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

Optimal. Leaf size=248 \[ \frac{2 c (d+e x)^{15/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{5 e^7}-\frac{2 (d+e x)^{13/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{13 e^7}+\frac{6 d (d+e x)^{11/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{11 e^7}-\frac{6 c^2 (d+e x)^{17/2} (2 c d-b e)}{17 e^7}-\frac{2 d^2 (d+e x)^{9/2} (c d-b e)^2 (2 c d-b e)}{3 e^7}+\frac{2 d^3 (d+e x)^{7/2} (c d-b e)^3}{7 e^7}+\frac{2 c^3 (d+e x)^{19/2}}{19 e^7} \]

[Out]

(2*d^3*(c*d - b*e)^3*(d + e*x)^(7/2))/(7*e^7) - (2*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x)^(9/2))/(3*e^7) +
(6*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(11/2))/(11*e^7) - (2*(2*c*d - b*e)*(10*c^2*d^2 -
10*b*c*d*e + b^2*e^2)*(d + e*x)^(13/2))/(13*e^7) + (2*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(15/2))/(
5*e^7) - (6*c^2*(2*c*d - b*e)*(d + e*x)^(17/2))/(17*e^7) + (2*c^3*(d + e*x)^(19/2))/(19*e^7)

________________________________________________________________________________________

Rubi [A]  time = 0.107545, antiderivative size = 248, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {698} \[ \frac{2 c (d+e x)^{15/2} \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{5 e^7}-\frac{2 (d+e x)^{13/2} (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{13 e^7}+\frac{6 d (d+e x)^{11/2} (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{11 e^7}-\frac{6 c^2 (d+e x)^{17/2} (2 c d-b e)}{17 e^7}-\frac{2 d^2 (d+e x)^{9/2} (c d-b e)^2 (2 c d-b e)}{3 e^7}+\frac{2 d^3 (d+e x)^{7/2} (c d-b e)^3}{7 e^7}+\frac{2 c^3 (d+e x)^{19/2}}{19 e^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*d^3*(c*d - b*e)^3*(d + e*x)^(7/2))/(7*e^7) - (2*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x)^(9/2))/(3*e^7) +
(6*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(11/2))/(11*e^7) - (2*(2*c*d - b*e)*(10*c^2*d^2 -
10*b*c*d*e + b^2*e^2)*(d + e*x)^(13/2))/(13*e^7) + (2*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(15/2))/(
5*e^7) - (6*c^2*(2*c*d - b*e)*(d + e*x)^(17/2))/(17*e^7) + (2*c^3*(d + e*x)^(19/2))/(19*e^7)

Rule 698

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

Rubi steps

\begin{align*} \int (d+e x)^{5/2} \left (b x+c x^2\right )^3 \, dx &=\int \left (\frac{d^3 (c d-b e)^3 (d+e x)^{5/2}}{e^6}-\frac{3 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)^{7/2}}{e^6}+\frac{3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{9/2}}{e^6}+\frac{(2 c d-b e) \left (-10 c^2 d^2+10 b c d e-b^2 e^2\right ) (d+e x)^{11/2}}{e^6}+\frac{3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{13/2}}{e^6}-\frac{3 c^2 (2 c d-b e) (d+e x)^{15/2}}{e^6}+\frac{c^3 (d+e x)^{17/2}}{e^6}\right ) \, dx\\ &=\frac{2 d^3 (c d-b e)^3 (d+e x)^{7/2}}{7 e^7}-\frac{2 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)^{9/2}}{3 e^7}+\frac{6 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{11/2}}{11 e^7}-\frac{2 (2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) (d+e x)^{13/2}}{13 e^7}+\frac{2 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{15/2}}{5 e^7}-\frac{6 c^2 (2 c d-b e) (d+e x)^{17/2}}{17 e^7}+\frac{2 c^3 (d+e x)^{19/2}}{19 e^7}\\ \end{align*}

Mathematica [A]  time = 0.160991, size = 206, normalized size = 0.83 \[ \frac{2 (d+e x)^{7/2} \left (969969 c (d+e x)^4 \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-373065 (d+e x)^3 (2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )+1322685 d (d+e x)^2 (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-855855 c^2 (d+e x)^5 (2 c d-b e)-1616615 d^2 (d+e x) (c d-b e)^2 (2 c d-b e)+692835 d^3 (c d-b e)^3+255255 c^3 (d+e x)^6\right )}{4849845 e^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(7/2)*(692835*d^3*(c*d - b*e)^3 - 1616615*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x) + 1322685*d*(
c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^2 - 373065*(2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2
*e^2)*(d + e*x)^3 + 969969*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^4 - 855855*c^2*(2*c*d - b*e)*(d + e*x
)^5 + 255255*c^3*(d + e*x)^6))/(4849845*e^7)

________________________________________________________________________________________

Maple [A]  time = 0.048, size = 286, normalized size = 1.2 \begin{align*} -{\frac{-510510\,{c}^{3}{x}^{6}{e}^{6}-1711710\,b{c}^{2}{e}^{6}{x}^{5}+360360\,{c}^{3}d{e}^{5}{x}^{5}-1939938\,{b}^{2}c{e}^{6}{x}^{4}+1141140\,b{c}^{2}d{e}^{5}{x}^{4}-240240\,{c}^{3}{d}^{2}{e}^{4}{x}^{4}-746130\,{b}^{3}{e}^{6}{x}^{3}+1193808\,{b}^{2}cd{e}^{5}{x}^{3}-702240\,b{c}^{2}{d}^{2}{e}^{4}{x}^{3}+147840\,{c}^{3}{d}^{3}{e}^{3}{x}^{3}+406980\,{b}^{3}d{e}^{5}{x}^{2}-651168\,{b}^{2}c{d}^{2}{e}^{4}{x}^{2}+383040\,b{c}^{2}{d}^{3}{e}^{3}{x}^{2}-80640\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}-180880\,{b}^{3}{d}^{2}{e}^{4}x+289408\,{b}^{2}c{d}^{3}{e}^{3}x-170240\,b{c}^{2}{d}^{4}{e}^{2}x+35840\,{c}^{3}{d}^{5}ex+51680\,{b}^{3}{d}^{3}{e}^{3}-82688\,{b}^{2}c{d}^{4}{e}^{2}+48640\,b{c}^{2}{d}^{5}e-10240\,{c}^{3}{d}^{6}}{4849845\,{e}^{7}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/4849845*(e*x+d)^(7/2)*(-255255*c^3*e^6*x^6-855855*b*c^2*e^6*x^5+180180*c^3*d*e^5*x^5-969969*b^2*c*e^6*x^4+5
70570*b*c^2*d*e^5*x^4-120120*c^3*d^2*e^4*x^4-373065*b^3*e^6*x^3+596904*b^2*c*d*e^5*x^3-351120*b*c^2*d^2*e^4*x^
3+73920*c^3*d^3*e^3*x^3+203490*b^3*d*e^5*x^2-325584*b^2*c*d^2*e^4*x^2+191520*b*c^2*d^3*e^3*x^2-40320*c^3*d^4*e
^2*x^2-90440*b^3*d^2*e^4*x+144704*b^2*c*d^3*e^3*x-85120*b*c^2*d^4*e^2*x+17920*c^3*d^5*e*x+25840*b^3*d^3*e^3-41
344*b^2*c*d^4*e^2+24320*b*c^2*d^5*e-5120*c^3*d^6)/e^7

________________________________________________________________________________________

Maxima [A]  time = 1.13156, size = 366, normalized size = 1.48 \begin{align*} \frac{2 \,{\left (255255 \,{\left (e x + d\right )}^{\frac{19}{2}} c^{3} - 855855 \,{\left (2 \, c^{3} d - b c^{2} e\right )}{\left (e x + d\right )}^{\frac{17}{2}} + 969969 \,{\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2}\right )}{\left (e x + d\right )}^{\frac{15}{2}} - 373065 \,{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, b^{2} c d e^{2} - b^{3} e^{3}\right )}{\left (e x + d\right )}^{\frac{13}{2}} + 1322685 \,{\left (5 \, c^{3} d^{4} - 10 \, b c^{2} d^{3} e + 6 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 1616615 \,{\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3}\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 692835 \,{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )}{\left (e x + d\right )}^{\frac{7}{2}}\right )}}{4849845 \, e^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/4849845*(255255*(e*x + d)^(19/2)*c^3 - 855855*(2*c^3*d - b*c^2*e)*(e*x + d)^(17/2) + 969969*(5*c^3*d^2 - 5*b
*c^2*d*e + b^2*c*e^2)*(e*x + d)^(15/2) - 373065*(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*b^2*c*d*e^2 - b^3*e^3)*(e*x
+ d)^(13/2) + 1322685*(5*c^3*d^4 - 10*b*c^2*d^3*e + 6*b^2*c*d^2*e^2 - b^3*d*e^3)*(e*x + d)^(11/2) - 1616615*(2
*c^3*d^5 - 5*b*c^2*d^4*e + 4*b^2*c*d^3*e^2 - b^3*d^2*e^3)*(e*x + d)^(9/2) + 692835*(c^3*d^6 - 3*b*c^2*d^5*e +
3*b^2*c*d^4*e^2 - b^3*d^3*e^3)*(e*x + d)^(7/2))/e^7

________________________________________________________________________________________

Fricas [A]  time = 1.94932, size = 1002, normalized size = 4.04 \begin{align*} \frac{2 \,{\left (255255 \, c^{3} e^{9} x^{9} + 5120 \, c^{3} d^{9} - 24320 \, b c^{2} d^{8} e + 41344 \, b^{2} c d^{7} e^{2} - 25840 \, b^{3} d^{6} e^{3} + 45045 \,{\left (13 \, c^{3} d e^{8} + 19 \, b c^{2} e^{9}\right )} x^{8} + 3003 \,{\left (115 \, c^{3} d^{2} e^{7} + 665 \, b c^{2} d e^{8} + 323 \, b^{2} c e^{9}\right )} x^{7} + 231 \,{\left (5 \, c^{3} d^{3} e^{6} + 5225 \, b c^{2} d^{2} e^{7} + 10013 \, b^{2} c d e^{8} + 1615 \, b^{3} e^{9}\right )} x^{6} - 63 \,{\left (20 \, c^{3} d^{4} e^{5} - 95 \, b c^{2} d^{3} e^{6} - 22933 \, b^{2} c d^{2} e^{7} - 14535 \, b^{3} d e^{8}\right )} x^{5} + 35 \,{\left (40 \, c^{3} d^{5} e^{4} - 190 \, b c^{2} d^{4} e^{5} + 323 \, b^{2} c d^{3} e^{6} + 17119 \, b^{3} d^{2} e^{7}\right )} x^{4} - 5 \,{\left (320 \, c^{3} d^{6} e^{3} - 1520 \, b c^{2} d^{5} e^{4} + 2584 \, b^{2} c d^{4} e^{5} - 1615 \, b^{3} d^{3} e^{6}\right )} x^{3} + 6 \,{\left (320 \, c^{3} d^{7} e^{2} - 1520 \, b c^{2} d^{6} e^{3} + 2584 \, b^{2} c d^{5} e^{4} - 1615 \, b^{3} d^{4} e^{5}\right )} x^{2} - 8 \,{\left (320 \, c^{3} d^{8} e - 1520 \, b c^{2} d^{7} e^{2} + 2584 \, b^{2} c d^{6} e^{3} - 1615 \, b^{3} d^{5} e^{4}\right )} x\right )} \sqrt{e x + d}}{4849845 \, e^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/4849845*(255255*c^3*e^9*x^9 + 5120*c^3*d^9 - 24320*b*c^2*d^8*e + 41344*b^2*c*d^7*e^2 - 25840*b^3*d^6*e^3 + 4
5045*(13*c^3*d*e^8 + 19*b*c^2*e^9)*x^8 + 3003*(115*c^3*d^2*e^7 + 665*b*c^2*d*e^8 + 323*b^2*c*e^9)*x^7 + 231*(5
*c^3*d^3*e^6 + 5225*b*c^2*d^2*e^7 + 10013*b^2*c*d*e^8 + 1615*b^3*e^9)*x^6 - 63*(20*c^3*d^4*e^5 - 95*b*c^2*d^3*
e^6 - 22933*b^2*c*d^2*e^7 - 14535*b^3*d*e^8)*x^5 + 35*(40*c^3*d^5*e^4 - 190*b*c^2*d^4*e^5 + 323*b^2*c*d^3*e^6
+ 17119*b^3*d^2*e^7)*x^4 - 5*(320*c^3*d^6*e^3 - 1520*b*c^2*d^5*e^4 + 2584*b^2*c*d^4*e^5 - 1615*b^3*d^3*e^6)*x^
3 + 6*(320*c^3*d^7*e^2 - 1520*b*c^2*d^6*e^3 + 2584*b^2*c*d^5*e^4 - 1615*b^3*d^4*e^5)*x^2 - 8*(320*c^3*d^8*e -
1520*b*c^2*d^7*e^2 + 2584*b^2*c*d^6*e^3 - 1615*b^3*d^5*e^4)*x)*sqrt(e*x + d)/e^7

________________________________________________________________________________________

Sympy [B]  time = 56.6808, size = 1207, normalized size = 4.87 \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)*(c*x**2+b*x)**3,x)

[Out]

2*b**3*d**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**4 + 4*b**3*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**4 + 2*b**3*(-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**4 + 6
*b**2*c*d**2*(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 + 12*b**2*c*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**5 + 6*
b**2*c*(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**5 + 6*b*c**2*d**2
*(-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*b*c**2*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**6 + 6*b*c**2*(-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**6 + 2*c**3*d**2*(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 + 4*c**3*d*(-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 + 2*c**3*(d**8*(d + e*x)**(3/2)/3 -
8*d**7*(d + e*x)**(5/2)/5 + 4*d**6*(d + e*x)**(7/2) - 56*d**5*(d + e*x)**(9/2)/9 + 70*d**4*(d + e*x)**(11/2)/
11 - 56*d**3*(d + e*x)**(13/2)/13 + 28*d**2*(d + e*x)**(15/2)/15 - 8*d*(d + e*x)**(17/2)/17 + (d + e*x)**(19/2
)/19)/e**7

________________________________________________________________________________________

Giac [B]  time = 1.46863, size = 1454, normalized size = 5.86 \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)*(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

2/14549535*(46189*(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)*b^3*d^2*e^(-3) + 12597*(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)*b^2*c*d^2*e^(-4) + 4845*(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*c^2*d^2*e^(-5) + 323*(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)*c^3*d^2*e^(-6) + 8398*(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)*b^3*d*e^(-3) + 9690*(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^2*c*d*e^(-4) + 1938*(3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61425*(x*e + d)^(11/2)*d^2 - 1
00100*(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*c^2*d*e^(-5) + 266*(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)*c^3*d*e^(-6) + 1615*(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^3*e^(-3) +
969*(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^2*c*e^(-4) + 399*(643
5*(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 + 42
5425*(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*c^2*e^(-5) + 7*(109395*(x*e + d)^(19/2) - 978120*(x*e + d)^(17/2)*d + 3879876*(x*e + d)^(15/2)*d^2 - 89535
60*(x*e + d)^(13/2)*d^3 + 13226850*(x*e + d)^(11/2)*d^4 - 12932920*(x*e + d)^(9/2)*d^5 + 8314020*(x*e + d)^(7/
2)*d^6 - 3325608*(x*e + d)^(5/2)*d^7 + 692835*(x*e + d)^(3/2)*d^8)*c^3*e^(-6))*e^(-1)