d - b*e))^5*((c*d*e^2)/(c*d*e + e*(c*d -
b*e)) + (e^2*(c*d - b*e))/(c*d*e + e*(c*d - b*e)))^5) + (2*Sqrt[c]*e*Sqrt[d + e*x]*ArcSin[(Sqrt[c]*e*Sqrt[d +
e*x])/(Sqrt[c*d*e + e*(c*d - b*e)]*Sqrt[(c*d*e^2)/(c*d*e + e*(c*d - b*e)) + (e^2*(c*d - b*e))/(c*d*e + e*(c*d
- b*e))])])/(Sqrt[c*d*e + e*(c*d - b*e)]*Sqrt[(c*d*e^2)/(c*d*e + e*(c*d - b*e)) + (e^2*(c*d - b*e))/(c*d*e +
e*(c*d - b*e))]*Sqrt[1 - (c*e^2*(d + e*x))/((c*d*e + e*(c*d - b*e))*((c*d*e^2)/(c*d*e + e*(c*d - b*e)) + (e^2*
(c*d - b*e))/(c*d*e + e*(c*d - b*e))))])))/(65536*c^6*e^12*(d + e*x)^6*(1 - (c*e^2*(d + e*x))/((c*d*e + e*(c*d
- b*e))*((c*d*e^2)/(c*d*e + e*(c*d - b*e)) + (e^2*(c*d - b*e))/(c*d*e + e*(c*d - b*e)))))^3)))/(99*c*e^5*(e/(
(c*d*e^2)/(c*d*e + e*(c*d - b*e)) + (e^2*(c*d - b*e))/(c*d*e + e*(c*d - b*e))))^(5/2)*(c*d - b*e - c*e*x)^2*Sq
rt[(e*(c*d - b*e - c*e*x))/(c*d*e + e*(c*d - b*e))])

________________________________________________________________________________________

Maple [B]  time = 0.013, size = 4332, normalized size = 8.9 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^2*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)

[Out]

-2/7*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(7/2)/c/e*d*f+15/64*c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*d^4*f-135
/256*g*b*c^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d^8+625/20
48*e^3*g*b^5/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^3+85/4096*e^5*g*b^7/c^5*(-c*e^2*x^2-b*e^2*x-b*d*e+c*
d^2)^(1/2)*d-885/8192*e^4*g*b^6/c^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^2-525/256*e^2*g*b^3/(c*e^2)^(1/2)
*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d^6-55/16384*e^6*g*b^7/c^5*(-c*e^2*x
^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x-55/6144*e^4*g*b^5/c^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x-55/65536*e^8*g*
b^9/c^6/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))+15/16*e*g*b^2*(
-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^5-185/1536*e^2*g*b^4/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d^2-
1025/2048*e^2*g*b^4/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^4-5/32*g*b/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)
^(5/2)*x*d^2-115/256*g*b*c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^6+97/504/e*g/c^2*(-c*e^2*x^2-b*e^2*x-b*d
*e+c*d^2)^(7/2)*b*d+1/16*e*g*b^3/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*d+15/32*e*g*b^3/c*(-c*e^2*x^2-b*e^
2*x-b*d*e+c*d^2)^(1/2)*d^5+35/192*e*g*b^3/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d^3+115/3072*e^3*g*b^5/c^
4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d-11/384*e^2*g*b^3/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*x+5/64*
c^2/e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^7*g-1/4*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(7/2)/c/e*d*g+45/8
192*b^6/c^4*e^6*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*f-3/32/c^2*e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*b
^2*d*f+5/64*c^3/e/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d^9*g
+5/128*c/e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b*d^7*g+675/1024*b^3/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2
)*d^4*e^2*f+3/64*b^2/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*x*e^2*f-15/64/c*e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*
d^2)^(3/2)*b^2*d^3*f+1/48/c/e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*b*d^3*g+5/96*c/e*(-c*e^2*x^2-b*e^2*x-b*d*
e+c*d^2)^(3/2)*x*d^5*g-15/256*b^4/c^3*e^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d*f-225/512*b^4/c^2*e^3*(-c*e
^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^3*f+15/1024*b^4/c^3*e^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*f+45/3276
8*b^8/c^5*e^8/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*f-315/128
*b^3*e^3/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d^5*f+45/256*b
^3/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d^2*e^2*f+675/4096*b^5/c^3*e^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^
(1/2)*d^2*f-15/32*e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*d^3*b*f-135/4096*b^6/c^4*e^5*(-c*e^2*x^2-b*e^2*x-
b*d*e+c*d^2)^(1/2)*d*f+675/512*b^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^4*e^2*f+9/112*b/c^2*(-c*e^2*x^2-
b*e^2*x-b*d*e+c*d^2)^(7/2)*f+3/16*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*x*d^2*f+15/128*(-c*e^2*x^2-b*e^2*x-b*
d*e+c*d^2)^(3/2)*b*d^4*f-11/224*g*b^2/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(7/2)-115/512*g*b^2*(-c*e^2*x^2-b*e
^2*x-b*d*e+c*d^2)^(1/2)*d^6-1/9*g*x^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(7/2)/c-1/8*x*(-c*e^2*x^2-b*e^2*x-b*d*e
+c*d^2)^(7/2)/c*f+1/24/e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*x*d^3*g+5/192/e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^
2)^(3/2)*b*d^5*g+45/128*c^3/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1
/2))*d^8*f+45/256*c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b*d^6*f+3/32/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/
2)*b*d^2*f-55/12288*e^4*g*b^6/c^5*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)-55/32768*e^6*g*b^8/c^6*(-c*e^2*x^2-b*
e^2*x-b*d*e+c*d^2)^(1/2)-11/768*e^2*g*b^4/c^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)-11/63/e^2*g/c*(-c*e^2*x^2
-b*e^2*x-b*d*e+c*d^2)^(7/2)*d^2-95/384*g*b*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*d^4-95/768*g*b^2/c*(-c*e^2
*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d^4+11/144*g*b/c^2*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(7/2)-5/64*g*b^2/c^2*(-c
*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*d^2+45/128*c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^6*f+45/16384*b^7
/c^5*e^6*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*f+15/2048*b^5/c^4*e^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*f
+3/128*b^3/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*e^2*f-135/256*e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b
^2*d^5*f+45/128*b^2/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*d^2*e^2*f-315/512*b^5/c^2*e^5/(c*e^2)^(1/2)*arc
tan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d^3*f-225/256*b^3/c*e^3*(-c*e^2*x^2-b*e^
2*x-b*d*e+c*d^2)^(1/2)*x*d^3*f+675/2048*b^4/c^2*e^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^2*f+1575/1024*b
^4/c*e^4/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d^4*f-45/2048*
b^7/c^4*e^7/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d*f+315/204
8*b^6/c^3*e^6/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d^2*f-45/
32*c^2*e/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d^7*b*f-3/16/c
*e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*x*b*d*f-135/2048*b^5/c^3*e^5*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*
x*d*f-15/128*b^3/c^2*e^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*d*f-2205/2048*e^4*g*b^5/c^2/(c*e^2)^(1/2)*ar
ctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d^4+115/1536*e^3*g*b^4/c^3*(-c*e^2*x^2-
b*e^2*x-b*d*e+c*d^2)^(3/2)*x*d+945/512*e^3*g*b^4/c/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-
b*e^2*x-b*d*e+c*d^2)^(1/2))*d^5+85/2048*e^5*g*b^6/c^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d-885/4096*e^4*
g*b^5/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^2-405/4096*e^6*g*b^7/c^4/(c*e^2)^(1/2)*arctan((c*e^2)^(1/
2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d^2+625/1024*e^3*g*b^4/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*
d^2)^(1/2)*x*d^3+105/256*e^5*g*b^6/c^3/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*
e+c*d^2)^(1/2))*d^3+225/16384*e^7*g*b^8/c^5/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x
-b*d*e+c*d^2)^(1/2))*d-185/768*e^2*g*b^3/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*d^2+1/8*e*g*b^2/c^2*(-c*
e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*x*d-1025/1024*e^2*g*b^3/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^4+45/3
2*e*g*b^2*c/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d^7+35/96*e
*g*b^2/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*d^3+315/128*b^2*c/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*
b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))*d^6*e^2*f-135/128*c*e*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^
5*b*f

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 77.7231, size = 6593, normalized size = 13.54 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="fricas")

[Out]

[1/8257536*(315*(18*(256*c^9*d^8*e - 1024*b*c^8*d^7*e^2 + 1792*b^2*c^7*d^6*e^3 - 1792*b^3*c^6*d^5*e^4 + 1120*b
^4*c^5*d^4*e^5 - 448*b^5*c^4*d^3*e^6 + 112*b^6*c^3*d^2*e^7 - 16*b^7*c^2*d*e^8 + b^8*c*e^9)*f + (1024*c^9*d^9 -
6912*b*c^8*d^8*e + 18432*b^2*c^7*d^7*e^2 - 26880*b^3*c^6*d^6*e^3 + 24192*b^4*c^5*d^5*e^4 - 14112*b^5*c^4*d^4*
e^5 + 5376*b^6*c^3*d^3*e^6 - 1296*b^7*c^2*d^2*e^7 + 180*b^8*c*d*e^8 - 11*b^9*e^9)*g)*sqrt(-c)*log(8*c^2*e^2*x^
2 + 8*b*c*e^2*x - 4*c^2*d^2 + 4*b*c*d*e + b^2*e^2 + 4*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b*
e)*sqrt(-c)) + 4*(229376*c^9*e^8*g*x^8 + 14336*(18*c^9*e^8*f + (36*c^9*d*e^7 + 37*b*c^8*e^8)*g)*x^7 + 1024*(18
*(32*c^9*d*e^7 + 33*b*c^8*e^8)*f - (320*c^9*d^2*e^6 - 1796*b*c^8*d*e^7 - 309*b^2*c^7*e^8)*g)*x^6 - 256*(54*(28
*c^9*d^2*e^6 - 156*b*c^8*d*e^7 - 27*b^2*c^7*e^8)*f + (5712*c^9*d^3*e^5 - 5484*b*c^8*d^2*e^6 - 5928*b^2*c^7*d*e
^7 - 5*b^3*c^6*e^8)*g)*x^5 - 128*(18*(768*c^9*d^3*e^5 - 732*b*c^8*d^2*e^6 - 804*b^2*c^7*d*e^7 - b^3*c^6*e^8)*f
+ (3072*c^9*d^4*e^4 + 15504*b*c^8*d^3*e^5 - 22044*b^2*c^7*d^2*e^6 - 120*b^3*c^6*d*e^7 + 11*b^4*c^5*e^8)*g)*x^
4 - 16*(18*(1680*c^9*d^4*e^4 + 8928*b*c^8*d^3*e^5 - 12552*b^2*c^7*d^2*e^6 - 104*b^3*c^6*d*e^7 + 9*b^4*c^5*e^8)
*f - (79296*c^9*d^5*e^3 - 232272*b*c^8*d^4*e^4 + 148416*b^2*c^7*d^3*e^5 + 5704*b^3*c^6*d^2*e^6 - 1180*b^4*c^5*
d*e^7 + 99*b^5*c^4*e^8)*g)*x^3 + 8*(54*(4096*c^9*d^5*e^3 - 11920*b*c^8*d^4*e^4 + 7456*b^2*c^7*d^3*e^5 + 456*b^
3*c^6*d^2*e^6 - 88*b^4*c^5*d*e^7 + 7*b^5*c^4*e^8)*f + (106496*c^9*d^6*e^2 - 192192*b*c^8*d^5*e^3 + 52752*b^2*c
^7*d^4*e^4 + 46144*b^3*c^6*d^3*e^5 - 16104*b^4*c^5*d^2*e^6 + 2988*b^5*c^4*d*e^7 - 231*b^6*c^3*e^8)*g)*x^2 - 18
*(32768*c^9*d^7*e - 151872*b*c^8*d^6*e^2 + 259008*b^2*c^7*d^5*e^3 - 218000*b^3*c^6*d^4*e^4 + 102624*b^4*c^5*d^
3*e^5 - 29148*b^5*c^4*d^2*e^6 + 4620*b^6*c^3*d*e^7 - 315*b^7*c^2*e^8)*f - (360448*c^9*d^8 - 1656064*b*c^8*d^7*
e + 3394752*b^2*c^7*d^6*e^2 - 3950464*b^3*c^6*d^5*e^3 + 2808496*b^4*c^5*d^4*e^4 - 1245456*b^5*c^4*d^3*e^5 + 33
9108*b^6*c^3*d^2*e^6 - 52080*b^7*c^2*d*e^7 + 3465*b^8*c*e^8)*g + 2*(18*(37184*c^9*d^6*e^2 - 62400*b*c^8*d^5*e^
3 + 6480*b^2*c^7*d^4*e^4 + 25504*b^3*c^6*d^3*e^5 - 8196*b^4*c^5*d^2*e^6 + 1428*b^5*c^4*d*e^7 - 105*b^6*c^3*e^8
)*f - (80640*c^9*d^7*e - 373568*b*c^8*d^6*e^2 + 663936*b^2*c^7*d^5*e^3 - 604176*b^3*c^6*d^4*e^4 + 313328*b^4*c
^5*d^3*e^5 - 95868*b^5*c^4*d^2*e^6 + 16128*b^6*c^3*d*e^7 - 1155*b^7*c^2*e^8)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x +
c*d^2 - b*d*e))/(c^7*e^2), -1/4128768*(315*(18*(256*c^9*d^8*e - 1024*b*c^8*d^7*e^2 + 1792*b^2*c^7*d^6*e^3 - 1
792*b^3*c^6*d^5*e^4 + 1120*b^4*c^5*d^4*e^5 - 448*b^5*c^4*d^3*e^6 + 112*b^6*c^3*d^2*e^7 - 16*b^7*c^2*d*e^8 + b^
8*c*e^9)*f + (1024*c^9*d^9 - 6912*b*c^8*d^8*e + 18432*b^2*c^7*d^7*e^2 - 26880*b^3*c^6*d^6*e^3 + 24192*b^4*c^5*
d^5*e^4 - 14112*b^5*c^4*d^4*e^5 + 5376*b^6*c^3*d^3*e^6 - 1296*b^7*c^2*d^2*e^7 + 180*b^8*c*d*e^8 - 11*b^9*e^9)*
g)*sqrt(c)*arctan(1/2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b*e)*sqrt(c)/(c^2*e^2*x^2 + b*c*e^
2*x - c^2*d^2 + b*c*d*e)) - 2*(229376*c^9*e^8*g*x^8 + 14336*(18*c^9*e^8*f + (36*c^9*d*e^7 + 37*b*c^8*e^8)*g)*x
^7 + 1024*(18*(32*c^9*d*e^7 + 33*b*c^8*e^8)*f - (320*c^9*d^2*e^6 - 1796*b*c^8*d*e^7 - 309*b^2*c^7*e^8)*g)*x^6
- 256*(54*(28*c^9*d^2*e^6 - 156*b*c^8*d*e^7 - 27*b^2*c^7*e^8)*f + (5712*c^9*d^3*e^5 - 5484*b*c^8*d^2*e^6 - 592
8*b^2*c^7*d*e^7 - 5*b^3*c^6*e^8)*g)*x^5 - 128*(18*(768*c^9*d^3*e^5 - 732*b*c^8*d^2*e^6 - 804*b^2*c^7*d*e^7 - b
^3*c^6*e^8)*f + (3072*c^9*d^4*e^4 + 15504*b*c^8*d^3*e^5 - 22044*b^2*c^7*d^2*e^6 - 120*b^3*c^6*d*e^7 + 11*b^4*c
^5*e^8)*g)*x^4 - 16*(18*(1680*c^9*d^4*e^4 + 8928*b*c^8*d^3*e^5 - 12552*b^2*c^7*d^2*e^6 - 104*b^3*c^6*d*e^7 + 9
*b^4*c^5*e^8)*f - (79296*c^9*d^5*e^3 - 232272*b*c^8*d^4*e^4 + 148416*b^2*c^7*d^3*e^5 + 5704*b^3*c^6*d^2*e^6 -
1180*b^4*c^5*d*e^7 + 99*b^5*c^4*e^8)*g)*x^3 + 8*(54*(4096*c^9*d^5*e^3 - 11920*b*c^8*d^4*e^4 + 7456*b^2*c^7*d^3
*e^5 + 456*b^3*c^6*d^2*e^6 - 88*b^4*c^5*d*e^7 + 7*b^5*c^4*e^8)*f + (106496*c^9*d^6*e^2 - 192192*b*c^8*d^5*e^3
+ 52752*b^2*c^7*d^4*e^4 + 46144*b^3*c^6*d^3*e^5 - 16104*b^4*c^5*d^2*e^6 + 2988*b^5*c^4*d*e^7 - 231*b^6*c^3*e^8
)*g)*x^2 - 18*(32768*c^9*d^7*e - 151872*b*c^8*d^6*e^2 + 259008*b^2*c^7*d^5*e^3 - 218000*b^3*c^6*d^4*e^4 + 1026
24*b^4*c^5*d^3*e^5 - 29148*b^5*c^4*d^2*e^6 + 4620*b^6*c^3*d*e^7 - 315*b^7*c^2*e^8)*f - (360448*c^9*d^8 - 16560
64*b*c^8*d^7*e + 3394752*b^2*c^7*d^6*e^2 - 3950464*b^3*c^6*d^5*e^3 + 2808496*b^4*c^5*d^4*e^4 - 1245456*b^5*c^4
*d^3*e^5 + 339108*b^6*c^3*d^2*e^6 - 52080*b^7*c^2*d*e^7 + 3465*b^8*c*e^8)*g + 2*(18*(37184*c^9*d^6*e^2 - 62400
*b*c^8*d^5*e^3 + 6480*b^2*c^7*d^4*e^4 + 25504*b^3*c^6*d^3*e^5 - 8196*b^4*c^5*d^2*e^6 + 1428*b^5*c^4*d*e^7 - 10
5*b^6*c^3*e^8)*f - (80640*c^9*d^7*e - 373568*b*c^8*d^6*e^2 + 663936*b^2*c^7*d^5*e^3 - 604176*b^3*c^6*d^4*e^4 +
313328*b^4*c^5*d^3*e^5 - 95868*b^5*c^4*d^2*e^6 + 16128*b^6*c^3*d*e^7 - 1155*b^7*c^2*e^8)*g)*x)*sqrt(-c*e^2*x^
2 - b*e^2*x + c*d^2 - b*d*e))/(c^7*e^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac{5}{2}} \left (d + e x\right )^{2} \left (f + g x\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**2*(g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x)

[Out]

Integral((-(d + e*x)*(b*e - c*d + c*e*x))**(5/2)*(d + e*x)**2*(f + g*x), x)

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Giac [B]  time = 1.48034, size = 1893, normalized size = 3.89 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="giac")

[Out]

1/2064384*sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)*(2*(4*(2*(8*(2*(4*(14*(16*c^2*g*x*e^6 + (36*c^10*d*g*e^19
+ 18*c^10*f*e^20 + 37*b*c^9*g*e^20)*e^(-14)/c^8)*x - (320*c^10*d^2*g*e^18 - 576*c^10*d*f*e^19 - 1796*b*c^9*d*
g*e^19 - 594*b*c^9*f*e^20 - 309*b^2*c^8*g*e^20)*e^(-14)/c^8)*x - (5712*c^10*d^3*g*e^17 + 1512*c^10*d^2*f*e^18
- 5484*b*c^9*d^2*g*e^18 - 8424*b*c^9*d*f*e^19 - 5928*b^2*c^8*d*g*e^19 - 1458*b^2*c^8*f*e^20 - 5*b^3*c^7*g*e^20
)*e^(-14)/c^8)*x - (3072*c^10*d^4*g*e^16 + 13824*c^10*d^3*f*e^17 + 15504*b*c^9*d^3*g*e^17 - 13176*b*c^9*d^2*f*
e^18 - 22044*b^2*c^8*d^2*g*e^18 - 14472*b^2*c^8*d*f*e^19 - 120*b^3*c^7*d*g*e^19 - 18*b^3*c^7*f*e^20 + 11*b^4*c
^6*g*e^20)*e^(-14)/c^8)*x + (79296*c^10*d^5*g*e^15 - 30240*c^10*d^4*f*e^16 - 232272*b*c^9*d^4*g*e^16 - 160704*
b*c^9*d^3*f*e^17 + 148416*b^2*c^8*d^3*g*e^17 + 225936*b^2*c^8*d^2*f*e^18 + 5704*b^3*c^7*d^2*g*e^18 + 1872*b^3*
c^7*d*f*e^19 - 1180*b^4*c^6*d*g*e^19 - 162*b^4*c^6*f*e^20 + 99*b^5*c^5*g*e^20)*e^(-14)/c^8)*x + (106496*c^10*d
^6*g*e^14 + 221184*c^10*d^5*f*e^15 - 192192*b*c^9*d^5*g*e^15 - 643680*b*c^9*d^4*f*e^16 + 52752*b^2*c^8*d^4*g*e
^16 + 402624*b^2*c^8*d^3*f*e^17 + 46144*b^3*c^7*d^3*g*e^17 + 24624*b^3*c^7*d^2*f*e^18 - 16104*b^4*c^6*d^2*g*e^
18 - 4752*b^4*c^6*d*f*e^19 + 2988*b^5*c^5*d*g*e^19 + 378*b^5*c^5*f*e^20 - 231*b^6*c^4*g*e^20)*e^(-14)/c^8)*x -
(80640*c^10*d^7*g*e^13 - 669312*c^10*d^6*f*e^14 - 373568*b*c^9*d^6*g*e^14 + 1123200*b*c^9*d^5*f*e^15 + 663936
*b^2*c^8*d^5*g*e^15 - 116640*b^2*c^8*d^4*f*e^16 - 604176*b^3*c^7*d^4*g*e^16 - 459072*b^3*c^7*d^3*f*e^17 + 3133
28*b^4*c^6*d^3*g*e^17 + 147528*b^4*c^6*d^2*f*e^18 - 95868*b^5*c^5*d^2*g*e^18 - 25704*b^5*c^5*d*f*e^19 + 16128*
b^6*c^4*d*g*e^19 + 1890*b^6*c^4*f*e^20 - 1155*b^7*c^3*g*e^20)*e^(-14)/c^8)*x - (360448*c^10*d^8*g*e^12 + 58982
4*c^10*d^7*f*e^13 - 1656064*b*c^9*d^7*g*e^13 - 2733696*b*c^9*d^6*f*e^14 + 3394752*b^2*c^8*d^6*g*e^14 + 4662144
*b^2*c^8*d^5*f*e^15 - 3950464*b^3*c^7*d^5*g*e^15 - 3924000*b^3*c^7*d^4*f*e^16 + 2808496*b^4*c^6*d^4*g*e^16 + 1
847232*b^4*c^6*d^3*f*e^17 - 1245456*b^5*c^5*d^3*g*e^17 - 524664*b^5*c^5*d^2*f*e^18 + 339108*b^6*c^4*d^2*g*e^18
+ 83160*b^6*c^4*d*f*e^19 - 52080*b^7*c^3*d*g*e^19 - 5670*b^7*c^3*f*e^20 + 3465*b^8*c^2*g*e^20)*e^(-14)/c^8) +
5/65536*(1024*c^9*d^9*g + 4608*c^9*d^8*f*e - 6912*b*c^8*d^8*g*e - 18432*b*c^8*d^7*f*e^2 + 18432*b^2*c^7*d^7*g
*e^2 + 32256*b^2*c^7*d^6*f*e^3 - 26880*b^3*c^6*d^6*g*e^3 - 32256*b^3*c^6*d^5*f*e^4 + 24192*b^4*c^5*d^5*g*e^4 +
20160*b^4*c^5*d^4*f*e^5 - 14112*b^5*c^4*d^4*g*e^5 - 8064*b^5*c^4*d^3*f*e^6 + 5376*b^6*c^3*d^3*g*e^6 + 2016*b^
6*c^3*d^2*f*e^7 - 1296*b^7*c^2*d^2*g*e^7 - 288*b^7*c^2*d*f*e^8 + 180*b^8*c*d*g*e^8 + 18*b^8*c*f*e^9 - 11*b^9*g
*e^9)*sqrt(-c*e^2)*e^(-3)*log(abs(-2*(sqrt(-c*e^2)*x - sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e))*c - sqrt(-c
*e^2)*b))/c^7