3.2210 \(\int \frac{(d+e x)^5}{\left (a+b x+c x^2\right )^5} \, dx\)

Optimal. Leaf size=388 \[ \frac{5 (d+e x) (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac{10 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2}}-\frac{(b+2 c x) (d+e x)^5}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac{(d+e x)^4 \left (-8 a c e-5 b^2 e+14 c x (2 c d-b e)+14 b c d\right )}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}+\frac{(d+e x)^3 \left (-c x \left (-4 c e (35 b d-8 a e)+27 b^2 e^2+140 c^2 d^2\right )-10 b c \left (3 a e^2+7 c d^2\right )+28 a c^2 d e-10 b^3 e^2+63 b^2 c d e\right )}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2} \]

[Out]

-((b + 2*c*x)*(d + e*x)^5)/(4*(b^2 - 4*a*c)*(a + b*x + c*x^2)^4) + ((d + e*x)^4*
(14*b*c*d - 5*b^2*e - 8*a*c*e + 14*c*(2*c*d - b*e)*x))/(12*(b^2 - 4*a*c)^2*(a +
b*x + c*x^2)^3) + ((d + e*x)^3*(63*b^2*c*d*e + 28*a*c^2*d*e - 10*b^3*e^2 - 10*b*
c*(7*c*d^2 + 3*a*e^2) - c*(140*c^2*d^2 + 27*b^2*e^2 - 4*c*e*(35*b*d - 8*a*e))*x)
)/(12*(b^2 - 4*a*c)^3*(a + b*x + c*x^2)^2) + (5*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e
^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)*(b*d - 2*a*e + (2*c*d - b*e)*x))/(2*(b^2 - 4
*a*c)^4*(a + b*x + c*x^2)) - (10*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(7*c^2*d^
2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2
- 4*a*c)^(9/2)

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Rubi [A]  time = 1.30473, antiderivative size = 388, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{5 (d+e x) (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac{10 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2}}-\frac{(b+2 c x) (d+e x)^5}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac{(d+e x)^4 \left (-8 a c e-5 b^2 e+14 c x (2 c d-b e)+14 b c d\right )}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}+\frac{(d+e x)^3 \left (-c x \left (-4 c e (35 b d-8 a e)+27 b^2 e^2+140 c^2 d^2\right )-10 b c \left (3 a e^2+7 c d^2\right )+28 a c^2 d e-10 b^3 e^2+63 b^2 c d e\right )}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-((b + 2*c*x)*(d + e*x)^5)/(4*(b^2 - 4*a*c)*(a + b*x + c*x^2)^4) + ((d + e*x)^4*
(14*b*c*d - 5*b^2*e - 8*a*c*e + 14*c*(2*c*d - b*e)*x))/(12*(b^2 - 4*a*c)^2*(a +
b*x + c*x^2)^3) + ((d + e*x)^3*(63*b^2*c*d*e + 28*a*c^2*d*e - 10*b^3*e^2 - 10*b*
c*(7*c*d^2 + 3*a*e^2) - c*(140*c^2*d^2 + 27*b^2*e^2 - 4*c*e*(35*b*d - 8*a*e))*x)
)/(12*(b^2 - 4*a*c)^3*(a + b*x + c*x^2)^2) + (5*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e
^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)*(b*d - 2*a*e + (2*c*d - b*e)*x))/(2*(b^2 - 4
*a*c)^4*(a + b*x + c*x^2)) - (10*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(7*c^2*d^
2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2
- 4*a*c)^(9/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [B]  time = 5.21488, size = 985, normalized size = 2.54 \[ \frac{1}{12} \left (\frac{30 (2 c d-b e) \left (7 c^3 d^4-2 c^2 e (7 b d-5 a e) d^2+b^2 e^3 (a e-b d)+c e^2 \left (8 b^2 d^2-10 a b e d+3 a^2 e^2\right )\right ) (b+2 c x)}{c \left (b^2-4 a c\right )^4 (a+x (b+c x))}+\frac{3 e^5 b^6+5 c d e^4 b^5+c e^3 \left (10 c d (e x-5 d)-41 a e^2\right ) b^4+10 c^2 e^2 \left (5 c (3 d-2 e x) d^2+a e^2 (6 d-e x)\right ) b^3+c^2 e \left (129 a^2 e^4-30 a c d (5 d-4 e x) e^2-25 c^2 d^3 (7 d-12 e x)\right ) b^2+10 c^3 \left (7 c^2 (d-5 e x) d^4+10 a c e^2 (d-3 e x) d^2+3 a^2 e^4 (d-e x)\right ) b+4 c^3 \left (35 c^3 x d^5+50 a c^2 e^2 x d^3+15 a^2 c e^4 x d-48 a^3 e^5\right )}{c^3 \left (4 a c-b^2\right )^3 (a+x (b+c x))^2}-\frac{3 \left (b^5 x e^5+b^4 (a e-5 c d x) e^4-5 b^3 c \left (a e (d+e x)-2 c d^2 x\right ) e^3-2 b^2 c \left (5 c^2 x d^3-5 a c e (d+2 e x) d+2 a^2 e^3\right ) e^2+2 c^2 \left (-c^3 x d^5+5 a c^2 e (d+2 e x) d^3-5 a^2 c e^3 (2 d+e x) d+a^3 e^5\right )+b c^2 \left (-c^2 (d-5 e x) d^4-10 a c e^2 (d+3 e x) d^2+5 a^2 e^4 (3 d+e x)\right )\right )}{c^4 \left (4 a c-b^2\right ) (a+x (b+c x))^4}+\frac{-3 e^5 b^6+3 c e^4 (5 d+2 e x) b^5+c e^3 \left (27 a e^2-10 c d (3 d+e x)\right ) b^4-10 c^2 e^2 \left (c (2 e x-3 d) d^2+5 a e^2 (2 d+e x)\right ) b^3+c^2 e \left (-83 a^2 e^4+10 a c d (13 d+12 e x) e^2+5 c^2 d^3 (12 e x-7 d)\right ) b^2+2 c^3 \left (7 c^2 (d-5 e x) d^4+10 a c e^2 (d-3 e x) d^2+5 a^2 e^4 (23 d+9 e x)\right ) b+4 c^3 \left (7 c^3 x d^5+10 a c^2 e^2 x d^3-5 a^2 c e^3 (16 d+9 e x) d+16 a^3 e^5\right )}{c^4 \left (b^2-4 a c\right )^2 (a+x (b+c x))^3}+\frac{120 (2 c d-b e) \left (7 c^3 d^4-2 c^2 e (7 b d-5 a e) d^2+b^2 e^3 (a e-b d)+c e^2 \left (8 b^2 d^2-10 a b e d+3 a^2 e^2\right )\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{9/2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((30*(2*c*d - b*e)*(7*c^3*d^4 - 2*c^2*d^2*e*(7*b*d - 5*a*e) + b^2*e^3*(-(b*d) +
a*e) + c*e^2*(8*b^2*d^2 - 10*a*b*d*e + 3*a^2*e^2))*(b + 2*c*x))/(c*(b^2 - 4*a*c)
^4*(a + x*(b + c*x))) + (5*b^5*c*d*e^4 + 3*b^6*e^5 + 4*c^3*(-48*a^3*e^5 + 35*c^3
*d^5*x + 50*a*c^2*d^3*e^2*x + 15*a^2*c*d*e^4*x) + b^2*c^2*e*(129*a^2*e^4 - 25*c^
2*d^3*(7*d - 12*e*x) - 30*a*c*d*e^2*(5*d - 4*e*x)) + 10*b*c^3*(7*c^2*d^4*(d - 5*
e*x) + 10*a*c*d^2*e^2*(d - 3*e*x) + 3*a^2*e^4*(d - e*x)) + 10*b^3*c^2*e^2*(5*c*d
^2*(3*d - 2*e*x) + a*e^2*(6*d - e*x)) + b^4*c*e^3*(-41*a*e^2 + 10*c*d*(-5*d + e*
x)))/(c^3*(-b^2 + 4*a*c)^3*(a + x*(b + c*x))^2) - (3*(b^5*e^5*x + b^4*e^4*(a*e -
5*c*d*x) - 5*b^3*c*e^3*(-2*c*d^2*x + a*e*(d + e*x)) - 2*b^2*c*e^2*(2*a^2*e^3 +
5*c^2*d^3*x - 5*a*c*d*e*(d + 2*e*x)) + 2*c^2*(a^3*e^5 - c^3*d^5*x - 5*a^2*c*d*e^
3*(2*d + e*x) + 5*a*c^2*d^3*e*(d + 2*e*x)) + b*c^2*(-(c^2*d^4*(d - 5*e*x)) + 5*a
^2*e^4*(3*d + e*x) - 10*a*c*d^2*e^2*(d + 3*e*x))))/(c^4*(-b^2 + 4*a*c)*(a + x*(b
+ c*x))^4) + (-3*b^6*e^5 + 3*b^5*c*e^4*(5*d + 2*e*x) + b^4*c*e^3*(27*a*e^2 - 10
*c*d*(3*d + e*x)) - 10*b^3*c^2*e^2*(5*a*e^2*(2*d + e*x) + c*d^2*(-3*d + 2*e*x))
+ 4*c^3*(16*a^3*e^5 + 7*c^3*d^5*x + 10*a*c^2*d^3*e^2*x - 5*a^2*c*d*e^3*(16*d + 9
*e*x)) + 2*b*c^3*(7*c^2*d^4*(d - 5*e*x) + 10*a*c*d^2*e^2*(d - 3*e*x) + 5*a^2*e^4
*(23*d + 9*e*x)) + b^2*c^2*e*(-83*a^2*e^4 + 5*c^2*d^3*(-7*d + 12*e*x) + 10*a*c*d
*e^2*(13*d + 12*e*x)))/(c^4*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^3) + (120*(2*c*d -
b*e)*(7*c^3*d^4 - 2*c^2*d^2*e*(7*b*d - 5*a*e) + b^2*e^3*(-(b*d) + a*e) + c*e^2*
(8*b^2*d^2 - 10*a*b*d*e + 3*a^2*e^2))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-
b^2 + 4*a*c)^(9/2))/12

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Maple [B]  time = 0.03, size = 3092, normalized size = 8. \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-5*(3*a^2*b*c*e^5-6*a^2*c^2*d*e^4+a*b^3*e^5-12*a*b^2*c*d*e^4+30*a*b*c^2*d^2*e^3
-20*a*c^3*d^3*e^2-b^4*d*e^4+10*b^3*c*d^2*e^3-30*b^2*c^2*d^3*e^2+35*b*c^3*d^4*e-1
4*c^4*d^5)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*c^3*x^7-3
5/2*b*c^2*(3*a^2*b*c*e^5-6*a^2*c^2*d*e^4+a*b^3*e^5-12*a*b^2*c*d*e^4+30*a*b*c^2*d
^2*e^3-20*a*c^3*d^3*e^2-b^4*d*e^4+10*b^3*c*d^2*e^3-30*b^2*c^2*d^3*e^2+35*b*c^3*d
^4*e-14*c^4*d^5)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*x^6
-5/3*c*(11*a*c+13*b^2)*(3*a^2*b*c*e^5-6*a^2*c^2*d*e^4+a*b^3*e^5-12*a*b^2*c*d*e^4
+30*a*b*c^2*d^2*e^3-20*a*c^3*d^3*e^2-b^4*d*e^4+10*b^3*c*d^2*e^3-30*b^2*c^2*d^3*e
^2+35*b*c^3*d^4*e-14*c^4*d^5)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b
^6*c+b^8)*x^5-1/12*(768*a^4*c^4*e^5+882*a^3*b^2*c^3*e^5-3300*a^3*b*c^4*d*e^4+121
3*a^2*b^4*c^2*e^5-7350*a^2*b^3*c^3*d*e^4+16500*a^2*b^2*c^4*d^2*e^3-11000*a^2*b*c
^5*d^3*e^2+77*a*b^6*c*e^5-2050*a*b^5*c^2*d*e^4+9250*a*b^4*c^3*d^2*e^3-19000*a*b^
3*c^4*d^3*e^2+19250*a*b^2*c^5*d^4*e-7700*a*b*c^6*d^5+3*b^8*e^5-125*b^7*c*d*e^4+1
250*b^6*c^2*d^2*e^3-3750*b^5*c^3*d^3*e^2+4375*b^4*c^4*d^4*e-1750*b^3*c^5*d^5)/c/
(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*x^4-1/3*(219*a^4*b*c
^3*e^5+330*a^4*c^4*d*e^4+376*a^3*b^3*c^2*e^5-2250*a^3*b^2*c^3*d*e^4+2190*a^3*b*c
^4*d^2*e^3-1460*a^3*c^5*d^3*e^2+110*a^2*b^5*c*e^5-1015*a^2*b^4*c^2*d*e^4+3760*a^
2*b^3*c^3*d^2*e^3-4210*a^2*b^2*c^4*d^3*e^2+2555*a^2*b*c^5*d^4*e-1022*a^2*c^6*d^5
+3*a*b^7*e^5-185*a*b^6*c*d*e^4+1100*a*b^5*c^2*d^2*e^3-3090*a*b^4*c^3*d^3*e^2+353
5*a*b^3*c^4*d^4*e-1414*a*b^2*c^5*d^5+30*b^7*c*d^2*e^3-90*b^6*c^2*d^3*e^2+105*b^5
*c^3*d^4*e-42*b^4*c^4*d^5)/c/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^
6*c+b^8)*x^3-1/6*(256*a^5*c^3*e^5+401*a^4*b^2*c^2*e^5-1570*a^4*b*c^3*d*e^4+2560*
a^4*c^4*d^2*e^3+399*a^3*b^4*c*e^5-2540*a^3*b^3*c^2*d*e^4+4010*a^3*b^2*c^3*d^2*e^
3-4380*a^3*b*c^4*d^3*e^2+9*a^2*b^6*e^5-645*a^2*b^5*c*d*e^4+3990*a^2*b^4*c^2*d^2*
e^3-7130*a^2*b^3*c^3*d^3*e^2+7665*a^2*b^2*c^4*d^4*e-3066*a^2*b*c^5*d^5+90*a*b^6*
c*d^2*e^3-820*a*b^5*c^2*d^3*e^2+980*a*b^4*c^3*d^4*e-392*a*b^3*c^4*d^5+30*b^7*c*d
^3*e^2-35*b^6*c^2*d^4*e+14*b^5*c^3*d^5)/c/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^
4*c^2-16*a*b^6*c+b^8)*x^2-1/3*(83*a^5*b*c^2*e^5+90*a^5*c^3*d*e^4+151*a^4*b^3*c*e
^5-920*a^4*b^2*c^2*d*e^4+830*a^4*b*c^3*d^2*e^3+300*a^4*c^4*d^3*e^2+3*a^3*b^5*e^5
-235*a^3*b^4*c*d*e^4+1510*a^3*b^3*c^2*d^2*e^3-2790*a^3*b^2*c^3*d^3*e^2+1395*a^3*
b*c^4*d^4*e-558*a^3*c^5*d^5+30*a^2*b^5*c*d^2*e^3-280*a^2*b^4*c^2*d^3*e^2+870*a^2
*b^3*c^3*d^4*e-348*a^2*b^2*c^4*d^5+10*a*b^6*c*d^3*e^2-95*a*b^5*c^2*d^4*e+38*a*b^
4*c^3*d^5+5*b^7*c*d^4*e-2*b^6*c^2*d^5)/c/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4
*c^2-16*a*b^6*c+b^8)*x-1/12*(128*a^6*c^2*e^5+166*a^5*b^2*c*e^5-1100*a^5*b*c^2*d*
e^4+1280*a^5*c^3*d^2*e^3+3*a^4*b^4*e^5-250*a^4*b^3*c*d*e^4+1660*a^4*b^2*c^2*d^2*
e^3-3240*a^4*b*c^3*d^3*e^2+1920*a^4*c^4*d^4*e+30*a^3*b^4*c*d^2*e^3-280*a^3*b^3*c
^2*d^3*e^2+870*a^3*b^2*c^3*d^4*e-1116*a^3*b*c^4*d^5+10*a^2*b^5*c*d^3*e^2-95*a^2*
b^4*c^2*d^4*e+326*a^2*b^3*c^3*d^5+5*a*b^6*c*d^4*e-50*a*b^5*c^2*d^5+3*b^7*c*d^5)/
c/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8))/(c*x^2+b*x+a)^4-3
0/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/(4*a*c-b^2)^(1/2)*
arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^2*b*c*e^5+60/(256*a^4*c^4-256*a^3*b^2*c^3+
96*a^2*b^4*c^2-16*a*b^6*c+b^8)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1
/2))*a^2*d*e^4*c^2-10/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8
)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b^3*e^5+120/(256*a^4*c
^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/(4*a*c-b^2)^(1/2)*arctan((2*c*
x+b)/(4*a*c-b^2)^(1/2))*d*a*b^2*c*e^4-300/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^
4*c^2-16*a*b^6*c+b^8)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*d^2*
a*b*c^2*e^3+200/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/(4*a
*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*c^3*d^3*a*e^2+10/(256*a^4*c^4-
256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b
)/(4*a*c-b^2)^(1/2))*b^4*d*e^4-100/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-1
6*a*b^6*c+b^8)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^3*c*d^2*e
^3+300/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/(4*a*c-b^2)^(
1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^2*c^2*d^3*e^2-350/(256*a^4*c^4-256*a^
3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a
*c-b^2)^(1/2))*d^4*b*c^3*e+140/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*
b^6*c+b^8)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*c^4*d^5

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.268935, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/12*(60*(14*a^4*c^5*d^5 - 35*a^4*b*c^4*d^4*e + (14*c^9*d^5 - 35*b*c^8*d^4*e +
10*(3*b^2*c^7 + 2*a*c^8)*d^3*e^2 - 10*(b^3*c^6 + 3*a*b*c^7)*d^2*e^3 + (b^4*c^5
+ 12*a*b^2*c^6 + 6*a^2*c^7)*d*e^4 - (a*b^3*c^5 + 3*a^2*b*c^6)*e^5)*x^8 + 4*(14*b
*c^8*d^5 - 35*b^2*c^7*d^4*e + 10*(3*b^3*c^6 + 2*a*b*c^7)*d^3*e^2 - 10*(b^4*c^5 +
3*a*b^2*c^6)*d^2*e^3 + (b^5*c^4 + 12*a*b^3*c^5 + 6*a^2*b*c^6)*d*e^4 - (a*b^4*c^
4 + 3*a^2*b^2*c^5)*e^5)*x^7 + 2*(14*(3*b^2*c^7 + 2*a*c^8)*d^5 - 35*(3*b^3*c^6 +
2*a*b*c^7)*d^4*e + 10*(9*b^4*c^5 + 12*a*b^2*c^6 + 4*a^2*c^7)*d^3*e^2 - 10*(3*b^5
*c^4 + 11*a*b^3*c^5 + 6*a^2*b*c^6)*d^2*e^3 + (3*b^6*c^3 + 38*a*b^4*c^4 + 42*a^2*
b^2*c^5 + 12*a^3*c^6)*d*e^4 - (3*a*b^5*c^3 + 11*a^2*b^3*c^4 + 6*a^3*b*c^5)*e^5)*
x^6 + 10*(3*a^4*b^2*c^3 + 2*a^5*c^4)*d^3*e^2 - 10*(a^4*b^3*c^2 + 3*a^5*b*c^3)*d^
2*e^3 + (a^4*b^4*c + 12*a^5*b^2*c^2 + 6*a^6*c^3)*d*e^4 - (a^5*b^3*c + 3*a^6*b*c^
2)*e^5 + 4*(14*(b^3*c^6 + 3*a*b*c^7)*d^5 - 35*(b^4*c^5 + 3*a*b^2*c^6)*d^4*e + 10
*(3*b^5*c^4 + 11*a*b^3*c^5 + 6*a^2*b*c^6)*d^3*e^2 - 10*(b^6*c^3 + 6*a*b^4*c^4 +
9*a^2*b^2*c^5)*d^2*e^3 + (b^7*c^2 + 15*a*b^5*c^3 + 42*a^2*b^3*c^4 + 18*a^3*b*c^5
)*d*e^4 - (a*b^6*c^2 + 6*a^2*b^4*c^3 + 9*a^3*b^2*c^4)*e^5)*x^5 + (14*(b^4*c^5 +
12*a*b^2*c^6 + 6*a^2*c^7)*d^5 - 35*(b^5*c^4 + 12*a*b^3*c^5 + 6*a^2*b*c^6)*d^4*e
+ 10*(3*b^6*c^3 + 38*a*b^4*c^4 + 42*a^2*b^2*c^5 + 12*a^3*c^6)*d^3*e^2 - 10*(b^7*
c^2 + 15*a*b^5*c^3 + 42*a^2*b^3*c^4 + 18*a^3*b*c^5)*d^2*e^3 + (b^8*c + 24*a*b^6*
c^2 + 156*a^2*b^4*c^3 + 144*a^3*b^2*c^4 + 36*a^4*c^5)*d*e^4 - (a*b^7*c + 15*a^2*
b^5*c^2 + 42*a^3*b^3*c^3 + 18*a^4*b*c^4)*e^5)*x^4 + 4*(14*(a*b^3*c^5 + 3*a^2*b*c
^6)*d^5 - 35*(a*b^4*c^4 + 3*a^2*b^2*c^5)*d^4*e + 10*(3*a*b^5*c^3 + 11*a^2*b^3*c^
4 + 6*a^3*b*c^5)*d^3*e^2 - 10*(a*b^6*c^2 + 6*a^2*b^4*c^3 + 9*a^3*b^2*c^4)*d^2*e^
3 + (a*b^7*c + 15*a^2*b^5*c^2 + 42*a^3*b^3*c^3 + 18*a^4*b*c^4)*d*e^4 - (a^2*b^6*
c + 6*a^3*b^4*c^2 + 9*a^4*b^2*c^3)*e^5)*x^3 + 2*(14*(3*a^2*b^2*c^5 + 2*a^3*c^6)*
d^5 - 35*(3*a^2*b^3*c^4 + 2*a^3*b*c^5)*d^4*e + 10*(9*a^2*b^4*c^3 + 12*a^3*b^2*c^
4 + 4*a^4*c^5)*d^3*e^2 - 10*(3*a^2*b^5*c^2 + 11*a^3*b^3*c^3 + 6*a^4*b*c^4)*d^2*e
^3 + (3*a^2*b^6*c + 38*a^3*b^4*c^2 + 42*a^4*b^2*c^3 + 12*a^5*c^4)*d*e^4 - (3*a^3
*b^5*c + 11*a^4*b^3*c^2 + 6*a^5*b*c^3)*e^5)*x^2 + 4*(14*a^3*b*c^5*d^5 - 35*a^3*b
^2*c^4*d^4*e + 10*(3*a^3*b^3*c^3 + 2*a^4*b*c^4)*d^3*e^2 - 10*(a^3*b^4*c^2 + 3*a^
4*b^2*c^3)*d^2*e^3 + (a^3*b^5*c + 12*a^4*b^3*c^2 + 6*a^5*b*c^3)*d*e^4 - (a^4*b^4
*c + 3*a^5*b^2*c^2)*e^5)*x)*log((b^3 - 4*a*b*c + 2*(b^2*c - 4*a*c^2)*x + (2*c^2*
x^2 + 2*b*c*x + b^2 - 2*a*c)*sqrt(b^2 - 4*a*c))/(c*x^2 + b*x + a)) - (60*(14*c^8
*d^5 - 35*b*c^7*d^4*e + 10*(3*b^2*c^6 + 2*a*c^7)*d^3*e^2 - 10*(b^3*c^5 + 3*a*b*c
^6)*d^2*e^3 + (b^4*c^4 + 12*a*b^2*c^5 + 6*a^2*c^6)*d*e^4 - (a*b^3*c^4 + 3*a^2*b*
c^5)*e^5)*x^7 + 210*(14*b*c^7*d^5 - 35*b^2*c^6*d^4*e + 10*(3*b^3*c^5 + 2*a*b*c^6
)*d^3*e^2 - 10*(b^4*c^4 + 3*a*b^2*c^5)*d^2*e^3 + (b^5*c^3 + 12*a*b^3*c^4 + 6*a^2
*b*c^5)*d*e^4 - (a*b^4*c^3 + 3*a^2*b^2*c^4)*e^5)*x^6 - (3*b^7*c - 50*a*b^5*c^2 +
326*a^2*b^3*c^3 - 1116*a^3*b*c^4)*d^5 - 5*(a*b^6*c - 19*a^2*b^4*c^2 + 174*a^3*b
^2*c^3 + 384*a^4*c^4)*d^4*e - 10*(a^2*b^5*c - 28*a^3*b^3*c^2 - 324*a^4*b*c^3)*d^
3*e^2 - 10*(3*a^3*b^4*c + 166*a^4*b^2*c^2 + 128*a^5*c^3)*d^2*e^3 + 50*(5*a^4*b^3
*c + 22*a^5*b*c^2)*d*e^4 - (3*a^4*b^4 + 166*a^5*b^2*c + 128*a^6*c^2)*e^5 + 20*(1
4*(13*b^2*c^6 + 11*a*c^7)*d^5 - 35*(13*b^3*c^5 + 11*a*b*c^6)*d^4*e + 10*(39*b^4*
c^4 + 59*a*b^2*c^5 + 22*a^2*c^6)*d^3*e^2 - 10*(13*b^5*c^3 + 50*a*b^3*c^4 + 33*a^
2*b*c^5)*d^2*e^3 + (13*b^6*c^2 + 167*a*b^4*c^3 + 210*a^2*b^2*c^4 + 66*a^3*c^5)*d
*e^4 - (13*a*b^5*c^2 + 50*a^2*b^3*c^3 + 33*a^3*b*c^4)*e^5)*x^5 + (350*(5*b^3*c^5
+ 22*a*b*c^6)*d^5 - 875*(5*b^4*c^4 + 22*a*b^2*c^5)*d^4*e + 250*(15*b^5*c^3 + 76
*a*b^3*c^4 + 44*a^2*b*c^5)*d^3*e^2 - 250*(5*b^6*c^2 + 37*a*b^4*c^3 + 66*a^2*b^2*
c^4)*d^2*e^3 + 25*(5*b^7*c + 82*a*b^5*c^2 + 294*a^2*b^3*c^3 + 132*a^3*b*c^4)*d*e
^4 - (3*b^8 + 77*a*b^6*c + 1213*a^2*b^4*c^2 + 882*a^3*b^2*c^3 + 768*a^4*c^4)*e^5
)*x^4 + 4*(14*(3*b^4*c^4 + 101*a*b^2*c^5 + 73*a^2*c^6)*d^5 - 35*(3*b^5*c^3 + 101
*a*b^3*c^4 + 73*a^2*b*c^5)*d^4*e + 10*(9*b^6*c^2 + 309*a*b^4*c^3 + 421*a^2*b^2*c
^4 + 146*a^3*c^5)*d^3*e^2 - 10*(3*b^7*c + 110*a*b^5*c^2 + 376*a^2*b^3*c^3 + 219*
a^3*b*c^4)*d^2*e^3 + 5*(37*a*b^6*c + 203*a^2*b^4*c^2 + 450*a^3*b^2*c^3 - 66*a^4*
c^4)*d*e^4 - (3*a*b^7 + 110*a^2*b^5*c + 376*a^3*b^3*c^2 + 219*a^4*b*c^3)*e^5)*x^
3 - 2*(14*(b^5*c^3 - 28*a*b^3*c^4 - 219*a^2*b*c^5)*d^5 - 35*(b^6*c^2 - 28*a*b^4*
c^3 - 219*a^2*b^2*c^4)*d^4*e + 10*(3*b^7*c - 82*a*b^5*c^2 - 713*a^2*b^3*c^3 - 43
8*a^3*b*c^4)*d^3*e^2 + 10*(9*a*b^6*c + 399*a^2*b^4*c^2 + 401*a^3*b^2*c^3 + 256*a
^4*c^4)*d^2*e^3 - 5*(129*a^2*b^5*c + 508*a^3*b^3*c^2 + 314*a^4*b*c^3)*d*e^4 + (9
*a^2*b^6 + 399*a^3*b^4*c + 401*a^4*b^2*c^2 + 256*a^5*c^3)*e^5)*x^2 + 4*(2*(b^6*c
^2 - 19*a*b^4*c^3 + 174*a^2*b^2*c^4 + 279*a^3*c^5)*d^5 - 5*(b^7*c - 19*a*b^5*c^2
+ 174*a^2*b^3*c^3 + 279*a^3*b*c^4)*d^4*e - 10*(a*b^6*c - 28*a^2*b^4*c^2 - 279*a
^3*b^2*c^3 + 30*a^4*c^4)*d^3*e^2 - 10*(3*a^2*b^5*c + 151*a^3*b^3*c^2 + 83*a^4*b*
c^3)*d^2*e^3 + 5*(47*a^3*b^4*c + 184*a^4*b^2*c^2 - 18*a^5*c^3)*d*e^4 - (3*a^3*b^
5 + 151*a^4*b^3*c + 83*a^5*b*c^2)*e^5)*x)*sqrt(b^2 - 4*a*c))/((a^4*b^8*c - 16*a^
5*b^6*c^2 + 96*a^6*b^4*c^3 - 256*a^7*b^2*c^4 + 256*a^8*c^5 + (b^8*c^5 - 16*a*b^6
*c^6 + 96*a^2*b^4*c^7 - 256*a^3*b^2*c^8 + 256*a^4*c^9)*x^8 + 4*(b^9*c^4 - 16*a*b
^7*c^5 + 96*a^2*b^5*c^6 - 256*a^3*b^3*c^7 + 256*a^4*b*c^8)*x^7 + 2*(3*b^10*c^3 -
46*a*b^8*c^4 + 256*a^2*b^6*c^5 - 576*a^3*b^4*c^6 + 256*a^4*b^2*c^7 + 512*a^5*c^
8)*x^6 + 4*(b^11*c^2 - 13*a*b^9*c^3 + 48*a^2*b^7*c^4 + 32*a^3*b^5*c^5 - 512*a^4*
b^3*c^6 + 768*a^5*b*c^7)*x^5 + (b^12*c - 4*a*b^10*c^2 - 90*a^2*b^8*c^3 + 800*a^3
*b^6*c^4 - 2240*a^4*b^4*c^5 + 1536*a^5*b^2*c^6 + 1536*a^6*c^7)*x^4 + 4*(a*b^11*c
- 13*a^2*b^9*c^2 + 48*a^3*b^7*c^3 + 32*a^4*b^5*c^4 - 512*a^5*b^3*c^5 + 768*a^6*
b*c^6)*x^3 + 2*(3*a^2*b^10*c - 46*a^3*b^8*c^2 + 256*a^4*b^6*c^3 - 576*a^5*b^4*c^
4 + 256*a^6*b^2*c^5 + 512*a^7*c^6)*x^2 + 4*(a^3*b^9*c - 16*a^4*b^7*c^2 + 96*a^5*
b^5*c^3 - 256*a^6*b^3*c^4 + 256*a^7*b*c^5)*x)*sqrt(b^2 - 4*a*c)), 1/12*(120*(14*
a^4*c^5*d^5 - 35*a^4*b*c^4*d^4*e + (14*c^9*d^5 - 35*b*c^8*d^4*e + 10*(3*b^2*c^7
+ 2*a*c^8)*d^3*e^2 - 10*(b^3*c^6 + 3*a*b*c^7)*d^2*e^3 + (b^4*c^5 + 12*a*b^2*c^6
+ 6*a^2*c^7)*d*e^4 - (a*b^3*c^5 + 3*a^2*b*c^6)*e^5)*x^8 + 4*(14*b*c^8*d^5 - 35*b
^2*c^7*d^4*e + 10*(3*b^3*c^6 + 2*a*b*c^7)*d^3*e^2 - 10*(b^4*c^5 + 3*a*b^2*c^6)*d
^2*e^3 + (b^5*c^4 + 12*a*b^3*c^5 + 6*a^2*b*c^6)*d*e^4 - (a*b^4*c^4 + 3*a^2*b^2*c
^5)*e^5)*x^7 + 2*(14*(3*b^2*c^7 + 2*a*c^8)*d^5 - 35*(3*b^3*c^6 + 2*a*b*c^7)*d^4*
e + 10*(9*b^4*c^5 + 12*a*b^2*c^6 + 4*a^2*c^7)*d^3*e^2 - 10*(3*b^5*c^4 + 11*a*b^3
*c^5 + 6*a^2*b*c^6)*d^2*e^3 + (3*b^6*c^3 + 38*a*b^4*c^4 + 42*a^2*b^2*c^5 + 12*a^
3*c^6)*d*e^4 - (3*a*b^5*c^3 + 11*a^2*b^3*c^4 + 6*a^3*b*c^5)*e^5)*x^6 + 10*(3*a^4
*b^2*c^3 + 2*a^5*c^4)*d^3*e^2 - 10*(a^4*b^3*c^2 + 3*a^5*b*c^3)*d^2*e^3 + (a^4*b^
4*c + 12*a^5*b^2*c^2 + 6*a^6*c^3)*d*e^4 - (a^5*b^3*c + 3*a^6*b*c^2)*e^5 + 4*(14*
(b^3*c^6 + 3*a*b*c^7)*d^5 - 35*(b^4*c^5 + 3*a*b^2*c^6)*d^4*e + 10*(3*b^5*c^4 + 1
1*a*b^3*c^5 + 6*a^2*b*c^6)*d^3*e^2 - 10*(b^6*c^3 + 6*a*b^4*c^4 + 9*a^2*b^2*c^5)*
d^2*e^3 + (b^7*c^2 + 15*a*b^5*c^3 + 42*a^2*b^3*c^4 + 18*a^3*b*c^5)*d*e^4 - (a*b^
6*c^2 + 6*a^2*b^4*c^3 + 9*a^3*b^2*c^4)*e^5)*x^5 + (14*(b^4*c^5 + 12*a*b^2*c^6 +
6*a^2*c^7)*d^5 - 35*(b^5*c^4 + 12*a*b^3*c^5 + 6*a^2*b*c^6)*d^4*e + 10*(3*b^6*c^3
+ 38*a*b^4*c^4 + 42*a^2*b^2*c^5 + 12*a^3*c^6)*d^3*e^2 - 10*(b^7*c^2 + 15*a*b^5*
c^3 + 42*a^2*b^3*c^4 + 18*a^3*b*c^5)*d^2*e^3 + (b^8*c + 24*a*b^6*c^2 + 156*a^2*b
^4*c^3 + 144*a^3*b^2*c^4 + 36*a^4*c^5)*d*e^4 - (a*b^7*c + 15*a^2*b^5*c^2 + 42*a^
3*b^3*c^3 + 18*a^4*b*c^4)*e^5)*x^4 + 4*(14*(a*b^3*c^5 + 3*a^2*b*c^6)*d^5 - 35*(a
*b^4*c^4 + 3*a^2*b^2*c^5)*d^4*e + 10*(3*a*b^5*c^3 + 11*a^2*b^3*c^4 + 6*a^3*b*c^5
)*d^3*e^2 - 10*(a*b^6*c^2 + 6*a^2*b^4*c^3 + 9*a^3*b^2*c^4)*d^2*e^3 + (a*b^7*c +
15*a^2*b^5*c^2 + 42*a^3*b^3*c^3 + 18*a^4*b*c^4)*d*e^4 - (a^2*b^6*c + 6*a^3*b^4*c
^2 + 9*a^4*b^2*c^3)*e^5)*x^3 + 2*(14*(3*a^2*b^2*c^5 + 2*a^3*c^6)*d^5 - 35*(3*a^2
*b^3*c^4 + 2*a^3*b*c^5)*d^4*e + 10*(9*a^2*b^4*c^3 + 12*a^3*b^2*c^4 + 4*a^4*c^5)*
d^3*e^2 - 10*(3*a^2*b^5*c^2 + 11*a^3*b^3*c^3 + 6*a^4*b*c^4)*d^2*e^3 + (3*a^2*b^6
*c + 38*a^3*b^4*c^2 + 42*a^4*b^2*c^3 + 12*a^5*c^4)*d*e^4 - (3*a^3*b^5*c + 11*a^4
*b^3*c^2 + 6*a^5*b*c^3)*e^5)*x^2 + 4*(14*a^3*b*c^5*d^5 - 35*a^3*b^2*c^4*d^4*e +
10*(3*a^3*b^3*c^3 + 2*a^4*b*c^4)*d^3*e^2 - 10*(a^3*b^4*c^2 + 3*a^4*b^2*c^3)*d^2*
e^3 + (a^3*b^5*c + 12*a^4*b^3*c^2 + 6*a^5*b*c^3)*d*e^4 - (a^4*b^4*c + 3*a^5*b^2*
c^2)*e^5)*x)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + (60*(14*c^8
*d^5 - 35*b*c^7*d^4*e + 10*(3*b^2*c^6 + 2*a*c^7)*d^3*e^2 - 10*(b^3*c^5 + 3*a*b*c
^6)*d^2*e^3 + (b^4*c^4 + 12*a*b^2*c^5 + 6*a^2*c^6)*d*e^4 - (a*b^3*c^4 + 3*a^2*b*
c^5)*e^5)*x^7 + 210*(14*b*c^7*d^5 - 35*b^2*c^6*d^4*e + 10*(3*b^3*c^5 + 2*a*b*c^6
)*d^3*e^2 - 10*(b^4*c^4 + 3*a*b^2*c^5)*d^2*e^3 + (b^5*c^3 + 12*a*b^3*c^4 + 6*a^2
*b*c^5)*d*e^4 - (a*b^4*c^3 + 3*a^2*b^2*c^4)*e^5)*x^6 - (3*b^7*c - 50*a*b^5*c^2 +
326*a^2*b^3*c^3 - 1116*a^3*b*c^4)*d^5 - 5*(a*b^6*c - 19*a^2*b^4*c^2 + 174*a^3*b
^2*c^3 + 384*a^4*c^4)*d^4*e - 10*(a^2*b^5*c - 28*a^3*b^3*c^2 - 324*a^4*b*c^3)*d^
3*e^2 - 10*(3*a^3*b^4*c + 166*a^4*b^2*c^2 + 128*a^5*c^3)*d^2*e^3 + 50*(5*a^4*b^3
*c + 22*a^5*b*c^2)*d*e^4 - (3*a^4*b^4 + 166*a^5*b^2*c + 128*a^6*c^2)*e^5 + 20*(1
4*(13*b^2*c^6 + 11*a*c^7)*d^5 - 35*(13*b^3*c^5 + 11*a*b*c^6)*d^4*e + 10*(39*b^4*
c^4 + 59*a*b^2*c^5 + 22*a^2*c^6)*d^3*e^2 - 10*(13*b^5*c^3 + 50*a*b^3*c^4 + 33*a^
2*b*c^5)*d^2*e^3 + (13*b^6*c^2 + 167*a*b^4*c^3 + 210*a^2*b^2*c^4 + 66*a^3*c^5)*d
*e^4 - (13*a*b^5*c^2 + 50*a^2*b^3*c^3 + 33*a^3*b*c^4)*e^5)*x^5 + (350*(5*b^3*c^5
+ 22*a*b*c^6)*d^5 - 875*(5*b^4*c^4 + 22*a*b^2*c^5)*d^4*e + 250*(15*b^5*c^3 + 76
*a*b^3*c^4 + 44*a^2*b*c^5)*d^3*e^2 - 250*(5*b^6*c^2 + 37*a*b^4*c^3 + 66*a^2*b^2*
c^4)*d^2*e^3 + 25*(5*b^7*c + 82*a*b^5*c^2 + 294*a^2*b^3*c^3 + 132*a^3*b*c^4)*d*e
^4 - (3*b^8 + 77*a*b^6*c + 1213*a^2*b^4*c^2 + 882*a^3*b^2*c^3 + 768*a^4*c^4)*e^5
)*x^4 + 4*(14*(3*b^4*c^4 + 101*a*b^2*c^5 + 73*a^2*c^6)*d^5 - 35*(3*b^5*c^3 + 101
*a*b^3*c^4 + 73*a^2*b*c^5)*d^4*e + 10*(9*b^6*c^2 + 309*a*b^4*c^3 + 421*a^2*b^2*c
^4 + 146*a^3*c^5)*d^3*e^2 - 10*(3*b^7*c + 110*a*b^5*c^2 + 376*a^2*b^3*c^3 + 219*
a^3*b*c^4)*d^2*e^3 + 5*(37*a*b^6*c + 203*a^2*b^4*c^2 + 450*a^3*b^2*c^3 - 66*a^4*
c^4)*d*e^4 - (3*a*b^7 + 110*a^2*b^5*c + 376*a^3*b^3*c^2 + 219*a^4*b*c^3)*e^5)*x^
3 - 2*(14*(b^5*c^3 - 28*a*b^3*c^4 - 219*a^2*b*c^5)*d^5 - 35*(b^6*c^2 - 28*a*b^4*
c^3 - 219*a^2*b^2*c^4)*d^4*e + 10*(3*b^7*c - 82*a*b^5*c^2 - 713*a^2*b^3*c^3 - 43
8*a^3*b*c^4)*d^3*e^2 + 10*(9*a*b^6*c + 399*a^2*b^4*c^2 + 401*a^3*b^2*c^3 + 256*a
^4*c^4)*d^2*e^3 - 5*(129*a^2*b^5*c + 508*a^3*b^3*c^2 + 314*a^4*b*c^3)*d*e^4 + (9
*a^2*b^6 + 399*a^3*b^4*c + 401*a^4*b^2*c^2 + 256*a^5*c^3)*e^5)*x^2 + 4*(2*(b^6*c
^2 - 19*a*b^4*c^3 + 174*a^2*b^2*c^4 + 279*a^3*c^5)*d^5 - 5*(b^7*c - 19*a*b^5*c^2
+ 174*a^2*b^3*c^3 + 279*a^3*b*c^4)*d^4*e - 10*(a*b^6*c - 28*a^2*b^4*c^2 - 279*a
^3*b^2*c^3 + 30*a^4*c^4)*d^3*e^2 - 10*(3*a^2*b^5*c + 151*a^3*b^3*c^2 + 83*a^4*b*
c^3)*d^2*e^3 + 5*(47*a^3*b^4*c + 184*a^4*b^2*c^2 - 18*a^5*c^3)*d*e^4 - (3*a^3*b^
5 + 151*a^4*b^3*c + 83*a^5*b*c^2)*e^5)*x)*sqrt(-b^2 + 4*a*c))/((a^4*b^8*c - 16*a
^5*b^6*c^2 + 96*a^6*b^4*c^3 - 256*a^7*b^2*c^4 + 256*a^8*c^5 + (b^8*c^5 - 16*a*b^
6*c^6 + 96*a^2*b^4*c^7 - 256*a^3*b^2*c^8 + 256*a^4*c^9)*x^8 + 4*(b^9*c^4 - 16*a*
b^7*c^5 + 96*a^2*b^5*c^6 - 256*a^3*b^3*c^7 + 256*a^4*b*c^8)*x^7 + 2*(3*b^10*c^3
- 46*a*b^8*c^4 + 256*a^2*b^6*c^5 - 576*a^3*b^4*c^6 + 256*a^4*b^2*c^7 + 512*a^5*c
^8)*x^6 + 4*(b^11*c^2 - 13*a*b^9*c^3 + 48*a^2*b^7*c^4 + 32*a^3*b^5*c^5 - 512*a^4
*b^3*c^6 + 768*a^5*b*c^7)*x^5 + (b^12*c - 4*a*b^10*c^2 - 90*a^2*b^8*c^3 + 800*a^
3*b^6*c^4 - 2240*a^4*b^4*c^5 + 1536*a^5*b^2*c^6 + 1536*a^6*c^7)*x^4 + 4*(a*b^11*
c - 13*a^2*b^9*c^2 + 48*a^3*b^7*c^3 + 32*a^4*b^5*c^4 - 512*a^5*b^3*c^5 + 768*a^6
*b*c^6)*x^3 + 2*(3*a^2*b^10*c - 46*a^3*b^8*c^2 + 256*a^4*b^6*c^3 - 576*a^5*b^4*c
^4 + 256*a^6*b^2*c^5 + 512*a^7*c^6)*x^2 + 4*(a^3*b^9*c - 16*a^4*b^7*c^2 + 96*a^5
*b^5*c^3 - 256*a^6*b^3*c^4 + 256*a^7*b*c^5)*x)*sqrt(-b^2 + 4*a*c))]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.21596, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done