3.1692 \(\int (d+e x)^{5/2} (a^2+2 a b x+b^2 x^2)^{5/2} \, dx\)

Optimal. Leaf size=320 \[ \frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{17/2}}{17 e^6 (a+b x)}-\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15/2} (b d-a e)}{3 e^6 (a+b x)}+\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)^2}{13 e^6 (a+b x)}-\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^3}{11 e^6 (a+b x)}+\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^4}{9 e^6 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^5}{7 e^6 (a+b x)} \]

[Out]

(-2*(b*d - a*e)^5*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^6*(a + b*x)) + (10*b*(b*d - a*e)^4*(d +
e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^6*(a + b*x)) - (20*b^2*(b*d - a*e)^3*(d + e*x)^(11/2)*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(11*e^6*(a + b*x)) + (20*b^3*(b*d - a*e)^2*(d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2
])/(13*e^6*(a + b*x)) - (2*b^4*(b*d - a*e)*(d + e*x)^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^6*(a + b*x)) +
(2*b^5*(d + e*x)^(17/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(17*e^6*(a + b*x))

________________________________________________________________________________________

Rubi [A]  time = 0.116144, antiderivative size = 320, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {646, 43} \[ \frac{2 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{17/2}}{17 e^6 (a+b x)}-\frac{2 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15/2} (b d-a e)}{3 e^6 (a+b x)}+\frac{20 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)^2}{13 e^6 (a+b x)}-\frac{20 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^3}{11 e^6 (a+b x)}+\frac{10 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^4}{9 e^6 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^5}{7 e^6 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(b*d - a*e)^5*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^6*(a + b*x)) + (10*b*(b*d - a*e)^4*(d +
e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^6*(a + b*x)) - (20*b^2*(b*d - a*e)^3*(d + e*x)^(11/2)*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(11*e^6*(a + b*x)) + (20*b^3*(b*d - a*e)^2*(d + e*x)^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2
])/(13*e^6*(a + b*x)) - (2*b^4*(b*d - a*e)*(d + e*x)^(15/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^6*(a + b*x)) +
(2*b^5*(d + e*x)^(17/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(17*e^6*(a + b*x))

Rule 646

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

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 (d+e x)^{5/2} \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 \left (a b+b^2 x\right )^5 (d+e x)^{5/2} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (-\frac{b^5 (b d-a e)^5 (d+e x)^{5/2}}{e^5}+\frac{5 b^6 (b d-a e)^4 (d+e x)^{7/2}}{e^5}-\frac{10 b^7 (b d-a e)^3 (d+e x)^{9/2}}{e^5}+\frac{10 b^8 (b d-a e)^2 (d+e x)^{11/2}}{e^5}-\frac{5 b^9 (b d-a e) (d+e x)^{13/2}}{e^5}+\frac{b^{10} (d+e x)^{15/2}}{e^5}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac{2 (b d-a e)^5 (d+e x)^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^6 (a+b x)}+\frac{10 b (b d-a e)^4 (d+e x)^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}{9 e^6 (a+b x)}-\frac{20 b^2 (b d-a e)^3 (d+e x)^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}{11 e^6 (a+b x)}+\frac{20 b^3 (b d-a e)^2 (d+e x)^{13/2} \sqrt{a^2+2 a b x+b^2 x^2}}{13 e^6 (a+b x)}-\frac{2 b^4 (b d-a e) (d+e x)^{15/2} \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x)}+\frac{2 b^5 (d+e x)^{17/2} \sqrt{a^2+2 a b x+b^2 x^2}}{17 e^6 (a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.182802, size = 141, normalized size = 0.44 \[ \frac{2 \left ((a+b x)^2\right )^{5/2} (d+e x)^{7/2} \left (-139230 b^2 (d+e x)^2 (b d-a e)^3+117810 b^3 (d+e x)^3 (b d-a e)^2-51051 b^4 (d+e x)^4 (b d-a e)+85085 b (d+e x) (b d-a e)^4-21879 (b d-a e)^5+9009 b^5 (d+e x)^5\right )}{153153 e^6 (a+b x)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((a + b*x)^2)^(5/2)*(d + e*x)^(7/2)*(-21879*(b*d - a*e)^5 + 85085*b*(b*d - a*e)^4*(d + e*x) - 139230*b^2*(b
*d - a*e)^3*(d + e*x)^2 + 117810*b^3*(b*d - a*e)^2*(d + e*x)^3 - 51051*b^4*(b*d - a*e)*(d + e*x)^4 + 9009*b^5*
(d + e*x)^5))/(153153*e^6*(a + b*x)^5)

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Maple [A]  time = 0.155, size = 289, normalized size = 0.9 \begin{align*}{\frac{18018\,{x}^{5}{b}^{5}{e}^{5}+102102\,{x}^{4}a{b}^{4}{e}^{5}-12012\,{x}^{4}{b}^{5}d{e}^{4}+235620\,{x}^{3}{a}^{2}{b}^{3}{e}^{5}-62832\,{x}^{3}a{b}^{4}d{e}^{4}+7392\,{x}^{3}{b}^{5}{d}^{2}{e}^{3}+278460\,{x}^{2}{a}^{3}{b}^{2}{e}^{5}-128520\,{x}^{2}{a}^{2}{b}^{3}d{e}^{4}+34272\,{x}^{2}a{b}^{4}{d}^{2}{e}^{3}-4032\,{x}^{2}{b}^{5}{d}^{3}{e}^{2}+170170\,x{a}^{4}b{e}^{5}-123760\,x{a}^{3}{b}^{2}d{e}^{4}+57120\,x{a}^{2}{b}^{3}{d}^{2}{e}^{3}-15232\,xa{b}^{4}{d}^{3}{e}^{2}+1792\,x{b}^{5}{d}^{4}e+43758\,{a}^{5}{e}^{5}-48620\,d{e}^{4}{a}^{4}b+35360\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}-16320\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+4352\,a{b}^{4}{d}^{4}e-512\,{b}^{5}{d}^{5}}{153153\,{e}^{6} \left ( bx+a \right ) ^{5}} \left ( ex+d \right ) ^{{\frac{7}{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((e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

2/153153*(e*x+d)^(7/2)*(9009*b^5*e^5*x^5+51051*a*b^4*e^5*x^4-6006*b^5*d*e^4*x^4+117810*a^2*b^3*e^5*x^3-31416*a
*b^4*d*e^4*x^3+3696*b^5*d^2*e^3*x^3+139230*a^3*b^2*e^5*x^2-64260*a^2*b^3*d*e^4*x^2+17136*a*b^4*d^2*e^3*x^2-201
6*b^5*d^3*e^2*x^2+85085*a^4*b*e^5*x-61880*a^3*b^2*d*e^4*x+28560*a^2*b^3*d^2*e^3*x-7616*a*b^4*d^3*e^2*x+896*b^5
*d^4*e*x+21879*a^5*e^5-24310*a^4*b*d*e^4+17680*a^3*b^2*d^2*e^3-8160*a^2*b^3*d^3*e^2+2176*a*b^4*d^4*e-256*b^5*d
^5)*((b*x+a)^2)^(5/2)/e^6/(b*x+a)^5

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Maxima [B]  time = 1.1395, size = 671, normalized size = 2.1 \begin{align*} \frac{2 \,{\left (9009 \, b^{5} e^{8} x^{8} - 256 \, b^{5} d^{8} + 2176 \, a b^{4} d^{7} e - 8160 \, a^{2} b^{3} d^{6} e^{2} + 17680 \, a^{3} b^{2} d^{5} e^{3} - 24310 \, a^{4} b d^{4} e^{4} + 21879 \, a^{5} d^{3} e^{5} + 3003 \,{\left (7 \, b^{5} d e^{7} + 17 \, a b^{4} e^{8}\right )} x^{7} + 231 \,{\left (55 \, b^{5} d^{2} e^{6} + 527 \, a b^{4} d e^{7} + 510 \, a^{2} b^{3} e^{8}\right )} x^{6} + 63 \,{\left (b^{5} d^{3} e^{5} + 1207 \, a b^{4} d^{2} e^{6} + 4590 \, a^{2} b^{3} d e^{7} + 2210 \, a^{3} b^{2} e^{8}\right )} x^{5} - 35 \,{\left (2 \, b^{5} d^{4} e^{4} - 17 \, a b^{4} d^{3} e^{5} - 5406 \, a^{2} b^{3} d^{2} e^{6} - 10166 \, a^{3} b^{2} d e^{7} - 2431 \, a^{4} b e^{8}\right )} x^{4} +{\left (80 \, b^{5} d^{5} e^{3} - 680 \, a b^{4} d^{4} e^{4} + 2550 \, a^{2} b^{3} d^{3} e^{5} + 249730 \, a^{3} b^{2} d^{2} e^{6} + 230945 \, a^{4} b d e^{7} + 21879 \, a^{5} e^{8}\right )} x^{3} - 3 \,{\left (32 \, b^{5} d^{6} e^{2} - 272 \, a b^{4} d^{5} e^{3} + 1020 \, a^{2} b^{3} d^{4} e^{4} - 2210 \, a^{3} b^{2} d^{3} e^{5} - 60775 \, a^{4} b d^{2} e^{6} - 21879 \, a^{5} d e^{7}\right )} x^{2} +{\left (128 \, b^{5} d^{7} e - 1088 \, a b^{4} d^{6} e^{2} + 4080 \, a^{2} b^{3} d^{5} e^{3} - 8840 \, a^{3} b^{2} d^{4} e^{4} + 12155 \, a^{4} b d^{3} e^{5} + 65637 \, a^{5} d^{2} e^{6}\right )} x\right )} \sqrt{e x + d}}{153153 \, e^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/153153*(9009*b^5*e^8*x^8 - 256*b^5*d^8 + 2176*a*b^4*d^7*e - 8160*a^2*b^3*d^6*e^2 + 17680*a^3*b^2*d^5*e^3 - 2
4310*a^4*b*d^4*e^4 + 21879*a^5*d^3*e^5 + 3003*(7*b^5*d*e^7 + 17*a*b^4*e^8)*x^7 + 231*(55*b^5*d^2*e^6 + 527*a*b
^4*d*e^7 + 510*a^2*b^3*e^8)*x^6 + 63*(b^5*d^3*e^5 + 1207*a*b^4*d^2*e^6 + 4590*a^2*b^3*d*e^7 + 2210*a^3*b^2*e^8
)*x^5 - 35*(2*b^5*d^4*e^4 - 17*a*b^4*d^3*e^5 - 5406*a^2*b^3*d^2*e^6 - 10166*a^3*b^2*d*e^7 - 2431*a^4*b*e^8)*x^
4 + (80*b^5*d^5*e^3 - 680*a*b^4*d^4*e^4 + 2550*a^2*b^3*d^3*e^5 + 249730*a^3*b^2*d^2*e^6 + 230945*a^4*b*d*e^7 +
21879*a^5*e^8)*x^3 - 3*(32*b^5*d^6*e^2 - 272*a*b^4*d^5*e^3 + 1020*a^2*b^3*d^4*e^4 - 2210*a^3*b^2*d^3*e^5 - 60
775*a^4*b*d^2*e^6 - 21879*a^5*d*e^7)*x^2 + (128*b^5*d^7*e - 1088*a*b^4*d^6*e^2 + 4080*a^2*b^3*d^5*e^3 - 8840*a
^3*b^2*d^4*e^4 + 12155*a^4*b*d^3*e^5 + 65637*a^5*d^2*e^6)*x)*sqrt(e*x + d)/e^6

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Fricas [B]  time = 1.63599, size = 1148, normalized size = 3.59 \begin{align*} \frac{2 \,{\left (9009 \, b^{5} e^{8} x^{8} - 256 \, b^{5} d^{8} + 2176 \, a b^{4} d^{7} e - 8160 \, a^{2} b^{3} d^{6} e^{2} + 17680 \, a^{3} b^{2} d^{5} e^{3} - 24310 \, a^{4} b d^{4} e^{4} + 21879 \, a^{5} d^{3} e^{5} + 3003 \,{\left (7 \, b^{5} d e^{7} + 17 \, a b^{4} e^{8}\right )} x^{7} + 231 \,{\left (55 \, b^{5} d^{2} e^{6} + 527 \, a b^{4} d e^{7} + 510 \, a^{2} b^{3} e^{8}\right )} x^{6} + 63 \,{\left (b^{5} d^{3} e^{5} + 1207 \, a b^{4} d^{2} e^{6} + 4590 \, a^{2} b^{3} d e^{7} + 2210 \, a^{3} b^{2} e^{8}\right )} x^{5} - 35 \,{\left (2 \, b^{5} d^{4} e^{4} - 17 \, a b^{4} d^{3} e^{5} - 5406 \, a^{2} b^{3} d^{2} e^{6} - 10166 \, a^{3} b^{2} d e^{7} - 2431 \, a^{4} b e^{8}\right )} x^{4} +{\left (80 \, b^{5} d^{5} e^{3} - 680 \, a b^{4} d^{4} e^{4} + 2550 \, a^{2} b^{3} d^{3} e^{5} + 249730 \, a^{3} b^{2} d^{2} e^{6} + 230945 \, a^{4} b d e^{7} + 21879 \, a^{5} e^{8}\right )} x^{3} - 3 \,{\left (32 \, b^{5} d^{6} e^{2} - 272 \, a b^{4} d^{5} e^{3} + 1020 \, a^{2} b^{3} d^{4} e^{4} - 2210 \, a^{3} b^{2} d^{3} e^{5} - 60775 \, a^{4} b d^{2} e^{6} - 21879 \, a^{5} d e^{7}\right )} x^{2} +{\left (128 \, b^{5} d^{7} e - 1088 \, a b^{4} d^{6} e^{2} + 4080 \, a^{2} b^{3} d^{5} e^{3} - 8840 \, a^{3} b^{2} d^{4} e^{4} + 12155 \, a^{4} b d^{3} e^{5} + 65637 \, a^{5} d^{2} e^{6}\right )} x\right )} \sqrt{e x + d}}{153153 \, e^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/153153*(9009*b^5*e^8*x^8 - 256*b^5*d^8 + 2176*a*b^4*d^7*e - 8160*a^2*b^3*d^6*e^2 + 17680*a^3*b^2*d^5*e^3 - 2
4310*a^4*b*d^4*e^4 + 21879*a^5*d^3*e^5 + 3003*(7*b^5*d*e^7 + 17*a*b^4*e^8)*x^7 + 231*(55*b^5*d^2*e^6 + 527*a*b
^4*d*e^7 + 510*a^2*b^3*e^8)*x^6 + 63*(b^5*d^3*e^5 + 1207*a*b^4*d^2*e^6 + 4590*a^2*b^3*d*e^7 + 2210*a^3*b^2*e^8
)*x^5 - 35*(2*b^5*d^4*e^4 - 17*a*b^4*d^3*e^5 - 5406*a^2*b^3*d^2*e^6 - 10166*a^3*b^2*d*e^7 - 2431*a^4*b*e^8)*x^
4 + (80*b^5*d^5*e^3 - 680*a*b^4*d^4*e^4 + 2550*a^2*b^3*d^3*e^5 + 249730*a^3*b^2*d^2*e^6 + 230945*a^4*b*d*e^7 +
21879*a^5*e^8)*x^3 - 3*(32*b^5*d^6*e^2 - 272*a*b^4*d^5*e^3 + 1020*a^2*b^3*d^4*e^4 - 2210*a^3*b^2*d^3*e^5 - 60
775*a^4*b*d^2*e^6 - 21879*a^5*d*e^7)*x^2 + (128*b^5*d^7*e - 1088*a*b^4*d^6*e^2 + 4080*a^2*b^3*d^5*e^3 - 8840*a
^3*b^2*d^4*e^4 + 12155*a^4*b*d^3*e^5 + 65637*a^5*d^2*e^6)*x)*sqrt(e*x + d)/e^6

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

[Out]

Timed out

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Giac [B]  time = 1.28014, size = 1702, normalized size = 5.32 \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)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")

[Out]

2/765765*(255255*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^4*b*d^2*e^(-1)*sgn(b*x + a) + 72930*(15*(x*e + d)
^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^3*b^2*d^2*e^(-2)*sgn(b*x + a) + 24310*(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^2*b^3*d^2*e^(-3)*sgn(b*x
+ a) + 1105*(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*b^4*d^2*e^(-4)*sgn(b*x + a) + 85*(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)*b^5*d^2*e^(-5)*sgn(b*x + a) + 255255*(x*e + d)^(3/2)*a^5*d^2*sgn(b*x + a) + 72930*(15*(x*e + d)^(7/2) - 4
2*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^4*b*d*e^(-1)*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^2*d*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^2*b^3*d*e^(-3)*sgn(b*x + a) + 850*(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^4*d*e^
(-4)*sgn(b*x + a) + 34*(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^5*
d*e^(-5)*sgn(b*x + a) + 102102*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^5*d*sgn(b*x + a) + 12155*(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*e^(-1)*sgn(b*x +
a) + 2210*(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^2*e^(-2)*sgn(b*x + a) + 850*(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^3*e^(-3)*sgn(b*x + a) + 85*(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^4*e^(-4)*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 - 36465*(x*e + d)^(3/2)*d^7)*b^5*e^(-5)*sgn(b*x + a) + 7293*(15*(x*e + d)^(7/2) - 42*(x*e + d)^
(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^5*sgn(b*x + a))*e^(-1)