3.2197 \(\int (d+e x) (f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2} \, dx\)

Optimal. Leaf size=371 \[ \frac{5 (b+2 c x) (2 c d-b e)^5 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-9 b e g+2 c d g+16 c e f)}{16384 c^5 e}+\frac{5 (b+2 c x) (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-9 b e g+2 c d g+16 c e f)}{6144 c^4 e}+\frac{(b+2 c x) (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-9 b e g+2 c d g+16 c e f)}{384 c^3 e}+\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (9 b e g-16 c (d g+e f)-14 c e g x)}{112 c^2 e^2}+\frac{5 (2 c d-b e)^7 (-9 b e g+2 c d g+16 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{32768 c^{11/2} e^2} \]

[Out]

(5*(2*c*d - b*e)^5*(16*c*e*f + 2*c*d*g - 9*b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(1638
4*c^5*e) + (5*(2*c*d - b*e)^3*(16*c*e*f + 2*c*d*g - 9*b*e*g)*(b + 2*c*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)
^(3/2))/(6144*c^4*e) + ((2*c*d - b*e)*(16*c*e*f + 2*c*d*g - 9*b*e*g)*(b + 2*c*x)*(d*(c*d - b*e) - b*e^2*x - c*
e^2*x^2)^(5/2))/(384*c^3*e) + ((9*b*e*g - 16*c*(e*f + d*g) - 14*c*e*g*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)
^(7/2))/(112*c^2*e^2) + (5*(2*c*d - b*e)^7*(16*c*e*f + 2*c*d*g - 9*b*e*g)*ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*Sq
rt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])])/(32768*c^(11/2)*e^2)

________________________________________________________________________________________

Rubi [A]  time = 0.618632, antiderivative size = 371, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {779, 612, 621, 204} \[ \frac{5 (b+2 c x) (2 c d-b e)^5 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-9 b e g+2 c d g+16 c e f)}{16384 c^5 e}+\frac{5 (b+2 c x) (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-9 b e g+2 c d g+16 c e f)}{6144 c^4 e}+\frac{(b+2 c x) (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-9 b e g+2 c d g+16 c e f)}{384 c^3 e}+\frac{\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (9 b e g-16 c (d g+e f)-14 c e g x)}{112 c^2 e^2}+\frac{5 (2 c d-b e)^7 (-9 b e g+2 c d g+16 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{32768 c^{11/2} e^2} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)*(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)^5*(16*c*e*f + 2*c*d*g - 9*b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(1638
4*c^5*e) + (5*(2*c*d - b*e)^3*(16*c*e*f + 2*c*d*g - 9*b*e*g)*(b + 2*c*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)
^(3/2))/(6144*c^4*e) + ((2*c*d - b*e)*(16*c*e*f + 2*c*d*g - 9*b*e*g)*(b + 2*c*x)*(d*(c*d - b*e) - b*e^2*x - c*
e^2*x^2)^(5/2))/(384*c^3*e) + ((9*b*e*g - 16*c*(e*f + d*g) - 14*c*e*g*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)
^(7/2))/(112*c^2*e^2) + (5*(2*c*d - b*e)^7*(16*c*e*f + 2*c*d*g - 9*b*e*g)*ArcTan[(e*(b + 2*c*x))/(2*Sqrt[c]*Sq
rt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])])/(32768*c^(11/2)*e^2)

Rule 779

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x)*(a + b*x + c*x^2)^(p + 1))/(2*c^2*(p + 1)*(2*p + 3
)), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(a
+ b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[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) (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx &=\frac{(9 b e g-16 c (e f+d g)-14 c e g x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{112 c^2 e^2}+\frac{((2 c d-b e) (16 c e f+2 c d g-9 b e g)) \int \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) (16 c e f+2 c d g-9 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}}{384 c^3 e}+\frac{(9 b e g-16 c (e f+d g)-14 c e g x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{112 c^2 e^2}+\frac{\left (5 (2 c d-b e)^3 (16 c e f+2 c d g-9 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}{768 c^3 e}\\ &=\frac{5 (2 c d-b e)^3 (16 c e f+2 c d g-9 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}}{6144 c^4 e}+\frac{(2 c d-b e) (16 c e f+2 c d g-9 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}}{384 c^3 e}+\frac{(9 b e g-16 c (e f+d g)-14 c e g x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{112 c^2 e^2}+\frac{\left (5 (2 c d-b e)^5 (16 c e f+2 c d g-9 b e g)\right ) \int \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{4096 c^4 e}\\ &=\frac{5 (2 c d-b e)^5 (16 c e f+2 c d g-9 b e g) (b+2 c x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{16384 c^5 e}+\frac{5 (2 c d-b e)^3 (16 c e f+2 c d g-9 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}}{6144 c^4 e}+\frac{(2 c d-b e) (16 c e f+2 c d g-9 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}}{384 c^3 e}+\frac{(9 b e g-16 c (e f+d g)-14 c e g x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{112 c^2 e^2}+\frac{\left (5 (2 c d-b e)^7 (16 c e f+2 c d g-9 b e g)\right ) \int \frac{1}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{32768 c^5 e}\\ &=\frac{5 (2 c d-b e)^5 (16 c e f+2 c d g-9 b e g) (b+2 c x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{16384 c^5 e}+\frac{5 (2 c d-b e)^3 (16 c e f+2 c d g-9 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}}{6144 c^4 e}+\frac{(2 c d-b e) (16 c e f+2 c d g-9 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}}{384 c^3 e}+\frac{(9 b e g-16 c (e f+d g)-14 c e g x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{112 c^2 e^2}+\frac{\left (5 (2 c d-b e)^7 (16 c e f+2 c d g-9 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 )}{16384 c^5 e}\\ &=\frac{5 (2 c d-b e)^5 (16 c e f+2 c d g-9 b e g) (b+2 c x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{16384 c^5 e}+\frac{5 (2 c d-b e)^3 (16 c e f+2 c d g-9 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}}{6144 c^4 e}+\frac{(2 c d-b e) (16 c e f+2 c d g-9 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}}{384 c^3 e}+\frac{(9 b e g-16 c (e f+d g)-14 c e g x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{112 c^2 e^2}+\frac{5 (2 c d-b e)^7 (16 c e f+2 c d g-9 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 )}{32768 c^{11/2} e^2}\\ \end{align*}

Mathematica [B]  time = 6.43209, size = 1422, normalized size = 3.83 \[ -\frac{(c d e+(c d-b e) e)^2 \left (-8 c f e^2-\left (\frac{9}{2} e (c d-b e)-\frac{7 c d e}{2}\right ) g\right ) (d+e x)^2 ((d+e x) (c (d-e x)-b e))^{5/2} \left (\frac{45 (c d e+(c d-b e) e)^5 \left (-\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 )^5}{4096 c^5 e^{10} (d+e x)^5 \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{9}{14} \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}{12 \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{1}{8 \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}}{36 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)^2 (c d-b e-c e x) ((d+e x) (c (d-e x)-b e))^{5/2}}{8 c e^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)*(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)^2*(c*d - b*e - c*e*x)*((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2))/(8*c*e^2) - ((c*d*e + e*(c*d - b
*e))^2*(-8*c*e^2*f - ((-7*c*d*e)/2 + (9*e*(c*d - b*e))/2)*g)*(d + e*x)^2*((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)*((9*(1/(8*(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/(12*(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)))/14 + (45*(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*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) + (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)]*Sqr
t[(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))))]
)))/(4096*c^5*e^10*(d + e*x)^5*(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)))/(36*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*Sqrt[(e*(c*d - b*e - c*e*x))/(c*d*e + e*(c
*d - b*e))])

________________________________________________________________________________________

Maple [B]  time = 0.01, size = 3472, normalized size = 9.4 \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)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)

[Out]

5/16*d^7*f*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))+1/12*d*f
/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*b+15/1024*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+45/
8192*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-5/16*g*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^7*b-35/128*g*c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^5*
b-5/48*g/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*x*b*d+475/4096*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^2+45/32768*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))-115/4096*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-35/32*e^2*g*b^3/(c*e^2)^(1/2)*arct
an((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+25/256*e*g*b^3/c^2*(-c*e^2*x^2-b*e^2*
x-b*d*e+c*d^2)^(3/2)*d^2+5/256/e*g*c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b*d^6+5/128/e*g*c^2*(-c*e^2*x^2-b*
e^2*x-b*d*e+c*d^2)^(1/2)*x*d^6+275/512*e*g*b^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^4-35/768*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-125/512*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^3-2
5/128*b^4/c^2*e^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^2*f-1/12*b/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)
*x*e*f-5/2048*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
))*f-5/192*b^3/c^2*e^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*f-5/512*b^5/c^3*e^5*(-c*e^2*x^2-b*e^2*x-b*d*e+
c*d^2)^(1/2)*x*f+275/1024*e*g*b^3/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^4+3/64*e*g*b^2/c^2*(-c*e^2*x^2-b*
e^2*x-b*d*e+c*d^2)^(5/2)*x+5/128/e*g*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+1/96/e*g/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*b*d^2+5/192/e*g*c*(-c*e^2*x^2-b*e^2*x-
b*d*e+c*d^2)^(3/2)*x*d^4+5/64*b^3/c^2*e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d*f+25/64*b^3/c*e^2*(-c*e^2*x
^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^3*f+25/32*b^2*e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x*d^3*f-175/128*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^4*f-5/32*b^2/c*(-c
*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*d^2*e*f+25/512*b^5/c^3*e^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d*f-5/16
*b*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*d^2*e*f-35/256*g*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b^2*d^5+5/
48*d^3*f*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*b+1/6*d*f*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*x-1/7*(-c*e^2
*x^2-b*e^2*x-b*d*e+c*d^2)^(7/2)/c/e*f+9/112/e*g*b/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(7/2)+45/16384*e^5*g*b^
7/c^5*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-5/64*g/c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*b^2*d^3-1/7*(-c*e
^2*x^2-b*e^2*x-b*d*e+c*d^2)^(7/2)/c/e^2*d*g+3/128*e*g*b^3/c^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)+1/48/e*g*
(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*x*d^2-5/32*g*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*d^3*b-5/96*g/c^2*
(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*b^2*d+5/384/e*g*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*b*d^4-1/8/e*g*x*
(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(7/2)/c+15/2048*e^3*g*b^5/c^4*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)+5/16*d^5
*f*c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*x+5/24*d^3*f*c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x+5/32*d^5
*f*c*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*b-25/64*b^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*d^4*e*f-5/1024*
b^6/c^4*e^5*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)*f-5/384*b^4/c^3*e^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*
f-1/24*b^2/c^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2)*e*f-105/512*b^5/c^2*e^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^2*f+105/64*b^2*c*e^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^5*f-25/64*b^3/c*e^3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^
(1/2)*x*d^2*f+25/256*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*f+35/1024*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*f+25/128*e*g*b^2/c*(-c*e^2*x^2-b*
e^2*x-b*d*e+c*d^2)^(3/2)*x*d^2+105/128*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^6-105/256*e^4*g*b^5/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^3-115/2048*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+475/2048*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^2-125/256*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^3+245/2048*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^2+875/1024*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^4-5/256*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-35/384*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-25/32*b*c*(-c*e^2*x^2-b*e^2*x
-b*d*e+c*d^2)^(1/2)*x*d^4*e*f+175/256*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^3*f+5/32*b^2/c*e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)*x*d*f-35/32*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^6*e*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)*(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 = 36.4831, size = 5287, normalized size = 14.25 \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)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="fricas")

[Out]

[-1/1376256*(105*(16*(128*c^8*d^7*e - 448*b*c^7*d^6*e^2 + 672*b^2*c^6*d^5*e^3 - 560*b^3*c^5*d^4*e^4 + 280*b^4*
c^4*d^3*e^5 - 84*b^5*c^3*d^2*e^6 + 14*b^6*c^2*d*e^7 - b^7*c*e^8)*f + (256*c^8*d^8 - 2048*b*c^7*d^7*e + 5376*b^
2*c^6*d^6*e^2 - 7168*b^3*c^5*d^5*e^3 + 5600*b^4*c^4*d^4*e^4 - 2688*b^5*c^3*d^3*e^5 + 784*b^6*c^2*d^2*e^6 - 128
*b^7*c*d*e^7 + 9*b^8*e^8)*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*sq
rt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b*e)*sqrt(-c)) - 4*(43008*c^8*e^7*g*x^7 + 3072*(16*c^8*e^7
*f + (16*c^8*d*e^6 + 33*b*c^7*e^7)*g)*x^6 + 256*(16*(14*c^8*d*e^6 + 29*b*c^7*e^7)*f - (476*c^8*d^2*e^5 - 940*b
*c^7*d*e^6 - 243*b^2*c^6*e^7)*g)*x^5 - 128*(16*(72*c^8*d^2*e^5 - 142*b*c^7*d*e^6 - 37*b^2*c^6*e^7)*f + (1152*c
^8*d^3*e^4 + 76*b*c^7*d^2*e^5 - 1820*b^2*c^6*d*e^6 - 3*b^3*c^5*e^7)*g)*x^4 - 16*(16*(728*c^8*d^3*e^4 + 60*b*c^
7*d^2*e^5 - 1166*b^2*c^6*d*e^6 - 3*b^3*c^5*e^7)*f - (6608*c^8*d^4*e^3 - 25824*b*c^7*d^3*e^4 + 19000*b^2*c^6*d^
2*e^5 + 264*b^3*c^5*d*e^6 - 27*b^4*c^4*e^7)*g)*x^3 + 8*(16*(1152*c^8*d^4*e^3 - 4488*b*c^7*d^3*e^4 + 3276*b^2*c
^6*d^2*e^5 + 74*b^3*c^5*d*e^6 - 7*b^4*c^4*e^7)*f + (18432*c^8*d^5*e^2 - 35472*b*c^7*d^4*e^3 + 14688*b^2*c^6*d^
3*e^4 + 2920*b^3*c^5*d^2*e^5 - 680*b^4*c^4*d*e^6 + 63*b^5*c^3*e^7)*g)*x^2 - 16*(3072*c^8*d^6*e - 16608*b*c^7*d
^5*e^2 + 27696*b^2*c^6*d^4*e^3 - 20096*b^3*c^5*d^3*e^4 + 7056*b^4*c^4*d^2*e^5 - 1330*b^5*c^3*d*e^6 + 105*b^6*c
^2*e^7)*f - (49152*c^8*d^7 - 189888*b*c^7*d^6*e + 333888*b^2*c^6*d^5*e^2 - 332464*b^3*c^5*d^4*e^3 + 194976*b^4
*c^4*d^3*e^4 - 66164*b^5*c^3*d^2*e^5 + 12180*b^6*c^2*d*e^6 - 945*b^7*c*e^7)*g + 2*(16*(7392*c^8*d^5*e^2 - 1387
2*b*c^7*d^4*e^3 + 4896*b^2*c^6*d^3*e^4 + 1920*b^3*c^5*d^2*e^5 - 406*b^4*c^4*d*e^6 + 35*b^5*c^3*e^7)*f - (6720*
c^8*d^6*e - 34752*b*c^7*d^5*e^2 + 58896*b^2*c^6*d^4*e^3 - 45792*b^3*c^5*d^3*e^4 + 18092*b^4*c^4*d^2*e^5 - 3724
*b^5*c^3*d*e^6 + 315*b^6*c^2*e^7)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^6*e^2), -1/688128*(105*
(16*(128*c^8*d^7*e - 448*b*c^7*d^6*e^2 + 672*b^2*c^6*d^5*e^3 - 560*b^3*c^5*d^4*e^4 + 280*b^4*c^4*d^3*e^5 - 84*
b^5*c^3*d^2*e^6 + 14*b^6*c^2*d*e^7 - b^7*c*e^8)*f + (256*c^8*d^8 - 2048*b*c^7*d^7*e + 5376*b^2*c^6*d^6*e^2 - 7
168*b^3*c^5*d^5*e^3 + 5600*b^4*c^4*d^4*e^4 - 2688*b^5*c^3*d^3*e^5 + 784*b^6*c^2*d^2*e^6 - 128*b^7*c*d*e^7 + 9*
b^8*e^8)*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*(43008*c^8*e^7*g*x^7 + 3072*(16*c^8*e^7*f + (16*c^8*d*e^6 + 33*b*c^7*e^
7)*g)*x^6 + 256*(16*(14*c^8*d*e^6 + 29*b*c^7*e^7)*f - (476*c^8*d^2*e^5 - 940*b*c^7*d*e^6 - 243*b^2*c^6*e^7)*g)
*x^5 - 128*(16*(72*c^8*d^2*e^5 - 142*b*c^7*d*e^6 - 37*b^2*c^6*e^7)*f + (1152*c^8*d^3*e^4 + 76*b*c^7*d^2*e^5 -
1820*b^2*c^6*d*e^6 - 3*b^3*c^5*e^7)*g)*x^4 - 16*(16*(728*c^8*d^3*e^4 + 60*b*c^7*d^2*e^5 - 1166*b^2*c^6*d*e^6 -
3*b^3*c^5*e^7)*f - (6608*c^8*d^4*e^3 - 25824*b*c^7*d^3*e^4 + 19000*b^2*c^6*d^2*e^5 + 264*b^3*c^5*d*e^6 - 27*b
^4*c^4*e^7)*g)*x^3 + 8*(16*(1152*c^8*d^4*e^3 - 4488*b*c^7*d^3*e^4 + 3276*b^2*c^6*d^2*e^5 + 74*b^3*c^5*d*e^6 -
7*b^4*c^4*e^7)*f + (18432*c^8*d^5*e^2 - 35472*b*c^7*d^4*e^3 + 14688*b^2*c^6*d^3*e^4 + 2920*b^3*c^5*d^2*e^5 - 6
80*b^4*c^4*d*e^6 + 63*b^5*c^3*e^7)*g)*x^2 - 16*(3072*c^8*d^6*e - 16608*b*c^7*d^5*e^2 + 27696*b^2*c^6*d^4*e^3 -
20096*b^3*c^5*d^3*e^4 + 7056*b^4*c^4*d^2*e^5 - 1330*b^5*c^3*d*e^6 + 105*b^6*c^2*e^7)*f - (49152*c^8*d^7 - 189
888*b*c^7*d^6*e + 333888*b^2*c^6*d^5*e^2 - 332464*b^3*c^5*d^4*e^3 + 194976*b^4*c^4*d^3*e^4 - 66164*b^5*c^3*d^2
*e^5 + 12180*b^6*c^2*d*e^6 - 945*b^7*c*e^7)*g + 2*(16*(7392*c^8*d^5*e^2 - 13872*b*c^7*d^4*e^3 + 4896*b^2*c^6*d
^3*e^4 + 1920*b^3*c^5*d^2*e^5 - 406*b^4*c^4*d*e^6 + 35*b^5*c^3*e^7)*f - (6720*c^8*d^6*e - 34752*b*c^7*d^5*e^2
+ 58896*b^2*c^6*d^4*e^3 - 45792*b^3*c^5*d^3*e^4 + 18092*b^4*c^4*d^2*e^5 - 3724*b^5*c^3*d*e^6 + 315*b^6*c^2*e^7
)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^6*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 ) \left (f + g x\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(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)*(f + g*x), x)

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Giac [B]  time = 1.26541, size = 1544, normalized size = 4.16 \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)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="giac")

[Out]

1/344064*sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)*(2*(4*(2*(8*(2*(12*(14*c^2*g*x*e^5 + (16*c^9*d*g*e^16 + 16
*c^9*f*e^17 + 33*b*c^8*g*e^17)*e^(-12)/c^7)*x - (476*c^9*d^2*g*e^15 - 224*c^9*d*f*e^16 - 940*b*c^8*d*g*e^16 -
464*b*c^8*f*e^17 - 243*b^2*c^7*g*e^17)*e^(-12)/c^7)*x - (1152*c^9*d^3*g*e^14 + 1152*c^9*d^2*f*e^15 + 76*b*c^8*
d^2*g*e^15 - 2272*b*c^8*d*f*e^16 - 1820*b^2*c^7*d*g*e^16 - 592*b^2*c^7*f*e^17 - 3*b^3*c^6*g*e^17)*e^(-12)/c^7)
*x + (6608*c^9*d^4*g*e^13 - 11648*c^9*d^3*f*e^14 - 25824*b*c^8*d^3*g*e^14 - 960*b*c^8*d^2*f*e^15 + 19000*b^2*c
^7*d^2*g*e^15 + 18656*b^2*c^7*d*f*e^16 + 264*b^3*c^6*d*g*e^16 + 48*b^3*c^6*f*e^17 - 27*b^4*c^5*g*e^17)*e^(-12)
/c^7)*x + (18432*c^9*d^5*g*e^12 + 18432*c^9*d^4*f*e^13 - 35472*b*c^8*d^4*g*e^13 - 71808*b*c^8*d^3*f*e^14 + 146
88*b^2*c^7*d^3*g*e^14 + 52416*b^2*c^7*d^2*f*e^15 + 2920*b^3*c^6*d^2*g*e^15 + 1184*b^3*c^6*d*f*e^16 - 680*b^4*c
^5*d*g*e^16 - 112*b^4*c^5*f*e^17 + 63*b^5*c^4*g*e^17)*e^(-12)/c^7)*x - (6720*c^9*d^6*g*e^11 - 118272*c^9*d^5*f
*e^12 - 34752*b*c^8*d^5*g*e^12 + 221952*b*c^8*d^4*f*e^13 + 58896*b^2*c^7*d^4*g*e^13 - 78336*b^2*c^7*d^3*f*e^14
- 45792*b^3*c^6*d^3*g*e^14 - 30720*b^3*c^6*d^2*f*e^15 + 18092*b^4*c^5*d^2*g*e^15 + 6496*b^4*c^5*d*f*e^16 - 37
24*b^5*c^4*d*g*e^16 - 560*b^5*c^4*f*e^17 + 315*b^6*c^3*g*e^17)*e^(-12)/c^7)*x - (49152*c^9*d^7*g*e^10 + 49152*
c^9*d^6*f*e^11 - 189888*b*c^8*d^6*g*e^11 - 265728*b*c^8*d^5*f*e^12 + 333888*b^2*c^7*d^5*g*e^12 + 443136*b^2*c^
7*d^4*f*e^13 - 332464*b^3*c^6*d^4*g*e^13 - 321536*b^3*c^6*d^3*f*e^14 + 194976*b^4*c^5*d^3*g*e^14 + 112896*b^4*
c^5*d^2*f*e^15 - 66164*b^5*c^4*d^2*g*e^15 - 21280*b^5*c^4*d*f*e^16 + 12180*b^6*c^3*d*g*e^16 + 1680*b^6*c^3*f*e
^17 - 945*b^7*c^2*g*e^17)*e^(-12)/c^7) + 5/32768*(256*c^8*d^8*g + 2048*c^8*d^7*f*e - 2048*b*c^7*d^7*g*e - 7168
*b*c^7*d^6*f*e^2 + 5376*b^2*c^6*d^6*g*e^2 + 10752*b^2*c^6*d^5*f*e^3 - 7168*b^3*c^5*d^5*g*e^3 - 8960*b^3*c^5*d^
4*f*e^4 + 5600*b^4*c^4*d^4*g*e^4 + 4480*b^4*c^4*d^3*f*e^5 - 2688*b^5*c^3*d^3*g*e^5 - 1344*b^5*c^3*d^2*f*e^6 +
784*b^6*c^2*d^2*g*e^6 + 224*b^6*c^2*d*f*e^7 - 128*b^7*c*d*g*e^7 - 16*b^7*c*f*e^8 + 9*b^8*g*e^8)*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^6