3.2195 \(\int (d+e x)^3 (f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2} \, dx\)

Optimal. Leaf size=562 \[ \frac{11 (b+2 c x) (2 c d-b e)^7 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-13 b e g+6 c d g+20 c e f)}{131072 c^7 e}+\frac{11 (b+2 c x) (2 c d-b e)^5 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-13 b e g+6 c d g+20 c e f)}{49152 c^6 e}+\frac{11 (b+2 c x) (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-13 b e g+6 c d g+20 c e f)}{15360 c^5 e}-\frac{11 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-13 b e g+6 c d g+20 c e f)}{4480 c^4 e^2}-\frac{11 (d+e x) (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-13 b e g+6 c d g+20 c e f)}{2880 c^3 e^2}-\frac{(d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-13 b e g+6 c d g+20 c e f)}{180 c^2 e^2}+\frac{11 (2 c d-b e)^9 (-13 b e g+6 c d g+20 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{262144 c^{15/2} e^2}-\frac{g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{10 c e^2} \]

[Out]

(11*(2*c*d - b*e)^7*(20*c*e*f + 6*c*d*g - 13*b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(13
1072*c^7*e) + (11*(2*c*d - b*e)^5*(20*c*e*f + 6*c*d*g - 13*b*e*g)*(b + 2*c*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2
*x^2)^(3/2))/(49152*c^6*e) + (11*(2*c*d - b*e)^3*(20*c*e*f + 6*c*d*g - 13*b*e*g)*(b + 2*c*x)*(d*(c*d - b*e) -
b*e^2*x - c*e^2*x^2)^(5/2))/(15360*c^5*e) - (11*(2*c*d - b*e)^2*(20*c*e*f + 6*c*d*g - 13*b*e*g)*(d*(c*d - b*e)
- b*e^2*x - c*e^2*x^2)^(7/2))/(4480*c^4*e^2) - (11*(2*c*d - b*e)*(20*c*e*f + 6*c*d*g - 13*b*e*g)*(d + e*x)*(d
*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(2880*c^3*e^2) - ((20*c*e*f + 6*c*d*g - 13*b*e*g)*(d + e*x)^2*(d*(c
*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(180*c^2*e^2) - (g*(d + e*x)^3*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(
7/2))/(10*c*e^2) + (11*(2*c*d - b*e)^9*(20*c*e*f + 6*c*d*g - 13*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])])/(262144*c^(15/2)*e^2)

________________________________________________________________________________________

Rubi [A]  time = 1.21463, antiderivative size = 562, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {794, 670, 640, 612, 621, 204} \[ \frac{11 (b+2 c x) (2 c d-b e)^7 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-13 b e g+6 c d g+20 c e f)}{131072 c^7 e}+\frac{11 (b+2 c x) (2 c d-b e)^5 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2} (-13 b e g+6 c d g+20 c e f)}{49152 c^6 e}+\frac{11 (b+2 c x) (2 c d-b e)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-13 b e g+6 c d g+20 c e f)}{15360 c^5 e}-\frac{11 (2 c d-b e)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-13 b e g+6 c d g+20 c e f)}{4480 c^4 e^2}-\frac{11 (d+e x) (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-13 b e g+6 c d g+20 c e f)}{2880 c^3 e^2}-\frac{(d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (-13 b e g+6 c d g+20 c e f)}{180 c^2 e^2}+\frac{11 (2 c d-b e)^9 (-13 b e g+6 c d g+20 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{262144 c^{15/2} e^2}-\frac{g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{10 c e^2} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3*(f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(5/2),x]

[Out]

(11*(2*c*d - b*e)^7*(20*c*e*f + 6*c*d*g - 13*b*e*g)*(b + 2*c*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(13
1072*c^7*e) + (11*(2*c*d - b*e)^5*(20*c*e*f + 6*c*d*g - 13*b*e*g)*(b + 2*c*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2
*x^2)^(3/2))/(49152*c^6*e) + (11*(2*c*d - b*e)^3*(20*c*e*f + 6*c*d*g - 13*b*e*g)*(b + 2*c*x)*(d*(c*d - b*e) -
b*e^2*x - c*e^2*x^2)^(5/2))/(15360*c^5*e) - (11*(2*c*d - b*e)^2*(20*c*e*f + 6*c*d*g - 13*b*e*g)*(d*(c*d - b*e)
- b*e^2*x - c*e^2*x^2)^(7/2))/(4480*c^4*e^2) - (11*(2*c*d - b*e)*(20*c*e*f + 6*c*d*g - 13*b*e*g)*(d + e*x)*(d
*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(2880*c^3*e^2) - ((20*c*e*f + 6*c*d*g - 13*b*e*g)*(d + e*x)^2*(d*(c
*d - b*e) - b*e^2*x - c*e^2*x^2)^(7/2))/(180*c^2*e^2) - (g*(d + e*x)^3*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(
7/2))/(10*c*e^2) + (11*(2*c*d - b*e)^9*(20*c*e*f + 6*c*d*g - 13*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])])/(262144*c^(15/2)*e^2)

Rule 794

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)
*(2*c*f - b*g))/(c*e*(m + 2*p + 2)), Int[(d + e*x)^m*(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] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[m + 2*p + 2, 0] && (NeQ[m, 2] || Eq
Q[d, 0])

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)^3 (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)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{10 c e^2}-\frac{\left (\frac{7}{2} e \left (-2 c e^2 f+b e^2 g\right )+3 \left (-c e^3 f+\left (-c d e^2+b e^3\right ) g\right )\right ) \int (d+e x)^3 \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx}{10 c e^3}\\ &=-\frac{(20 c e f+6 c d g-13 b e g) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{180 c^2 e^2}-\frac{g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{10 c e^2}+\frac{(11 (2 c d-b e) (20 c e f+6 c d g-13 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}{360 c^2 e}\\ &=-\frac{11 (2 c d-b e) (20 c e f+6 c d g-13 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2880 c^3 e^2}-\frac{(20 c e f+6 c d g-13 b e g) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{180 c^2 e^2}-\frac{g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{10 c e^2}+\frac{\left (11 (2 c d-b e)^2 (20 c e f+6 c d g-13 b e g)\right ) \int (d+e x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx}{640 c^3 e}\\ &=-\frac{11 (2 c d-b e)^2 (20 c e f+6 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{4480 c^4 e^2}-\frac{11 (2 c d-b e) (20 c e f+6 c d g-13 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2880 c^3 e^2}-\frac{(20 c e f+6 c d g-13 b e g) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{180 c^2 e^2}-\frac{g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{10 c e^2}+\frac{\left (11 (2 c d-b e)^3 (20 c e f+6 c d g-13 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}{1280 c^4 e}\\ &=\frac{11 (2 c d-b e)^3 (20 c e f+6 c d g-13 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}}{15360 c^5 e}-\frac{11 (2 c d-b e)^2 (20 c e f+6 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{4480 c^4 e^2}-\frac{11 (2 c d-b e) (20 c e f+6 c d g-13 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2880 c^3 e^2}-\frac{(20 c e f+6 c d g-13 b e g) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{180 c^2 e^2}-\frac{g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{10 c e^2}+\frac{\left (11 (2 c d-b e)^5 (20 c e f+6 c d g-13 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}{6144 c^5 e}\\ &=\frac{11 (2 c d-b e)^5 (20 c e f+6 c d g-13 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}}{49152 c^6 e}+\frac{11 (2 c d-b e)^3 (20 c e f+6 c d g-13 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}}{15360 c^5 e}-\frac{11 (2 c d-b e)^2 (20 c e f+6 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{4480 c^4 e^2}-\frac{11 (2 c d-b e) (20 c e f+6 c d g-13 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2880 c^3 e^2}-\frac{(20 c e f+6 c d g-13 b e g) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{180 c^2 e^2}-\frac{g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{10 c e^2}+\frac{\left (11 (2 c d-b e)^7 (20 c e f+6 c d g-13 b e g)\right ) \int \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2} \, dx}{32768 c^6 e}\\ &=\frac{11 (2 c d-b e)^7 (20 c e f+6 c d g-13 b e g) (b+2 c x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{131072 c^7 e}+\frac{11 (2 c d-b e)^5 (20 c e f+6 c d g-13 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}}{49152 c^6 e}+\frac{11 (2 c d-b e)^3 (20 c e f+6 c d g-13 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}}{15360 c^5 e}-\frac{11 (2 c d-b e)^2 (20 c e f+6 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{4480 c^4 e^2}-\frac{11 (2 c d-b e) (20 c e f+6 c d g-13 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2880 c^3 e^2}-\frac{(20 c e f+6 c d g-13 b e g) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{180 c^2 e^2}-\frac{g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{10 c e^2}+\frac{\left (11 (2 c d-b e)^9 (20 c e f+6 c d g-13 b e g)\right ) \int \frac{1}{\sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx}{262144 c^7 e}\\ &=\frac{11 (2 c d-b e)^7 (20 c e f+6 c d g-13 b e g) (b+2 c x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{131072 c^7 e}+\frac{11 (2 c d-b e)^5 (20 c e f+6 c d g-13 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}}{49152 c^6 e}+\frac{11 (2 c d-b e)^3 (20 c e f+6 c d g-13 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}}{15360 c^5 e}-\frac{11 (2 c d-b e)^2 (20 c e f+6 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{4480 c^4 e^2}-\frac{11 (2 c d-b e) (20 c e f+6 c d g-13 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2880 c^3 e^2}-\frac{(20 c e f+6 c d g-13 b e g) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{180 c^2 e^2}-\frac{g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{10 c e^2}+\frac{\left (11 (2 c d-b e)^9 (20 c e f+6 c d g-13 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 )}{131072 c^7 e}\\ &=\frac{11 (2 c d-b e)^7 (20 c e f+6 c d g-13 b e g) (b+2 c x) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{131072 c^7 e}+\frac{11 (2 c d-b e)^5 (20 c e f+6 c d g-13 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}}{49152 c^6 e}+\frac{11 (2 c d-b e)^3 (20 c e f+6 c d g-13 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}}{15360 c^5 e}-\frac{11 (2 c d-b e)^2 (20 c e f+6 c d g-13 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{4480 c^4 e^2}-\frac{11 (2 c d-b e) (20 c e f+6 c d g-13 b e g) (d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{2880 c^3 e^2}-\frac{(20 c e f+6 c d g-13 b e g) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{180 c^2 e^2}-\frac{g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{10 c e^2}+\frac{11 (2 c d-b e)^9 (20 c e f+6 c d g-13 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 )}{262144 c^{15/2} e^2}\\ \end{align*}

Mathematica [B]  time = 6.88685, size = 1600, normalized size = 2.85 \[ -\frac{g (c d-b e-c e x) ((d+e x) (c (d-e x)-b e))^{5/2} (d+e x)^4}{10 c e^2}-\frac{(c d e+(c d-b e) e)^2 \left (-10 c f e^2-\left (\frac{13}{2} e (c d-b e)-\frac{7 c d e}{2}\right ) g\right ) ((d+e x) (c (d-e x)-b e))^{5/2} \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} \left (\frac{715 (c d e+(c d-b e) e)^7 \left (-\frac{512 c^6 (d+e x)^6 e^{12}}{693 (c d e+(c d-b e) e)^6 \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}-\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 )^7}{131072 c^7 e^{14} (d+e x)^7 \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{13}{18} \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}{16 \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{15}{224 \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 ) (d+e x)^4}{65 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}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^3*(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)^4*(c*d - b*e - c*e*x)*((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2))/(10*c*e^2) - ((c*d*e + e*(c*d -
b*e))^2*(-10*c*e^2*f - ((-7*c*d*e)/2 + (13*e*(c*d - b*e))/2)*g)*(d + e*x)^4*((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)*((13*(15/(224*(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/(16*(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)))/18 + (715*(c*d*e + e*(c*d - b*e))^7*((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*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) - (512*c^6*e^12*(d + e*x)^6)/(693*(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*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))))])))/(131072*c^7*e^14*(d + e*x)^7*(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)))/(65*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 = 0.03, size = 5287, 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)^3*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x)

[Out]

result too large to display

________________________________________________________________________________________

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)^3*(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

________________________________________________________________________________________

Fricas [B]  time = 174.987, size = 8292, normalized size = 14.75 \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)^3*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="fricas")

[Out]

[-1/165150720*(3465*(20*(512*c^10*d^9*e - 2304*b*c^9*d^8*e^2 + 4608*b^2*c^8*d^7*e^3 - 5376*b^3*c^7*d^6*e^4 + 4
032*b^4*c^6*d^5*e^5 - 2016*b^5*c^5*d^4*e^6 + 672*b^6*c^4*d^3*e^7 - 144*b^7*c^3*d^2*e^8 + 18*b^8*c^2*d*e^9 - b^
9*c*e^10)*f + (3072*c^10*d^10 - 20480*b*c^9*d^9*e + 57600*b^2*c^8*d^8*e^2 - 92160*b^3*c^7*d^7*e^3 + 94080*b^4*
c^6*d^6*e^4 - 64512*b^5*c^5*d^5*e^5 + 30240*b^6*c^4*d^4*e^6 - 9600*b^7*c^3*d^3*e^7 + 1980*b^8*c^2*d^2*e^8 - 24
0*b^9*c*d*e^9 + 13*b^10*e^10)*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*(4128768*c^10*e^9*g*x^9 + 229376*(2
0*c^10*e^9*f + (60*c^10*d*e^8 + 41*b*c^9*e^9)*g)*x^8 + 14336*(20*(54*c^10*d*e^8 + 37*b*c^9*e^9)*f + (324*c^10*
d^2*e^7 + 2976*b*c^9*d*e^8 + 383*b^2*c^8*e^9)*g)*x^7 + 1024*(20*(256*c^10*d^2*e^7 + 2390*b*c^9*d*e^8 + 309*b^2
*c^8*e^9)*f - (30720*c^10*d^3*e^6 - 59396*b*c^9*d^2*e^7 - 31264*b^2*c^8*d*e^8 - 15*b^3*c^7*e^9)*g)*x^6 - 256*(
20*(7224*c^10*d^3*e^6 - 13908*b*c^9*d^2*e^7 - 7386*b^2*c^8*d*e^8 - 5*b^3*c^7*e^9)*f + (140112*c^10*d^4*e^5 + 1
6176*b*c^9*d^3*e^6 - 298968*b^2*c^8*d^2*e^7 - 780*b^3*c^7*d*e^8 + 65*b^4*c^6*e^9)*g)*x^5 - 128*(20*(16896*c^10
*d^4*e^5 + 2328*b*c^9*d^3*e^6 - 36516*b^2*c^8*d^2*e^7 - 138*b^3*c^7*d*e^8 + 11*b^4*c^6*e^9)*f - (92160*c^10*d^
5*e^4 - 762000*b*c^9*d^4*e^5 + 733200*b^2*c^8*d^3*e^6 + 9960*b^3*c^7*d^2*e^7 - 1860*b^4*c^6*d*e^8 + 143*b^5*c^
5*e^9)*g)*x^4 + 16*(20*(49056*c^10*d^5*e^4 - 392976*b*c^9*d^4*e^5 + 374352*b^2*c^8*d^3*e^6 + 7576*b^3*c^7*d^2*
e^7 - 1342*b^4*c^6*d*e^8 + 99*b^5*c^5*e^9)*f + (2358720*c^10*d^6*e^3 - 6092160*b*c^9*d^5*e^4 + 3484080*b^2*c^8
*d^4*e^5 + 339840*b^3*c^7*d^3*e^6 - 106540*b^4*c^6*d^2*e^7 + 18040*b^5*c^5*d*e^8 - 1287*b^6*c^4*e^9)*g)*x^3 +
8*(20*(327680*c^10*d^6*e^3 - 835872*b*c^9*d^5*e^4 + 455376*b^2*c^8*d^4*e^5 + 70768*b^3*c^7*d^3*e^6 - 20856*b^4
*c^6*d^2*e^7 + 3366*b^5*c^5*d*e^8 - 231*b^6*c^4*e^9)*f + (1966080*c^10*d^7*e^2 - 3081920*b*c^9*d^6*e^3 - 33600
0*b^2*c^8*d^5*e^4 + 2246160*b^3*c^7*d^4*e^5 - 1045120*b^4*c^6*d^3*e^6 + 291324*b^5*c^5*d^2*e^7 - 45144*b^6*c^4
*d*e^8 + 3003*b^7*c^3*e^9)*g)*x^2 - 20*(950272*c^10*d^8*e - 4389760*b*c^9*d^7*e^2 + 8056896*b^2*c^8*d^6*e^3 -
7874464*b^3*c^7*d^5*e^4 + 4655728*b^4*c^6*d^4*e^5 - 1770120*b^5*c^5*d^3*e^6 + 422268*b^6*c^4*d^2*e^7 - 57750*b
^7*c^3*d*e^8 + 3465*b^8*c^2*e^9)*f - (9830400*c^10*d^9 - 51078400*b*c^9*d^8*e + 117794560*b^2*c^8*d^7*e^2 - 15
6115200*b^3*c^7*d^6*e^3 + 130302400*b^4*c^6*d^5*e^4 - 71145184*b^5*c^5*d^4*e^5 + 25545168*b^6*c^4*d^3*e^6 - 58
35984*b^7*c^3*d^2*e^7 + 771540*b^8*c^2*d*e^8 - 45045*b^9*c*e^9)*g + 2*(20*(588672*c^10*d^7*e^2 - 749632*b*c^9*
d^6*e^3 - 547296*b^2*c^8*d^5*e^4 + 1063248*b^3*c^7*d^4*e^5 - 460856*b^4*c^6*d^3*e^6 + 121572*b^5*c^5*d^2*e^7 -
18018*b^6*c^4*d*e^8 + 1155*b^7*c^3*e^9)*f - (2661120*c^10*d^8*e - 12622080*b*c^9*d^7*e^2 + 24504320*b^2*c^8*d
^6*e^3 - 25880640*b^3*c^7*d^5*e^4 + 16587360*b^4*c^6*d^4*e^5 - 6720560*b^5*c^5*d^3*e^6 + 1688544*b^6*c^4*d^2*e
^7 - 241164*b^7*c^3*d*e^8 + 15015*b^8*c^2*e^9)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^8*e^2), -1
/82575360*(3465*(20*(512*c^10*d^9*e - 2304*b*c^9*d^8*e^2 + 4608*b^2*c^8*d^7*e^3 - 5376*b^3*c^7*d^6*e^4 + 4032*
b^4*c^6*d^5*e^5 - 2016*b^5*c^5*d^4*e^6 + 672*b^6*c^4*d^3*e^7 - 144*b^7*c^3*d^2*e^8 + 18*b^8*c^2*d*e^9 - b^9*c*
e^10)*f + (3072*c^10*d^10 - 20480*b*c^9*d^9*e + 57600*b^2*c^8*d^8*e^2 - 92160*b^3*c^7*d^7*e^3 + 94080*b^4*c^6*
d^6*e^4 - 64512*b^5*c^5*d^5*e^5 + 30240*b^6*c^4*d^4*e^6 - 9600*b^7*c^3*d^3*e^7 + 1980*b^8*c^2*d^2*e^8 - 240*b^
9*c*d*e^9 + 13*b^10*e^10)*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)*sqr
t(c)/(c^2*e^2*x^2 + b*c*e^2*x - c^2*d^2 + b*c*d*e)) - 2*(4128768*c^10*e^9*g*x^9 + 229376*(20*c^10*e^9*f + (60*
c^10*d*e^8 + 41*b*c^9*e^9)*g)*x^8 + 14336*(20*(54*c^10*d*e^8 + 37*b*c^9*e^9)*f + (324*c^10*d^2*e^7 + 2976*b*c^
9*d*e^8 + 383*b^2*c^8*e^9)*g)*x^7 + 1024*(20*(256*c^10*d^2*e^7 + 2390*b*c^9*d*e^8 + 309*b^2*c^8*e^9)*f - (3072
0*c^10*d^3*e^6 - 59396*b*c^9*d^2*e^7 - 31264*b^2*c^8*d*e^8 - 15*b^3*c^7*e^9)*g)*x^6 - 256*(20*(7224*c^10*d^3*e
^6 - 13908*b*c^9*d^2*e^7 - 7386*b^2*c^8*d*e^8 - 5*b^3*c^7*e^9)*f + (140112*c^10*d^4*e^5 + 16176*b*c^9*d^3*e^6
- 298968*b^2*c^8*d^2*e^7 - 780*b^3*c^7*d*e^8 + 65*b^4*c^6*e^9)*g)*x^5 - 128*(20*(16896*c^10*d^4*e^5 + 2328*b*c
^9*d^3*e^6 - 36516*b^2*c^8*d^2*e^7 - 138*b^3*c^7*d*e^8 + 11*b^4*c^6*e^9)*f - (92160*c^10*d^5*e^4 - 762000*b*c^
9*d^4*e^5 + 733200*b^2*c^8*d^3*e^6 + 9960*b^3*c^7*d^2*e^7 - 1860*b^4*c^6*d*e^8 + 143*b^5*c^5*e^9)*g)*x^4 + 16*
(20*(49056*c^10*d^5*e^4 - 392976*b*c^9*d^4*e^5 + 374352*b^2*c^8*d^3*e^6 + 7576*b^3*c^7*d^2*e^7 - 1342*b^4*c^6*
d*e^8 + 99*b^5*c^5*e^9)*f + (2358720*c^10*d^6*e^3 - 6092160*b*c^9*d^5*e^4 + 3484080*b^2*c^8*d^4*e^5 + 339840*b
^3*c^7*d^3*e^6 - 106540*b^4*c^6*d^2*e^7 + 18040*b^5*c^5*d*e^8 - 1287*b^6*c^4*e^9)*g)*x^3 + 8*(20*(327680*c^10*
d^6*e^3 - 835872*b*c^9*d^5*e^4 + 455376*b^2*c^8*d^4*e^5 + 70768*b^3*c^7*d^3*e^6 - 20856*b^4*c^6*d^2*e^7 + 3366
*b^5*c^5*d*e^8 - 231*b^6*c^4*e^9)*f + (1966080*c^10*d^7*e^2 - 3081920*b*c^9*d^6*e^3 - 336000*b^2*c^8*d^5*e^4 +
2246160*b^3*c^7*d^4*e^5 - 1045120*b^4*c^6*d^3*e^6 + 291324*b^5*c^5*d^2*e^7 - 45144*b^6*c^4*d*e^8 + 3003*b^7*c
^3*e^9)*g)*x^2 - 20*(950272*c^10*d^8*e - 4389760*b*c^9*d^7*e^2 + 8056896*b^2*c^8*d^6*e^3 - 7874464*b^3*c^7*d^5
*e^4 + 4655728*b^4*c^6*d^4*e^5 - 1770120*b^5*c^5*d^3*e^6 + 422268*b^6*c^4*d^2*e^7 - 57750*b^7*c^3*d*e^8 + 3465
*b^8*c^2*e^9)*f - (9830400*c^10*d^9 - 51078400*b*c^9*d^8*e + 117794560*b^2*c^8*d^7*e^2 - 156115200*b^3*c^7*d^6
*e^3 + 130302400*b^4*c^6*d^5*e^4 - 71145184*b^5*c^5*d^4*e^5 + 25545168*b^6*c^4*d^3*e^6 - 5835984*b^7*c^3*d^2*e
^7 + 771540*b^8*c^2*d*e^8 - 45045*b^9*c*e^9)*g + 2*(20*(588672*c^10*d^7*e^2 - 749632*b*c^9*d^6*e^3 - 547296*b^
2*c^8*d^5*e^4 + 1063248*b^3*c^7*d^4*e^5 - 460856*b^4*c^6*d^3*e^6 + 121572*b^5*c^5*d^2*e^7 - 18018*b^6*c^4*d*e^
8 + 1155*b^7*c^3*e^9)*f - (2661120*c^10*d^8*e - 12622080*b*c^9*d^7*e^2 + 24504320*b^2*c^8*d^6*e^3 - 25880640*b
^3*c^7*d^5*e^4 + 16587360*b^4*c^6*d^4*e^5 - 6720560*b^5*c^5*d^3*e^6 + 1688544*b^6*c^4*d^2*e^7 - 241164*b^7*c^3
*d*e^8 + 15015*b^8*c^2*e^9)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^8*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 )^{3} \left (f + g x\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*(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)**3*(f + g*x), x)

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Giac [B]  time = 1.30784, size = 2276, normalized size = 4.05 \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)^3*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algorithm="giac")

[Out]

1/41287680*sqrt(-c*x^2*e^2 + c*d^2 - b*x*e^2 - b*d*e)*(2*(4*(2*(8*(2*(4*(14*(16*(18*c^2*g*x*e^7 + (60*c^11*d*g
*e^22 + 20*c^11*f*e^23 + 41*b*c^10*g*e^23)*e^(-16)/c^9)*x + (324*c^11*d^2*g*e^21 + 1080*c^11*d*f*e^22 + 2976*b
*c^10*d*g*e^22 + 740*b*c^10*f*e^23 + 383*b^2*c^9*g*e^23)*e^(-16)/c^9)*x - (30720*c^11*d^3*g*e^20 - 5120*c^11*d
^2*f*e^21 - 59396*b*c^10*d^2*g*e^21 - 47800*b*c^10*d*f*e^22 - 31264*b^2*c^9*d*g*e^22 - 6180*b^2*c^9*f*e^23 - 1
5*b^3*c^8*g*e^23)*e^(-16)/c^9)*x - (140112*c^11*d^4*g*e^19 + 144480*c^11*d^3*f*e^20 + 16176*b*c^10*d^3*g*e^20
- 278160*b*c^10*d^2*f*e^21 - 298968*b^2*c^9*d^2*g*e^21 - 147720*b^2*c^9*d*f*e^22 - 780*b^3*c^8*d*g*e^22 - 100*
b^3*c^8*f*e^23 + 65*b^4*c^7*g*e^23)*e^(-16)/c^9)*x + (92160*c^11*d^5*g*e^18 - 337920*c^11*d^4*f*e^19 - 762000*
b*c^10*d^4*g*e^19 - 46560*b*c^10*d^3*f*e^20 + 733200*b^2*c^9*d^3*g*e^20 + 730320*b^2*c^9*d^2*f*e^21 + 9960*b^3
*c^8*d^2*g*e^21 + 2760*b^3*c^8*d*f*e^22 - 1860*b^4*c^7*d*g*e^22 - 220*b^4*c^7*f*e^23 + 143*b^5*c^6*g*e^23)*e^(
-16)/c^9)*x + (2358720*c^11*d^6*g*e^17 + 981120*c^11*d^5*f*e^18 - 6092160*b*c^10*d^5*g*e^18 - 7859520*b*c^10*d
^4*f*e^19 + 3484080*b^2*c^9*d^4*g*e^19 + 7487040*b^2*c^9*d^3*f*e^20 + 339840*b^3*c^8*d^3*g*e^20 + 151520*b^3*c
^8*d^2*f*e^21 - 106540*b^4*c^7*d^2*g*e^21 - 26840*b^4*c^7*d*f*e^22 + 18040*b^5*c^6*d*g*e^22 + 1980*b^5*c^6*f*e
^23 - 1287*b^6*c^5*g*e^23)*e^(-16)/c^9)*x + (1966080*c^11*d^7*g*e^16 + 6553600*c^11*d^6*f*e^17 - 3081920*b*c^1
0*d^6*g*e^17 - 16717440*b*c^10*d^5*f*e^18 - 336000*b^2*c^9*d^5*g*e^18 + 9107520*b^2*c^9*d^4*f*e^19 + 2246160*b
^3*c^8*d^4*g*e^19 + 1415360*b^3*c^8*d^3*f*e^20 - 1045120*b^4*c^7*d^3*g*e^20 - 417120*b^4*c^7*d^2*f*e^21 + 2913
24*b^5*c^6*d^2*g*e^21 + 67320*b^5*c^6*d*f*e^22 - 45144*b^6*c^5*d*g*e^22 - 4620*b^6*c^5*f*e^23 + 3003*b^7*c^4*g
*e^23)*e^(-16)/c^9)*x - (2661120*c^11*d^8*g*e^15 - 11773440*c^11*d^7*f*e^16 - 12622080*b*c^10*d^7*g*e^16 + 149
92640*b*c^10*d^6*f*e^17 + 24504320*b^2*c^9*d^6*g*e^17 + 10945920*b^2*c^9*d^5*f*e^18 - 25880640*b^3*c^8*d^5*g*e
^18 - 21264960*b^3*c^8*d^4*f*e^19 + 16587360*b^4*c^7*d^4*g*e^19 + 9217120*b^4*c^7*d^3*f*e^20 - 6720560*b^5*c^6
*d^3*g*e^20 - 2431440*b^5*c^6*d^2*f*e^21 + 1688544*b^6*c^5*d^2*g*e^21 + 360360*b^6*c^5*d*f*e^22 - 241164*b^7*c
^4*d*g*e^22 - 23100*b^7*c^4*f*e^23 + 15015*b^8*c^3*g*e^23)*e^(-16)/c^9)*x - (9830400*c^11*d^9*g*e^14 + 1900544
0*c^11*d^8*f*e^15 - 51078400*b*c^10*d^8*g*e^15 - 87795200*b*c^10*d^7*f*e^16 + 117794560*b^2*c^9*d^7*g*e^16 + 1
61137920*b^2*c^9*d^6*f*e^17 - 156115200*b^3*c^8*d^6*g*e^17 - 157489280*b^3*c^8*d^5*f*e^18 + 130302400*b^4*c^7*
d^5*g*e^18 + 93114560*b^4*c^7*d^4*f*e^19 - 71145184*b^5*c^6*d^4*g*e^19 - 35402400*b^5*c^6*d^3*f*e^20 + 2554516
8*b^6*c^5*d^3*g*e^20 + 8445360*b^6*c^5*d^2*f*e^21 - 5835984*b^7*c^4*d^2*g*e^21 - 1155000*b^7*c^4*d*f*e^22 + 77
1540*b^8*c^3*d*g*e^22 + 69300*b^8*c^3*f*e^23 - 45045*b^9*c^2*g*e^23)*e^(-16)/c^9) + 11/262144*(3072*c^10*d^10*
g + 10240*c^10*d^9*f*e - 20480*b*c^9*d^9*g*e - 46080*b*c^9*d^8*f*e^2 + 57600*b^2*c^8*d^8*g*e^2 + 92160*b^2*c^8
*d^7*f*e^3 - 92160*b^3*c^7*d^7*g*e^3 - 107520*b^3*c^7*d^6*f*e^4 + 94080*b^4*c^6*d^6*g*e^4 + 80640*b^4*c^6*d^5*
f*e^5 - 64512*b^5*c^5*d^5*g*e^5 - 40320*b^5*c^5*d^4*f*e^6 + 30240*b^6*c^4*d^4*g*e^6 + 13440*b^6*c^4*d^3*f*e^7
- 9600*b^7*c^3*d^3*g*e^7 - 2880*b^7*c^3*d^2*f*e^8 + 1980*b^8*c^2*d^2*g*e^8 + 360*b^8*c^2*d*f*e^9 - 240*b^9*c*d
*g*e^9 - 20*b^9*c*f*e^10 + 13*b^10*g*e^10)*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^8