3.2291 \(\int \frac{(d+e x)^{7/2}}{\left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=751 \[ \frac{\sqrt{d+e x} \left (x (2 c d-b e) \left (-4 c e (3 b d-2 a e)+b^2 e^2+12 c^2 d^2\right )+b^2 \left (-\left (a e^3+11 c d^2 e\right )\right )+12 b c d \left (3 a e^2+c d^2\right )-4 a c e \left (5 a e^2+7 c d^2\right )\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{(d+e x)^{5/2} (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{\left (-8 c^3 d^2 e \left (-3 d \sqrt{b^2-4 a c}-19 a e+24 b d\right )+2 c^2 e^2 \left (-2 b d \left (9 d \sqrt{b^2-4 a c}+38 a e\right )+4 a e \left (4 d \sqrt{b^2-4 a c}+5 a e\right )+53 b^2 d^2\right )-2 b c e^3 \left (-5 b d \sqrt{b^2-4 a c}+8 a e \sqrt{b^2-4 a c}-9 a b e+5 b^2 d\right )-b^3 e^4 \left (b-\sqrt{b^2-4 a c}\right )+96 c^4 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{4 \sqrt{2} c^{3/2} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}+\frac{\left (-8 c^3 d^2 e \left (3 d \sqrt{b^2-4 a c}-19 a e+24 b d\right )+2 c^2 e^2 \left (2 b d \left (9 d \sqrt{b^2-4 a c}-38 a e\right )-4 a e \left (4 d \sqrt{b^2-4 a c}-5 a e\right )+53 b^2 d^2\right )-2 b c e^3 \left (5 b d \sqrt{b^2-4 a c}-8 a e \sqrt{b^2-4 a c}-9 a b e+5 b^2 d\right )-b^3 e^4 \left (\sqrt{b^2-4 a c}+b\right )+96 c^4 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{4 \sqrt{2} c^{3/2} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} \]

[Out]

-((d + e*x)^(5/2)*(b*d - 2*a*e + (2*c*d - b*e)*x))/(2*(b^2 - 4*a*c)*(a + b*x + c
*x^2)^2) + (Sqrt[d + e*x]*(12*b*c*d*(c*d^2 + 3*a*e^2) - 4*a*c*e*(7*c*d^2 + 5*a*e
^2) - b^2*(11*c*d^2*e + a*e^3) + (2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*
b*d - 2*a*e))*x))/(4*c*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)) - ((96*c^4*d^4 - b^3*(
b - Sqrt[b^2 - 4*a*c])*e^4 - 8*c^3*d^2*e*(24*b*d - 3*Sqrt[b^2 - 4*a*c]*d - 19*a*
e) - 2*b*c*e^3*(5*b^2*d - 5*b*Sqrt[b^2 - 4*a*c]*d - 9*a*b*e + 8*a*Sqrt[b^2 - 4*a
*c]*e) + 2*c^2*e^2*(53*b^2*d^2 + 4*a*e*(4*Sqrt[b^2 - 4*a*c]*d + 5*a*e) - 2*b*d*(
9*Sqrt[b^2 - 4*a*c]*d + 38*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2
*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(4*Sqrt[2]*c^(3/2)*(b^2 - 4*a*c)^(5/2)*Sqrt[
2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) + ((96*c^4*d^4 - b^3*(b + Sqrt[b^2 - 4*a*c])
*e^4 - 8*c^3*d^2*e*(24*b*d + 3*Sqrt[b^2 - 4*a*c]*d - 19*a*e) - 2*b*c*e^3*(5*b^2*
d + 5*b*Sqrt[b^2 - 4*a*c]*d - 9*a*b*e - 8*a*Sqrt[b^2 - 4*a*c]*e) + 2*c^2*e^2*(53
*b^2*d^2 + 2*b*d*(9*Sqrt[b^2 - 4*a*c]*d - 38*a*e) - 4*a*e*(4*Sqrt[b^2 - 4*a*c]*d
- 5*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 -
4*a*c])*e]])/(4*Sqrt[2]*c^(3/2)*(b^2 - 4*a*c)^(5/2)*Sqrt[2*c*d - (b + Sqrt[b^2
- 4*a*c])*e])

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Rubi [A]  time = 21.6158, antiderivative size = 751, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{\sqrt{d+e x} \left (x (2 c d-b e) \left (-4 c e (3 b d-2 a e)+b^2 e^2+12 c^2 d^2\right )+b^2 \left (-\left (a e^3+11 c d^2 e\right )\right )+12 b c d \left (3 a e^2+c d^2\right )-4 a c e \left (5 a e^2+7 c d^2\right )\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{(d+e x)^{5/2} (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{\left (-8 c^3 d^2 e \left (-3 d \sqrt{b^2-4 a c}-19 a e+24 b d\right )+2 c^2 e^2 \left (-2 b d \left (9 d \sqrt{b^2-4 a c}+38 a e\right )+4 a e \left (4 d \sqrt{b^2-4 a c}+5 a e\right )+53 b^2 d^2\right )-2 b c e^3 \left (-5 b d \sqrt{b^2-4 a c}+8 a e \sqrt{b^2-4 a c}-9 a b e+5 b^2 d\right )-b^3 e^4 \left (b-\sqrt{b^2-4 a c}\right )+96 c^4 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{4 \sqrt{2} c^{3/2} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}+\frac{\left (-8 c^3 d^2 e \left (3 d \sqrt{b^2-4 a c}-19 a e+24 b d\right )+2 c^2 e^2 \left (2 b d \left (9 d \sqrt{b^2-4 a c}-38 a e\right )-4 a e \left (4 d \sqrt{b^2-4 a c}-5 a e\right )+53 b^2 d^2\right )-2 b c e^3 \left (5 b d \sqrt{b^2-4 a c}-8 a e \sqrt{b^2-4 a c}-9 a b e+5 b^2 d\right )-b^3 e^4 \left (\sqrt{b^2-4 a c}+b\right )+96 c^4 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{4 \sqrt{2} c^{3/2} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(7/2)/(a + b*x + c*x^2)^3,x]

[Out]

-((d + e*x)^(5/2)*(b*d - 2*a*e + (2*c*d - b*e)*x))/(2*(b^2 - 4*a*c)*(a + b*x + c
*x^2)^2) + (Sqrt[d + e*x]*(12*b*c*d*(c*d^2 + 3*a*e^2) - 4*a*c*e*(7*c*d^2 + 5*a*e
^2) - b^2*(11*c*d^2*e + a*e^3) + (2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*
b*d - 2*a*e))*x))/(4*c*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)) - ((96*c^4*d^4 - b^3*(
b - Sqrt[b^2 - 4*a*c])*e^4 - 8*c^3*d^2*e*(24*b*d - 3*Sqrt[b^2 - 4*a*c]*d - 19*a*
e) - 2*b*c*e^3*(5*b^2*d - 5*b*Sqrt[b^2 - 4*a*c]*d - 9*a*b*e + 8*a*Sqrt[b^2 - 4*a
*c]*e) + 2*c^2*e^2*(53*b^2*d^2 + 4*a*e*(4*Sqrt[b^2 - 4*a*c]*d + 5*a*e) - 2*b*d*(
9*Sqrt[b^2 - 4*a*c]*d + 38*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2
*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(4*Sqrt[2]*c^(3/2)*(b^2 - 4*a*c)^(5/2)*Sqrt[
2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]) + ((96*c^4*d^4 - b^3*(b + Sqrt[b^2 - 4*a*c])
*e^4 - 8*c^3*d^2*e*(24*b*d + 3*Sqrt[b^2 - 4*a*c]*d - 19*a*e) - 2*b*c*e^3*(5*b^2*
d + 5*b*Sqrt[b^2 - 4*a*c]*d - 9*a*b*e - 8*a*Sqrt[b^2 - 4*a*c]*e) + 2*c^2*e^2*(53
*b^2*d^2 + 2*b*d*(9*Sqrt[b^2 - 4*a*c]*d - 38*a*e) - 4*a*e*(4*Sqrt[b^2 - 4*a*c]*d
- 5*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 -
4*a*c])*e]])/(4*Sqrt[2]*c^(3/2)*(b^2 - 4*a*c)^(5/2)*Sqrt[2*c*d - (b + Sqrt[b^2
- 4*a*c])*e])

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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)**(7/2)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 6.39948, size = 975, normalized size = 1.3 \[ \sqrt{d+e x} \left (\frac{b c^2 d^3+2 c^3 x d^3-6 a c^2 e d^2-3 b c^2 e x d^2+3 a b c e^2 d-6 a c^2 e^2 x d+3 b^2 c e^2 x d-a b^2 e^3+2 a^2 c e^3-b^3 e^3 x+3 a b c e^3 x}{2 c^2 \left (4 a c-b^2\right ) \left (c x^2+b x+a\right )^2}+\frac{-2 e^3 b^4+6 c d e^2 b^3+c e^3 x b^3+11 a c e^3 b^2-19 c^2 d^2 e b^2+10 c^2 d e^2 x b^2+12 c^3 d^3 b+12 a c^2 d e^2 b-16 a c^2 e^3 x b-36 c^3 d^2 e x b-36 a^2 c^2 e^3+4 a c^3 d^2 e+24 c^4 d^3 x+32 a c^3 d e^2 x}{4 c^2 \left (4 a c-b^2\right )^2 \left (c x^2+b x+a\right )}\right )-\frac{\left (-96 c^4 d^4+192 b c^3 e d^3+24 c^3 \sqrt{b^2-4 a c} e d^3-152 a c^3 e^2 d^2-106 b^2 c^2 e^2 d^2-36 b c^2 \sqrt{b^2-4 a c} e^2 d^2+152 a b c^2 e^3 d+10 b^3 c e^3 d+32 a c^2 \sqrt{b^2-4 a c} e^3 d+10 b^2 c \sqrt{b^2-4 a c} e^3 d+b^4 e^4-40 a^2 c^2 e^4-18 a b^2 c e^4+b^3 \sqrt{b^2-4 a c} e^4-16 a b c \sqrt{b^2-4 a c} e^4\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-b e-\sqrt{b^2-4 a c} e}}\right )}{4 \sqrt{2} c^{3/2} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-b e-\sqrt{b^2-4 a c} e}}-\frac{\left (96 c^4 d^4-192 b c^3 e d^3+24 c^3 \sqrt{b^2-4 a c} e d^3+152 a c^3 e^2 d^2+106 b^2 c^2 e^2 d^2-36 b c^2 \sqrt{b^2-4 a c} e^2 d^2-152 a b c^2 e^3 d-10 b^3 c e^3 d+32 a c^2 \sqrt{b^2-4 a c} e^3 d+10 b^2 c \sqrt{b^2-4 a c} e^3 d-b^4 e^4+40 a^2 c^2 e^4+18 a b^2 c e^4+b^3 \sqrt{b^2-4 a c} e^4-16 a b c \sqrt{b^2-4 a c} e^4\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-b e+\sqrt{b^2-4 a c} e}}\right )}{4 \sqrt{2} c^{3/2} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-b e+\sqrt{b^2-4 a c} e}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(7/2)/(a + b*x + c*x^2)^3,x]

[Out]

Sqrt[d + e*x]*((b*c^2*d^3 - 6*a*c^2*d^2*e + 3*a*b*c*d*e^2 - a*b^2*e^3 + 2*a^2*c*
e^3 + 2*c^3*d^3*x - 3*b*c^2*d^2*e*x + 3*b^2*c*d*e^2*x - 6*a*c^2*d*e^2*x - b^3*e^
3*x + 3*a*b*c*e^3*x)/(2*c^2*(-b^2 + 4*a*c)*(a + b*x + c*x^2)^2) + (12*b*c^3*d^3
- 19*b^2*c^2*d^2*e + 4*a*c^3*d^2*e + 6*b^3*c*d*e^2 + 12*a*b*c^2*d*e^2 - 2*b^4*e^
3 + 11*a*b^2*c*e^3 - 36*a^2*c^2*e^3 + 24*c^4*d^3*x - 36*b*c^3*d^2*e*x + 10*b^2*c
^2*d*e^2*x + 32*a*c^3*d*e^2*x + b^3*c*e^3*x - 16*a*b*c^2*e^3*x)/(4*c^2*(-b^2 + 4
*a*c)^2*(a + b*x + c*x^2))) - ((-96*c^4*d^4 + 192*b*c^3*d^3*e + 24*c^3*Sqrt[b^2
- 4*a*c]*d^3*e - 106*b^2*c^2*d^2*e^2 - 152*a*c^3*d^2*e^2 - 36*b*c^2*Sqrt[b^2 - 4
*a*c]*d^2*e^2 + 10*b^3*c*d*e^3 + 152*a*b*c^2*d*e^3 + 10*b^2*c*Sqrt[b^2 - 4*a*c]*
d*e^3 + 32*a*c^2*Sqrt[b^2 - 4*a*c]*d*e^3 + b^4*e^4 - 18*a*b^2*c*e^4 - 40*a^2*c^2
*e^4 + b^3*Sqrt[b^2 - 4*a*c]*e^4 - 16*a*b*c*Sqrt[b^2 - 4*a*c]*e^4)*ArcTanh[(Sqrt
[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e]])/(4*Sqrt[2]*
c^(3/2)*(b^2 - 4*a*c)^(5/2)*Sqrt[2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e]) - ((96*c^4*
d^4 - 192*b*c^3*d^3*e + 24*c^3*Sqrt[b^2 - 4*a*c]*d^3*e + 106*b^2*c^2*d^2*e^2 + 1
52*a*c^3*d^2*e^2 - 36*b*c^2*Sqrt[b^2 - 4*a*c]*d^2*e^2 - 10*b^3*c*d*e^3 - 152*a*b
*c^2*d*e^3 + 10*b^2*c*Sqrt[b^2 - 4*a*c]*d*e^3 + 32*a*c^2*Sqrt[b^2 - 4*a*c]*d*e^3
- b^4*e^4 + 18*a*b^2*c*e^4 + 40*a^2*c^2*e^4 + b^3*Sqrt[b^2 - 4*a*c]*e^4 - 16*a*
b*c*Sqrt[b^2 - 4*a*c]*e^4)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d -
b*e + Sqrt[b^2 - 4*a*c]*e]])/(4*Sqrt[2]*c^(3/2)*(b^2 - 4*a*c)^(5/2)*Sqrt[2*c*d -
b*e + Sqrt[b^2 - 4*a*c]*e])

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Maple [B]  time = 0.255, size = 31230, normalized size = 41.6 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{\frac{7}{2}}}{{\left (c x^{2} + b x + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(7/2)/(c*x^2 + b*x + a)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.83463, size = 11853, normalized size = 15.78 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(7/2)/(c*x^2 + b*x + a)^3,x, algorithm="fricas")

[Out]

-1/8*(sqrt(1/2)*(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3 + (b^4*c^3 - 8*a*b^2*c^4
+ 16*a^2*c^5)*x^4 + 2*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*x^3 + (b^6*c - 6*a
*b^4*c^2 + 32*a^3*c^4)*x^2 + 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*x)*sqrt(
(4608*c^7*d^7 - 16128*b*c^6*d^6*e + 672*(31*b^2*c^5 + 20*a*c^6)*d^5*e^2 - 1680*(
7*b^3*c^4 + 20*a*b*c^5)*d^4*e^3 + 70*(35*b^4*c^3 + 392*a*b^2*c^4 + 176*a^2*c^5)*
d^3*e^4 + 21*(b^5*c^2 - 360*a*b^3*c^3 - 880*a^2*b*c^4)*d^2*e^5 - 21*(b^6*c - 10*
a*b^4*c^2 - 320*a^2*b^2*c^3 - 160*a^3*c^4)*d*e^6 - (b^7 - 35*a*b^5*c + 280*a^2*b
^3*c^2 + 1680*a^3*b*c^3)*e^7 + (b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*
a^3*b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8)*sqrt((441*c^4*d^4*e^10 - 882*b*c^
3*d^3*e^11 + 21*(19*b^2*c^2 + 50*a*c^3)*d^2*e^12 + 42*(b^3*c - 25*a*b*c^2)*d*e^1
3 + (b^4 - 50*a*b^2*c + 625*a^2*c^2)*e^14)/(b^10*c^6 - 20*a*b^8*c^7 + 160*a^2*b^
6*c^8 - 640*a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)))/(b^10*c^3 - 20*a*
b^8*c^4 + 160*a^2*b^6*c^5 - 640*a^3*b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8))*
log(1/2*sqrt(1/2)*(504*(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)*d^
4*e^6 - 1008*(b^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*d^3*e^7 +
3*(167*b^8*c^2 - 1664*a*b^6*c^3 + 3936*a^2*b^4*c^4 + 5632*a^3*b^2*c^5 - 21760*a^
4*c^6)*d^2*e^8 + 3*(b^9*c - 352*a*b^7*c^2 + 4128*a^2*b^5*c^3 - 16384*a^3*b^3*c^4
+ 21760*a^4*b*c^5)*d*e^9 - (b^10 - 17*a*b^8*c - 392*a^2*b^6*c^2 + 5696*a^3*b^4*
c^3 - 23680*a^4*b^2*c^4 + 32000*a^5*c^5)*e^10 - (96*(b^10*c^6 - 20*a*b^8*c^7 + 1
60*a^2*b^6*c^8 - 640*a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)*d^3 - 144*
(b^11*c^5 - 20*a*b^9*c^6 + 160*a^2*b^7*c^7 - 640*a^3*b^5*c^8 + 1280*a^4*b^3*c^9
- 1024*a^5*b*c^10)*d^2*e + 2*(23*b^12*c^4 - 408*a*b^10*c^5 + 2640*a^2*b^8*c^6 -
6400*a^3*b^6*c^7 - 3840*a^4*b^4*c^8 + 43008*a^5*b^2*c^9 - 53248*a^6*c^10)*d*e^2
+ (b^13*c^3 - 72*a*b^11*c^4 + 1200*a^2*b^9*c^5 - 8960*a^3*b^7*c^6 + 34560*a^4*b^
5*c^7 - 67584*a^5*b^3*c^8 + 53248*a^6*b*c^9)*e^3)*sqrt((441*c^4*d^4*e^10 - 882*b
*c^3*d^3*e^11 + 21*(19*b^2*c^2 + 50*a*c^3)*d^2*e^12 + 42*(b^3*c - 25*a*b*c^2)*d*
e^13 + (b^4 - 50*a*b^2*c + 625*a^2*c^2)*e^14)/(b^10*c^6 - 20*a*b^8*c^7 + 160*a^2
*b^6*c^8 - 640*a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)))*sqrt((4608*c^7
*d^7 - 16128*b*c^6*d^6*e + 672*(31*b^2*c^5 + 20*a*c^6)*d^5*e^2 - 1680*(7*b^3*c^4
+ 20*a*b*c^5)*d^4*e^3 + 70*(35*b^4*c^3 + 392*a*b^2*c^4 + 176*a^2*c^5)*d^3*e^4 +
21*(b^5*c^2 - 360*a*b^3*c^3 - 880*a^2*b*c^4)*d^2*e^5 - 21*(b^6*c - 10*a*b^4*c^2
- 320*a^2*b^2*c^3 - 160*a^3*c^4)*d*e^6 - (b^7 - 35*a*b^5*c + 280*a^2*b^3*c^2 +
1680*a^3*b*c^3)*e^7 + (b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*a^3*b^4*c
^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8)*sqrt((441*c^4*d^4*e^10 - 882*b*c^3*d^3*e^1
1 + 21*(19*b^2*c^2 + 50*a*c^3)*d^2*e^12 + 42*(b^3*c - 25*a*b*c^2)*d*e^13 + (b^4
- 50*a*b^2*c + 625*a^2*c^2)*e^14)/(b^10*c^6 - 20*a*b^8*c^7 + 160*a^2*b^6*c^8 - 6
40*a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)))/(b^10*c^3 - 20*a*b^8*c^4 +
160*a^2*b^6*c^5 - 640*a^3*b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8)) + (48384*
c^7*d^8*e^5 - 193536*b*c^6*d^7*e^6 + 432*(683*b^2*c^5 + 404*a*c^6)*d^6*e^7 - 432
*(481*b^3*c^4 + 1212*a*b*c^5)*d^5*e^8 + 9*(6841*b^4*c^3 + 60712*a*b^2*c^4 + 2401
6*a^2*c^5)*d^4*e^9 - 18*(145*b^5*c^2 + 12232*a*b^3*c^3 + 24016*a^2*b*c^4)*d^3*e^
10 - 2*(518*b^6*c - 10131*a*b^4*c^2 - 124608*a^2*b^2*c^3 - 50000*a^3*c^4)*d^2*e^
11 - (35*b^7 - 2562*a*b^5*c + 33072*a^2*b^3*c^2 + 100000*a^3*b*c^3)*d*e^12 + (35
*a*b^6 - 1491*a^2*b^4*c + 15000*a^3*b^2*c^2 + 10000*a^4*c^3)*e^13)*sqrt(e*x + d)
) - sqrt(1/2)*(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3 + (b^4*c^3 - 8*a*b^2*c^4 +
16*a^2*c^5)*x^4 + 2*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*x^3 + (b^6*c - 6*a*b
^4*c^2 + 32*a^3*c^4)*x^2 + 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*x)*sqrt((4
608*c^7*d^7 - 16128*b*c^6*d^6*e + 672*(31*b^2*c^5 + 20*a*c^6)*d^5*e^2 - 1680*(7*
b^3*c^4 + 20*a*b*c^5)*d^4*e^3 + 70*(35*b^4*c^3 + 392*a*b^2*c^4 + 176*a^2*c^5)*d^
3*e^4 + 21*(b^5*c^2 - 360*a*b^3*c^3 - 880*a^2*b*c^4)*d^2*e^5 - 21*(b^6*c - 10*a*
b^4*c^2 - 320*a^2*b^2*c^3 - 160*a^3*c^4)*d*e^6 - (b^7 - 35*a*b^5*c + 280*a^2*b^3
*c^2 + 1680*a^3*b*c^3)*e^7 + (b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*a^
3*b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8)*sqrt((441*c^4*d^4*e^10 - 882*b*c^3*
d^3*e^11 + 21*(19*b^2*c^2 + 50*a*c^3)*d^2*e^12 + 42*(b^3*c - 25*a*b*c^2)*d*e^13
+ (b^4 - 50*a*b^2*c + 625*a^2*c^2)*e^14)/(b^10*c^6 - 20*a*b^8*c^7 + 160*a^2*b^6*
c^8 - 640*a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)))/(b^10*c^3 - 20*a*b^
8*c^4 + 160*a^2*b^6*c^5 - 640*a^3*b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8))*lo
g(-1/2*sqrt(1/2)*(504*(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)*d^4
*e^6 - 1008*(b^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*d^3*e^7 + 3
*(167*b^8*c^2 - 1664*a*b^6*c^3 + 3936*a^2*b^4*c^4 + 5632*a^3*b^2*c^5 - 21760*a^4
*c^6)*d^2*e^8 + 3*(b^9*c - 352*a*b^7*c^2 + 4128*a^2*b^5*c^3 - 16384*a^3*b^3*c^4
+ 21760*a^4*b*c^5)*d*e^9 - (b^10 - 17*a*b^8*c - 392*a^2*b^6*c^2 + 5696*a^3*b^4*c
^3 - 23680*a^4*b^2*c^4 + 32000*a^5*c^5)*e^10 - (96*(b^10*c^6 - 20*a*b^8*c^7 + 16
0*a^2*b^6*c^8 - 640*a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)*d^3 - 144*(
b^11*c^5 - 20*a*b^9*c^6 + 160*a^2*b^7*c^7 - 640*a^3*b^5*c^8 + 1280*a^4*b^3*c^9 -
1024*a^5*b*c^10)*d^2*e + 2*(23*b^12*c^4 - 408*a*b^10*c^5 + 2640*a^2*b^8*c^6 - 6
400*a^3*b^6*c^7 - 3840*a^4*b^4*c^8 + 43008*a^5*b^2*c^9 - 53248*a^6*c^10)*d*e^2 +
(b^13*c^3 - 72*a*b^11*c^4 + 1200*a^2*b^9*c^5 - 8960*a^3*b^7*c^6 + 34560*a^4*b^5
*c^7 - 67584*a^5*b^3*c^8 + 53248*a^6*b*c^9)*e^3)*sqrt((441*c^4*d^4*e^10 - 882*b*
c^3*d^3*e^11 + 21*(19*b^2*c^2 + 50*a*c^3)*d^2*e^12 + 42*(b^3*c - 25*a*b*c^2)*d*e
^13 + (b^4 - 50*a*b^2*c + 625*a^2*c^2)*e^14)/(b^10*c^6 - 20*a*b^8*c^7 + 160*a^2*
b^6*c^8 - 640*a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)))*sqrt((4608*c^7*
d^7 - 16128*b*c^6*d^6*e + 672*(31*b^2*c^5 + 20*a*c^6)*d^5*e^2 - 1680*(7*b^3*c^4
+ 20*a*b*c^5)*d^4*e^3 + 70*(35*b^4*c^3 + 392*a*b^2*c^4 + 176*a^2*c^5)*d^3*e^4 +
21*(b^5*c^2 - 360*a*b^3*c^3 - 880*a^2*b*c^4)*d^2*e^5 - 21*(b^6*c - 10*a*b^4*c^2
- 320*a^2*b^2*c^3 - 160*a^3*c^4)*d*e^6 - (b^7 - 35*a*b^5*c + 280*a^2*b^3*c^2 + 1
680*a^3*b*c^3)*e^7 + (b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*a^3*b^4*c^
6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8)*sqrt((441*c^4*d^4*e^10 - 882*b*c^3*d^3*e^11
+ 21*(19*b^2*c^2 + 50*a*c^3)*d^2*e^12 + 42*(b^3*c - 25*a*b*c^2)*d*e^13 + (b^4 -
50*a*b^2*c + 625*a^2*c^2)*e^14)/(b^10*c^6 - 20*a*b^8*c^7 + 160*a^2*b^6*c^8 - 64
0*a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)))/(b^10*c^3 - 20*a*b^8*c^4 +
160*a^2*b^6*c^5 - 640*a^3*b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8)) + (48384*c
^7*d^8*e^5 - 193536*b*c^6*d^7*e^6 + 432*(683*b^2*c^5 + 404*a*c^6)*d^6*e^7 - 432*
(481*b^3*c^4 + 1212*a*b*c^5)*d^5*e^8 + 9*(6841*b^4*c^3 + 60712*a*b^2*c^4 + 24016
*a^2*c^5)*d^4*e^9 - 18*(145*b^5*c^2 + 12232*a*b^3*c^3 + 24016*a^2*b*c^4)*d^3*e^1
0 - 2*(518*b^6*c - 10131*a*b^4*c^2 - 124608*a^2*b^2*c^3 - 50000*a^3*c^4)*d^2*e^1
1 - (35*b^7 - 2562*a*b^5*c + 33072*a^2*b^3*c^2 + 100000*a^3*b*c^3)*d*e^12 + (35*
a*b^6 - 1491*a^2*b^4*c + 15000*a^3*b^2*c^2 + 10000*a^4*c^3)*e^13)*sqrt(e*x + d))
+ sqrt(1/2)*(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3 + (b^4*c^3 - 8*a*b^2*c^4 +
16*a^2*c^5)*x^4 + 2*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*x^3 + (b^6*c - 6*a*b^
4*c^2 + 32*a^3*c^4)*x^2 + 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*x)*sqrt((46
08*c^7*d^7 - 16128*b*c^6*d^6*e + 672*(31*b^2*c^5 + 20*a*c^6)*d^5*e^2 - 1680*(7*b
^3*c^4 + 20*a*b*c^5)*d^4*e^3 + 70*(35*b^4*c^3 + 392*a*b^2*c^4 + 176*a^2*c^5)*d^3
*e^4 + 21*(b^5*c^2 - 360*a*b^3*c^3 - 880*a^2*b*c^4)*d^2*e^5 - 21*(b^6*c - 10*a*b
^4*c^2 - 320*a^2*b^2*c^3 - 160*a^3*c^4)*d*e^6 - (b^7 - 35*a*b^5*c + 280*a^2*b^3*
c^2 + 1680*a^3*b*c^3)*e^7 - (b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*a^3
*b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8)*sqrt((441*c^4*d^4*e^10 - 882*b*c^3*d
^3*e^11 + 21*(19*b^2*c^2 + 50*a*c^3)*d^2*e^12 + 42*(b^3*c - 25*a*b*c^2)*d*e^13 +
(b^4 - 50*a*b^2*c + 625*a^2*c^2)*e^14)/(b^10*c^6 - 20*a*b^8*c^7 + 160*a^2*b^6*c
^8 - 640*a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)))/(b^10*c^3 - 20*a*b^8
*c^4 + 160*a^2*b^6*c^5 - 640*a^3*b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8))*log
(1/2*sqrt(1/2)*(504*(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)*d^4*e
^6 - 1008*(b^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*d^3*e^7 + 3*(
167*b^8*c^2 - 1664*a*b^6*c^3 + 3936*a^2*b^4*c^4 + 5632*a^3*b^2*c^5 - 21760*a^4*c
^6)*d^2*e^8 + 3*(b^9*c - 352*a*b^7*c^2 + 4128*a^2*b^5*c^3 - 16384*a^3*b^3*c^4 +
21760*a^4*b*c^5)*d*e^9 - (b^10 - 17*a*b^8*c - 392*a^2*b^6*c^2 + 5696*a^3*b^4*c^3
- 23680*a^4*b^2*c^4 + 32000*a^5*c^5)*e^10 + (96*(b^10*c^6 - 20*a*b^8*c^7 + 160*
a^2*b^6*c^8 - 640*a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)*d^3 - 144*(b^
11*c^5 - 20*a*b^9*c^6 + 160*a^2*b^7*c^7 - 640*a^3*b^5*c^8 + 1280*a^4*b^3*c^9 - 1
024*a^5*b*c^10)*d^2*e + 2*(23*b^12*c^4 - 408*a*b^10*c^5 + 2640*a^2*b^8*c^6 - 640
0*a^3*b^6*c^7 - 3840*a^4*b^4*c^8 + 43008*a^5*b^2*c^9 - 53248*a^6*c^10)*d*e^2 + (
b^13*c^3 - 72*a*b^11*c^4 + 1200*a^2*b^9*c^5 - 8960*a^3*b^7*c^6 + 34560*a^4*b^5*c
^7 - 67584*a^5*b^3*c^8 + 53248*a^6*b*c^9)*e^3)*sqrt((441*c^4*d^4*e^10 - 882*b*c^
3*d^3*e^11 + 21*(19*b^2*c^2 + 50*a*c^3)*d^2*e^12 + 42*(b^3*c - 25*a*b*c^2)*d*e^1
3 + (b^4 - 50*a*b^2*c + 625*a^2*c^2)*e^14)/(b^10*c^6 - 20*a*b^8*c^7 + 160*a^2*b^
6*c^8 - 640*a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)))*sqrt((4608*c^7*d^
7 - 16128*b*c^6*d^6*e + 672*(31*b^2*c^5 + 20*a*c^6)*d^5*e^2 - 1680*(7*b^3*c^4 +
20*a*b*c^5)*d^4*e^3 + 70*(35*b^4*c^3 + 392*a*b^2*c^4 + 176*a^2*c^5)*d^3*e^4 + 21
*(b^5*c^2 - 360*a*b^3*c^3 - 880*a^2*b*c^4)*d^2*e^5 - 21*(b^6*c - 10*a*b^4*c^2 -
320*a^2*b^2*c^3 - 160*a^3*c^4)*d*e^6 - (b^7 - 35*a*b^5*c + 280*a^2*b^3*c^2 + 168
0*a^3*b*c^3)*e^7 - (b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*a^3*b^4*c^6
+ 1280*a^4*b^2*c^7 - 1024*a^5*c^8)*sqrt((441*c^4*d^4*e^10 - 882*b*c^3*d^3*e^11 +
21*(19*b^2*c^2 + 50*a*c^3)*d^2*e^12 + 42*(b^3*c - 25*a*b*c^2)*d*e^13 + (b^4 - 5
0*a*b^2*c + 625*a^2*c^2)*e^14)/(b^10*c^6 - 20*a*b^8*c^7 + 160*a^2*b^6*c^8 - 640*
a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)))/(b^10*c^3 - 20*a*b^8*c^4 + 16
0*a^2*b^6*c^5 - 640*a^3*b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8)) + (48384*c^7
*d^8*e^5 - 193536*b*c^6*d^7*e^6 + 432*(683*b^2*c^5 + 404*a*c^6)*d^6*e^7 - 432*(4
81*b^3*c^4 + 1212*a*b*c^5)*d^5*e^8 + 9*(6841*b^4*c^3 + 60712*a*b^2*c^4 + 24016*a
^2*c^5)*d^4*e^9 - 18*(145*b^5*c^2 + 12232*a*b^3*c^3 + 24016*a^2*b*c^4)*d^3*e^10
- 2*(518*b^6*c - 10131*a*b^4*c^2 - 124608*a^2*b^2*c^3 - 50000*a^3*c^4)*d^2*e^11
- (35*b^7 - 2562*a*b^5*c + 33072*a^2*b^3*c^2 + 100000*a^3*b*c^3)*d*e^12 + (35*a*
b^6 - 1491*a^2*b^4*c + 15000*a^3*b^2*c^2 + 10000*a^4*c^3)*e^13)*sqrt(e*x + d)) -
sqrt(1/2)*(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3 + (b^4*c^3 - 8*a*b^2*c^4 + 16
*a^2*c^5)*x^4 + 2*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*x^3 + (b^6*c - 6*a*b^4*
c^2 + 32*a^3*c^4)*x^2 + 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*x)*sqrt((4608
*c^7*d^7 - 16128*b*c^6*d^6*e + 672*(31*b^2*c^5 + 20*a*c^6)*d^5*e^2 - 1680*(7*b^3
*c^4 + 20*a*b*c^5)*d^4*e^3 + 70*(35*b^4*c^3 + 392*a*b^2*c^4 + 176*a^2*c^5)*d^3*e
^4 + 21*(b^5*c^2 - 360*a*b^3*c^3 - 880*a^2*b*c^4)*d^2*e^5 - 21*(b^6*c - 10*a*b^4
*c^2 - 320*a^2*b^2*c^3 - 160*a^3*c^4)*d*e^6 - (b^7 - 35*a*b^5*c + 280*a^2*b^3*c^
2 + 1680*a^3*b*c^3)*e^7 - (b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*a^3*b
^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8)*sqrt((441*c^4*d^4*e^10 - 882*b*c^3*d^3
*e^11 + 21*(19*b^2*c^2 + 50*a*c^3)*d^2*e^12 + 42*(b^3*c - 25*a*b*c^2)*d*e^13 + (
b^4 - 50*a*b^2*c + 625*a^2*c^2)*e^14)/(b^10*c^6 - 20*a*b^8*c^7 + 160*a^2*b^6*c^8
- 640*a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)))/(b^10*c^3 - 20*a*b^8*c
^4 + 160*a^2*b^6*c^5 - 640*a^3*b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8))*log(-
1/2*sqrt(1/2)*(504*(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)*d^4*e^
6 - 1008*(b^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64*a^3*b*c^6)*d^3*e^7 + 3*(1
67*b^8*c^2 - 1664*a*b^6*c^3 + 3936*a^2*b^4*c^4 + 5632*a^3*b^2*c^5 - 21760*a^4*c^
6)*d^2*e^8 + 3*(b^9*c - 352*a*b^7*c^2 + 4128*a^2*b^5*c^3 - 16384*a^3*b^3*c^4 + 2
1760*a^4*b*c^5)*d*e^9 - (b^10 - 17*a*b^8*c - 392*a^2*b^6*c^2 + 5696*a^3*b^4*c^3
- 23680*a^4*b^2*c^4 + 32000*a^5*c^5)*e^10 + (96*(b^10*c^6 - 20*a*b^8*c^7 + 160*a
^2*b^6*c^8 - 640*a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)*d^3 - 144*(b^1
1*c^5 - 20*a*b^9*c^6 + 160*a^2*b^7*c^7 - 640*a^3*b^5*c^8 + 1280*a^4*b^3*c^9 - 10
24*a^5*b*c^10)*d^2*e + 2*(23*b^12*c^4 - 408*a*b^10*c^5 + 2640*a^2*b^8*c^6 - 6400
*a^3*b^6*c^7 - 3840*a^4*b^4*c^8 + 43008*a^5*b^2*c^9 - 53248*a^6*c^10)*d*e^2 + (b
^13*c^3 - 72*a*b^11*c^4 + 1200*a^2*b^9*c^5 - 8960*a^3*b^7*c^6 + 34560*a^4*b^5*c^
7 - 67584*a^5*b^3*c^8 + 53248*a^6*b*c^9)*e^3)*sqrt((441*c^4*d^4*e^10 - 882*b*c^3
*d^3*e^11 + 21*(19*b^2*c^2 + 50*a*c^3)*d^2*e^12 + 42*(b^3*c - 25*a*b*c^2)*d*e^13
+ (b^4 - 50*a*b^2*c + 625*a^2*c^2)*e^14)/(b^10*c^6 - 20*a*b^8*c^7 + 160*a^2*b^6
*c^8 - 640*a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)))*sqrt((4608*c^7*d^7
- 16128*b*c^6*d^6*e + 672*(31*b^2*c^5 + 20*a*c^6)*d^5*e^2 - 1680*(7*b^3*c^4 + 2
0*a*b*c^5)*d^4*e^3 + 70*(35*b^4*c^3 + 392*a*b^2*c^4 + 176*a^2*c^5)*d^3*e^4 + 21*
(b^5*c^2 - 360*a*b^3*c^3 - 880*a^2*b*c^4)*d^2*e^5 - 21*(b^6*c - 10*a*b^4*c^2 - 3
20*a^2*b^2*c^3 - 160*a^3*c^4)*d*e^6 - (b^7 - 35*a*b^5*c + 280*a^2*b^3*c^2 + 1680
*a^3*b*c^3)*e^7 - (b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*a^3*b^4*c^6 +
1280*a^4*b^2*c^7 - 1024*a^5*c^8)*sqrt((441*c^4*d^4*e^10 - 882*b*c^3*d^3*e^11 +
21*(19*b^2*c^2 + 50*a*c^3)*d^2*e^12 + 42*(b^3*c - 25*a*b*c^2)*d*e^13 + (b^4 - 50
*a*b^2*c + 625*a^2*c^2)*e^14)/(b^10*c^6 - 20*a*b^8*c^7 + 160*a^2*b^6*c^8 - 640*a
^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)))/(b^10*c^3 - 20*a*b^8*c^4 + 160
*a^2*b^6*c^5 - 640*a^3*b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8)) + (48384*c^7*
d^8*e^5 - 193536*b*c^6*d^7*e^6 + 432*(683*b^2*c^5 + 404*a*c^6)*d^6*e^7 - 432*(48
1*b^3*c^4 + 1212*a*b*c^5)*d^5*e^8 + 9*(6841*b^4*c^3 + 60712*a*b^2*c^4 + 24016*a^
2*c^5)*d^4*e^9 - 18*(145*b^5*c^2 + 12232*a*b^3*c^3 + 24016*a^2*b*c^4)*d^3*e^10 -
2*(518*b^6*c - 10131*a*b^4*c^2 - 124608*a^2*b^2*c^3 - 50000*a^3*c^4)*d^2*e^11 -
(35*b^7 - 2562*a*b^5*c + 33072*a^2*b^3*c^2 + 100000*a^3*b*c^3)*d*e^12 + (35*a*b
^6 - 1491*a^2*b^4*c + 15000*a^3*b^2*c^2 + 10000*a^4*c^3)*e^13)*sqrt(e*x + d)) -
2*(36*a^2*b*c*d*e^2 - 2*(b^3*c - 10*a*b*c^2)*d^3 - (7*a*b^2*c + 44*a^2*c^2)*d^2*
e - (a^2*b^2 + 20*a^3*c)*e^3 + (24*c^4*d^3 - 36*b*c^3*d^2*e + 2*(5*b^2*c^2 + 16*
a*c^3)*d*e^2 + (b^3*c - 16*a*b*c^2)*e^3)*x^3 + (36*b*c^3*d^3 - (55*b^2*c^2 - 4*a
*c^3)*d^2*e + 4*(4*b^3*c + 11*a*b*c^2)*d*e^2 - (b^4 + 5*a*b^2*c + 36*a^2*c^2)*e^
3)*x^2 + (8*(b^2*c^2 + 5*a*c^3)*d^3 - (13*b^3*c + 56*a*b*c^2)*d^2*e + 2*(29*a*b^
2*c - 8*a^2*c^2)*d*e^2 - 2*(a*b^3 + 14*a^2*b*c)*e^3)*x)*sqrt(e*x + d))/(a^2*b^4*
c - 8*a^3*b^2*c^2 + 16*a^4*c^3 + (b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*x^4 + 2*(b
^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*x^3 + (b^6*c - 6*a*b^4*c^2 + 32*a^3*c^4)*x^
2 + 2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*x)

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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)**(7/2)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [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)^(7/2)/(c*x^2 + b*x + a)^3,x, algorithm="giac")

[Out]

Timed out