3.2118 \(\int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx\)

Optimal. Leaf size=370 \[ -\frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^7 (a+b x) (d+e x)^{5/2}}+\frac{12 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{7 e^7 (a+b x) (d+e x)^{7/2}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{9 e^7 (a+b x) (d+e x)^{9/2}}+\frac{2 b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2}}{3 e^7 (a+b x)}-\frac{12 b^5 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)}{e^7 (a+b x)}-\frac{30 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^7 (a+b x) \sqrt{d+e x}}+\frac{40 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^7 (a+b x) (d+e x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.397259, antiderivative size = 370, 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 \[ -\frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4}{e^7 (a+b x) (d+e x)^{5/2}}+\frac{12 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5}{7 e^7 (a+b x) (d+e x)^{7/2}}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^6}{9 e^7 (a+b x) (d+e x)^{9/2}}+\frac{2 b^6 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2}}{3 e^7 (a+b x)}-\frac{12 b^5 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)}{e^7 (a+b x)}-\frac{30 b^4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2}{e^7 (a+b x) \sqrt{d+e x}}+\frac{40 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3}{3 e^7 (a+b x) (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(11/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 49.9211, size = 299, normalized size = 0.81 \[ \frac{1024 b^{5} \sqrt{d + e x} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{63 e^{6}} + \frac{2048 b^{5} \sqrt{d + e x} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{63 e^{7} \left (a + b x\right )} - \frac{256 b^{4} \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{63 e^{5} \sqrt{d + e x}} - \frac{128 b^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{63 e^{4} \left (d + e x\right )^{\frac{3}{2}}} - \frac{16 b^{2} \left (5 a + 5 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{105 e^{3} \left (d + e x\right )^{\frac{5}{2}}} - \frac{8 b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{21 e^{2} \left (d + e x\right )^{\frac{7}{2}}} - \frac{2 \left (a + b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{9 e \left (d + e x\right )^{\frac{9}{2}}} \]

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)**(11/2),x)

[Out]

1024*b**5*sqrt(d + e*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(63*e**6) + 2048*b**5*s
qrt(d + e*x)*(a*e - b*d)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(63*e**7*(a + b*x)) -
256*b**4*(3*a + 3*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(63*e**5*sqrt(d + e*x))
- 128*b**3*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(63*e**4*(d + e*x)**(3/2)) - 16*b
**2*(5*a + 5*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)/(105*e**3*(d + e*x)**(5/2)
) - 8*b*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(21*e**2*(d + e*x)**(7/2)) - 2*(a +
b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(9*e*(d + e*x)**(9/2))

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Mathematica [A]  time = 0.372061, size = 161, normalized size = 0.44 \[ \frac{2 \sqrt{(a+b x)^2} \sqrt{d+e x} \left (-21 b^5 (17 b d-18 a e)-\frac{945 b^4 (b d-a e)^2}{d+e x}+\frac{420 b^3 (b d-a e)^3}{(d+e x)^2}-\frac{189 b^2 (b d-a e)^4}{(d+e x)^3}+\frac{54 b (b d-a e)^5}{(d+e x)^4}-\frac{7 (b d-a e)^6}{(d+e x)^5}+21 b^6 e x\right )}{63 e^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^(11/2),x]

[Out]

(2*Sqrt[(a + b*x)^2]*Sqrt[d + e*x]*(-21*b^5*(17*b*d - 18*a*e) + 21*b^6*e*x - (7*
(b*d - a*e)^6)/(d + e*x)^5 + (54*b*(b*d - a*e)^5)/(d + e*x)^4 - (189*b^2*(b*d -
a*e)^4)/(d + e*x)^3 + (420*b^3*(b*d - a*e)^3)/(d + e*x)^2 - (945*b^4*(b*d - a*e)
^2)/(d + e*x)))/(63*e^7*(a + b*x))

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Maple [A]  time = 0.013, size = 393, normalized size = 1.1 \[ -{\frac{-42\,{x}^{6}{b}^{6}{e}^{6}-756\,{x}^{5}a{b}^{5}{e}^{6}+504\,{x}^{5}{b}^{6}d{e}^{5}+1890\,{x}^{4}{a}^{2}{b}^{4}{e}^{6}-7560\,{x}^{4}a{b}^{5}d{e}^{5}+5040\,{x}^{4}{b}^{6}{d}^{2}{e}^{4}+840\,{x}^{3}{a}^{3}{b}^{3}{e}^{6}+5040\,{x}^{3}{a}^{2}{b}^{4}d{e}^{5}-20160\,{x}^{3}a{b}^{5}{d}^{2}{e}^{4}+13440\,{x}^{3}{b}^{6}{d}^{3}{e}^{3}+378\,{x}^{2}{a}^{4}{b}^{2}{e}^{6}+1008\,{x}^{2}{a}^{3}{b}^{3}d{e}^{5}+6048\,{x}^{2}{a}^{2}{b}^{4}{d}^{2}{e}^{4}-24192\,{x}^{2}a{b}^{5}{d}^{3}{e}^{3}+16128\,{x}^{2}{b}^{6}{d}^{4}{e}^{2}+108\,x{a}^{5}b{e}^{6}+216\,x{a}^{4}{b}^{2}d{e}^{5}+576\,x{a}^{3}{b}^{3}{d}^{2}{e}^{4}+3456\,x{a}^{2}{b}^{4}{d}^{3}{e}^{3}-13824\,xa{b}^{5}{d}^{4}{e}^{2}+9216\,x{b}^{6}{d}^{5}e+14\,{a}^{6}{e}^{6}+24\,{a}^{5}bd{e}^{5}+48\,{b}^{2}{a}^{4}{d}^{2}{e}^{4}+128\,{a}^{3}{b}^{3}{d}^{3}{e}^{3}+768\,{d}^{4}{e}^{2}{a}^{2}{b}^{4}-3072\,{d}^{5}a{b}^{5}e+2048\,{b}^{6}{d}^{6}}{63\, \left ( bx+a \right ) ^{5}{e}^{7}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{9}{2}}}} \]

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)^(11/2),x)

[Out]

-2/63/(e*x+d)^(9/2)*(-21*b^6*e^6*x^6-378*a*b^5*e^6*x^5+252*b^6*d*e^5*x^5+945*a^2
*b^4*e^6*x^4-3780*a*b^5*d*e^5*x^4+2520*b^6*d^2*e^4*x^4+420*a^3*b^3*e^6*x^3+2520*
a^2*b^4*d*e^5*x^3-10080*a*b^5*d^2*e^4*x^3+6720*b^6*d^3*e^3*x^3+189*a^4*b^2*e^6*x
^2+504*a^3*b^3*d*e^5*x^2+3024*a^2*b^4*d^2*e^4*x^2-12096*a*b^5*d^3*e^3*x^2+8064*b
^6*d^4*e^2*x^2+54*a^5*b*e^6*x+108*a^4*b^2*d*e^5*x+288*a^3*b^3*d^2*e^4*x+1728*a^2
*b^4*d^3*e^3*x-6912*a*b^5*d^4*e^2*x+4608*b^6*d^5*e*x+7*a^6*e^6+12*a^5*b*d*e^5+24
*a^4*b^2*d^2*e^4+64*a^3*b^3*d^3*e^3+384*a^2*b^4*d^4*e^2-1536*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 = 0.759711, size = 927, normalized size = 2.51 \[ \frac{2 \,{\left (63 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} - 128 \, a b^{4} d^{4} e - 32 \, a^{2} b^{3} d^{3} e^{2} - 16 \, a^{3} b^{2} d^{2} e^{3} - 10 \, a^{4} b d e^{4} - 7 \, a^{5} e^{5} + 315 \,{\left (2 \, b^{5} d e^{4} - a b^{4} e^{5}\right )} x^{4} + 210 \,{\left (8 \, b^{5} d^{2} e^{3} - 4 \, a b^{4} d e^{4} - a^{2} b^{3} e^{5}\right )} x^{3} + 126 \,{\left (16 \, b^{5} d^{3} e^{2} - 8 \, a b^{4} d^{2} e^{3} - 2 \, a^{2} b^{3} d e^{4} - a^{3} b^{2} e^{5}\right )} x^{2} + 9 \,{\left (128 \, b^{5} d^{4} e - 64 \, a b^{4} d^{3} e^{2} - 16 \, a^{2} b^{3} d^{2} e^{3} - 8 \, a^{3} b^{2} d e^{4} - 5 \, a^{4} b e^{5}\right )} x\right )} a}{63 \,{\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )} \sqrt{e x + d}} + \frac{2 \,{\left (21 \, b^{5} e^{6} x^{6} - 1024 \, b^{5} d^{6} + 1280 \, a b^{4} d^{5} e - 256 \, a^{2} b^{3} d^{4} e^{2} - 32 \, a^{3} b^{2} d^{3} e^{3} - 8 \, a^{4} b d^{2} e^{4} - 2 \, a^{5} d e^{5} - 63 \,{\left (4 \, b^{5} d e^{5} - 5 \, a b^{4} e^{6}\right )} x^{5} - 630 \,{\left (4 \, b^{5} d^{2} e^{4} - 5 \, a b^{4} d e^{5} + a^{2} b^{3} e^{6}\right )} x^{4} - 210 \,{\left (32 \, b^{5} d^{3} e^{3} - 40 \, a b^{4} d^{2} e^{4} + 8 \, a^{2} b^{3} d e^{5} + a^{3} b^{2} e^{6}\right )} x^{3} - 63 \,{\left (128 \, b^{5} d^{4} e^{2} - 160 \, a b^{4} d^{3} e^{3} + 32 \, a^{2} b^{3} d^{2} e^{4} + 4 \, a^{3} b^{2} d e^{5} + a^{4} b e^{6}\right )} x^{2} - 9 \,{\left (512 \, b^{5} d^{5} e - 640 \, a b^{4} d^{4} e^{2} + 128 \, a^{2} b^{3} d^{3} e^{3} + 16 \, a^{3} b^{2} d^{2} e^{4} + 4 \, a^{4} b d e^{5} + a^{5} e^{6}\right )} x\right )} b}{63 \,{\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} e^{7}\right )} \sqrt{e x + d}} \]

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)/(e*x + d)^(11/2),x, algorithm="maxima")

[Out]

2/63*(63*b^5*e^5*x^5 + 256*b^5*d^5 - 128*a*b^4*d^4*e - 32*a^2*b^3*d^3*e^2 - 16*a
^3*b^2*d^2*e^3 - 10*a^4*b*d*e^4 - 7*a^5*e^5 + 315*(2*b^5*d*e^4 - a*b^4*e^5)*x^4
+ 210*(8*b^5*d^2*e^3 - 4*a*b^4*d*e^4 - a^2*b^3*e^5)*x^3 + 126*(16*b^5*d^3*e^2 -
8*a*b^4*d^2*e^3 - 2*a^2*b^3*d*e^4 - a^3*b^2*e^5)*x^2 + 9*(128*b^5*d^4*e - 64*a*b
^4*d^3*e^2 - 16*a^2*b^3*d^2*e^3 - 8*a^3*b^2*d*e^4 - 5*a^4*b*e^5)*x)*a/((e^10*x^4
+ 4*d*e^9*x^3 + 6*d^2*e^8*x^2 + 4*d^3*e^7*x + d^4*e^6)*sqrt(e*x + d)) + 2/63*(2
1*b^5*e^6*x^6 - 1024*b^5*d^6 + 1280*a*b^4*d^5*e - 256*a^2*b^3*d^4*e^2 - 32*a^3*b
^2*d^3*e^3 - 8*a^4*b*d^2*e^4 - 2*a^5*d*e^5 - 63*(4*b^5*d*e^5 - 5*a*b^4*e^6)*x^5
- 630*(4*b^5*d^2*e^4 - 5*a*b^4*d*e^5 + a^2*b^3*e^6)*x^4 - 210*(32*b^5*d^3*e^3 -
40*a*b^4*d^2*e^4 + 8*a^2*b^3*d*e^5 + a^3*b^2*e^6)*x^3 - 63*(128*b^5*d^4*e^2 - 16
0*a*b^4*d^3*e^3 + 32*a^2*b^3*d^2*e^4 + 4*a^3*b^2*d*e^5 + a^4*b*e^6)*x^2 - 9*(512
*b^5*d^5*e - 640*a*b^4*d^4*e^2 + 128*a^2*b^3*d^3*e^3 + 16*a^3*b^2*d^2*e^4 + 4*a^
4*b*d*e^5 + a^5*e^6)*x)*b/((e^11*x^4 + 4*d*e^10*x^3 + 6*d^2*e^9*x^2 + 4*d^3*e^8*
x + d^4*e^7)*sqrt(e*x + d))

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Fricas [A]  time = 0.306194, size = 537, normalized size = 1.45 \[ \frac{2 \,{\left (21 \, b^{6} e^{6} x^{6} - 1024 \, b^{6} d^{6} + 1536 \, a b^{5} d^{5} e - 384 \, a^{2} b^{4} d^{4} e^{2} - 64 \, a^{3} b^{3} d^{3} e^{3} - 24 \, a^{4} b^{2} d^{2} e^{4} - 12 \, a^{5} b d e^{5} - 7 \, a^{6} e^{6} - 126 \,{\left (2 \, b^{6} d e^{5} - 3 \, a b^{5} e^{6}\right )} x^{5} - 315 \,{\left (8 \, b^{6} d^{2} e^{4} - 12 \, a b^{5} d e^{5} + 3 \, a^{2} b^{4} e^{6}\right )} x^{4} - 420 \,{\left (16 \, b^{6} d^{3} e^{3} - 24 \, a b^{5} d^{2} e^{4} + 6 \, a^{2} b^{4} d e^{5} + a^{3} b^{3} e^{6}\right )} x^{3} - 63 \,{\left (128 \, b^{6} d^{4} e^{2} - 192 \, a b^{5} d^{3} e^{3} + 48 \, a^{2} b^{4} d^{2} e^{4} + 8 \, a^{3} b^{3} d e^{5} + 3 \, a^{4} b^{2} e^{6}\right )} x^{2} - 18 \,{\left (256 \, b^{6} d^{5} e - 384 \, a b^{5} d^{4} e^{2} + 96 \, a^{2} b^{4} d^{3} e^{3} + 16 \, a^{3} b^{3} d^{2} e^{4} + 6 \, a^{4} b^{2} d e^{5} + 3 \, a^{5} b e^{6}\right )} x\right )}}{63 \,{\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} e^{7}\right )} \sqrt{e x + d}} \]

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)/(e*x + d)^(11/2),x, algorithm="fricas")

[Out]

2/63*(21*b^6*e^6*x^6 - 1024*b^6*d^6 + 1536*a*b^5*d^5*e - 384*a^2*b^4*d^4*e^2 - 6
4*a^3*b^3*d^3*e^3 - 24*a^4*b^2*d^2*e^4 - 12*a^5*b*d*e^5 - 7*a^6*e^6 - 126*(2*b^6
*d*e^5 - 3*a*b^5*e^6)*x^5 - 315*(8*b^6*d^2*e^4 - 12*a*b^5*d*e^5 + 3*a^2*b^4*e^6)
*x^4 - 420*(16*b^6*d^3*e^3 - 24*a*b^5*d^2*e^4 + 6*a^2*b^4*d*e^5 + a^3*b^3*e^6)*x
^3 - 63*(128*b^6*d^4*e^2 - 192*a*b^5*d^3*e^3 + 48*a^2*b^4*d^2*e^4 + 8*a^3*b^3*d*
e^5 + 3*a^4*b^2*e^6)*x^2 - 18*(256*b^6*d^5*e - 384*a*b^5*d^4*e^2 + 96*a^2*b^4*d^
3*e^3 + 16*a^3*b^3*d^2*e^4 + 6*a^4*b^2*d*e^5 + 3*a^5*b*e^6)*x)/((e^11*x^4 + 4*d*
e^10*x^3 + 6*d^2*e^9*x^2 + 4*d^3*e^8*x + d^4*e^7)*sqrt(e*x + d))

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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)**(11/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.320137, size = 841, normalized size = 2.27 \[ \text{result too large to display} \]

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)/(e*x + d)^(11/2),x, algorithm="giac")

[Out]

2/3*((x*e + d)^(3/2)*b^6*e^14*sign(b*x + a) - 18*sqrt(x*e + d)*b^6*d*e^14*sign(b
*x + a) + 18*sqrt(x*e + d)*a*b^5*e^15*sign(b*x + a))*e^(-21) - 2/63*(945*(x*e +
d)^4*b^6*d^2*sign(b*x + a) - 420*(x*e + d)^3*b^6*d^3*sign(b*x + a) + 189*(x*e +
d)^2*b^6*d^4*sign(b*x + a) - 54*(x*e + d)*b^6*d^5*sign(b*x + a) + 7*b^6*d^6*sign
(b*x + a) - 1890*(x*e + d)^4*a*b^5*d*e*sign(b*x + a) + 1260*(x*e + d)^3*a*b^5*d^
2*e*sign(b*x + a) - 756*(x*e + d)^2*a*b^5*d^3*e*sign(b*x + a) + 270*(x*e + d)*a*
b^5*d^4*e*sign(b*x + a) - 42*a*b^5*d^5*e*sign(b*x + a) + 945*(x*e + d)^4*a^2*b^4
*e^2*sign(b*x + a) - 1260*(x*e + d)^3*a^2*b^4*d*e^2*sign(b*x + a) + 1134*(x*e +
d)^2*a^2*b^4*d^2*e^2*sign(b*x + a) - 540*(x*e + d)*a^2*b^4*d^3*e^2*sign(b*x + a)
+ 105*a^2*b^4*d^4*e^2*sign(b*x + a) + 420*(x*e + d)^3*a^3*b^3*e^3*sign(b*x + a)
- 756*(x*e + d)^2*a^3*b^3*d*e^3*sign(b*x + a) + 540*(x*e + d)*a^3*b^3*d^2*e^3*s
ign(b*x + a) - 140*a^3*b^3*d^3*e^3*sign(b*x + a) + 189*(x*e + d)^2*a^4*b^2*e^4*s
ign(b*x + a) - 270*(x*e + d)*a^4*b^2*d*e^4*sign(b*x + a) + 105*a^4*b^2*d^2*e^4*s
ign(b*x + a) + 54*(x*e + d)*a^5*b*e^5*sign(b*x + a) - 42*a^5*b*d*e^5*sign(b*x +
a) + 7*a^6*e^6*sign(b*x + a))*e^(-7)/(x*e + d)^(9/2)