3.2114 \(\int \frac{(a+b x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=368 \[ \frac{2 b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^7 (a+b x)}-\frac{4 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)}{3 e^7 (a+b x)}+\frac{30 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^2}{7 e^7 (a+b x)}-\frac{8 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}{e^7 (a+b x)}+\frac{10 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}{e^7 (a+b x)}-\frac{12 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^5}{e^7 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{e^7 (a+b x) \sqrt{d+e x}} \]

[Out]

(-2*(b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*Sqrt[d + e*x]) - (12*b*(b*d - a*e)^5*Sqrt[d +
e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) + (10*b^2*(b*d - a*e)^4*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2])/(e^7*(a + b*x)) - (8*b^3*(b*d - a*e)^3*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b
*x)) + (30*b^4*(b*d - a*e)^2*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)) - (4*b^5*(b*d -
a*e)*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)) + (2*b^6*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a
*b*x + b^2*x^2])/(11*e^7*(a + b*x))

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Rubi [A]  time = 0.138727, antiderivative size = 368, 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} \[ \frac{2 b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2}}{11 e^7 (a+b x)}-\frac{4 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)}{3 e^7 (a+b x)}+\frac{30 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^2}{7 e^7 (a+b x)}-\frac{8 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3}{e^7 (a+b x)}+\frac{10 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}{e^7 (a+b x)}-\frac{12 b \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^5}{e^7 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{e^7 (a+b x) \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b*d - a*e)^6*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)*Sqrt[d + e*x]) - (12*b*(b*d - a*e)^5*Sqrt[d +
e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b*x)) + (10*b^2*(b*d - a*e)^4*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2])/(e^7*(a + b*x)) - (8*b^3*(b*d - a*e)^3*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(e^7*(a + b
*x)) + (30*b^4*(b*d - a*e)^2*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)) - (4*b^5*(b*d -
a*e)*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)) + (2*b^6*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a
*b*x + b^2*x^2])/(11*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)^{3/2}} \, 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)^{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 \frac{(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}{e^6 (d+e x)^{3/2}}-\frac{6 b (b d-a e)^5}{e^6 \sqrt{d+e x}}+\frac{15 b^2 (b d-a e)^4 \sqrt{d+e x}}{e^6}-\frac{20 b^3 (b d-a e)^3 (d+e x)^{3/2}}{e^6}+\frac{15 b^4 (b d-a e)^2 (d+e x)^{5/2}}{e^6}-\frac{6 b^5 (b d-a e) (d+e x)^{7/2}}{e^6}+\frac{b^6 (d+e x)^{9/2}}{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}}{e^7 (a+b x) \sqrt{d+e x}}-\frac{12 b (b d-a e)^5 \sqrt{d+e x} \sqrt{a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}+\frac{10 b^2 (b d-a e)^4 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}-\frac{8 b^3 (b d-a e)^3 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}+\frac{30 b^4 (b d-a e)^2 (d+e x)^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x)}-\frac{4 b^5 (b d-a e) (d+e x)^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}+\frac{2 b^6 (d+e x)^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.155951, size = 163, normalized size = 0.44 \[ \frac{2 \sqrt{(a+b x)^2} \left (1155 b^2 (d+e x)^2 (b d-a e)^4-924 b^3 (d+e x)^3 (b d-a e)^3+495 b^4 (d+e x)^4 (b d-a e)^2-154 b^5 (d+e x)^5 (b d-a e)-1386 b (d+e x) (b d-a e)^5-231 (b d-a e)^6+21 b^6 (d+e x)^6\right )}{231 e^7 (a+b x) \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(a + b*x)^2]*(-231*(b*d - a*e)^6 - 1386*b*(b*d - a*e)^5*(d + e*x) + 1155*b^2*(b*d - a*e)^4*(d + e*x)^2
- 924*b^3*(b*d - a*e)^3*(d + e*x)^3 + 495*b^4*(b*d - a*e)^2*(d + e*x)^4 - 154*b^5*(b*d - a*e)*(d + e*x)^5 + 2
1*b^6*(d + e*x)^6))/(231*e^7*(a + b*x)*Sqrt[d + e*x])

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Maple [A]  time = 0.007, size = 393, normalized size = 1.1 \begin{align*} -{\frac{-42\,{x}^{6}{b}^{6}{e}^{6}-308\,{x}^{5}a{b}^{5}{e}^{6}+56\,{x}^{5}{b}^{6}d{e}^{5}-990\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}+440\,{x}^{4}a{b}^{5}d{e}^{5}-80\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}-1848\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}+1584\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}-704\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}+128\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}-2310\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}+3696\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}-3168\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}+1408\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}-256\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}-2772\,x{a}^{5}b{e}^{6}+9240\,x{a}^{4}{b}^{2}d{e}^{5}-14784\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}+12672\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}-5632\,xa{b}^{5}{d}^{4}{e}^{2}+1024\,x{b}^{6}{d}^{5}e+462\,{a}^{6}{e}^{6}-5544\,d{e}^{5}{a}^{5}b+18480\,{a}^{4}{b}^{2}{d}^{2}{e}^{4}-29568\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+25344\,{a}^{2}{b}^{4}{d}^{4}{e}^{2}-11264\,a{b}^{5}{d}^{5}e+2048\,{b}^{6}{d}^{6}}{231\, \left ( bx+a \right ) ^{5}{e}^{7}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}{\frac{1}{\sqrt{ex+d}}}} \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)^(3/2),x)

[Out]

-2/231/(e*x+d)^(1/2)*(-21*b^6*e^6*x^6-154*a*b^5*e^6*x^5+28*b^6*d*e^5*x^5-495*a^2*b^4*e^6*x^4+220*a*b^5*d*e^5*x
^4-40*b^6*d^2*e^4*x^4-924*a^3*b^3*e^6*x^3+792*a^2*b^4*d*e^5*x^3-352*a*b^5*d^2*e^4*x^3+64*b^6*d^3*e^3*x^3-1155*
a^4*b^2*e^6*x^2+1848*a^3*b^3*d*e^5*x^2-1584*a^2*b^4*d^2*e^4*x^2+704*a*b^5*d^3*e^3*x^2-128*b^6*d^4*e^2*x^2-1386
*a^5*b*e^6*x+4620*a^4*b^2*d*e^5*x-7392*a^3*b^3*d^2*e^4*x+6336*a^2*b^4*d^3*e^3*x-2816*a*b^5*d^4*e^2*x+512*b^6*d
^5*e*x+231*a^6*e^6-2772*a^5*b*d*e^5+9240*a^4*b^2*d^2*e^4-14784*a^3*b^3*d^3*e^3+12672*a^2*b^4*d^4*e^2-5632*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 = 1.1505, size = 814, normalized size = 2.21 \begin{align*} \frac{2 \,{\left (7 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} - 1152 \, a b^{4} d^{4} e + 2016 \, a^{2} b^{3} d^{3} e^{2} - 1680 \, a^{3} b^{2} d^{2} e^{3} + 630 \, a^{4} b d e^{4} - 63 \, a^{5} e^{5} - 5 \,{\left (2 \, b^{5} d e^{4} - 9 \, a b^{4} e^{5}\right )} x^{4} + 2 \,{\left (8 \, b^{5} d^{2} e^{3} - 36 \, a b^{4} d e^{4} + 63 \, a^{2} b^{3} e^{5}\right )} x^{3} - 2 \,{\left (16 \, b^{5} d^{3} e^{2} - 72 \, a b^{4} d^{2} e^{3} + 126 \, a^{2} b^{3} d e^{4} - 105 \, a^{3} b^{2} e^{5}\right )} x^{2} +{\left (128 \, b^{5} d^{4} e - 576 \, a b^{4} d^{3} e^{2} + 1008 \, a^{2} b^{3} d^{2} e^{3} - 840 \, a^{3} b^{2} d e^{4} + 315 \, a^{4} b e^{5}\right )} x\right )} a}{63 \, \sqrt{e x + d} e^{6}} + \frac{2 \,{\left (63 \, b^{5} e^{6} x^{6} - 3072 \, b^{5} d^{6} + 14080 \, a b^{4} d^{5} e - 25344 \, a^{2} b^{3} d^{4} e^{2} + 22176 \, a^{3} b^{2} d^{3} e^{3} - 9240 \, a^{4} b d^{2} e^{4} + 1386 \, a^{5} d e^{5} - 7 \,{\left (12 \, b^{5} d e^{5} - 55 \, a b^{4} e^{6}\right )} x^{5} + 10 \,{\left (12 \, b^{5} d^{2} e^{4} - 55 \, a b^{4} d e^{5} + 99 \, a^{2} b^{3} e^{6}\right )} x^{4} - 2 \,{\left (96 \, b^{5} d^{3} e^{3} - 440 \, a b^{4} d^{2} e^{4} + 792 \, a^{2} b^{3} d e^{5} - 693 \, a^{3} b^{2} e^{6}\right )} x^{3} +{\left (384 \, b^{5} d^{4} e^{2} - 1760 \, a b^{4} d^{3} e^{3} + 3168 \, a^{2} b^{3} d^{2} e^{4} - 2772 \, a^{3} b^{2} d e^{5} + 1155 \, a^{4} b e^{6}\right )} x^{2} -{\left (1536 \, b^{5} d^{5} e - 7040 \, a b^{4} d^{4} e^{2} + 12672 \, a^{2} b^{3} d^{3} e^{3} - 11088 \, a^{3} b^{2} d^{2} e^{4} + 4620 \, a^{4} b d e^{5} - 693 \, a^{5} e^{6}\right )} x\right )} b}{693 \, \sqrt{e x + d} e^{7}} \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)^(3/2),x, algorithm="maxima")

[Out]

2/63*(7*b^5*e^5*x^5 + 256*b^5*d^5 - 1152*a*b^4*d^4*e + 2016*a^2*b^3*d^3*e^2 - 1680*a^3*b^2*d^2*e^3 + 630*a^4*b
*d*e^4 - 63*a^5*e^5 - 5*(2*b^5*d*e^4 - 9*a*b^4*e^5)*x^4 + 2*(8*b^5*d^2*e^3 - 36*a*b^4*d*e^4 + 63*a^2*b^3*e^5)*
x^3 - 2*(16*b^5*d^3*e^2 - 72*a*b^4*d^2*e^3 + 126*a^2*b^3*d*e^4 - 105*a^3*b^2*e^5)*x^2 + (128*b^5*d^4*e - 576*a
*b^4*d^3*e^2 + 1008*a^2*b^3*d^2*e^3 - 840*a^3*b^2*d*e^4 + 315*a^4*b*e^5)*x)*a/(sqrt(e*x + d)*e^6) + 2/693*(63*
b^5*e^6*x^6 - 3072*b^5*d^6 + 14080*a*b^4*d^5*e - 25344*a^2*b^3*d^4*e^2 + 22176*a^3*b^2*d^3*e^3 - 9240*a^4*b*d^
2*e^4 + 1386*a^5*d*e^5 - 7*(12*b^5*d*e^5 - 55*a*b^4*e^6)*x^5 + 10*(12*b^5*d^2*e^4 - 55*a*b^4*d*e^5 + 99*a^2*b^
3*e^6)*x^4 - 2*(96*b^5*d^3*e^3 - 440*a*b^4*d^2*e^4 + 792*a^2*b^3*d*e^5 - 693*a^3*b^2*e^6)*x^3 + (384*b^5*d^4*e
^2 - 1760*a*b^4*d^3*e^3 + 3168*a^2*b^3*d^2*e^4 - 2772*a^3*b^2*d*e^5 + 1155*a^4*b*e^6)*x^2 - (1536*b^5*d^5*e -
7040*a*b^4*d^4*e^2 + 12672*a^2*b^3*d^3*e^3 - 11088*a^3*b^2*d^2*e^4 + 4620*a^4*b*d*e^5 - 693*a^5*e^6)*x)*b/(sqr
t(e*x + d)*e^7)

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Fricas [A]  time = 1.01984, size = 814, normalized size = 2.21 \begin{align*} \frac{2 \,{\left (21 \, b^{6} e^{6} x^{6} - 1024 \, b^{6} d^{6} + 5632 \, a b^{5} d^{5} e - 12672 \, a^{2} b^{4} d^{4} e^{2} + 14784 \, a^{3} b^{3} d^{3} e^{3} - 9240 \, a^{4} b^{2} d^{2} e^{4} + 2772 \, a^{5} b d e^{5} - 231 \, a^{6} e^{6} - 14 \,{\left (2 \, b^{6} d e^{5} - 11 \, a b^{5} e^{6}\right )} x^{5} + 5 \,{\left (8 \, b^{6} d^{2} e^{4} - 44 \, a b^{5} d e^{5} + 99 \, a^{2} b^{4} e^{6}\right )} x^{4} - 4 \,{\left (16 \, b^{6} d^{3} e^{3} - 88 \, a b^{5} d^{2} e^{4} + 198 \, a^{2} b^{4} d e^{5} - 231 \, a^{3} b^{3} e^{6}\right )} x^{3} +{\left (128 \, b^{6} d^{4} e^{2} - 704 \, a b^{5} d^{3} e^{3} + 1584 \, a^{2} b^{4} d^{2} e^{4} - 1848 \, a^{3} b^{3} d e^{5} + 1155 \, a^{4} b^{2} e^{6}\right )} x^{2} - 2 \,{\left (256 \, b^{6} d^{5} e - 1408 \, a b^{5} d^{4} e^{2} + 3168 \, a^{2} b^{4} d^{3} e^{3} - 3696 \, a^{3} b^{3} d^{2} e^{4} + 2310 \, a^{4} b^{2} d e^{5} - 693 \, a^{5} b e^{6}\right )} x\right )} \sqrt{e x + d}}{231 \,{\left (e^{8} x + d 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)^(3/2),x, algorithm="fricas")

[Out]

2/231*(21*b^6*e^6*x^6 - 1024*b^6*d^6 + 5632*a*b^5*d^5*e - 12672*a^2*b^4*d^4*e^2 + 14784*a^3*b^3*d^3*e^3 - 9240
*a^4*b^2*d^2*e^4 + 2772*a^5*b*d*e^5 - 231*a^6*e^6 - 14*(2*b^6*d*e^5 - 11*a*b^5*e^6)*x^5 + 5*(8*b^6*d^2*e^4 - 4
4*a*b^5*d*e^5 + 99*a^2*b^4*e^6)*x^4 - 4*(16*b^6*d^3*e^3 - 88*a*b^5*d^2*e^4 + 198*a^2*b^4*d*e^5 - 231*a^3*b^3*e
^6)*x^3 + (128*b^6*d^4*e^2 - 704*a*b^5*d^3*e^3 + 1584*a^2*b^4*d^2*e^4 - 1848*a^3*b^3*d*e^5 + 1155*a^4*b^2*e^6)
*x^2 - 2*(256*b^6*d^5*e - 1408*a*b^5*d^4*e^2 + 3168*a^2*b^4*d^3*e^3 - 3696*a^3*b^3*d^2*e^4 + 2310*a^4*b^2*d*e^
5 - 693*a^5*b*e^6)*x)*sqrt(e*x + d)/(e^8*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)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**(3/2),x)

[Out]

Timed out

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Giac [B]  time = 1.22776, size = 867, normalized size = 2.36 \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)^(3/2),x, algorithm="giac")

[Out]

2/231*(21*(x*e + d)^(11/2)*b^6*e^70*sgn(b*x + a) - 154*(x*e + d)^(9/2)*b^6*d*e^70*sgn(b*x + a) + 495*(x*e + d)
^(7/2)*b^6*d^2*e^70*sgn(b*x + a) - 924*(x*e + d)^(5/2)*b^6*d^3*e^70*sgn(b*x + a) + 1155*(x*e + d)^(3/2)*b^6*d^
4*e^70*sgn(b*x + a) - 1386*sqrt(x*e + d)*b^6*d^5*e^70*sgn(b*x + a) + 154*(x*e + d)^(9/2)*a*b^5*e^71*sgn(b*x +
a) - 990*(x*e + d)^(7/2)*a*b^5*d*e^71*sgn(b*x + a) + 2772*(x*e + d)^(5/2)*a*b^5*d^2*e^71*sgn(b*x + a) - 4620*(
x*e + d)^(3/2)*a*b^5*d^3*e^71*sgn(b*x + a) + 6930*sqrt(x*e + d)*a*b^5*d^4*e^71*sgn(b*x + a) + 495*(x*e + d)^(7
/2)*a^2*b^4*e^72*sgn(b*x + a) - 2772*(x*e + d)^(5/2)*a^2*b^4*d*e^72*sgn(b*x + a) + 6930*(x*e + d)^(3/2)*a^2*b^
4*d^2*e^72*sgn(b*x + a) - 13860*sqrt(x*e + d)*a^2*b^4*d^3*e^72*sgn(b*x + a) + 924*(x*e + d)^(5/2)*a^3*b^3*e^73
*sgn(b*x + a) - 4620*(x*e + d)^(3/2)*a^3*b^3*d*e^73*sgn(b*x + a) + 13860*sqrt(x*e + d)*a^3*b^3*d^2*e^73*sgn(b*
x + a) + 1155*(x*e + d)^(3/2)*a^4*b^2*e^74*sgn(b*x + a) - 6930*sqrt(x*e + d)*a^4*b^2*d*e^74*sgn(b*x + a) + 138
6*sqrt(x*e + d)*a^5*b*e^75*sgn(b*x + a))*e^(-77) - 2*(b^6*d^6*sgn(b*x + a) - 6*a*b^5*d^5*e*sgn(b*x + a) + 15*a
^2*b^4*d^4*e^2*sgn(b*x + a) - 20*a^3*b^3*d^3*e^3*sgn(b*x + a) + 15*a^4*b^2*d^2*e^4*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)/sqrt(x*e + d)