3.2112 \(\int (a+b x) \sqrt{d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=376 \[ \frac{30 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^4}{7 e^7 (a+b x)}-\frac{12 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^5}{5 e^7 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^6}{3 e^7 (a+b x)}+\frac{2 b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15/2}}{15 e^7 (a+b x)}-\frac{12 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)}{13 e^7 (a+b x)}+\frac{30 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^2}{11 e^7 (a+b x)}-\frac{40 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^3}{9 e^7 (a+b x)} \]

[Out]

(2*(b*d - a*e)^6*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)
) - (12*b*(b*d - a*e)^5*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a
+ b*x)) + (30*b^2*(b*d - a*e)^4*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/
(7*e^7*(a + b*x)) - (40*b^3*(b*d - a*e)^3*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b
^2*x^2])/(9*e^7*(a + b*x)) + (30*b^4*(b*d - a*e)^2*(d + e*x)^(11/2)*Sqrt[a^2 + 2
*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)) - (12*b^5*(b*d - a*e)*(d + e*x)^(13/2)*Sqr
t[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x)) + (2*b^6*(d + e*x)^(15/2)*Sqrt[a^
2 + 2*a*b*x + b^2*x^2])/(15*e^7*(a + b*x))

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Rubi [A]  time = 0.410516, antiderivative size = 376, 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 \[ \frac{30 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^4}{7 e^7 (a+b x)}-\frac{12 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^5}{5 e^7 (a+b x)}+\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^6}{3 e^7 (a+b x)}+\frac{2 b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{15/2}}{15 e^7 (a+b x)}-\frac{12 b^5 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)}{13 e^7 (a+b x)}+\frac{30 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^2}{11 e^7 (a+b x)}-\frac{40 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^3}{9 e^7 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b*d - a*e)^6*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^7*(a + b*x)
) - (12*b*(b*d - a*e)^5*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^7*(a
+ b*x)) + (30*b^2*(b*d - a*e)^4*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/
(7*e^7*(a + b*x)) - (40*b^3*(b*d - a*e)^3*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b
^2*x^2])/(9*e^7*(a + b*x)) + (30*b^4*(b*d - a*e)^2*(d + e*x)^(11/2)*Sqrt[a^2 + 2
*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)) - (12*b^5*(b*d - a*e)*(d + e*x)^(13/2)*Sqr
t[a^2 + 2*a*b*x + b^2*x^2])/(13*e^7*(a + b*x)) + (2*b^6*(d + e*x)^(15/2)*Sqrt[a^
2 + 2*a*b*x + b^2*x^2])/(15*e^7*(a + b*x))

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Rubi in Sympy [A]  time = 62.5565, size = 323, normalized size = 0.86 \[ \frac{2 \left (a + b x\right ) \left (d + e x\right )^{\frac{3}{2}} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{15 e} + \frac{8 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{65 e^{2}} + \frac{16 \left (5 a + 5 b x\right ) \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{715 e^{3}} + \frac{128 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{1287 e^{4}} + \frac{256 \left (3 a + 3 b x\right ) \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{9009 e^{5}} + \frac{1024 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{15015 e^{6}} + \frac{2048 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{6} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{45045 e^{7} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**(5/2)*(e*x+d)**(1/2),x)

[Out]

2*(a + b*x)*(d + e*x)**(3/2)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(15*e) + 8*(d +
e*x)**(3/2)*(a*e - b*d)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(65*e**2) + 16*(5*a
+ 5*b*x)*(d + e*x)**(3/2)*(a*e - b*d)**2*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(7
15*e**3) + 128*(d + e*x)**(3/2)*(a*e - b*d)**3*(a**2 + 2*a*b*x + b**2*x**2)**(3/
2)/(1287*e**4) + 256*(3*a + 3*b*x)*(d + e*x)**(3/2)*(a*e - b*d)**4*sqrt(a**2 + 2
*a*b*x + b**2*x**2)/(9009*e**5) + 1024*(d + e*x)**(3/2)*(a*e - b*d)**5*sqrt(a**2
+ 2*a*b*x + b**2*x**2)/(15015*e**6) + 2048*(d + e*x)**(3/2)*(a*e - b*d)**6*sqrt
(a**2 + 2*a*b*x + b**2*x**2)/(45045*e**7*(a + b*x))

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Mathematica [A]  time = 0.29969, size = 309, normalized size = 0.82 \[ \frac{2 \sqrt{(a+b x)^2} (d+e x)^{3/2} \left (15015 a^6 e^6+18018 a^5 b e^5 (3 e x-2 d)+6435 a^4 b^2 e^4 \left (8 d^2-12 d e x+15 e^2 x^2\right )+2860 a^3 b^3 e^3 \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+195 a^2 b^4 e^2 \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )+30 a b^5 e \left (-256 d^5+384 d^4 e x-480 d^3 e^2 x^2+560 d^2 e^3 x^3-630 d e^4 x^4+693 e^5 x^5\right )+b^6 \left (1024 d^6-1536 d^5 e x+1920 d^4 e^2 x^2-2240 d^3 e^3 x^3+2520 d^2 e^4 x^4-2772 d e^5 x^5+3003 e^6 x^6\right )\right )}{45045 e^7 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[(a + b*x)^2]*(d + e*x)^(3/2)*(15015*a^6*e^6 + 18018*a^5*b*e^5*(-2*d + 3*
e*x) + 6435*a^4*b^2*e^4*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + 2860*a^3*b^3*e^3*(-16*
d^3 + 24*d^2*e*x - 30*d*e^2*x^2 + 35*e^3*x^3) + 195*a^2*b^4*e^2*(128*d^4 - 192*d
^3*e*x + 240*d^2*e^2*x^2 - 280*d*e^3*x^3 + 315*e^4*x^4) + 30*a*b^5*e*(-256*d^5 +
384*d^4*e*x - 480*d^3*e^2*x^2 + 560*d^2*e^3*x^3 - 630*d*e^4*x^4 + 693*e^5*x^5)
+ b^6*(1024*d^6 - 1536*d^5*e*x + 1920*d^4*e^2*x^2 - 2240*d^3*e^3*x^3 + 2520*d^2*
e^4*x^4 - 2772*d*e^5*x^5 + 3003*e^6*x^6)))/(45045*e^7*(a + b*x))

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Maple [A]  time = 0.013, size = 393, normalized size = 1.1 \[{\frac{6006\,{x}^{6}{b}^{6}{e}^{6}+41580\,{x}^{5}a{b}^{5}{e}^{6}-5544\,{x}^{5}{b}^{6}d{e}^{5}+122850\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}-37800\,{x}^{4}a{b}^{5}d{e}^{5}+5040\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+200200\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}-109200\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}+33600\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}-4480\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+193050\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}-171600\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+93600\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}-28800\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+3840\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+108108\,x{a}^{5}b{e}^{6}-154440\,x{a}^{4}{b}^{2}d{e}^{5}+137280\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}-74880\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}+23040\,xa{b}^{5}{d}^{4}{e}^{2}-3072\,x{b}^{6}{d}^{5}e+30030\,{a}^{6}{e}^{6}-72072\,{a}^{5}bd{e}^{5}+102960\,{b}^{2}{a}^{4}{d}^{2}{e}^{4}-91520\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+49920\,{d}^{4}{e}^{2}{a}^{2}{b}^{4}-15360\,{d}^{5}a{b}^{5}e+2048\,{b}^{6}{d}^{6}}{45045\,{e}^{7} \left ( bx+a \right ) ^{5}} \left ( ex+d \right ) ^{{\frac{3}{2}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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)^(1/2),x)

[Out]

2/45045*(e*x+d)^(3/2)*(3003*b^6*e^6*x^6+20790*a*b^5*e^6*x^5-2772*b^6*d*e^5*x^5+6
1425*a^2*b^4*e^6*x^4-18900*a*b^5*d*e^5*x^4+2520*b^6*d^2*e^4*x^4+100100*a^3*b^3*e
^6*x^3-54600*a^2*b^4*d*e^5*x^3+16800*a*b^5*d^2*e^4*x^3-2240*b^6*d^3*e^3*x^3+9652
5*a^4*b^2*e^6*x^2-85800*a^3*b^3*d*e^5*x^2+46800*a^2*b^4*d^2*e^4*x^2-14400*a*b^5*
d^3*e^3*x^2+1920*b^6*d^4*e^2*x^2+54054*a^5*b*e^6*x-77220*a^4*b^2*d*e^5*x+68640*a
^3*b^3*d^2*e^4*x-37440*a^2*b^4*d^3*e^3*x+11520*a*b^5*d^4*e^2*x-1536*b^6*d^5*e*x+
15015*a^6*e^6-36036*a^5*b*d*e^5+51480*a^4*b^2*d^2*e^4-45760*a^3*b^3*d^3*e^3+2496
0*a^2*b^4*d^4*e^2-7680*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 [A]  time = 0.72148, size = 1026, normalized size = 2.73 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9009*(693*b^5*e^6*x^6 - 256*b^5*d^6 + 1664*a*b^4*d^5*e - 4576*a^2*b^3*d^4*e^2
+ 6864*a^3*b^2*d^3*e^3 - 6006*a^4*b*d^2*e^4 + 3003*a^5*d*e^5 + 63*(b^5*d*e^5 + 6
5*a*b^4*e^6)*x^5 - 35*(2*b^5*d^2*e^4 - 13*a*b^4*d*e^5 - 286*a^2*b^3*e^6)*x^4 + 1
0*(8*b^5*d^3*e^3 - 52*a*b^4*d^2*e^4 + 143*a^2*b^3*d*e^5 + 1287*a^3*b^2*e^6)*x^3
- 3*(32*b^5*d^4*e^2 - 208*a*b^4*d^3*e^3 + 572*a^2*b^3*d^2*e^4 - 858*a^3*b^2*d*e^
5 - 3003*a^4*b*e^6)*x^2 + (128*b^5*d^5*e - 832*a*b^4*d^4*e^2 + 2288*a^2*b^3*d^3*
e^3 - 3432*a^3*b^2*d^2*e^4 + 3003*a^4*b*d*e^5 + 3003*a^5*e^6)*x)*sqrt(e*x + d)*a
/e^6 + 2/45045*(3003*b^5*e^7*x^7 + 1024*b^5*d^7 - 6400*a*b^4*d^6*e + 16640*a^2*b
^3*d^5*e^2 - 22880*a^3*b^2*d^4*e^3 + 17160*a^4*b*d^3*e^4 - 6006*a^5*d^2*e^5 + 23
1*(b^5*d*e^6 + 75*a*b^4*e^7)*x^6 - 63*(4*b^5*d^2*e^5 - 25*a*b^4*d*e^6 - 650*a^2*
b^3*e^7)*x^5 + 70*(4*b^5*d^3*e^4 - 25*a*b^4*d^2*e^5 + 65*a^2*b^3*d*e^6 + 715*a^3
*b^2*e^7)*x^4 - 5*(64*b^5*d^4*e^3 - 400*a*b^4*d^3*e^4 + 1040*a^2*b^3*d^2*e^5 - 1
430*a^3*b^2*d*e^6 - 6435*a^4*b*e^7)*x^3 + 3*(128*b^5*d^5*e^2 - 800*a*b^4*d^4*e^3
+ 2080*a^2*b^3*d^3*e^4 - 2860*a^3*b^2*d^2*e^5 + 2145*a^4*b*d*e^6 + 3003*a^5*e^7
)*x^2 - (512*b^5*d^6*e - 3200*a*b^4*d^5*e^2 + 8320*a^2*b^3*d^4*e^3 - 11440*a^3*b
^2*d^3*e^4 + 8580*a^4*b*d^2*e^5 - 3003*a^5*d*e^6)*x)*sqrt(e*x + d)*b/e^7

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Fricas [A]  time = 0.301851, size = 603, normalized size = 1.6 \[ \frac{2 \,{\left (3003 \, b^{6} e^{7} x^{7} + 1024 \, b^{6} d^{7} - 7680 \, a b^{5} d^{6} e + 24960 \, a^{2} b^{4} d^{5} e^{2} - 45760 \, a^{3} b^{3} d^{4} e^{3} + 51480 \, a^{4} b^{2} d^{3} e^{4} - 36036 \, a^{5} b d^{2} e^{5} + 15015 \, a^{6} d e^{6} + 231 \,{\left (b^{6} d e^{6} + 90 \, a b^{5} e^{7}\right )} x^{6} - 63 \,{\left (4 \, b^{6} d^{2} e^{5} - 30 \, a b^{5} d e^{6} - 975 \, a^{2} b^{4} e^{7}\right )} x^{5} + 35 \,{\left (8 \, b^{6} d^{3} e^{4} - 60 \, a b^{5} d^{2} e^{5} + 195 \, a^{2} b^{4} d e^{6} + 2860 \, a^{3} b^{3} e^{7}\right )} x^{4} - 5 \,{\left (64 \, b^{6} d^{4} e^{3} - 480 \, a b^{5} d^{3} e^{4} + 1560 \, a^{2} b^{4} d^{2} e^{5} - 2860 \, a^{3} b^{3} d e^{6} - 19305 \, a^{4} b^{2} e^{7}\right )} x^{3} + 3 \,{\left (128 \, b^{6} d^{5} e^{2} - 960 \, a b^{5} d^{4} e^{3} + 3120 \, a^{2} b^{4} d^{3} e^{4} - 5720 \, a^{3} b^{3} d^{2} e^{5} + 6435 \, a^{4} b^{2} d e^{6} + 18018 \, a^{5} b e^{7}\right )} x^{2} -{\left (512 \, b^{6} d^{6} e - 3840 \, a b^{5} d^{5} e^{2} + 12480 \, a^{2} b^{4} d^{4} e^{3} - 22880 \, a^{3} b^{3} d^{3} e^{4} + 25740 \, a^{4} b^{2} d^{2} e^{5} - 18018 \, a^{5} b d e^{6} - 15015 \, a^{6} e^{7}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/45045*(3003*b^6*e^7*x^7 + 1024*b^6*d^7 - 7680*a*b^5*d^6*e + 24960*a^2*b^4*d^5*
e^2 - 45760*a^3*b^3*d^4*e^3 + 51480*a^4*b^2*d^3*e^4 - 36036*a^5*b*d^2*e^5 + 1501
5*a^6*d*e^6 + 231*(b^6*d*e^6 + 90*a*b^5*e^7)*x^6 - 63*(4*b^6*d^2*e^5 - 30*a*b^5*
d*e^6 - 975*a^2*b^4*e^7)*x^5 + 35*(8*b^6*d^3*e^4 - 60*a*b^5*d^2*e^5 + 195*a^2*b^
4*d*e^6 + 2860*a^3*b^3*e^7)*x^4 - 5*(64*b^6*d^4*e^3 - 480*a*b^5*d^3*e^4 + 1560*a
^2*b^4*d^2*e^5 - 2860*a^3*b^3*d*e^6 - 19305*a^4*b^2*e^7)*x^3 + 3*(128*b^6*d^5*e^
2 - 960*a*b^5*d^4*e^3 + 3120*a^2*b^4*d^3*e^4 - 5720*a^3*b^3*d^2*e^5 + 6435*a^4*b
^2*d*e^6 + 18018*a^5*b*e^7)*x^2 - (512*b^6*d^6*e - 3840*a*b^5*d^5*e^2 + 12480*a^
2*b^4*d^4*e^3 - 22880*a^3*b^3*d^3*e^4 + 25740*a^4*b^2*d^2*e^5 - 18018*a^5*b*d*e^
6 - 15015*a^6*e^7)*x)*sqrt(e*x + d)/e^7

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.335101, size = 660, normalized size = 1.76 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(b*x + a)*sqrt(e*x + d),x, algorithm="giac")

[Out]

2/45045*(18018*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^5*b*e^(-1)*sign(b*x +
a) + 6435*(15*(x*e + d)^(7/2)*e^12 - 42*(x*e + d)^(5/2)*d*e^12 + 35*(x*e + d)^(
3/2)*d^2*e^12)*a^4*b^2*e^(-14)*sign(b*x + a) + 2860*(35*(x*e + d)^(9/2)*e^24 - 1
35*(x*e + d)^(7/2)*d*e^24 + 189*(x*e + d)^(5/2)*d^2*e^24 - 105*(x*e + d)^(3/2)*d
^3*e^24)*a^3*b^3*e^(-27)*sign(b*x + a) + 195*(315*(x*e + d)^(11/2)*e^40 - 1540*(
x*e + d)^(9/2)*d*e^40 + 2970*(x*e + d)^(7/2)*d^2*e^40 - 2772*(x*e + d)^(5/2)*d^3
*e^40 + 1155*(x*e + d)^(3/2)*d^4*e^40)*a^2*b^4*e^(-44)*sign(b*x + a) + 30*(693*(
x*e + d)^(13/2)*e^60 - 4095*(x*e + d)^(11/2)*d*e^60 + 10010*(x*e + d)^(9/2)*d^2*
e^60 - 12870*(x*e + d)^(7/2)*d^3*e^60 + 9009*(x*e + d)^(5/2)*d^4*e^60 - 3003*(x*
e + d)^(3/2)*d^5*e^60)*a*b^5*e^(-65)*sign(b*x + a) + (3003*(x*e + d)^(15/2)*e^84
- 20790*(x*e + d)^(13/2)*d*e^84 + 61425*(x*e + d)^(11/2)*d^2*e^84 - 100100*(x*e
+ d)^(9/2)*d^3*e^84 + 96525*(x*e + d)^(7/2)*d^4*e^84 - 54054*(x*e + d)^(5/2)*d^
5*e^84 + 15015*(x*e + d)^(3/2)*d^6*e^84)*b^6*e^(-90)*sign(b*x + a) + 15015*(x*e
+ d)^(3/2)*a^6*sign(b*x + a))*e^(-1)