3.1621 \(\int \frac{b+2 c x}{\sqrt{d+e x} \left (a+b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=364 \[ -\frac{\sqrt{d+e x} \left (\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}+\frac{\sqrt{c} e \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{c} e \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )} \]

[Out]

-((Sqrt[d + e*x]*((b^2 - 4*a*c)*(c*d - b*e) - c*(b^2 - 4*a*c)*e*x))/((b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2))) + (Sqrt[c]*e*(2*c*d - (b + Sqrt[b
^2 - 4*a*c])*e)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b
^2 - 4*a*c])*e]])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c]
)*e]*(c*d^2 - b*d*e + a*e^2)) - (Sqrt[c]*e*(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)*A
rcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])
/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*(c*d^2 - b*d
*e + a*e^2))

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Rubi [A]  time = 1.78735, antiderivative size = 364, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{\sqrt{c} e \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{c} e \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )}-\frac{\sqrt{d+e x} (-b e+c d-c e x)}{\left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[d + e*x]*(c*d - b*e - c*e*x))/((c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)
)) + (Sqrt[c]*e*(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqr
t[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]
*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*(c*d^2 - b*d*e + a*e^2)) - (Sqrt[c]*e*(
2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[
2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b
+ Sqrt[b^2 - 4*a*c])*e]*(c*d^2 - b*d*e + a*e^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((2*c*x+b)/(c*x**2+b*x+a)**2/(e*x+d)**(1/2),x)

[Out]

Timed out

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Mathematica [A]  time = 0.671649, size = 330, normalized size = 0.91 \[ \frac{\sqrt{c} e \left (e \left (\sqrt{b^2-4 a c}+b\right )-2 c d\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right )}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{e \left (\sqrt{b^2-4 a c}-b\right )+2 c d} \left (e (b d-a e)-c d^2\right )}+\frac{\sqrt{c} e \left (e \left (\sqrt{b^2-4 a c}-b\right )+2 c d\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )} \left (e (b d-a e)-c d^2\right )}+\frac{\sqrt{d+e x} (b e-c d+c e x)}{(a+x (b+c x)) \left (e (a e-b d)+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d + e*x]*(-(c*d) + b*e + c*e*x))/((c*d^2 + e*(-(b*d) + a*e))*(a + x*(b + c
*x))) + (Sqrt[c]*e*(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)*ArcTanh[(Sqrt[2]*Sqrt[c]
*Sqrt[d + e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[2]*Sqrt[b^2 - 4*
a*c]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e]*(-(c*d^2) + e*(b*d - a*e))) + (Sqr
t[c]*e*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*
x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[2*
c*d - (b + Sqrt[b^2 - 4*a*c])*e]*(-(c*d^2) + e*(b*d - a*e)))

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Maple [A]  time = 0.049, size = 469, normalized size = 1.3 \[ 4\,{\frac{{c}^{2}\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) }\sqrt{ex+d}}{ \left ( 4\,ac-{b}^{2} \right ) \left ( be-2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) \left ( 2\,cex+be+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) }}+2\,{\frac{{c}^{2}\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) }\sqrt{2}}{ \left ( 4\,ac-{b}^{2} \right ) \left ( be-2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) \sqrt{ \left ( be-2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}\sqrt{2}}{\sqrt{ \left ( be-2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}} \right ) }-4\,{\frac{{c}^{2}\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) }\sqrt{ex+d}}{ \left ( 4\,ac-{b}^{2} \right ) \left ( -be+2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) \left ( -2\,cex-be+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) }}-2\,{\frac{{c}^{2}\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) }\sqrt{2}}{ \left ( 4\,ac-{b}^{2} \right ) \left ( -be+2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) \sqrt{ \left ( -be+2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}{\it Artanh} \left ({\frac{c\sqrt{ex+d}\sqrt{2}}{\sqrt{ \left ( -be+2\,cd+\sqrt{-{e}^{2} \left ( 4\,ac-{b}^{2} \right ) } \right ) c}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*x+b)/(c*x^2+b*x+a)^2/(e*x+d)^(1/2),x)

[Out]

4*c^2*(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)*(e*x+d)^(1/2)/(b*e-2*c*d+(-e^2*(4*a*c
-b^2))^(1/2))/(2*c*e*x+b*e+(-e^2*(4*a*c-b^2))^(1/2))+2*c^2*(-e^2*(4*a*c-b^2))^(1
/2)/(4*a*c-b^2)/(b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*2^(1/2)/((b*e-2*c*d+(-e^2*(
4*a*c-b^2))^(1/2))*c)^(1/2)*arctan(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*
a*c-b^2))^(1/2))*c)^(1/2))-4*c^2*(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)*(e*x+d)^(1
/2)/(-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))/(-2*c*e*x-b*e+(-e^2*(4*a*c-b^2))^(1/2)
)-2*c^2*(-e^2*(4*a*c-b^2))^(1/2)/(4*a*c-b^2)/(-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2
))*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/
2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.478056, size = 15501, normalized size = 42.59 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(sqrt(1/2)*(a*c*d^2 - a*b*d*e + a^2*e^2 + (c^2*d^2 - b*c*d*e + a*c*e^2)*x^2
+ (b*c*d^2 - b^2*d*e + a*b*e^2)*x)*sqrt((2*c^3*d^3*e^2 - 3*b*c^2*d^2*e^3 + 3*(b
^2*c - 2*a*c^2)*d*e^4 - (b^3 - 3*a*b*c)*e^5 + ((b^2*c^3 - 4*a*c^4)*d^6 - 3*(b^3*
c^2 - 4*a*b*c^3)*d^5*e + 3*(b^4*c - 3*a*b^2*c^2 - 4*a^2*c^3)*d^4*e^2 - (b^5 + 2*
a*b^3*c - 24*a^2*b*c^2)*d^3*e^3 + 3*(a*b^4 - 3*a^2*b^2*c - 4*a^3*c^2)*d^2*e^4 -
3*(a^2*b^3 - 4*a^3*b*c)*d*e^5 + (a^3*b^2 - 4*a^4*c)*e^6)*sqrt((9*c^4*d^4*e^6 - 1
8*b*c^3*d^3*e^7 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^8 - 6*(b^3*c - a*b*c^2)*d*e^9 +
(b^4 - 2*a*b^2*c + a^2*c^2)*e^10)/((b^2*c^6 - 4*a*c^7)*d^12 - 6*(b^3*c^5 - 4*a*b
*c^6)*d^11*e + 3*(5*b^4*c^4 - 18*a*b^2*c^5 - 8*a^2*c^6)*d^10*e^2 - 10*(2*b^5*c^3
- 5*a*b^3*c^4 - 12*a^2*b*c^5)*d^9*e^3 + 15*(b^6*c^2 - 15*a^2*b^2*c^4 - 4*a^3*c^
5)*d^8*e^4 - 6*(b^7*c + 6*a*b^5*c^2 - 30*a^2*b^3*c^3 - 40*a^3*b*c^4)*d^7*e^5 + (
b^8 + 26*a*b^6*c - 30*a^2*b^4*c^2 - 340*a^3*b^2*c^3 - 80*a^4*c^4)*d^6*e^6 - 6*(a
*b^7 + 6*a^2*b^5*c - 30*a^3*b^3*c^2 - 40*a^4*b*c^3)*d^5*e^7 + 15*(a^2*b^6 - 15*a
^4*b^2*c^2 - 4*a^5*c^3)*d^4*e^8 - 10*(2*a^3*b^5 - 5*a^4*b^3*c - 12*a^5*b*c^2)*d^
3*e^9 + 3*(5*a^4*b^4 - 18*a^5*b^2*c - 8*a^6*c^2)*d^2*e^10 - 6*(a^5*b^3 - 4*a^6*b
*c)*d*e^11 + (a^6*b^2 - 4*a^7*c)*e^12)))/((b^2*c^3 - 4*a*c^4)*d^6 - 3*(b^3*c^2 -
4*a*b*c^3)*d^5*e + 3*(b^4*c - 3*a*b^2*c^2 - 4*a^2*c^3)*d^4*e^2 - (b^5 + 2*a*b^3
*c - 24*a^2*b*c^2)*d^3*e^3 + 3*(a*b^4 - 3*a^2*b^2*c - 4*a^3*c^2)*d^2*e^4 - 3*(a^
2*b^3 - 4*a^3*b*c)*d*e^5 + (a^3*b^2 - 4*a^4*c)*e^6))*log(sqrt(1/2)*(6*(b^2*c^3 -
4*a*c^4)*d^3*e^4 - 9*(b^3*c^2 - 4*a*b*c^3)*d^2*e^5 + (5*b^4*c - 22*a*b^2*c^2 +
8*a^2*c^3)*d*e^6 - (b^5 - 5*a*b^3*c + 4*a^2*b*c^2)*e^7 - (2*(b^2*c^5 - 4*a*c^6)*
d^8 - 8*(b^3*c^4 - 4*a*b*c^5)*d^7*e + (13*b^4*c^3 - 48*a*b^2*c^4 - 16*a^2*c^5)*d
^6*e^2 - (11*b^5*c^2 - 32*a*b^3*c^3 - 48*a^2*b*c^4)*d^5*e^3 + 5*(b^6*c - a*b^4*c
^2 - 12*a^2*b^2*c^3)*d^4*e^4 - (b^7 + 6*a*b^5*c - 40*a^2*b^3*c^2)*d^3*e^5 + (3*a
*b^6 - 9*a^2*b^4*c - 16*a^3*b^2*c^2 + 16*a^4*c^3)*d^2*e^6 - (3*a^2*b^5 - 16*a^3*
b^3*c + 16*a^4*b*c^2)*d*e^7 + (a^3*b^4 - 6*a^4*b^2*c + 8*a^5*c^2)*e^8)*sqrt((9*c
^4*d^4*e^6 - 18*b*c^3*d^3*e^7 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^8 - 6*(b^3*c - a*b
*c^2)*d*e^9 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^10)/((b^2*c^6 - 4*a*c^7)*d^12 - 6*(b
^3*c^5 - 4*a*b*c^6)*d^11*e + 3*(5*b^4*c^4 - 18*a*b^2*c^5 - 8*a^2*c^6)*d^10*e^2 -
10*(2*b^5*c^3 - 5*a*b^3*c^4 - 12*a^2*b*c^5)*d^9*e^3 + 15*(b^6*c^2 - 15*a^2*b^2*
c^4 - 4*a^3*c^5)*d^8*e^4 - 6*(b^7*c + 6*a*b^5*c^2 - 30*a^2*b^3*c^3 - 40*a^3*b*c^
4)*d^7*e^5 + (b^8 + 26*a*b^6*c - 30*a^2*b^4*c^2 - 340*a^3*b^2*c^3 - 80*a^4*c^4)*
d^6*e^6 - 6*(a*b^7 + 6*a^2*b^5*c - 30*a^3*b^3*c^2 - 40*a^4*b*c^3)*d^5*e^7 + 15*(
a^2*b^6 - 15*a^4*b^2*c^2 - 4*a^5*c^3)*d^4*e^8 - 10*(2*a^3*b^5 - 5*a^4*b^3*c - 12
*a^5*b*c^2)*d^3*e^9 + 3*(5*a^4*b^4 - 18*a^5*b^2*c - 8*a^6*c^2)*d^2*e^10 - 6*(a^5
*b^3 - 4*a^6*b*c)*d*e^11 + (a^6*b^2 - 4*a^7*c)*e^12)))*sqrt((2*c^3*d^3*e^2 - 3*b
*c^2*d^2*e^3 + 3*(b^2*c - 2*a*c^2)*d*e^4 - (b^3 - 3*a*b*c)*e^5 + ((b^2*c^3 - 4*a
*c^4)*d^6 - 3*(b^3*c^2 - 4*a*b*c^3)*d^5*e + 3*(b^4*c - 3*a*b^2*c^2 - 4*a^2*c^3)*
d^4*e^2 - (b^5 + 2*a*b^3*c - 24*a^2*b*c^2)*d^3*e^3 + 3*(a*b^4 - 3*a^2*b^2*c - 4*
a^3*c^2)*d^2*e^4 - 3*(a^2*b^3 - 4*a^3*b*c)*d*e^5 + (a^3*b^2 - 4*a^4*c)*e^6)*sqrt
((9*c^4*d^4*e^6 - 18*b*c^3*d^3*e^7 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^8 - 6*(b^3*c
- a*b*c^2)*d*e^9 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^10)/((b^2*c^6 - 4*a*c^7)*d^12 -
6*(b^3*c^5 - 4*a*b*c^6)*d^11*e + 3*(5*b^4*c^4 - 18*a*b^2*c^5 - 8*a^2*c^6)*d^10*
e^2 - 10*(2*b^5*c^3 - 5*a*b^3*c^4 - 12*a^2*b*c^5)*d^9*e^3 + 15*(b^6*c^2 - 15*a^2
*b^2*c^4 - 4*a^3*c^5)*d^8*e^4 - 6*(b^7*c + 6*a*b^5*c^2 - 30*a^2*b^3*c^3 - 40*a^3
*b*c^4)*d^7*e^5 + (b^8 + 26*a*b^6*c - 30*a^2*b^4*c^2 - 340*a^3*b^2*c^3 - 80*a^4*
c^4)*d^6*e^6 - 6*(a*b^7 + 6*a^2*b^5*c - 30*a^3*b^3*c^2 - 40*a^4*b*c^3)*d^5*e^7 +
15*(a^2*b^6 - 15*a^4*b^2*c^2 - 4*a^5*c^3)*d^4*e^8 - 10*(2*a^3*b^5 - 5*a^4*b^3*c
- 12*a^5*b*c^2)*d^3*e^9 + 3*(5*a^4*b^4 - 18*a^5*b^2*c - 8*a^6*c^2)*d^2*e^10 - 6
*(a^5*b^3 - 4*a^6*b*c)*d*e^11 + (a^6*b^2 - 4*a^7*c)*e^12)))/((b^2*c^3 - 4*a*c^4)
*d^6 - 3*(b^3*c^2 - 4*a*b*c^3)*d^5*e + 3*(b^4*c - 3*a*b^2*c^2 - 4*a^2*c^3)*d^4*e
^2 - (b^5 + 2*a*b^3*c - 24*a^2*b*c^2)*d^3*e^3 + 3*(a*b^4 - 3*a^2*b^2*c - 4*a^3*c
^2)*d^2*e^4 - 3*(a^2*b^3 - 4*a^3*b*c)*d*e^5 + (a^3*b^2 - 4*a^4*c)*e^6)) - 2*(3*c
^4*d^2*e^4 - 3*b*c^3*d*e^5 + (b^2*c^2 - a*c^3)*e^6)*sqrt(e*x + d)) - sqrt(1/2)*(
a*c*d^2 - a*b*d*e + a^2*e^2 + (c^2*d^2 - b*c*d*e + a*c*e^2)*x^2 + (b*c*d^2 - b^2
*d*e + a*b*e^2)*x)*sqrt((2*c^3*d^3*e^2 - 3*b*c^2*d^2*e^3 + 3*(b^2*c - 2*a*c^2)*d
*e^4 - (b^3 - 3*a*b*c)*e^5 + ((b^2*c^3 - 4*a*c^4)*d^6 - 3*(b^3*c^2 - 4*a*b*c^3)*
d^5*e + 3*(b^4*c - 3*a*b^2*c^2 - 4*a^2*c^3)*d^4*e^2 - (b^5 + 2*a*b^3*c - 24*a^2*
b*c^2)*d^3*e^3 + 3*(a*b^4 - 3*a^2*b^2*c - 4*a^3*c^2)*d^2*e^4 - 3*(a^2*b^3 - 4*a^
3*b*c)*d*e^5 + (a^3*b^2 - 4*a^4*c)*e^6)*sqrt((9*c^4*d^4*e^6 - 18*b*c^3*d^3*e^7 +
3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^8 - 6*(b^3*c - a*b*c^2)*d*e^9 + (b^4 - 2*a*b^2*c
+ a^2*c^2)*e^10)/((b^2*c^6 - 4*a*c^7)*d^12 - 6*(b^3*c^5 - 4*a*b*c^6)*d^11*e + 3*
(5*b^4*c^4 - 18*a*b^2*c^5 - 8*a^2*c^6)*d^10*e^2 - 10*(2*b^5*c^3 - 5*a*b^3*c^4 -
12*a^2*b*c^5)*d^9*e^3 + 15*(b^6*c^2 - 15*a^2*b^2*c^4 - 4*a^3*c^5)*d^8*e^4 - 6*(b
^7*c + 6*a*b^5*c^2 - 30*a^2*b^3*c^3 - 40*a^3*b*c^4)*d^7*e^5 + (b^8 + 26*a*b^6*c
- 30*a^2*b^4*c^2 - 340*a^3*b^2*c^3 - 80*a^4*c^4)*d^6*e^6 - 6*(a*b^7 + 6*a^2*b^5*
c - 30*a^3*b^3*c^2 - 40*a^4*b*c^3)*d^5*e^7 + 15*(a^2*b^6 - 15*a^4*b^2*c^2 - 4*a^
5*c^3)*d^4*e^8 - 10*(2*a^3*b^5 - 5*a^4*b^3*c - 12*a^5*b*c^2)*d^3*e^9 + 3*(5*a^4*
b^4 - 18*a^5*b^2*c - 8*a^6*c^2)*d^2*e^10 - 6*(a^5*b^3 - 4*a^6*b*c)*d*e^11 + (a^6
*b^2 - 4*a^7*c)*e^12)))/((b^2*c^3 - 4*a*c^4)*d^6 - 3*(b^3*c^2 - 4*a*b*c^3)*d^5*e
+ 3*(b^4*c - 3*a*b^2*c^2 - 4*a^2*c^3)*d^4*e^2 - (b^5 + 2*a*b^3*c - 24*a^2*b*c^2
)*d^3*e^3 + 3*(a*b^4 - 3*a^2*b^2*c - 4*a^3*c^2)*d^2*e^4 - 3*(a^2*b^3 - 4*a^3*b*c
)*d*e^5 + (a^3*b^2 - 4*a^4*c)*e^6))*log(-sqrt(1/2)*(6*(b^2*c^3 - 4*a*c^4)*d^3*e^
4 - 9*(b^3*c^2 - 4*a*b*c^3)*d^2*e^5 + (5*b^4*c - 22*a*b^2*c^2 + 8*a^2*c^3)*d*e^6
- (b^5 - 5*a*b^3*c + 4*a^2*b*c^2)*e^7 - (2*(b^2*c^5 - 4*a*c^6)*d^8 - 8*(b^3*c^4
- 4*a*b*c^5)*d^7*e + (13*b^4*c^3 - 48*a*b^2*c^4 - 16*a^2*c^5)*d^6*e^2 - (11*b^5
*c^2 - 32*a*b^3*c^3 - 48*a^2*b*c^4)*d^5*e^3 + 5*(b^6*c - a*b^4*c^2 - 12*a^2*b^2*
c^3)*d^4*e^4 - (b^7 + 6*a*b^5*c - 40*a^2*b^3*c^2)*d^3*e^5 + (3*a*b^6 - 9*a^2*b^4
*c - 16*a^3*b^2*c^2 + 16*a^4*c^3)*d^2*e^6 - (3*a^2*b^5 - 16*a^3*b^3*c + 16*a^4*b
*c^2)*d*e^7 + (a^3*b^4 - 6*a^4*b^2*c + 8*a^5*c^2)*e^8)*sqrt((9*c^4*d^4*e^6 - 18*
b*c^3*d^3*e^7 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^8 - 6*(b^3*c - a*b*c^2)*d*e^9 + (b
^4 - 2*a*b^2*c + a^2*c^2)*e^10)/((b^2*c^6 - 4*a*c^7)*d^12 - 6*(b^3*c^5 - 4*a*b*c
^6)*d^11*e + 3*(5*b^4*c^4 - 18*a*b^2*c^5 - 8*a^2*c^6)*d^10*e^2 - 10*(2*b^5*c^3 -
5*a*b^3*c^4 - 12*a^2*b*c^5)*d^9*e^3 + 15*(b^6*c^2 - 15*a^2*b^2*c^4 - 4*a^3*c^5)
*d^8*e^4 - 6*(b^7*c + 6*a*b^5*c^2 - 30*a^2*b^3*c^3 - 40*a^3*b*c^4)*d^7*e^5 + (b^
8 + 26*a*b^6*c - 30*a^2*b^4*c^2 - 340*a^3*b^2*c^3 - 80*a^4*c^4)*d^6*e^6 - 6*(a*b
^7 + 6*a^2*b^5*c - 30*a^3*b^3*c^2 - 40*a^4*b*c^3)*d^5*e^7 + 15*(a^2*b^6 - 15*a^4
*b^2*c^2 - 4*a^5*c^3)*d^4*e^8 - 10*(2*a^3*b^5 - 5*a^4*b^3*c - 12*a^5*b*c^2)*d^3*
e^9 + 3*(5*a^4*b^4 - 18*a^5*b^2*c - 8*a^6*c^2)*d^2*e^10 - 6*(a^5*b^3 - 4*a^6*b*c
)*d*e^11 + (a^6*b^2 - 4*a^7*c)*e^12)))*sqrt((2*c^3*d^3*e^2 - 3*b*c^2*d^2*e^3 + 3
*(b^2*c - 2*a*c^2)*d*e^4 - (b^3 - 3*a*b*c)*e^5 + ((b^2*c^3 - 4*a*c^4)*d^6 - 3*(b
^3*c^2 - 4*a*b*c^3)*d^5*e + 3*(b^4*c - 3*a*b^2*c^2 - 4*a^2*c^3)*d^4*e^2 - (b^5 +
2*a*b^3*c - 24*a^2*b*c^2)*d^3*e^3 + 3*(a*b^4 - 3*a^2*b^2*c - 4*a^3*c^2)*d^2*e^4
- 3*(a^2*b^3 - 4*a^3*b*c)*d*e^5 + (a^3*b^2 - 4*a^4*c)*e^6)*sqrt((9*c^4*d^4*e^6
- 18*b*c^3*d^3*e^7 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^8 - 6*(b^3*c - a*b*c^2)*d*e^9
+ (b^4 - 2*a*b^2*c + a^2*c^2)*e^10)/((b^2*c^6 - 4*a*c^7)*d^12 - 6*(b^3*c^5 - 4*
a*b*c^6)*d^11*e + 3*(5*b^4*c^4 - 18*a*b^2*c^5 - 8*a^2*c^6)*d^10*e^2 - 10*(2*b^5*
c^3 - 5*a*b^3*c^4 - 12*a^2*b*c^5)*d^9*e^3 + 15*(b^6*c^2 - 15*a^2*b^2*c^4 - 4*a^3
*c^5)*d^8*e^4 - 6*(b^7*c + 6*a*b^5*c^2 - 30*a^2*b^3*c^3 - 40*a^3*b*c^4)*d^7*e^5
+ (b^8 + 26*a*b^6*c - 30*a^2*b^4*c^2 - 340*a^3*b^2*c^3 - 80*a^4*c^4)*d^6*e^6 - 6
*(a*b^7 + 6*a^2*b^5*c - 30*a^3*b^3*c^2 - 40*a^4*b*c^3)*d^5*e^7 + 15*(a^2*b^6 - 1
5*a^4*b^2*c^2 - 4*a^5*c^3)*d^4*e^8 - 10*(2*a^3*b^5 - 5*a^4*b^3*c - 12*a^5*b*c^2)
*d^3*e^9 + 3*(5*a^4*b^4 - 18*a^5*b^2*c - 8*a^6*c^2)*d^2*e^10 - 6*(a^5*b^3 - 4*a^
6*b*c)*d*e^11 + (a^6*b^2 - 4*a^7*c)*e^12)))/((b^2*c^3 - 4*a*c^4)*d^6 - 3*(b^3*c^
2 - 4*a*b*c^3)*d^5*e + 3*(b^4*c - 3*a*b^2*c^2 - 4*a^2*c^3)*d^4*e^2 - (b^5 + 2*a*
b^3*c - 24*a^2*b*c^2)*d^3*e^3 + 3*(a*b^4 - 3*a^2*b^2*c - 4*a^3*c^2)*d^2*e^4 - 3*
(a^2*b^3 - 4*a^3*b*c)*d*e^5 + (a^3*b^2 - 4*a^4*c)*e^6)) - 2*(3*c^4*d^2*e^4 - 3*b
*c^3*d*e^5 + (b^2*c^2 - a*c^3)*e^6)*sqrt(e*x + d)) + sqrt(1/2)*(a*c*d^2 - a*b*d*
e + a^2*e^2 + (c^2*d^2 - b*c*d*e + a*c*e^2)*x^2 + (b*c*d^2 - b^2*d*e + a*b*e^2)*
x)*sqrt((2*c^3*d^3*e^2 - 3*b*c^2*d^2*e^3 + 3*(b^2*c - 2*a*c^2)*d*e^4 - (b^3 - 3*
a*b*c)*e^5 - ((b^2*c^3 - 4*a*c^4)*d^6 - 3*(b^3*c^2 - 4*a*b*c^3)*d^5*e + 3*(b^4*c
- 3*a*b^2*c^2 - 4*a^2*c^3)*d^4*e^2 - (b^5 + 2*a*b^3*c - 24*a^2*b*c^2)*d^3*e^3 +
3*(a*b^4 - 3*a^2*b^2*c - 4*a^3*c^2)*d^2*e^4 - 3*(a^2*b^3 - 4*a^3*b*c)*d*e^5 + (
a^3*b^2 - 4*a^4*c)*e^6)*sqrt((9*c^4*d^4*e^6 - 18*b*c^3*d^3*e^7 + 3*(5*b^2*c^2 -
2*a*c^3)*d^2*e^8 - 6*(b^3*c - a*b*c^2)*d*e^9 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^10)
/((b^2*c^6 - 4*a*c^7)*d^12 - 6*(b^3*c^5 - 4*a*b*c^6)*d^11*e + 3*(5*b^4*c^4 - 18*
a*b^2*c^5 - 8*a^2*c^6)*d^10*e^2 - 10*(2*b^5*c^3 - 5*a*b^3*c^4 - 12*a^2*b*c^5)*d^
9*e^3 + 15*(b^6*c^2 - 15*a^2*b^2*c^4 - 4*a^3*c^5)*d^8*e^4 - 6*(b^7*c + 6*a*b^5*c
^2 - 30*a^2*b^3*c^3 - 40*a^3*b*c^4)*d^7*e^5 + (b^8 + 26*a*b^6*c - 30*a^2*b^4*c^2
- 340*a^3*b^2*c^3 - 80*a^4*c^4)*d^6*e^6 - 6*(a*b^7 + 6*a^2*b^5*c - 30*a^3*b^3*c
^2 - 40*a^4*b*c^3)*d^5*e^7 + 15*(a^2*b^6 - 15*a^4*b^2*c^2 - 4*a^5*c^3)*d^4*e^8 -
10*(2*a^3*b^5 - 5*a^4*b^3*c - 12*a^5*b*c^2)*d^3*e^9 + 3*(5*a^4*b^4 - 18*a^5*b^2
*c - 8*a^6*c^2)*d^2*e^10 - 6*(a^5*b^3 - 4*a^6*b*c)*d*e^11 + (a^6*b^2 - 4*a^7*c)*
e^12)))/((b^2*c^3 - 4*a*c^4)*d^6 - 3*(b^3*c^2 - 4*a*b*c^3)*d^5*e + 3*(b^4*c - 3*
a*b^2*c^2 - 4*a^2*c^3)*d^4*e^2 - (b^5 + 2*a*b^3*c - 24*a^2*b*c^2)*d^3*e^3 + 3*(a
*b^4 - 3*a^2*b^2*c - 4*a^3*c^2)*d^2*e^4 - 3*(a^2*b^3 - 4*a^3*b*c)*d*e^5 + (a^3*b
^2 - 4*a^4*c)*e^6))*log(sqrt(1/2)*(6*(b^2*c^3 - 4*a*c^4)*d^3*e^4 - 9*(b^3*c^2 -
4*a*b*c^3)*d^2*e^5 + (5*b^4*c - 22*a*b^2*c^2 + 8*a^2*c^3)*d*e^6 - (b^5 - 5*a*b^3
*c + 4*a^2*b*c^2)*e^7 + (2*(b^2*c^5 - 4*a*c^6)*d^8 - 8*(b^3*c^4 - 4*a*b*c^5)*d^7
*e + (13*b^4*c^3 - 48*a*b^2*c^4 - 16*a^2*c^5)*d^6*e^2 - (11*b^5*c^2 - 32*a*b^3*c
^3 - 48*a^2*b*c^4)*d^5*e^3 + 5*(b^6*c - a*b^4*c^2 - 12*a^2*b^2*c^3)*d^4*e^4 - (b
^7 + 6*a*b^5*c - 40*a^2*b^3*c^2)*d^3*e^5 + (3*a*b^6 - 9*a^2*b^4*c - 16*a^3*b^2*c
^2 + 16*a^4*c^3)*d^2*e^6 - (3*a^2*b^5 - 16*a^3*b^3*c + 16*a^4*b*c^2)*d*e^7 + (a^
3*b^4 - 6*a^4*b^2*c + 8*a^5*c^2)*e^8)*sqrt((9*c^4*d^4*e^6 - 18*b*c^3*d^3*e^7 + 3
*(5*b^2*c^2 - 2*a*c^3)*d^2*e^8 - 6*(b^3*c - a*b*c^2)*d*e^9 + (b^4 - 2*a*b^2*c +
a^2*c^2)*e^10)/((b^2*c^6 - 4*a*c^7)*d^12 - 6*(b^3*c^5 - 4*a*b*c^6)*d^11*e + 3*(5
*b^4*c^4 - 18*a*b^2*c^5 - 8*a^2*c^6)*d^10*e^2 - 10*(2*b^5*c^3 - 5*a*b^3*c^4 - 12
*a^2*b*c^5)*d^9*e^3 + 15*(b^6*c^2 - 15*a^2*b^2*c^4 - 4*a^3*c^5)*d^8*e^4 - 6*(b^7
*c + 6*a*b^5*c^2 - 30*a^2*b^3*c^3 - 40*a^3*b*c^4)*d^7*e^5 + (b^8 + 26*a*b^6*c -
30*a^2*b^4*c^2 - 340*a^3*b^2*c^3 - 80*a^4*c^4)*d^6*e^6 - 6*(a*b^7 + 6*a^2*b^5*c
- 30*a^3*b^3*c^2 - 40*a^4*b*c^3)*d^5*e^7 + 15*(a^2*b^6 - 15*a^4*b^2*c^2 - 4*a^5*
c^3)*d^4*e^8 - 10*(2*a^3*b^5 - 5*a^4*b^3*c - 12*a^5*b*c^2)*d^3*e^9 + 3*(5*a^4*b^
4 - 18*a^5*b^2*c - 8*a^6*c^2)*d^2*e^10 - 6*(a^5*b^3 - 4*a^6*b*c)*d*e^11 + (a^6*b
^2 - 4*a^7*c)*e^12)))*sqrt((2*c^3*d^3*e^2 - 3*b*c^2*d^2*e^3 + 3*(b^2*c - 2*a*c^2
)*d*e^4 - (b^3 - 3*a*b*c)*e^5 - ((b^2*c^3 - 4*a*c^4)*d^6 - 3*(b^3*c^2 - 4*a*b*c^
3)*d^5*e + 3*(b^4*c - 3*a*b^2*c^2 - 4*a^2*c^3)*d^4*e^2 - (b^5 + 2*a*b^3*c - 24*a
^2*b*c^2)*d^3*e^3 + 3*(a*b^4 - 3*a^2*b^2*c - 4*a^3*c^2)*d^2*e^4 - 3*(a^2*b^3 - 4
*a^3*b*c)*d*e^5 + (a^3*b^2 - 4*a^4*c)*e^6)*sqrt((9*c^4*d^4*e^6 - 18*b*c^3*d^3*e^
7 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^8 - 6*(b^3*c - a*b*c^2)*d*e^9 + (b^4 - 2*a*b^2
*c + a^2*c^2)*e^10)/((b^2*c^6 - 4*a*c^7)*d^12 - 6*(b^3*c^5 - 4*a*b*c^6)*d^11*e +
3*(5*b^4*c^4 - 18*a*b^2*c^5 - 8*a^2*c^6)*d^10*e^2 - 10*(2*b^5*c^3 - 5*a*b^3*c^4
- 12*a^2*b*c^5)*d^9*e^3 + 15*(b^6*c^2 - 15*a^2*b^2*c^4 - 4*a^3*c^5)*d^8*e^4 - 6
*(b^7*c + 6*a*b^5*c^2 - 30*a^2*b^3*c^3 - 40*a^3*b*c^4)*d^7*e^5 + (b^8 + 26*a*b^6
*c - 30*a^2*b^4*c^2 - 340*a^3*b^2*c^3 - 80*a^4*c^4)*d^6*e^6 - 6*(a*b^7 + 6*a^2*b
^5*c - 30*a^3*b^3*c^2 - 40*a^4*b*c^3)*d^5*e^7 + 15*(a^2*b^6 - 15*a^4*b^2*c^2 - 4
*a^5*c^3)*d^4*e^8 - 10*(2*a^3*b^5 - 5*a^4*b^3*c - 12*a^5*b*c^2)*d^3*e^9 + 3*(5*a
^4*b^4 - 18*a^5*b^2*c - 8*a^6*c^2)*d^2*e^10 - 6*(a^5*b^3 - 4*a^6*b*c)*d*e^11 + (
a^6*b^2 - 4*a^7*c)*e^12)))/((b^2*c^3 - 4*a*c^4)*d^6 - 3*(b^3*c^2 - 4*a*b*c^3)*d^
5*e + 3*(b^4*c - 3*a*b^2*c^2 - 4*a^2*c^3)*d^4*e^2 - (b^5 + 2*a*b^3*c - 24*a^2*b*
c^2)*d^3*e^3 + 3*(a*b^4 - 3*a^2*b^2*c - 4*a^3*c^2)*d^2*e^4 - 3*(a^2*b^3 - 4*a^3*
b*c)*d*e^5 + (a^3*b^2 - 4*a^4*c)*e^6)) - 2*(3*c^4*d^2*e^4 - 3*b*c^3*d*e^5 + (b^2
*c^2 - a*c^3)*e^6)*sqrt(e*x + d)) - sqrt(1/2)*(a*c*d^2 - a*b*d*e + a^2*e^2 + (c^
2*d^2 - b*c*d*e + a*c*e^2)*x^2 + (b*c*d^2 - b^2*d*e + a*b*e^2)*x)*sqrt((2*c^3*d^
3*e^2 - 3*b*c^2*d^2*e^3 + 3*(b^2*c - 2*a*c^2)*d*e^4 - (b^3 - 3*a*b*c)*e^5 - ((b^
2*c^3 - 4*a*c^4)*d^6 - 3*(b^3*c^2 - 4*a*b*c^3)*d^5*e + 3*(b^4*c - 3*a*b^2*c^2 -
4*a^2*c^3)*d^4*e^2 - (b^5 + 2*a*b^3*c - 24*a^2*b*c^2)*d^3*e^3 + 3*(a*b^4 - 3*a^2
*b^2*c - 4*a^3*c^2)*d^2*e^4 - 3*(a^2*b^3 - 4*a^3*b*c)*d*e^5 + (a^3*b^2 - 4*a^4*c
)*e^6)*sqrt((9*c^4*d^4*e^6 - 18*b*c^3*d^3*e^7 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^8
- 6*(b^3*c - a*b*c^2)*d*e^9 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^10)/((b^2*c^6 - 4*a*
c^7)*d^12 - 6*(b^3*c^5 - 4*a*b*c^6)*d^11*e + 3*(5*b^4*c^4 - 18*a*b^2*c^5 - 8*a^2
*c^6)*d^10*e^2 - 10*(2*b^5*c^3 - 5*a*b^3*c^4 - 12*a^2*b*c^5)*d^9*e^3 + 15*(b^6*c
^2 - 15*a^2*b^2*c^4 - 4*a^3*c^5)*d^8*e^4 - 6*(b^7*c + 6*a*b^5*c^2 - 30*a^2*b^3*c
^3 - 40*a^3*b*c^4)*d^7*e^5 + (b^8 + 26*a*b^6*c - 30*a^2*b^4*c^2 - 340*a^3*b^2*c^
3 - 80*a^4*c^4)*d^6*e^6 - 6*(a*b^7 + 6*a^2*b^5*c - 30*a^3*b^3*c^2 - 40*a^4*b*c^3
)*d^5*e^7 + 15*(a^2*b^6 - 15*a^4*b^2*c^2 - 4*a^5*c^3)*d^4*e^8 - 10*(2*a^3*b^5 -
5*a^4*b^3*c - 12*a^5*b*c^2)*d^3*e^9 + 3*(5*a^4*b^4 - 18*a^5*b^2*c - 8*a^6*c^2)*d
^2*e^10 - 6*(a^5*b^3 - 4*a^6*b*c)*d*e^11 + (a^6*b^2 - 4*a^7*c)*e^12)))/((b^2*c^3
- 4*a*c^4)*d^6 - 3*(b^3*c^2 - 4*a*b*c^3)*d^5*e + 3*(b^4*c - 3*a*b^2*c^2 - 4*a^2
*c^3)*d^4*e^2 - (b^5 + 2*a*b^3*c - 24*a^2*b*c^2)*d^3*e^3 + 3*(a*b^4 - 3*a^2*b^2*
c - 4*a^3*c^2)*d^2*e^4 - 3*(a^2*b^3 - 4*a^3*b*c)*d*e^5 + (a^3*b^2 - 4*a^4*c)*e^6
))*log(-sqrt(1/2)*(6*(b^2*c^3 - 4*a*c^4)*d^3*e^4 - 9*(b^3*c^2 - 4*a*b*c^3)*d^2*e
^5 + (5*b^4*c - 22*a*b^2*c^2 + 8*a^2*c^3)*d*e^6 - (b^5 - 5*a*b^3*c + 4*a^2*b*c^2
)*e^7 + (2*(b^2*c^5 - 4*a*c^6)*d^8 - 8*(b^3*c^4 - 4*a*b*c^5)*d^7*e + (13*b^4*c^3
- 48*a*b^2*c^4 - 16*a^2*c^5)*d^6*e^2 - (11*b^5*c^2 - 32*a*b^3*c^3 - 48*a^2*b*c^
4)*d^5*e^3 + 5*(b^6*c - a*b^4*c^2 - 12*a^2*b^2*c^3)*d^4*e^4 - (b^7 + 6*a*b^5*c -
40*a^2*b^3*c^2)*d^3*e^5 + (3*a*b^6 - 9*a^2*b^4*c - 16*a^3*b^2*c^2 + 16*a^4*c^3)
*d^2*e^6 - (3*a^2*b^5 - 16*a^3*b^3*c + 16*a^4*b*c^2)*d*e^7 + (a^3*b^4 - 6*a^4*b^
2*c + 8*a^5*c^2)*e^8)*sqrt((9*c^4*d^4*e^6 - 18*b*c^3*d^3*e^7 + 3*(5*b^2*c^2 - 2*
a*c^3)*d^2*e^8 - 6*(b^3*c - a*b*c^2)*d*e^9 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^10)/(
(b^2*c^6 - 4*a*c^7)*d^12 - 6*(b^3*c^5 - 4*a*b*c^6)*d^11*e + 3*(5*b^4*c^4 - 18*a*
b^2*c^5 - 8*a^2*c^6)*d^10*e^2 - 10*(2*b^5*c^3 - 5*a*b^3*c^4 - 12*a^2*b*c^5)*d^9*
e^3 + 15*(b^6*c^2 - 15*a^2*b^2*c^4 - 4*a^3*c^5)*d^8*e^4 - 6*(b^7*c + 6*a*b^5*c^2
- 30*a^2*b^3*c^3 - 40*a^3*b*c^4)*d^7*e^5 + (b^8 + 26*a*b^6*c - 30*a^2*b^4*c^2 -
340*a^3*b^2*c^3 - 80*a^4*c^4)*d^6*e^6 - 6*(a*b^7 + 6*a^2*b^5*c - 30*a^3*b^3*c^2
- 40*a^4*b*c^3)*d^5*e^7 + 15*(a^2*b^6 - 15*a^4*b^2*c^2 - 4*a^5*c^3)*d^4*e^8 - 1
0*(2*a^3*b^5 - 5*a^4*b^3*c - 12*a^5*b*c^2)*d^3*e^9 + 3*(5*a^4*b^4 - 18*a^5*b^2*c
- 8*a^6*c^2)*d^2*e^10 - 6*(a^5*b^3 - 4*a^6*b*c)*d*e^11 + (a^6*b^2 - 4*a^7*c)*e^
12)))*sqrt((2*c^3*d^3*e^2 - 3*b*c^2*d^2*e^3 + 3*(b^2*c - 2*a*c^2)*d*e^4 - (b^3 -
3*a*b*c)*e^5 - ((b^2*c^3 - 4*a*c^4)*d^6 - 3*(b^3*c^2 - 4*a*b*c^3)*d^5*e + 3*(b^
4*c - 3*a*b^2*c^2 - 4*a^2*c^3)*d^4*e^2 - (b^5 + 2*a*b^3*c - 24*a^2*b*c^2)*d^3*e^
3 + 3*(a*b^4 - 3*a^2*b^2*c - 4*a^3*c^2)*d^2*e^4 - 3*(a^2*b^3 - 4*a^3*b*c)*d*e^5
+ (a^3*b^2 - 4*a^4*c)*e^6)*sqrt((9*c^4*d^4*e^6 - 18*b*c^3*d^3*e^7 + 3*(5*b^2*c^2
- 2*a*c^3)*d^2*e^8 - 6*(b^3*c - a*b*c^2)*d*e^9 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^
10)/((b^2*c^6 - 4*a*c^7)*d^12 - 6*(b^3*c^5 - 4*a*b*c^6)*d^11*e + 3*(5*b^4*c^4 -
18*a*b^2*c^5 - 8*a^2*c^6)*d^10*e^2 - 10*(2*b^5*c^3 - 5*a*b^3*c^4 - 12*a^2*b*c^5)
*d^9*e^3 + 15*(b^6*c^2 - 15*a^2*b^2*c^4 - 4*a^3*c^5)*d^8*e^4 - 6*(b^7*c + 6*a*b^
5*c^2 - 30*a^2*b^3*c^3 - 40*a^3*b*c^4)*d^7*e^5 + (b^8 + 26*a*b^6*c - 30*a^2*b^4*
c^2 - 340*a^3*b^2*c^3 - 80*a^4*c^4)*d^6*e^6 - 6*(a*b^7 + 6*a^2*b^5*c - 30*a^3*b^
3*c^2 - 40*a^4*b*c^3)*d^5*e^7 + 15*(a^2*b^6 - 15*a^4*b^2*c^2 - 4*a^5*c^3)*d^4*e^
8 - 10*(2*a^3*b^5 - 5*a^4*b^3*c - 12*a^5*b*c^2)*d^3*e^9 + 3*(5*a^4*b^4 - 18*a^5*
b^2*c - 8*a^6*c^2)*d^2*e^10 - 6*(a^5*b^3 - 4*a^6*b*c)*d*e^11 + (a^6*b^2 - 4*a^7*
c)*e^12)))/((b^2*c^3 - 4*a*c^4)*d^6 - 3*(b^3*c^2 - 4*a*b*c^3)*d^5*e + 3*(b^4*c -
3*a*b^2*c^2 - 4*a^2*c^3)*d^4*e^2 - (b^5 + 2*a*b^3*c - 24*a^2*b*c^2)*d^3*e^3 + 3
*(a*b^4 - 3*a^2*b^2*c - 4*a^3*c^2)*d^2*e^4 - 3*(a^2*b^3 - 4*a^3*b*c)*d*e^5 + (a^
3*b^2 - 4*a^4*c)*e^6)) - 2*(3*c^4*d^2*e^4 - 3*b*c^3*d*e^5 + (b^2*c^2 - a*c^3)*e^
6)*sqrt(e*x + d)) - 2*(c*e*x - c*d + b*e)*sqrt(e*x + d))/(a*c*d^2 - a*b*d*e + a^
2*e^2 + (c^2*d^2 - b*c*d*e + a*c*e^2)*x^2 + (b*c*d^2 - b^2*d*e + a*b*e^2)*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((2*c*x+b)/(c*x**2+b*x+a)**2/(e*x+d)**(1/2),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((2*c*x + b)/((c*x^2 + b*x + a)^2*sqrt(e*x + d)),x, algorithm="giac")

[Out]

Timed out