3.2111 \(\int (a+b x) (d+e x)^{3/2} (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\)

Optimal. Leaf size=376 \[ \frac{2 b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{17/2}}{17 e^7 (a+b x)}-\frac{4 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15/2} (b d-a e)}{5 e^7 (a+b x)}+\frac{30 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)^2}{13 e^7 (a+b x)}-\frac{40 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^3}{11 e^7 (a+b x)}+\frac{10 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^4}{3 e^7 (a+b x)}-\frac{12 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^5}{7 e^7 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^6}{5 e^7 (a+b x)} \]

[Out]

(2*(b*d - a*e)^6*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)) - (12*b*(b*d - a*e)^5*(d + e
*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)) + (10*b^2*(b*d - a*e)^4*(d + e*x)^(9/2)*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)) - (40*b^3*(b*d - a*e)^3*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/
(11*e^7*(a + b*x)) + (30*b^4*(b*d - a*e)^2*(d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x))
- (4*b^5*(b*d - a*e)*(d + e*x)^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)) + (2*b^6*(d + e*x)^(17/
2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(17*e^7*(a + b*x))

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Rubi [A]  time = 0.136629, antiderivative size = 376, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086, Rules used = {770, 21, 43} \[ \frac{2 b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{17/2}}{17 e^7 (a+b x)}-\frac{4 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15/2} (b d-a e)}{5 e^7 (a+b x)}+\frac{30 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)^2}{13 e^7 (a+b x)}-\frac{40 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^3}{11 e^7 (a+b x)}+\frac{10 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^4}{3 e^7 (a+b x)}-\frac{12 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^5}{7 e^7 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^6}{5 e^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(2*(b*d - a*e)^6*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)) - (12*b*(b*d - a*e)^5*(d + e
*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)) + (10*b^2*(b*d - a*e)^4*(d + e*x)^(9/2)*Sqrt[a^2 +
2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)) - (40*b^3*(b*d - a*e)^3*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/
(11*e^7*(a + b*x)) + (30*b^4*(b*d - a*e)^2*(d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x))
- (4*b^5*(b*d - a*e)*(d + e*x)^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a + b*x)) + (2*b^6*(d + e*x)^(17/
2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(17*e^7*(a + b*x))

Rule 770

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

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int (a+b x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int (a+b x) \left (a b+b^2 x\right )^5 (d+e x)^{3/2} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int (a+b x)^6 (d+e x)^{3/2} \, dx}{a b+b^2 x}\\ &=\frac{\left (b \sqrt{a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac{(-b d+a e)^6 (d+e x)^{3/2}}{e^6}-\frac{6 b (b d-a e)^5 (d+e x)^{5/2}}{e^6}+\frac{15 b^2 (b d-a e)^4 (d+e x)^{7/2}}{e^6}-\frac{20 b^3 (b d-a e)^3 (d+e x)^{9/2}}{e^6}+\frac{15 b^4 (b d-a e)^2 (d+e x)^{11/2}}{e^6}-\frac{6 b^5 (b d-a e) (d+e x)^{13/2}}{e^6}+\frac{b^6 (d+e x)^{15/2}}{e^6}\right ) \, dx}{a b+b^2 x}\\ &=\frac{2 (b d-a e)^6 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x)}-\frac{12 b (b d-a e)^5 (d+e x)^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x)}+\frac{10 b^2 (b d-a e)^4 (d+e x)^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}-\frac{40 b^3 (b d-a e)^3 (d+e x)^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x)}+\frac{30 b^4 (b d-a e)^2 (d+e x)^{13/2} \sqrt{a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x)}-\frac{4 b^5 (b d-a e) (d+e x)^{15/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x)}+\frac{2 b^6 (d+e x)^{17/2} \sqrt{a^2+2 a b x+b^2 x^2}}{17 e^7 (a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.18252, size = 163, normalized size = 0.43 \[ \frac{2 \sqrt{(a+b x)^2} (d+e x)^{5/2} \left (425425 b^2 (d+e x)^2 (b d-a e)^4-464100 b^3 (d+e x)^3 (b d-a e)^3+294525 b^4 (d+e x)^4 (b d-a e)^2-102102 b^5 (d+e x)^5 (b d-a e)-218790 b (d+e x) (b d-a e)^5+51051 (b d-a e)^6+15015 b^6 (d+e x)^6\right )}{255255 e^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]

[Out]

(2*Sqrt[(a + b*x)^2]*(d + e*x)^(5/2)*(51051*(b*d - a*e)^6 - 218790*b*(b*d - a*e)^5*(d + e*x) + 425425*b^2*(b*d
- a*e)^4*(d + e*x)^2 - 464100*b^3*(b*d - a*e)^3*(d + e*x)^3 + 294525*b^4*(b*d - a*e)^2*(d + e*x)^4 - 102102*b
^5*(b*d - a*e)*(d + e*x)^5 + 15015*b^6*(d + e*x)^6))/(255255*e^7*(a + b*x))

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Maple [A]  time = 0.007, size = 393, normalized size = 1.1 \begin{align*}{\frac{30030\,{x}^{6}{b}^{6}{e}^{6}+204204\,{x}^{5}a{b}^{5}{e}^{6}-24024\,{x}^{5}{b}^{6}d{e}^{5}+589050\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}-157080\,{x}^{4}a{b}^{5}d{e}^{5}+18480\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+928200\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}-428400\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+114240\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}-13440\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+850850\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}-618800\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+285600\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}-76160\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+8960\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+437580\,x{a}^{5}b{e}^{6}-486200\,x{a}^{4}{b}^{2}d{e}^{5}+353600\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}-163200\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+43520\,xa{b}^{5}{d}^{4}{e}^{2}-5120\,x{b}^{6}{d}^{5}e+102102\,{a}^{6}{e}^{6}-175032\,d{e}^{5}{a}^{5}b+194480\,{a}^{4}{b}^{2}{d}^{2}{e}^{4}-141440\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+65280\,{a}^{2}{b}^{4}{d}^{4}{e}^{2}-17408\,a{b}^{5}{d}^{5}e+2048\,{b}^{6}{d}^{6}}{255255\,{e}^{7} \left ( bx+a \right ) ^{5}} \left ( ex+d \right ) ^{{\frac{5}{2}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

2/255255*(e*x+d)^(5/2)*(15015*b^6*e^6*x^6+102102*a*b^5*e^6*x^5-12012*b^6*d*e^5*x^5+294525*a^2*b^4*e^6*x^4-7854
0*a*b^5*d*e^5*x^4+9240*b^6*d^2*e^4*x^4+464100*a^3*b^3*e^6*x^3-214200*a^2*b^4*d*e^5*x^3+57120*a*b^5*d^2*e^4*x^3
-6720*b^6*d^3*e^3*x^3+425425*a^4*b^2*e^6*x^2-309400*a^3*b^3*d*e^5*x^2+142800*a^2*b^4*d^2*e^4*x^2-38080*a*b^5*d
^3*e^3*x^2+4480*b^6*d^4*e^2*x^2+218790*a^5*b*e^6*x-243100*a^4*b^2*d*e^5*x+176800*a^3*b^3*d^2*e^4*x-81600*a^2*b
^4*d^3*e^3*x+21760*a*b^5*d^4*e^2*x-2560*b^6*d^5*e*x+51051*a^6*e^6-87516*a^5*b*d*e^5+97240*a^4*b^2*d^2*e^4-7072
0*a^3*b^3*d^3*e^3+32640*a^2*b^4*d^4*e^2-8704*a*b^5*d^5*e+1024*b^6*d^6)*((b*x+a)^2)^(5/2)/e^7/(b*x+a)^5

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Maxima [B]  time = 1.21785, size = 1243, normalized size = 3.31 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")

[Out]

2/45045*(3003*b^5*e^7*x^7 - 256*b^5*d^7 + 1920*a*b^4*d^6*e - 6240*a^2*b^3*d^5*e^2 + 11440*a^3*b^2*d^4*e^3 - 12
870*a^4*b*d^3*e^4 + 9009*a^5*d^2*e^5 + 231*(16*b^5*d*e^6 + 75*a*b^4*e^7)*x^6 + 63*(b^5*d^2*e^5 + 350*a*b^4*d*e
^6 + 650*a^2*b^3*e^7)*x^5 - 35*(2*b^5*d^3*e^4 - 15*a*b^4*d^2*e^5 - 1560*a^2*b^3*d*e^6 - 1430*a^3*b^2*e^7)*x^4
+ 5*(16*b^5*d^4*e^3 - 120*a*b^4*d^3*e^4 + 390*a^2*b^3*d^2*e^5 + 14300*a^3*b^2*d*e^6 + 6435*a^4*b*e^7)*x^3 - 3*
(32*b^5*d^5*e^2 - 240*a*b^4*d^4*e^3 + 780*a^2*b^3*d^3*e^4 - 1430*a^3*b^2*d^2*e^5 - 17160*a^4*b*d*e^6 - 3003*a^
5*e^7)*x^2 + (128*b^5*d^6*e - 960*a*b^4*d^5*e^2 + 3120*a^2*b^3*d^4*e^3 - 5720*a^3*b^2*d^3*e^4 + 6435*a^4*b*d^2
*e^5 + 18018*a^5*d*e^6)*x)*sqrt(e*x + d)*a/e^6 + 2/765765*(45045*b^5*e^8*x^8 + 3072*b^5*d^8 - 21760*a*b^4*d^7*
e + 65280*a^2*b^3*d^6*e^2 - 106080*a^3*b^2*d^5*e^3 + 97240*a^4*b*d^4*e^4 - 43758*a^5*d^3*e^5 + 3003*(18*b^5*d*
e^7 + 85*a*b^4*e^8)*x^7 + 231*(3*b^5*d^2*e^6 + 1360*a*b^4*d*e^7 + 2550*a^2*b^3*e^8)*x^6 - 63*(12*b^5*d^3*e^5 -
85*a*b^4*d^2*e^6 - 11900*a^2*b^3*d*e^7 - 11050*a^3*b^2*e^8)*x^5 + 35*(24*b^5*d^4*e^4 - 170*a*b^4*d^3*e^5 + 51
0*a^2*b^3*d^2*e^6 + 26520*a^3*b^2*d*e^7 + 12155*a^4*b*e^8)*x^4 - 5*(192*b^5*d^5*e^3 - 1360*a*b^4*d^4*e^4 + 408
0*a^2*b^3*d^3*e^5 - 6630*a^3*b^2*d^2*e^6 - 121550*a^4*b*d*e^7 - 21879*a^5*e^8)*x^3 + 3*(384*b^5*d^6*e^2 - 2720
*a*b^4*d^5*e^3 + 8160*a^2*b^3*d^4*e^4 - 13260*a^3*b^2*d^3*e^5 + 12155*a^4*b*d^2*e^6 + 58344*a^5*d*e^7)*x^2 - (
1536*b^5*d^7*e - 10880*a*b^4*d^6*e^2 + 32640*a^2*b^3*d^5*e^3 - 53040*a^3*b^2*d^4*e^4 + 48620*a^4*b*d^3*e^5 - 2
1879*a^5*d^2*e^6)*x)*sqrt(e*x + d)*b/e^7

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Fricas [A]  time = 1.03751, size = 1256, normalized size = 3.34 \begin{align*} \frac{2 \,{\left (15015 \, b^{6} e^{8} x^{8} + 1024 \, b^{6} d^{8} - 8704 \, a b^{5} d^{7} e + 32640 \, a^{2} b^{4} d^{6} e^{2} - 70720 \, a^{3} b^{3} d^{5} e^{3} + 97240 \, a^{4} b^{2} d^{4} e^{4} - 87516 \, a^{5} b d^{3} e^{5} + 51051 \, a^{6} d^{2} e^{6} + 6006 \,{\left (3 \, b^{6} d e^{7} + 17 \, a b^{5} e^{8}\right )} x^{7} + 231 \,{\left (b^{6} d^{2} e^{6} + 544 \, a b^{5} d e^{7} + 1275 \, a^{2} b^{4} e^{8}\right )} x^{6} - 42 \,{\left (6 \, b^{6} d^{3} e^{5} - 51 \, a b^{5} d^{2} e^{6} - 8925 \, a^{2} b^{4} d e^{7} - 11050 \, a^{3} b^{3} e^{8}\right )} x^{5} + 35 \,{\left (8 \, b^{6} d^{4} e^{4} - 68 \, a b^{5} d^{3} e^{5} + 255 \, a^{2} b^{4} d^{2} e^{6} + 17680 \, a^{3} b^{3} d e^{7} + 12155 \, a^{4} b^{2} e^{8}\right )} x^{4} - 10 \,{\left (32 \, b^{6} d^{5} e^{3} - 272 \, a b^{5} d^{4} e^{4} + 1020 \, a^{2} b^{4} d^{3} e^{5} - 2210 \, a^{3} b^{3} d^{2} e^{6} - 60775 \, a^{4} b^{2} d e^{7} - 21879 \, a^{5} b e^{8}\right )} x^{3} + 3 \,{\left (128 \, b^{6} d^{6} e^{2} - 1088 \, a b^{5} d^{5} e^{3} + 4080 \, a^{2} b^{4} d^{4} e^{4} - 8840 \, a^{3} b^{3} d^{3} e^{5} + 12155 \, a^{4} b^{2} d^{2} e^{6} + 116688 \, a^{5} b d e^{7} + 17017 \, a^{6} e^{8}\right )} x^{2} - 2 \,{\left (256 \, b^{6} d^{7} e - 2176 \, a b^{5} d^{6} e^{2} + 8160 \, a^{2} b^{4} d^{5} e^{3} - 17680 \, a^{3} b^{3} d^{4} e^{4} + 24310 \, a^{4} b^{2} d^{3} e^{5} - 21879 \, a^{5} b d^{2} e^{6} - 51051 \, a^{6} d e^{7}\right )} x\right )} \sqrt{e x + d}}{255255 \, e^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")

[Out]

2/255255*(15015*b^6*e^8*x^8 + 1024*b^6*d^8 - 8704*a*b^5*d^7*e + 32640*a^2*b^4*d^6*e^2 - 70720*a^3*b^3*d^5*e^3
+ 97240*a^4*b^2*d^4*e^4 - 87516*a^5*b*d^3*e^5 + 51051*a^6*d^2*e^6 + 6006*(3*b^6*d*e^7 + 17*a*b^5*e^8)*x^7 + 23
1*(b^6*d^2*e^6 + 544*a*b^5*d*e^7 + 1275*a^2*b^4*e^8)*x^6 - 42*(6*b^6*d^3*e^5 - 51*a*b^5*d^2*e^6 - 8925*a^2*b^4
*d*e^7 - 11050*a^3*b^3*e^8)*x^5 + 35*(8*b^6*d^4*e^4 - 68*a*b^5*d^3*e^5 + 255*a^2*b^4*d^2*e^6 + 17680*a^3*b^3*d
*e^7 + 12155*a^4*b^2*e^8)*x^4 - 10*(32*b^6*d^5*e^3 - 272*a*b^5*d^4*e^4 + 1020*a^2*b^4*d^3*e^5 - 2210*a^3*b^3*d
^2*e^6 - 60775*a^4*b^2*d*e^7 - 21879*a^5*b*e^8)*x^3 + 3*(128*b^6*d^6*e^2 - 1088*a*b^5*d^5*e^3 + 4080*a^2*b^4*d
^4*e^4 - 8840*a^3*b^3*d^3*e^5 + 12155*a^4*b^2*d^2*e^6 + 116688*a^5*b*d*e^7 + 17017*a^6*e^8)*x^2 - 2*(256*b^6*d
^7*e - 2176*a*b^5*d^6*e^2 + 8160*a^2*b^4*d^5*e^3 - 17680*a^3*b^3*d^4*e^4 + 24310*a^4*b^2*d^3*e^5 - 21879*a^5*b
*d^2*e^6 - 51051*a^6*d*e^7)*x)*sqrt(e*x + d)/e^7

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)**(3/2)*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 1.2781, size = 1315, normalized size = 3.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

2/765765*(306306*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^5*b*d*e^(-1)*sgn(b*x + a) + 109395*(15*(x*e + d)^
(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^4*b^2*d*e^(-2)*sgn(b*x + a) + 48620*(35*(x*e + d)^(9/
2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*a^3*b^3*d*e^(-3)*sgn(b*x + a)
+ 3315*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 +
1155*(x*e + d)^(3/2)*d^4)*a^2*b^4*d*e^(-4)*sgn(b*x + a) + 510*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d
+ 10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)
*a*b^5*d*e^(-5)*sgn(b*x + a) + 17*(3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61425*(x*e + d)^(11/2)*d
^2 - 100100*(x*e + d)^(9/2)*d^3 + 96525*(x*e + d)^(7/2)*d^4 - 54054*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2
)*d^6)*b^6*d*e^(-6)*sgn(b*x + a) + 255255*(x*e + d)^(3/2)*a^6*d*sgn(b*x + a) + 43758*(15*(x*e + d)^(7/2) - 42*
(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^5*b*e^(-1)*sgn(b*x + a) + 36465*(35*(x*e + d)^(9/2) - 135*(x*e +
d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*a^4*b^2*e^(-2)*sgn(b*x + a) + 4420*(315*(x*e
+ d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3
/2)*d^4)*a^3*b^3*e^(-3)*sgn(b*x + a) + 1275*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d + 10010*(x*e + d)^
(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*a^2*b^4*e^(-4)*sg
n(b*x + a) + 102*(3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61425*(x*e + d)^(11/2)*d^2 - 100100*(x*e
+ d)^(9/2)*d^3 + 96525*(x*e + d)^(7/2)*d^4 - 54054*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6)*a*b^5*e^(-
5)*sgn(b*x + a) + 7*(6435*(x*e + d)^(17/2) - 51051*(x*e + d)^(15/2)*d + 176715*(x*e + d)^(13/2)*d^2 - 348075*(
x*e + d)^(11/2)*d^3 + 425425*(x*e + d)^(9/2)*d^4 - 328185*(x*e + d)^(7/2)*d^5 + 153153*(x*e + d)^(5/2)*d^6 - 3
6465*(x*e + d)^(3/2)*d^7)*b^6*e^(-6)*sgn(b*x + a) + 51051*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^6*sgn(b*
x + a))*e^(-1)