3.1639 \(\int (d+e x)^{3/2} (a^2+2 a b x+b^2 x^2)^3 \, dx\)

Optimal. Leaf size=187 \[ -\frac{4 b^5 (d+e x)^{15/2} (b d-a e)}{5 e^7}+\frac{30 b^4 (d+e x)^{13/2} (b d-a e)^2}{13 e^7}-\frac{40 b^3 (d+e x)^{11/2} (b d-a e)^3}{11 e^7}+\frac{10 b^2 (d+e x)^{9/2} (b d-a e)^4}{3 e^7}-\frac{12 b (d+e x)^{7/2} (b d-a e)^5}{7 e^7}+\frac{2 (d+e x)^{5/2} (b d-a e)^6}{5 e^7}+\frac{2 b^6 (d+e x)^{17/2}}{17 e^7} \]

[Out]

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

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Rubi [A]  time = 0.0609793, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {27, 43} \[ -\frac{4 b^5 (d+e x)^{15/2} (b d-a e)}{5 e^7}+\frac{30 b^4 (d+e x)^{13/2} (b d-a e)^2}{13 e^7}-\frac{40 b^3 (d+e x)^{11/2} (b d-a e)^3}{11 e^7}+\frac{10 b^2 (d+e x)^{9/2} (b d-a e)^4}{3 e^7}-\frac{12 b (d+e x)^{7/2} (b d-a e)^5}{7 e^7}+\frac{2 (d+e x)^{5/2} (b d-a e)^6}{5 e^7}+\frac{2 b^6 (d+e x)^{17/2}}{17 e^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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

Mathematica [A]  time = 0.104006, size = 145, normalized size = 0.78 \[ \frac{2 (d+e x)^{5/2} \left (425425 b^2 (d+e x)^2 (b d-a e)^4-464100 b^3 (d+e x)^3 (b d-a e)^3+294525 b^4 (d+e x)^4 (b d-a e)^2-102102 b^5 (d+e x)^5 (b d-a e)-218790 b (d+e x) (b d-a e)^5+51051 (b d-a e)^6+15015 b^6 (d+e x)^6\right )}{255255 e^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(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)

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Maple [B]  time = 0.047, size = 377, normalized size = 2. \begin{align*}{\frac{30030\,{b}^{6}{x}^{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\,{a}^{5}bd{e}^{5}+194480\,{d}^{2}{e}^{4}{a}^{4}{b}^{2}-141440\,{b}^{3}{a}^{3}{d}^{3}{e}^{3}+65280\,{a}^{2}{b}^{4}{d}^{4}{e}^{2}-17408\,a{b}^{5}{d}^{5}e+2048\,{d}^{6}{b}^{6}}{255255\,{e}^{7}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^3,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)/e^7

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Maxima [B]  time = 1.07083, size = 473, normalized size = 2.53 \begin{align*} \frac{2 \,{\left (15015 \,{\left (e x + d\right )}^{\frac{17}{2}} b^{6} - 102102 \,{\left (b^{6} d - a b^{5} e\right )}{\left (e x + d\right )}^{\frac{15}{2}} + 294525 \,{\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )}{\left (e x + d\right )}^{\frac{13}{2}} - 464100 \,{\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )}{\left (e x + d\right )}^{\frac{11}{2}} + 425425 \,{\left (b^{6} d^{4} - 4 \, a b^{5} d^{3} e + 6 \, a^{2} b^{4} d^{2} e^{2} - 4 \, a^{3} b^{3} d e^{3} + a^{4} b^{2} e^{4}\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 218790 \,{\left (b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 51051 \,{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{255255 \, e^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/255255*(15015*(e*x + d)^(17/2)*b^6 - 102102*(b^6*d - a*b^5*e)*(e*x + d)^(15/2) + 294525*(b^6*d^2 - 2*a*b^5*d
*e + a^2*b^4*e^2)*(e*x + d)^(13/2) - 464100*(b^6*d^3 - 3*a*b^5*d^2*e + 3*a^2*b^4*d*e^2 - a^3*b^3*e^3)*(e*x + d
)^(11/2) + 425425*(b^6*d^4 - 4*a*b^5*d^3*e + 6*a^2*b^4*d^2*e^2 - 4*a^3*b^3*d*e^3 + a^4*b^2*e^4)*(e*x + d)^(9/2
) - 218790*(b^6*d^5 - 5*a*b^5*d^4*e + 10*a^2*b^4*d^3*e^2 - 10*a^3*b^3*d^2*e^3 + 5*a^4*b^2*d*e^4 - a^5*b*e^5)*(
e*x + d)^(7/2) + 51051*(b^6*d^6 - 6*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 + 15*a^4*b^2*d^2*e^4
- 6*a^5*b*d*e^5 + a^6*e^6)*(e*x + d)^(5/2))/e^7

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Fricas [B]  time = 1.61811, size = 1256, normalized size = 6.72 \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((e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^3,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 [A]  time = 33.1206, size = 1000, normalized size = 5.35 \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)**(3/2)*(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

a**6*d*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 2*a**6*(-d*(d + e*x)**(3/2)/3 + (d
+ e*x)**(5/2)/5)/e + 12*a**5*b*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 12*a**5*b*(d**2*(d + e*x
)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 30*a**4*b**2*d*(d**2*(d + e*x)**(3/2)/3 - 2*d
*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 30*a**4*b**2*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5
/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**3 + 40*a**3*b**3*d*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*
(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**4 + 40*a**3*b**3*(d**4*(d + e*x)**(3/2)/3
- 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**4
+ 30*a**2*b**4*d*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d +
e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**5 + 30*a**2*b**4*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) -
10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**
5 + 12*a*b**5*d*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d +
e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**6 + 12*a*b**5*(d**6*(d + e*x)**(3/2)/3 - 6
*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)
/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**6 + 2*b**6*d*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d +
e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(
d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**7 + 2*b**6*(-d**7*(d + e*x)**(3/2)/3 + 7*d**6*(d + e*x)**(5/2)/
5 - 3*d**5*(d + e*x)**(7/2) + 35*d**4*(d + e*x)**(9/2)/9 - 35*d**3*(d + e*x)**(11/2)/11 + 21*d**2*(d + e*x)**(
13/2)/13 - 7*d*(d + e*x)**(15/2)/15 + (d + e*x)**(17/2)/17)/e**7

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Giac [B]  time = 1.24026, size = 1202, normalized size = 6.43 \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)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^3,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) + 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) + 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) + 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) + 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) + 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 - 540
54*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6)*b^6*d*e^(-6) + 255255*(x*e + d)^(3/2)*a^6*d + 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) + 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) + 4420*(315*(x*e + d)^(1
1/2) - 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) + 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) + 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) + 7*(6435*(x*e + d)^(17/2) - 5
1051*(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^6*e^(-6) + 51051*(
3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^6)*e^(-1)