3.2115 \(\int \frac{(a+b x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=370 \[ \frac{2 b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2}}{9 e^7 (a+b x)}-\frac{12 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)}{7 e^7 (a+b x)}+\frac{6 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}{e^7 (a+b x)}-\frac{40 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}{3 e^7 (a+b x)}+\frac{30 b^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^4}{e^7 (a+b x)}+\frac{12 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^7 (a+b x) \sqrt{d+e x}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{3 e^7 (a+b x) (d+e x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.144315, antiderivative size = 370, 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} \[ \frac{2 b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2}}{9 e^7 (a+b x)}-\frac{12 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)}{7 e^7 (a+b x)}+\frac{6 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}{e^7 (a+b x)}-\frac{40 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}{3 e^7 (a+b x)}+\frac{30 b^2 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^4}{e^7 (a+b x)}+\frac{12 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{e^7 (a+b x) \sqrt{d+e x}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{3 e^7 (a+b x) (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.159504, size = 163, normalized size = 0.44 \[ \frac{2 \sqrt{(a+b x)^2} \left (945 b^2 (d+e x)^2 (b d-a e)^4-420 b^3 (d+e x)^3 (b d-a e)^3+189 b^4 (d+e x)^4 (b d-a e)^2-54 b^5 (d+e x)^5 (b d-a e)+378 b (d+e x) (b d-a e)^5-21 (b d-a e)^6+7 b^6 (d+e x)^6\right )}{63 e^7 (a+b x) (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(a + b*x)^2]*(-21*(b*d - a*e)^6 + 378*b*(b*d - a*e)^5*(d + e*x) + 945*b^2*(b*d - a*e)^4*(d + e*x)^2 -
420*b^3*(b*d - a*e)^3*(d + e*x)^3 + 189*b^4*(b*d - a*e)^2*(d + e*x)^4 - 54*b^5*(b*d - a*e)*(d + e*x)^5 + 7*b^6
*(d + e*x)^6))/(63*e^7*(a + b*x)*(d + e*x)^(3/2))

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Maple [A]  time = 0.007, size = 393, normalized size = 1.1 \begin{align*} -{\frac{-14\,{x}^{6}{b}^{6}{e}^{6}-108\,{x}^{5}a{b}^{5}{e}^{6}+24\,{x}^{5}{b}^{6}d{e}^{5}-378\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}+216\,{x}^{4}a{b}^{5}d{e}^{5}-48\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}-840\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}+1008\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}-576\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}+128\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}-1890\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}+5040\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}-6048\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}+3456\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}-768\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+756\,x{a}^{5}b{e}^{6}-7560\,x{a}^{4}{b}^{2}d{e}^{5}+20160\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}-24192\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+13824\,xa{b}^{5}{d}^{4}{e}^{2}-3072\,x{b}^{6}{d}^{5}e+42\,{a}^{6}{e}^{6}+504\,d{e}^{5}{a}^{5}b-5040\,{a}^{4}{b}^{2}{d}^{2}{e}^{4}+13440\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}-16128\,{a}^{2}{b}^{4}{d}^{4}{e}^{2}+9216\,a{b}^{5}{d}^{5}e-2048\,{b}^{6}{d}^{6}}{63\, \left ( bx+a \right ) ^{5}{e}^{7}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/63/(e*x+d)^(3/2)*(-7*b^6*e^6*x^6-54*a*b^5*e^6*x^5+12*b^6*d*e^5*x^5-189*a^2*b^4*e^6*x^4+108*a*b^5*d*e^5*x^4-
24*b^6*d^2*e^4*x^4-420*a^3*b^3*e^6*x^3+504*a^2*b^4*d*e^5*x^3-288*a*b^5*d^2*e^4*x^3+64*b^6*d^3*e^3*x^3-945*a^4*
b^2*e^6*x^2+2520*a^3*b^3*d*e^5*x^2-3024*a^2*b^4*d^2*e^4*x^2+1728*a*b^5*d^3*e^3*x^2-384*b^6*d^4*e^2*x^2+378*a^5
*b*e^6*x-3780*a^4*b^2*d*e^5*x+10080*a^3*b^3*d^2*e^4*x-12096*a^2*b^4*d^3*e^3*x+6912*a*b^5*d^4*e^2*x-1536*b^6*d^
5*e*x+21*a^6*e^6+252*a^5*b*d*e^5-2520*a^4*b^2*d^2*e^4+6720*a^3*b^3*d^3*e^3-8064*a^2*b^4*d^4*e^2+4608*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 = 1.30323, size = 844, normalized size = 2.28 \begin{align*} \frac{2 \,{\left (3 \, b^{5} e^{5} x^{5} - 256 \, b^{5} d^{5} + 896 \, a b^{4} d^{4} e - 1120 \, a^{2} b^{3} d^{3} e^{2} + 560 \, a^{3} b^{2} d^{2} e^{3} - 70 \, a^{4} b d e^{4} - 7 \, a^{5} e^{5} - 3 \,{\left (2 \, b^{5} d e^{4} - 7 \, a b^{4} e^{5}\right )} x^{4} + 2 \,{\left (8 \, b^{5} d^{2} e^{3} - 28 \, a b^{4} d e^{4} + 35 \, a^{2} b^{3} e^{5}\right )} x^{3} - 6 \,{\left (16 \, b^{5} d^{3} e^{2} - 56 \, a b^{4} d^{2} e^{3} + 70 \, a^{2} b^{3} d e^{4} - 35 \, a^{3} b^{2} e^{5}\right )} x^{2} - 3 \,{\left (128 \, b^{5} d^{4} e - 448 \, a b^{4} d^{3} e^{2} + 560 \, a^{2} b^{3} d^{2} e^{3} - 280 \, a^{3} b^{2} d e^{4} + 35 \, a^{4} b e^{5}\right )} x\right )} a}{21 \,{\left (e^{7} x + d e^{6}\right )} \sqrt{e x + d}} + \frac{2 \,{\left (7 \, b^{5} e^{6} x^{6} + 1024 \, b^{5} d^{6} - 3840 \, a b^{4} d^{5} e + 5376 \, a^{2} b^{3} d^{4} e^{2} - 3360 \, a^{3} b^{2} d^{3} e^{3} + 840 \, a^{4} b d^{2} e^{4} - 42 \, a^{5} d e^{5} - 3 \,{\left (4 \, b^{5} d e^{5} - 15 \, a b^{4} e^{6}\right )} x^{5} + 6 \,{\left (4 \, b^{5} d^{2} e^{4} - 15 \, a b^{4} d e^{5} + 21 \, a^{2} b^{3} e^{6}\right )} x^{4} - 2 \,{\left (32 \, b^{5} d^{3} e^{3} - 120 \, a b^{4} d^{2} e^{4} + 168 \, a^{2} b^{3} d e^{5} - 105 \, a^{3} b^{2} e^{6}\right )} x^{3} + 3 \,{\left (128 \, b^{5} d^{4} e^{2} - 480 \, a b^{4} d^{3} e^{3} + 672 \, a^{2} b^{3} d^{2} e^{4} - 420 \, a^{3} b^{2} d e^{5} + 105 \, a^{4} b e^{6}\right )} x^{2} + 3 \,{\left (512 \, b^{5} d^{5} e - 1920 \, a b^{4} d^{4} e^{2} + 2688 \, a^{2} b^{3} d^{3} e^{3} - 1680 \, a^{3} b^{2} d^{2} e^{4} + 420 \, a^{4} b d e^{5} - 21 \, a^{5} e^{6}\right )} x\right )} b}{63 \,{\left (e^{8} x + d e^{7}\right )} \sqrt{e x + d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*(3*b^5*e^5*x^5 - 256*b^5*d^5 + 896*a*b^4*d^4*e - 1120*a^2*b^3*d^3*e^2 + 560*a^3*b^2*d^2*e^3 - 70*a^4*b*d*
e^4 - 7*a^5*e^5 - 3*(2*b^5*d*e^4 - 7*a*b^4*e^5)*x^4 + 2*(8*b^5*d^2*e^3 - 28*a*b^4*d*e^4 + 35*a^2*b^3*e^5)*x^3
- 6*(16*b^5*d^3*e^2 - 56*a*b^4*d^2*e^3 + 70*a^2*b^3*d*e^4 - 35*a^3*b^2*e^5)*x^2 - 3*(128*b^5*d^4*e - 448*a*b^4
*d^3*e^2 + 560*a^2*b^3*d^2*e^3 - 280*a^3*b^2*d*e^4 + 35*a^4*b*e^5)*x)*a/((e^7*x + d*e^6)*sqrt(e*x + d)) + 2/63
*(7*b^5*e^6*x^6 + 1024*b^5*d^6 - 3840*a*b^4*d^5*e + 5376*a^2*b^3*d^4*e^2 - 3360*a^3*b^2*d^3*e^3 + 840*a^4*b*d^
2*e^4 - 42*a^5*d*e^5 - 3*(4*b^5*d*e^5 - 15*a*b^4*e^6)*x^5 + 6*(4*b^5*d^2*e^4 - 15*a*b^4*d*e^5 + 21*a^2*b^3*e^6
)*x^4 - 2*(32*b^5*d^3*e^3 - 120*a*b^4*d^2*e^4 + 168*a^2*b^3*d*e^5 - 105*a^3*b^2*e^6)*x^3 + 3*(128*b^5*d^4*e^2
- 480*a*b^4*d^3*e^3 + 672*a^2*b^3*d^2*e^4 - 420*a^3*b^2*d*e^5 + 105*a^4*b*e^6)*x^2 + 3*(512*b^5*d^5*e - 1920*a
*b^4*d^4*e^2 + 2688*a^2*b^3*d^3*e^3 - 1680*a^3*b^2*d^2*e^4 + 420*a^4*b*d*e^5 - 21*a^5*e^6)*x)*b/((e^8*x + d*e^
7)*sqrt(e*x + d))

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Fricas [A]  time = 1.01634, size = 822, normalized size = 2.22 \begin{align*} \frac{2 \,{\left (7 \, b^{6} e^{6} x^{6} + 1024 \, b^{6} d^{6} - 4608 \, a b^{5} d^{5} e + 8064 \, a^{2} b^{4} d^{4} e^{2} - 6720 \, a^{3} b^{3} d^{3} e^{3} + 2520 \, a^{4} b^{2} d^{2} e^{4} - 252 \, a^{5} b d e^{5} - 21 \, a^{6} e^{6} - 6 \,{\left (2 \, b^{6} d e^{5} - 9 \, a b^{5} e^{6}\right )} x^{5} + 3 \,{\left (8 \, b^{6} d^{2} e^{4} - 36 \, a b^{5} d e^{5} + 63 \, a^{2} b^{4} e^{6}\right )} x^{4} - 4 \,{\left (16 \, b^{6} d^{3} e^{3} - 72 \, a b^{5} d^{2} e^{4} + 126 \, a^{2} b^{4} d e^{5} - 105 \, a^{3} b^{3} e^{6}\right )} x^{3} + 3 \,{\left (128 \, b^{6} d^{4} e^{2} - 576 \, a b^{5} d^{3} e^{3} + 1008 \, a^{2} b^{4} d^{2} e^{4} - 840 \, a^{3} b^{3} d e^{5} + 315 \, a^{4} b^{2} e^{6}\right )} x^{2} + 6 \,{\left (256 \, b^{6} d^{5} e - 1152 \, a b^{5} d^{4} e^{2} + 2016 \, a^{2} b^{4} d^{3} e^{3} - 1680 \, a^{3} b^{3} d^{2} e^{4} + 630 \, a^{4} b^{2} d e^{5} - 63 \, a^{5} b e^{6}\right )} x\right )} \sqrt{e x + d}}{63 \,{\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/63*(7*b^6*e^6*x^6 + 1024*b^6*d^6 - 4608*a*b^5*d^5*e + 8064*a^2*b^4*d^4*e^2 - 6720*a^3*b^3*d^3*e^3 + 2520*a^4
*b^2*d^2*e^4 - 252*a^5*b*d*e^5 - 21*a^6*e^6 - 6*(2*b^6*d*e^5 - 9*a*b^5*e^6)*x^5 + 3*(8*b^6*d^2*e^4 - 36*a*b^5*
d*e^5 + 63*a^2*b^4*e^6)*x^4 - 4*(16*b^6*d^3*e^3 - 72*a*b^5*d^2*e^4 + 126*a^2*b^4*d*e^5 - 105*a^3*b^3*e^6)*x^3
+ 3*(128*b^6*d^4*e^2 - 576*a*b^5*d^3*e^3 + 1008*a^2*b^4*d^2*e^4 - 840*a^3*b^3*d*e^5 + 315*a^4*b^2*e^6)*x^2 + 6
*(256*b^6*d^5*e - 1152*a*b^5*d^4*e^2 + 2016*a^2*b^4*d^3*e^3 - 1680*a^3*b^3*d^2*e^4 + 630*a^4*b^2*d*e^5 - 63*a^
5*b*e^6)*x)*sqrt(e*x + d)/(e^9*x^2 + 2*d*e^8*x + d^2*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)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 1.22505, size = 851, normalized size = 2.3 \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)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^(5/2),x, algorithm="giac")

[Out]

2/63*(7*(x*e + d)^(9/2)*b^6*e^56*sgn(b*x + a) - 54*(x*e + d)^(7/2)*b^6*d*e^56*sgn(b*x + a) + 189*(x*e + d)^(5/
2)*b^6*d^2*e^56*sgn(b*x + a) - 420*(x*e + d)^(3/2)*b^6*d^3*e^56*sgn(b*x + a) + 945*sqrt(x*e + d)*b^6*d^4*e^56*
sgn(b*x + a) + 54*(x*e + d)^(7/2)*a*b^5*e^57*sgn(b*x + a) - 378*(x*e + d)^(5/2)*a*b^5*d*e^57*sgn(b*x + a) + 12
60*(x*e + d)^(3/2)*a*b^5*d^2*e^57*sgn(b*x + a) - 3780*sqrt(x*e + d)*a*b^5*d^3*e^57*sgn(b*x + a) + 189*(x*e + d
)^(5/2)*a^2*b^4*e^58*sgn(b*x + a) - 1260*(x*e + d)^(3/2)*a^2*b^4*d*e^58*sgn(b*x + a) + 5670*sqrt(x*e + d)*a^2*
b^4*d^2*e^58*sgn(b*x + a) + 420*(x*e + d)^(3/2)*a^3*b^3*e^59*sgn(b*x + a) - 3780*sqrt(x*e + d)*a^3*b^3*d*e^59*
sgn(b*x + a) + 945*sqrt(x*e + d)*a^4*b^2*e^60*sgn(b*x + a))*e^(-63) + 2/3*(18*(x*e + d)*b^6*d^5*sgn(b*x + a) -
b^6*d^6*sgn(b*x + a) - 90*(x*e + d)*a*b^5*d^4*e*sgn(b*x + a) + 6*a*b^5*d^5*e*sgn(b*x + a) + 180*(x*e + d)*a^2
*b^4*d^3*e^2*sgn(b*x + a) - 15*a^2*b^4*d^4*e^2*sgn(b*x + a) - 180*(x*e + d)*a^3*b^3*d^2*e^3*sgn(b*x + a) + 20*
a^3*b^3*d^3*e^3*sgn(b*x + a) + 90*(x*e + d)*a^4*b^2*d*e^4*sgn(b*x + a) - 15*a^4*b^2*d^2*e^4*sgn(b*x + a) - 18*
(x*e + d)*a^5*b*e^5*sgn(b*x + a) + 6*a^5*b*d*e^5*sgn(b*x + a) - a^6*e^6*sgn(b*x + a))*e^(-7)/(x*e + d)^(3/2)