3.18.8 \(\int \frac {(d+e x)^{3/2}}{(a d e+(c d^2+a e^2) x+c d e x^2)^3} \, dx\)

Optimal. Leaf size=176 \[ \frac {15 e^2}{4 \sqrt {d+e x} \left (c d^2-a e^2\right )^3}+\frac {5 e}{4 \sqrt {d+e x} \left (c d^2-a e^2\right )^2 (a e+c d x)}-\frac {1}{2 \sqrt {d+e x} \left (c d^2-a e^2\right ) (a e+c d x)^2}-\frac {15 \sqrt {c} \sqrt {d} e^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{4 \left (c d^2-a e^2\right )^{7/2}} \]

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Rubi [A]  time = 0.12, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {626, 51, 63, 208} \begin {gather*} \frac {15 e^2}{4 \sqrt {d+e x} \left (c d^2-a e^2\right )^3}+\frac {5 e}{4 \sqrt {d+e x} \left (c d^2-a e^2\right )^2 (a e+c d x)}-\frac {1}{2 \sqrt {d+e x} \left (c d^2-a e^2\right ) (a e+c d x)^2}-\frac {15 \sqrt {c} \sqrt {d} e^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{4 \left (c d^2-a e^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(15*e^2)/(4*(c*d^2 - a*e^2)^3*Sqrt[d + e*x]) - 1/(2*(c*d^2 - a*e^2)*(a*e + c*d*x)^2*Sqrt[d + e*x]) + (5*e)/(4*
(c*d^2 - a*e^2)^2*(a*e + c*d*x)*Sqrt[d + e*x]) - (15*Sqrt[c]*Sqrt[d]*e^2*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x
])/Sqrt[c*d^2 - a*e^2]])/(4*(c*d^2 - a*e^2)^(7/2))

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 626

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c*x)/e)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&
IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx &=\int \frac {1}{(a e+c d x)^3 (d+e x)^{3/2}} \, dx\\ &=-\frac {1}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2 \sqrt {d+e x}}-\frac {(5 e) \int \frac {1}{(a e+c d x)^2 (d+e x)^{3/2}} \, dx}{4 \left (c d^2-a e^2\right )}\\ &=-\frac {1}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2 \sqrt {d+e x}}+\frac {5 e}{4 \left (c d^2-a e^2\right )^2 (a e+c d x) \sqrt {d+e x}}+\frac {\left (15 e^2\right ) \int \frac {1}{(a e+c d x) (d+e x)^{3/2}} \, dx}{8 \left (c d^2-a e^2\right )^2}\\ &=\frac {15 e^2}{4 \left (c d^2-a e^2\right )^3 \sqrt {d+e x}}-\frac {1}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2 \sqrt {d+e x}}+\frac {5 e}{4 \left (c d^2-a e^2\right )^2 (a e+c d x) \sqrt {d+e x}}+\frac {\left (15 c d e^2\right ) \int \frac {1}{(a e+c d x) \sqrt {d+e x}} \, dx}{8 \left (c d^2-a e^2\right )^3}\\ &=\frac {15 e^2}{4 \left (c d^2-a e^2\right )^3 \sqrt {d+e x}}-\frac {1}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2 \sqrt {d+e x}}+\frac {5 e}{4 \left (c d^2-a e^2\right )^2 (a e+c d x) \sqrt {d+e x}}+\frac {(15 c d e) \operatorname {Subst}\left (\int \frac {1}{-\frac {c d^2}{e}+a e+\frac {c d x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{4 \left (c d^2-a e^2\right )^3}\\ &=\frac {15 e^2}{4 \left (c d^2-a e^2\right )^3 \sqrt {d+e x}}-\frac {1}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2 \sqrt {d+e x}}+\frac {5 e}{4 \left (c d^2-a e^2\right )^2 (a e+c d x) \sqrt {d+e x}}-\frac {15 \sqrt {c} \sqrt {d} e^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{4 \left (c d^2-a e^2\right )^{7/2}}\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 59, normalized size = 0.34 \begin {gather*} -\frac {2 e^2 \, _2F_1\left (-\frac {1}{2},3;\frac {1}{2};-\frac {c d (d+e x)}{a e^2-c d^2}\right )}{\sqrt {d+e x} \left (a e^2-c d^2\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*e^2*Hypergeometric2F1[-1/2, 3, 1/2, -((c*d*(d + e*x))/(-(c*d^2) + a*e^2))])/((-(c*d^2) + a*e^2)^3*Sqrt[d +
e*x])

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IntegrateAlgebraic [A]  time = 0.72, size = 212, normalized size = 1.20 \begin {gather*} \frac {8 a^2 e^6-16 a c d^2 e^4+25 a c d e^4 (d+e x)+8 c^2 d^4 e^2-25 c^2 d^3 e^2 (d+e x)+15 c^2 d^2 e^2 (d+e x)^2}{4 \sqrt {d+e x} \left (c d^2-a e^2\right )^3 \left (-a e^2+c d^2-c d (d+e x)\right )^2}+\frac {15 \sqrt {c} \sqrt {d} e^2 \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x} \sqrt {a e^2-c d^2}}{c d^2-a e^2}\right )}{4 \left (a e^2-c d^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(d + e*x)^(3/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

(8*c^2*d^4*e^2 - 16*a*c*d^2*e^4 + 8*a^2*e^6 - 25*c^2*d^3*e^2*(d + e*x) + 25*a*c*d*e^4*(d + e*x) + 15*c^2*d^2*e
^2*(d + e*x)^2)/(4*(c*d^2 - a*e^2)^3*Sqrt[d + e*x]*(c*d^2 - a*e^2 - c*d*(d + e*x))^2) + (15*Sqrt[c]*Sqrt[d]*e^
2*ArcTan[(Sqrt[c]*Sqrt[d]*Sqrt[-(c*d^2) + a*e^2]*Sqrt[d + e*x])/(c*d^2 - a*e^2)])/(4*(-(c*d^2) + a*e^2)^(7/2))

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fricas [B]  time = 0.44, size = 897, normalized size = 5.10 \begin {gather*} \left [\frac {15 \, {\left (c^{2} d^{2} e^{3} x^{3} + a^{2} d e^{4} + {\left (c^{2} d^{3} e^{2} + 2 \, a c d e^{4}\right )} x^{2} + {\left (2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x\right )} \sqrt {\frac {c d}{c d^{2} - a e^{2}}} \log \left (\frac {c d e x + 2 \, c d^{2} - a e^{2} - 2 \, {\left (c d^{2} - a e^{2}\right )} \sqrt {e x + d} \sqrt {\frac {c d}{c d^{2} - a e^{2}}}}{c d x + a e}\right ) + 2 \, {\left (15 \, c^{2} d^{2} e^{2} x^{2} - 2 \, c^{2} d^{4} + 9 \, a c d^{2} e^{2} + 8 \, a^{2} e^{4} + 5 \, {\left (c^{2} d^{3} e + 5 \, a c d e^{3}\right )} x\right )} \sqrt {e x + d}}{8 \, {\left (a^{2} c^{3} d^{7} e^{2} - 3 \, a^{3} c^{2} d^{5} e^{4} + 3 \, a^{4} c d^{3} e^{6} - a^{5} d e^{8} + {\left (c^{5} d^{8} e - 3 \, a c^{4} d^{6} e^{3} + 3 \, a^{2} c^{3} d^{4} e^{5} - a^{3} c^{2} d^{2} e^{7}\right )} x^{3} + {\left (c^{5} d^{9} - a c^{4} d^{7} e^{2} - 3 \, a^{2} c^{3} d^{5} e^{4} + 5 \, a^{3} c^{2} d^{3} e^{6} - 2 \, a^{4} c d e^{8}\right )} x^{2} + {\left (2 \, a c^{4} d^{8} e - 5 \, a^{2} c^{3} d^{6} e^{3} + 3 \, a^{3} c^{2} d^{4} e^{5} + a^{4} c d^{2} e^{7} - a^{5} e^{9}\right )} x\right )}}, -\frac {15 \, {\left (c^{2} d^{2} e^{3} x^{3} + a^{2} d e^{4} + {\left (c^{2} d^{3} e^{2} + 2 \, a c d e^{4}\right )} x^{2} + {\left (2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x\right )} \sqrt {-\frac {c d}{c d^{2} - a e^{2}}} \arctan \left (-\frac {{\left (c d^{2} - a e^{2}\right )} \sqrt {e x + d} \sqrt {-\frac {c d}{c d^{2} - a e^{2}}}}{c d e x + c d^{2}}\right ) - {\left (15 \, c^{2} d^{2} e^{2} x^{2} - 2 \, c^{2} d^{4} + 9 \, a c d^{2} e^{2} + 8 \, a^{2} e^{4} + 5 \, {\left (c^{2} d^{3} e + 5 \, a c d e^{3}\right )} x\right )} \sqrt {e x + d}}{4 \, {\left (a^{2} c^{3} d^{7} e^{2} - 3 \, a^{3} c^{2} d^{5} e^{4} + 3 \, a^{4} c d^{3} e^{6} - a^{5} d e^{8} + {\left (c^{5} d^{8} e - 3 \, a c^{4} d^{6} e^{3} + 3 \, a^{2} c^{3} d^{4} e^{5} - a^{3} c^{2} d^{2} e^{7}\right )} x^{3} + {\left (c^{5} d^{9} - a c^{4} d^{7} e^{2} - 3 \, a^{2} c^{3} d^{5} e^{4} + 5 \, a^{3} c^{2} d^{3} e^{6} - 2 \, a^{4} c d e^{8}\right )} x^{2} + {\left (2 \, a c^{4} d^{8} e - 5 \, a^{2} c^{3} d^{6} e^{3} + 3 \, a^{3} c^{2} d^{4} e^{5} + a^{4} c d^{2} e^{7} - a^{5} e^{9}\right )} x\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="fricas")

[Out]

[1/8*(15*(c^2*d^2*e^3*x^3 + a^2*d*e^4 + (c^2*d^3*e^2 + 2*a*c*d*e^4)*x^2 + (2*a*c*d^2*e^3 + a^2*e^5)*x)*sqrt(c*
d/(c*d^2 - a*e^2))*log((c*d*e*x + 2*c*d^2 - a*e^2 - 2*(c*d^2 - a*e^2)*sqrt(e*x + d)*sqrt(c*d/(c*d^2 - a*e^2)))
/(c*d*x + a*e)) + 2*(15*c^2*d^2*e^2*x^2 - 2*c^2*d^4 + 9*a*c*d^2*e^2 + 8*a^2*e^4 + 5*(c^2*d^3*e + 5*a*c*d*e^3)*
x)*sqrt(e*x + d))/(a^2*c^3*d^7*e^2 - 3*a^3*c^2*d^5*e^4 + 3*a^4*c*d^3*e^6 - a^5*d*e^8 + (c^5*d^8*e - 3*a*c^4*d^
6*e^3 + 3*a^2*c^3*d^4*e^5 - a^3*c^2*d^2*e^7)*x^3 + (c^5*d^9 - a*c^4*d^7*e^2 - 3*a^2*c^3*d^5*e^4 + 5*a^3*c^2*d^
3*e^6 - 2*a^4*c*d*e^8)*x^2 + (2*a*c^4*d^8*e - 5*a^2*c^3*d^6*e^3 + 3*a^3*c^2*d^4*e^5 + a^4*c*d^2*e^7 - a^5*e^9)
*x), -1/4*(15*(c^2*d^2*e^3*x^3 + a^2*d*e^4 + (c^2*d^3*e^2 + 2*a*c*d*e^4)*x^2 + (2*a*c*d^2*e^3 + a^2*e^5)*x)*sq
rt(-c*d/(c*d^2 - a*e^2))*arctan(-(c*d^2 - a*e^2)*sqrt(e*x + d)*sqrt(-c*d/(c*d^2 - a*e^2))/(c*d*e*x + c*d^2)) -
(15*c^2*d^2*e^2*x^2 - 2*c^2*d^4 + 9*a*c*d^2*e^2 + 8*a^2*e^4 + 5*(c^2*d^3*e + 5*a*c*d*e^3)*x)*sqrt(e*x + d))/(
a^2*c^3*d^7*e^2 - 3*a^3*c^2*d^5*e^4 + 3*a^4*c*d^3*e^6 - a^5*d*e^8 + (c^5*d^8*e - 3*a*c^4*d^6*e^3 + 3*a^2*c^3*d
^4*e^5 - a^3*c^2*d^2*e^7)*x^3 + (c^5*d^9 - a*c^4*d^7*e^2 - 3*a^2*c^3*d^5*e^4 + 5*a^3*c^2*d^3*e^6 - 2*a^4*c*d*e
^8)*x^2 + (2*a*c^4*d^8*e - 5*a^2*c^3*d^6*e^3 + 3*a^3*c^2*d^4*e^5 + a^4*c*d^2*e^7 - a^5*e^9)*x)]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: (12*a^6*c*d*exp(1)^2*exp(2)^6+6*a^6*sqrt
(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^2*ex
p(2)^6-48*a^5*c^2*d^3*exp(1)^4*exp(2)^4-24*a^5*c^2*d^3*exp(1)^2*exp(2)^5-12*a^5*c^2*d^2*exp(1)^2*exp(2)^5-24*a
^5*c*d^2*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2
)*exp(1)^4*exp(2)^4-12*a^5*c*d^2*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(
2))+a*c*d*exp(2))*sqrt(2)*exp(1)^2*exp(2)^5-12*a^5*c*d*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*e
xp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^2*exp(2)^5+6*a^5*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d
^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+
2*a*c*d^2*exp(2))*exp(1)^2*exp(2)^5-192*a^4*c^3*d^5*exp(1)^6*exp(2)^2+576*a^4*c^3*d^5*exp(1)^4*exp(2)^3-204*a^
4*c^3*d^5*exp(1)^2*exp(2)^4-96*a^4*c^3*d^4*exp(1)^4*exp(2)^3+156*a^4*c^3*d^4*exp(1)^2*exp(2)^4-96*a^4*c^2*d^4*
sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^
6*exp(2)^2+288*a^4*c^2*d^4*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*
c*d*exp(2))*sqrt(2)*exp(1)^4*exp(2)^3-102*a^4*c^2*d^4*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*ex
p(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^2*exp(2)^4+60*a^4*c^2*d^3*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4
-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^2*exp(2)^4+6*a^4*c^2*d^2*sqrt(
-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^2*exp
(2)^4+48*a^4*c*d^2*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(
2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^4*exp(2)^3-78*a^4*c*d^2*sqrt
(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^
4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^2*exp(2)^4-12*a^4*c*d*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^
4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*
exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^2*exp(2)^4-12*a^4*c*d*(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp
(2))*exp(1)^2*exp(2)^4+768*a^3*c^4*d^7*exp(1)^8-1920*a^3*c^4*d^7*exp(1)^6*exp(2)+1248*a^3*c^4*d^7*exp(1)^4*exp
(2)^2-336*a^3*c^4*d^7*exp(1)^2*exp(2)^3+576*a^3*c^4*d^6*exp(1)^6*exp(2)-864*a^3*c^4*d^6*exp(1)^4*exp(2)^2+168*
a^3*c^4*d^6*exp(1)^2*exp(2)^3+384*a^3*c^3*d^6*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2
*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^8-960*a^3*c^3*d^6*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1
)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^6*exp(2)+624*a^3*c^3*d^6*sqrt(-c^2*d^3-c*d*sqr
t(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^4*exp(2)^2-168*a^3*c^
3*d^6*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*e
xp(1)^2*exp(2)^3+192*a^3*c^3*d^5*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(
2))+a*c*d*exp(2))*sqrt(2)*exp(1)^6*exp(2)-384*a^3*c^3*d^5*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^
2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^4*exp(2)^2+72*a^3*c^3*d^5*sqrt(-c^2*d^3-c*d*sqrt(c^2
*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^2*exp(2)^3-24*a^3*c^3*d^4*
sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^
2*exp(2)^3-288*a^3*c^2*d^4*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*
c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^6*exp(2)+432*a^3*c^2
*d^4*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sq
rt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^4*exp(2)^2-84*a^3*c^2*d^4*sqrt(-c^2*d^3-c*
d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*
exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^2*exp(2)^3-144*a^3*c^2*d^3*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c
*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^
2+2*a*c*d^2*exp(2))*exp(1)^4*exp(2)^2+192*a^3*c^2*d^3*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*ex
p(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*
exp(1)^2*exp(2)^3+48*a^3*c^2*d^3*(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^2*exp(2)^3+
6*a^3*c^2*d^2*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*s
qrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^2*exp(2)^3+12*a^3*c^2*d^2*(c^2*d^
4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^2*exp(2)^3-192*a^2*c^5*d^9*exp(1)^6+576*a^2*c^5*d^9
*exp(1)^4*exp(2)-204*a^2*c^5*d^9*exp(1)^2*exp(2)^2-576*a^2*c^5*d^8*exp(1)^6+864*a^2*c^5*d^8*exp(1)^4*exp(2)-16
8*a^2*c^5*d^8*exp(1)^2*exp(2)^2-96*a^2*c^4*d^8*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+
2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^6+288*a^2*c^4*d^8*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(
1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^4*exp(2)-102*a^2*c^4*d^8*sqrt(-c^2*d^3-c*d*sq
rt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^2*exp(2)^2-192*a^2*c
^4*d^7*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*
exp(1)^6+384*a^2*c^4*d^7*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*
d*exp(2))*sqrt(2)*exp(1)^4*exp(2)-72*a^2*c^4*d^7*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^
2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^2*exp(2)^2-96*a^2*c^4*d^6*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*
c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^6+192*a^2*c^4*d^6*sqrt(-c^2*d^3-c*d
*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^4*exp(2)-60*a^2*c
^4*d^6*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*
exp(1)^2*exp(2)^2+288*a^2*c^3*d^6*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp
(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^6-432*a^2*c^3
*d^6*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sq
rt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^4*exp(2)+84*a^2*c^3*d^6*sqrt(-c^2*d^3-c*d*
sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*ex
p(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^2*exp(2)^2+288*a^2*c^3*d^5*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d
^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+
2*a*c*d^2*exp(2))*exp(1)^4*exp(2)-360*a^2*c^3*d^5*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)
^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(
1)^2*exp(2)^2+192*a^2*c^3*d^5*(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^6-384*a^2*c^3*
d^5*(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^4*exp(2)+120*a^2*c^3*d^5*(c^2*d^4-4*a*c*
d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^2*exp(2)^2+72*a^2*c^3*d^4*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*
a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(
2)^2+2*a*c*d^2*exp(2))*exp(1)^4*exp(2)-90*a^2*c^3*d^4*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*ex
p(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*
exp(1)^2*exp(2)^2+144*a^2*c^3*d^4*(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^4*exp(2)-1
80*a^2*c^3*d^4*(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^2*exp(2)^2-48*a*c^6*d^11*exp(
1)^4-24*a*c^6*d^11*exp(1)^2*exp(2)+96*a*c^6*d^10*exp(1)^4-156*a*c^6*d^10*exp(1)^2*exp(2)-24*a*c^5*d^10*sqrt(-c
^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^4-12*a*
c^5*d^10*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2
)*exp(1)^2*exp(2)-60*a*c^5*d^9*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2)
)+a*c*d*exp(2))*sqrt(2)*exp(1)^2*exp(2)-24*a*c^5*d^8*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp
(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^2*exp(2)-48*a*c^4*d^8*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*
c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)
^2+2*a*c*d^2*exp(2))*exp(1)^4+78*a*c^4*d^8*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*
c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^2*ex
p(2)-144*a*c^4*d^7*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(
2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^4+192*a*c^4*d^7*sqrt(-c^2*d^
3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*
d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^2*exp(2)+48*a*c^4*d^7*(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2
)^2+2*a*c*d^2*exp(2))*exp(1)^2*exp(2)-72*a*c^4*d^6*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2
)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp
(1)^4+90*a*c^4*d^6*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(
2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^2*exp(2)-144*a*c^4*d^6*(c^2*
d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^4+180*a*c^4*d^6*(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*e
xp(2)^2+2*a*c*d^2*exp(2))*exp(1)^2*exp(2)+12*c^7*d^13*exp(1)^2+12*c^7*d^12*exp(1)^2+6*c^6*d^12*sqrt(-c^2*d^3-c
*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^2+12*c^6*d^11*s
qrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^2
+6*c^6*d^10*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqr
t(2)*exp(1)^2-6*c^5*d^10*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*
d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^2-12*c^5*d^9*sqrt(-c^2
*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a
*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^2-12*c^5*d^9*(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*
a*c*d^2*exp(2))*exp(1)^2-6*c^5*d^8*sqrt(-c^2*d^3-c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*ex
p(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^2-12*c^5*d^8
*(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^2)/(32*a^9*d*exp(1)^2*exp(2)^8-32*a^9*d*exp
(2)^9-512*a^8*c*d^3*exp(1)^4*exp(2)^6+768*a^8*c*d^3*exp(1)^2*exp(2)^7-256*a^8*c*d^3*exp(2)^8-64*a^8*c*d^2*exp(
1)^2*exp(2)^7+64*a^8*c*d^2*exp(2)^8+3072*a^7*c^2*d^5*exp(1)^6*exp(2)^4-6144*a^7*c^2*d^5*exp(1)^4*exp(2)^5+3968
*a^7*c^2*d^5*exp(1)^2*exp(2)^6-896*a^7*c^2*d^5*exp(2)^7+768*a^7*c^2*d^4*exp(1)^4*exp(2)^5-1088*a^7*c^2*d^4*exp
(1)^2*exp(2)^6+320*a^7*c^2*d^4*exp(2)^7+32*a^7*c^2*d^3*exp(1)^2*exp(2)^6-32*a^7*c^2*d^3*exp(2)^7-8192*a^6*c^3*
d^7*exp(1)^8*exp(2)^2+20480*a^6*c^3*d^7*exp(1)^6*exp(2)^3-19968*a^6*c^3*d^7*exp(1)^4*exp(2)^4+9472*a^6*c^3*d^7
*exp(1)^2*exp(2)^5-1792*a^6*c^3*d^7*exp(2)^6-3072*a^6*c^3*d^6*exp(1)^6*exp(2)^3+5376*a^6*c^3*d^6*exp(1)^4*exp(
2)^4-2880*a^6*c^3*d^6*exp(1)^2*exp(2)^5+576*a^6*c^3*d^6*exp(2)^6-384*a^6*c^3*d^5*exp(1)^4*exp(2)^4+576*a^6*c^3
*d^5*exp(1)^2*exp(2)^5-192*a^6*c^3*d^5*exp(2)^6+8192*a^5*c^4*d^9*exp(1)^10-24576*a^5*c^4*d^9*exp(1)^8*exp(2)+3
4816*a^5*c^4*d^9*exp(1)^6*exp(2)^2-28672*a^5*c^4*d^9*exp(1)^4*exp(2)^3+12480*a^5*c^4*d^9*exp(1)^2*exp(2)^4-224
0*a^5*c^4*d^9*exp(2)^5+4096*a^5*c^4*d^8*exp(1)^8*exp(2)-7168*a^5*c^4*d^8*exp(1)^6*exp(2)^2+4608*a^5*c^4*d^8*ex
p(1)^4*exp(2)^3-1856*a^5*c^4*d^8*exp(1)^2*exp(2)^4+320*a^5*c^4*d^8*exp(2)^5+1536*a^5*c^4*d^7*exp(1)^6*exp(2)^2
-3072*a^5*c^4*d^7*exp(1)^4*exp(2)^3+2016*a^5*c^4*d^7*exp(1)^2*exp(2)^4-480*a^5*c^4*d^7*exp(2)^5-8192*a^4*c^5*d
^11*exp(1)^8+20480*a^4*c^5*d^11*exp(1)^6*exp(2)-19968*a^4*c^5*d^11*exp(1)^4*exp(2)^2+9472*a^4*c^5*d^11*exp(1)^
2*exp(2)^3-1792*a^4*c^5*d^11*exp(2)^4-4096*a^4*c^5*d^10*exp(1)^8+7168*a^4*c^5*d^10*exp(1)^6*exp(2)-4608*a^4*c^
5*d^10*exp(1)^4*exp(2)^2+1856*a^4*c^5*d^10*exp(1)^2*exp(2)^3-320*a^4*c^5*d^10*exp(2)^4-2048*a^4*c^5*d^9*exp(1)
^8+5120*a^4*c^5*d^9*exp(1)^6*exp(2)-5376*a^4*c^5*d^9*exp(1)^4*exp(2)^2+2944*a^4*c^5*d^9*exp(1)^2*exp(2)^3-640*
a^4*c^5*d^9*exp(2)^4+3072*a^3*c^6*d^13*exp(1)^6-6144*a^3*c^6*d^13*exp(1)^4*exp(2)+3968*a^3*c^6*d^13*exp(1)^2*e
xp(2)^2-896*a^3*c^6*d^13*exp(2)^3+3072*a^3*c^6*d^12*exp(1)^6-5376*a^3*c^6*d^12*exp(1)^4*exp(2)+2880*a^3*c^6*d^
12*exp(1)^2*exp(2)^2-576*a^3*c^6*d^12*exp(2)^3+1536*a^3*c^6*d^11*exp(1)^6-3072*a^3*c^6*d^11*exp(1)^4*exp(2)+20
16*a^3*c^6*d^11*exp(1)^2*exp(2)^2-480*a^3*c^6*d^11*exp(2)^3-512*a^2*c^7*d^15*exp(1)^4+768*a^2*c^7*d^15*exp(1)^
2*exp(2)-256*a^2*c^7*d^15*exp(2)^2-768*a^2*c^7*d^14*exp(1)^4+1088*a^2*c^7*d^14*exp(1)^2*exp(2)-320*a^2*c^7*d^1
4*exp(2)^2-384*a^2*c^7*d^13*exp(1)^4+576*a^2*c^7*d^13*exp(1)^2*exp(2)-192*a^2*c^7*d^13*exp(2)^2+32*a*c^8*d^17*
exp(1)^2-32*a*c^8*d^17*exp(2)+64*a*c^8*d^16*exp(1)^2-64*a*c^8*d^16*exp(2)+32*a*c^8*d^15*exp(1)^2-32*a*c^8*d^15
*exp(2))/abs(c)/abs(d)*atan(sqrt(d+x*exp(1))/sqrt(-(4*d^10*c^5-32*d^8*exp(1)^2*a*c^4+12*d^8*a*exp(2)*c^4+64*d^
6*exp(1)^4*a^2*c^3-32*d^6*exp(1)^2*a^2*exp(2)*c^3+8*d^6*a^2*exp(2)^2*c^3-64*d^4*exp(1)^4*a^3*exp(2)*c^2+32*d^4
*exp(1)^2*a^3*exp(2)^2*c^2-8*d^4*a^3*exp(2)^3*c^2+32*d^2*exp(1)^2*a^4*exp(2)^3*c-12*d^2*a^4*exp(2)^4*c-4*a^5*e
xp(2)^5+sqrt((-4*d^10*c^5+32*d^8*exp(1)^2*a*c^4-12*d^8*a*exp(2)*c^4-64*d^6*exp(1)^4*a^2*c^3+32*d^6*exp(1)^2*a^
2*exp(2)*c^3-8*d^6*a^2*exp(2)^2*c^3+64*d^4*exp(1)^4*a^3*exp(2)*c^2-32*d^4*exp(1)^2*a^3*exp(2)^2*c^2+8*d^4*a^3*
exp(2)^3*c^2-32*d^2*exp(1)^2*a^4*exp(2)^3*c+12*d^2*a^4*exp(2)^4*c+4*a^5*exp(2)^5)*(-4*d^10*c^5+32*d^8*exp(1)^2
*a*c^4-12*d^8*a*exp(2)*c^4-64*d^6*exp(1)^4*a^2*c^3+32*d^6*exp(1)^2*a^2*exp(2)*c^3-8*d^6*a^2*exp(2)^2*c^3+64*d^
4*exp(1)^4*a^3*exp(2)*c^2-32*d^4*exp(1)^2*a^3*exp(2)^2*c^2+8*d^4*a^3*exp(2)^3*c^2-32*d^2*exp(1)^2*a^4*exp(2)^3
*c+12*d^2*a^4*exp(2)^4*c+4*a^5*exp(2)^5)-4*(4*d^9*c^5-32*d^7*exp(1)^2*a*c^4+16*d^7*a*exp(2)*c^4+64*d^5*exp(1)^
4*a^2*c^3-64*d^5*exp(1)^2*a^2*exp(2)*c^3+24*d^5*a^2*exp(2)^2*c^3-32*d^3*exp(1)^2*a^3*exp(2)^2*c^2+16*d^3*a^3*e
xp(2)^3*c^2+4*d*a^4*exp(2)^4*c)*(4*d^9*exp(1)^2*a*c^4-4*d^9*a*exp(2)*c^4-32*d^7*exp(1)^4*a^2*c^3+48*d^7*exp(1)
^2*a^2*exp(2)*c^3-16*d^7*a^2*exp(2)^2*c^3+64*d^5*exp(1)^6*a^3*c^2-128*d^5*exp(1)^4*a^3*exp(2)*c^2+88*d^5*exp(1
)^2*a^3*exp(2)^2*c^2-24*d^5*a^3*exp(2)^3*c^2-32*d^3*exp(1)^4*a^4*exp(2)^2*c+48*d^3*exp(1)^2*a^4*exp(2)^3*c-16*
d^3*a^4*exp(2)^4*c+4*d*exp(1)^2*a^5*exp(2)^4-4*d*a^5*exp(2)^5)))/2/(4*d^9*c^5-32*d^7*exp(1)^2*a*c^4+16*d^7*a*e
xp(2)*c^4+64*d^5*exp(1)^4*a^2*c^3-64*d^5*exp(1)^2*a^2*exp(2)*c^3+24*d^5*a^2*exp(2)^2*c^3-32*d^3*exp(1)^2*a^3*e
xp(2)^2*c^2+16*d^3*a^3*exp(2)^3*c^2+4*d*a^4*exp(2)^4*c)))-(12*a^6*c*d*exp(1)^2*exp(2)^6-6*a^6*sqrt(-c^2*d^3+c*
d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^2*exp(2)^6-48*a^
5*c^2*d^3*exp(1)^4*exp(2)^4-24*a^5*c^2*d^3*exp(1)^2*exp(2)^5-12*a^5*c^2*d^2*exp(1)^2*exp(2)^5+24*a^5*c*d^2*sqr
t(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^4*e
xp(2)^4+12*a^5*c*d^2*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*ex
p(2))*sqrt(2)*exp(1)^2*exp(2)^5+12*a^5*c*d*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*
c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^2*exp(2)^5+6*a^5*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+
a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*ex
p(2))*exp(1)^2*exp(2)^5-192*a^4*c^3*d^5*exp(1)^6*exp(2)^2+576*a^4*c^3*d^5*exp(1)^4*exp(2)^3-204*a^4*c^3*d^5*ex
p(1)^2*exp(2)^4-96*a^4*c^3*d^4*exp(1)^4*exp(2)^3+156*a^4*c^3*d^4*exp(1)^2*exp(2)^4+96*a^4*c^2*d^4*sqrt(-c^2*d^
3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^6*exp(2)^2-2
88*a^4*c^2*d^4*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*
sqrt(2)*exp(1)^4*exp(2)^3+102*a^4*c^2*d^4*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c
*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^2*exp(2)^4-60*a^4*c^2*d^3*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*e
xp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^2*exp(2)^4-6*a^4*c^2*d^2*sqrt(-c^2*d^3+c*d
*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^2*exp(2)^4+48*a^4
*c*d^2*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*
sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^4*exp(2)^3-78*a^4*c*d^2*sqrt(-c^2*d^3+c*
d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*
exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^2*exp(2)^4-12*a^4*c*d*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*
exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a
*c*d^2*exp(2))*exp(1)^2*exp(2)^4-12*a^4*c*d*(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^
2*exp(2)^4+768*a^3*c^4*d^7*exp(1)^8-1920*a^3*c^4*d^7*exp(1)^6*exp(2)+1248*a^3*c^4*d^7*exp(1)^4*exp(2)^2-336*a^
3*c^4*d^7*exp(1)^2*exp(2)^3+576*a^3*c^4*d^6*exp(1)^6*exp(2)-864*a^3*c^4*d^6*exp(1)^4*exp(2)^2+168*a^3*c^4*d^6*
exp(1)^2*exp(2)^3-384*a^3*c^3*d^6*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp
(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^8+960*a^3*c^3*d^6*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(
2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^6*exp(2)-624*a^3*c^3*d^6*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*
a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^4*exp(2)^2+168*a^3*c^3*d^6*sqrt(-
c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^2*exp(
2)^3-192*a^3*c^3*d^5*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*ex
p(2))*sqrt(2)*exp(1)^6*exp(2)+384*a^3*c^3*d^5*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2
*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^4*exp(2)^2-72*a^3*c^3*d^5*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d
^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^2*exp(2)^3+24*a^3*c^3*d^4*sqrt(-c^2*d^
3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^2*exp(2)^3-2
88*a^3*c^2*d^4*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*
sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^6*exp(2)+432*a^3*c^2*d^4*sqrt(-c
^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4
*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^4*exp(2)^2-84*a^3*c^2*d^4*sqrt(-c^2*d^3+c*d*sqrt(c^2*d
^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2
*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^2*exp(2)^3-144*a^3*c^2*d^3*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^
2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*
exp(2))*exp(1)^4*exp(2)^2+192*a^3*c^2*d^3*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c
*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^2*exp
(2)^3+48*a^3*c^2*d^3*(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^2*exp(2)^3+6*a^3*c^2*d^
2*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(
c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^2*exp(2)^3+12*a^3*c^2*d^2*(c^2*d^4-4*a*c*d^2*
exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^2*exp(2)^3-192*a^2*c^5*d^9*exp(1)^6+576*a^2*c^5*d^9*exp(1)^4*ex
p(2)-204*a^2*c^5*d^9*exp(1)^2*exp(2)^2-576*a^2*c^5*d^8*exp(1)^6+864*a^2*c^5*d^8*exp(1)^4*exp(2)-168*a^2*c^5*d^
8*exp(1)^2*exp(2)^2+96*a^2*c^4*d^8*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*ex
p(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^6-288*a^2*c^4*d^8*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp
(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^4*exp(2)+102*a^2*c^4*d^8*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4
*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^2*exp(2)^2+192*a^2*c^4*d^7*sqrt(
-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^6-384
*a^2*c^4*d^7*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sq
rt(2)*exp(1)^4*exp(2)+72*a^2*c^4*d^7*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*
exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^2*exp(2)^2+96*a^2*c^4*d^6*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)
^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^6-192*a^2*c^4*d^6*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^
4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^4*exp(2)+60*a^2*c^4*d^6*sqrt(
-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^2*exp
(2)^2+288*a^2*c^3*d^6*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*e
xp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^6-432*a^2*c^3*d^6*sqrt(-c
^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4
*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^4*exp(2)+84*a^2*c^3*d^6*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4
-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*e
xp(2)^2+2*a*c*d^2*exp(2))*exp(1)^2*exp(2)^2+288*a^2*c^3*d^5*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+
a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*ex
p(2))*exp(1)^4*exp(2)-360*a^2*c^3*d^5*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2
*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^2*exp(2)^
2+192*a^2*c^3*d^5*(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^6-384*a^2*c^3*d^5*(c^2*d^4
-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^4*exp(2)+120*a^2*c^3*d^5*(c^2*d^4-4*a*c*d^2*exp(1)^2
+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^2*exp(2)^2+72*a^2*c^3*d^4*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(
1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d
^2*exp(2))*exp(1)^4*exp(2)-90*a^2*c^3*d^4*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c
*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^2*exp
(2)^2+144*a^2*c^3*d^4*(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^4*exp(2)-180*a^2*c^3*d
^4*(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^2*exp(2)^2-48*a*c^6*d^11*exp(1)^4-24*a*c^
6*d^11*exp(1)^2*exp(2)+96*a*c^6*d^10*exp(1)^4-156*a*c^6*d^10*exp(1)^2*exp(2)+24*a*c^5*d^10*sqrt(-c^2*d^3+c*d*s
qrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^4+12*a*c^5*d^10*sqr
t(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^2*e
xp(2)+60*a*c^5*d^9*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(
2))*sqrt(2)*exp(1)^2*exp(2)+24*a*c^5*d^8*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*
d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^2*exp(2)-48*a*c^4*d^8*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)
^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2
*exp(2))*exp(1)^4+78*a*c^4*d^8*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2)
)+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^2*exp(2)-144*a*c
^4*d^7*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*
sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^4+192*a*c^4*d^7*sqrt(-c^2*d^3+c*d*sqrt(c
^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2
+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^2*exp(2)+48*a*c^4*d^7*(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^
2*exp(2))*exp(1)^2*exp(2)-72*a*c^4*d^6*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^
2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^4+90*a*c
^4*d^6*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*
sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^2*exp(2)-144*a*c^4*d^6*(c^2*d^4-4*a*c*d^
2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^4+180*a*c^4*d^6*(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*
c*d^2*exp(2))*exp(1)^2*exp(2)+12*c^7*d^13*exp(1)^2+12*c^7*d^12*exp(1)^2-6*c^6*d^12*sqrt(-c^2*d^3+c*d*sqrt(c^2*
d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^2-12*c^6*d^11*sqrt(-c^2*d^3
+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^2-6*c^6*d^10*
sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^
2-6*c^5*d^10*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sq
rt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^2-12*c^5*d^9*sqrt(-c^2*d^3+c*d*sqr
t(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1
)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^2-12*c^5*d^9*(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(
2))*exp(1)^2-6*c^5*d^8*sqrt(-c^2*d^3+c*d*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))+a*c*d*
exp(2))*sqrt(2)*sqrt(c^2*d^4-4*a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^2-12*c^5*d^8*(c^2*d^4-4*
a*c*d^2*exp(1)^2+a^2*exp(2)^2+2*a*c*d^2*exp(2))*exp(1)^2)/(32*a^9*d*exp(1)^2*exp(2)^8-32*a^9*d*exp(2)^9-512*a^
8*c*d^3*exp(1)^4*exp(2)^6+768*a^8*c*d^3*exp(1)^2*exp(2)^7-256*a^8*c*d^3*exp(2)^8-64*a^8*c*d^2*exp(1)^2*exp(2)^
7+64*a^8*c*d^2*exp(2)^8+3072*a^7*c^2*d^5*exp(1)^6*exp(2)^4-6144*a^7*c^2*d^5*exp(1)^4*exp(2)^5+3968*a^7*c^2*d^5
*exp(1)^2*exp(2)^6-896*a^7*c^2*d^5*exp(2)^7+768*a^7*c^2*d^4*exp(1)^4*exp(2)^5-1088*a^7*c^2*d^4*exp(1)^2*exp(2)
^6+320*a^7*c^2*d^4*exp(2)^7+32*a^7*c^2*d^3*exp(1)^2*exp(2)^6-32*a^7*c^2*d^3*exp(2)^7-8192*a^6*c^3*d^7*exp(1)^8
*exp(2)^2+20480*a^6*c^3*d^7*exp(1)^6*exp(2)^3-19968*a^6*c^3*d^7*exp(1)^4*exp(2)^4+9472*a^6*c^3*d^7*exp(1)^2*ex
p(2)^5-1792*a^6*c^3*d^7*exp(2)^6-3072*a^6*c^3*d^6*exp(1)^6*exp(2)^3+5376*a^6*c^3*d^6*exp(1)^4*exp(2)^4-2880*a^
6*c^3*d^6*exp(1)^2*exp(2)^5+576*a^6*c^3*d^6*exp(2)^6-384*a^6*c^3*d^5*exp(1)^4*exp(2)^4+576*a^6*c^3*d^5*exp(1)^
2*exp(2)^5-192*a^6*c^3*d^5*exp(2)^6+8192*a^5*c^4*d^9*exp(1)^10-24576*a^5*c^4*d^9*exp(1)^8*exp(2)+34816*a^5*c^4
*d^9*exp(1)^6*exp(2)^2-28672*a^5*c^4*d^9*exp(1)^4*exp(2)^3+12480*a^5*c^4*d^9*exp(1)^2*exp(2)^4-2240*a^5*c^4*d^
9*exp(2)^5+4096*a^5*c^4*d^8*exp(1)^8*exp(2)-7168*a^5*c^4*d^8*exp(1)^6*exp(2)^2+4608*a^5*c^4*d^8*exp(1)^4*exp(2
)^3-1856*a^5*c^4*d^8*exp(1)^2*exp(2)^4+320*a^5*c^4*d^8*exp(2)^5+1536*a^5*c^4*d^7*exp(1)^6*exp(2)^2-3072*a^5*c^
4*d^7*exp(1)^4*exp(2)^3+2016*a^5*c^4*d^7*exp(1)^2*exp(2)^4-480*a^5*c^4*d^7*exp(2)^5-8192*a^4*c^5*d^11*exp(1)^8
+20480*a^4*c^5*d^11*exp(1)^6*exp(2)-19968*a^4*c^5*d^11*exp(1)^4*exp(2)^2+9472*a^4*c^5*d^11*exp(1)^2*exp(2)^3-1
792*a^4*c^5*d^11*exp(2)^4-4096*a^4*c^5*d^10*exp(1)^8+7168*a^4*c^5*d^10*exp(1)^6*exp(2)-4608*a^4*c^5*d^10*exp(1
)^4*exp(2)^2+1856*a^4*c^5*d^10*exp(1)^2*exp(2)^3-320*a^4*c^5*d^10*exp(2)^4-2048*a^4*c^5*d^9*exp(1)^8+5120*a^4*
c^5*d^9*exp(1)^6*exp(2)-5376*a^4*c^5*d^9*exp(1)^4*exp(2)^2+2944*a^4*c^5*d^9*exp(1)^2*exp(2)^3-640*a^4*c^5*d^9*
exp(2)^4+3072*a^3*c^6*d^13*exp(1)^6-6144*a^3*c^6*d^13*exp(1)^4*exp(2)+3968*a^3*c^6*d^13*exp(1)^2*exp(2)^2-896*
a^3*c^6*d^13*exp(2)^3+3072*a^3*c^6*d^12*exp(1)^6-5376*a^3*c^6*d^12*exp(1)^4*exp(2)+2880*a^3*c^6*d^12*exp(1)^2*
exp(2)^2-576*a^3*c^6*d^12*exp(2)^3+1536*a^3*c^6*d^11*exp(1)^6-3072*a^3*c^6*d^11*exp(1)^4*exp(2)+2016*a^3*c^6*d
^11*exp(1)^2*exp(2)^2-480*a^3*c^6*d^11*exp(2)^3-512*a^2*c^7*d^15*exp(1)^4+768*a^2*c^7*d^15*exp(1)^2*exp(2)-256
*a^2*c^7*d^15*exp(2)^2-768*a^2*c^7*d^14*exp(1)^4+1088*a^2*c^7*d^14*exp(1)^2*exp(2)-320*a^2*c^7*d^14*exp(2)^2-3
84*a^2*c^7*d^13*exp(1)^4+576*a^2*c^7*d^13*exp(1)^2*exp(2)-192*a^2*c^7*d^13*exp(2)^2+32*a*c^8*d^17*exp(1)^2-32*
a*c^8*d^17*exp(2)+64*a*c^8*d^16*exp(1)^2-64*a*c^8*d^16*exp(2)+32*a*c^8*d^15*exp(1)^2-32*a*c^8*d^15*exp(2))/abs
(c)/abs(d)*atan(sqrt(d+x*exp(1))/sqrt(-(4*d^10*c^5-32*d^8*exp(1)^2*a*c^4+12*d^8*a*exp(2)*c^4+64*d^6*exp(1)^4*a
^2*c^3-32*d^6*exp(1)^2*a^2*exp(2)*c^3+8*d^6*a^2*exp(2)^2*c^3-64*d^4*exp(1)^4*a^3*exp(2)*c^2+32*d^4*exp(1)^2*a^
3*exp(2)^2*c^2-8*d^4*a^3*exp(2)^3*c^2+32*d^2*exp(1)^2*a^4*exp(2)^3*c-12*d^2*a^4*exp(2)^4*c-4*a^5*exp(2)^5-sqrt
((-4*d^10*c^5+32*d^8*exp(1)^2*a*c^4-12*d^8*a*exp(2)*c^4-64*d^6*exp(1)^4*a^2*c^3+32*d^6*exp(1)^2*a^2*exp(2)*c^3
-8*d^6*a^2*exp(2)^2*c^3+64*d^4*exp(1)^4*a^3*exp(2)*c^2-32*d^4*exp(1)^2*a^3*exp(2)^2*c^2+8*d^4*a^3*exp(2)^3*c^2
-32*d^2*exp(1)^2*a^4*exp(2)^3*c+12*d^2*a^4*exp(2)^4*c+4*a^5*exp(2)^5)*(-4*d^10*c^5+32*d^8*exp(1)^2*a*c^4-12*d^
8*a*exp(2)*c^4-64*d^6*exp(1)^4*a^2*c^3+32*d^6*exp(1)^2*a^2*exp(2)*c^3-8*d^6*a^2*exp(2)^2*c^3+64*d^4*exp(1)^4*a
^3*exp(2)*c^2-32*d^4*exp(1)^2*a^3*exp(2)^2*c^2+8*d^4*a^3*exp(2)^3*c^2-32*d^2*exp(1)^2*a^4*exp(2)^3*c+12*d^2*a^
4*exp(2)^4*c+4*a^5*exp(2)^5)-4*(4*d^9*c^5-32*d^7*exp(1)^2*a*c^4+16*d^7*a*exp(2)*c^4+64*d^5*exp(1)^4*a^2*c^3-64
*d^5*exp(1)^2*a^2*exp(2)*c^3+24*d^5*a^2*exp(2)^2*c^3-32*d^3*exp(1)^2*a^3*exp(2)^2*c^2+16*d^3*a^3*exp(2)^3*c^2+
4*d*a^4*exp(2)^4*c)*(4*d^9*exp(1)^2*a*c^4-4*d^9*a*exp(2)*c^4-32*d^7*exp(1)^4*a^2*c^3+48*d^7*exp(1)^2*a^2*exp(2
)*c^3-16*d^7*a^2*exp(2)^2*c^3+64*d^5*exp(1)^6*a^3*c^2-128*d^5*exp(1)^4*a^3*exp(2)*c^2+88*d^5*exp(1)^2*a^3*exp(
2)^2*c^2-24*d^5*a^3*exp(2)^3*c^2-32*d^3*exp(1)^4*a^4*exp(2)^2*c+48*d^3*exp(1)^2*a^4*exp(2)^3*c-16*d^3*a^4*exp(
2)^4*c+4*d*exp(1)^2*a^5*exp(2)^4-4*d*a^5*exp(2)^5)))/2/(4*d^9*c^5-32*d^7*exp(1)^2*a*c^4+16*d^7*a*exp(2)*c^4+64
*d^5*exp(1)^4*a^2*c^3-64*d^5*exp(1)^2*a^2*exp(2)*c^3+24*d^5*a^2*exp(2)^2*c^3-32*d^3*exp(1)^2*a^3*exp(2)^2*c^2+
16*d^3*a^3*exp(2)^3*c^2+4*d*a^4*exp(2)^4*c)))+(-12*sqrt(d+x*exp(1))*(d+x*exp(1))^3*d^4*exp(1)^2*c^3+12*sqrt(d+
x*exp(1))*(d+x*exp(1))^3*d^2*exp(1)^2*a*exp(2)*c^2+19*sqrt(d+x*exp(1))*(d+x*exp(1))^2*d^5*exp(1)^2*c^3-4*sqrt(
d+x*exp(1))*(d+x*exp(1))^2*d^3*exp(1)^4*a*c^2-34*sqrt(d+x*exp(1))*(d+x*exp(1))^2*d^3*exp(1)^2*a*exp(2)*c^2+19*
sqrt(d+x*exp(1))*(d+x*exp(1))^2*d*exp(1)^2*a^2*exp(2)^2*c-5*sqrt(d+x*exp(1))*(d+x*exp(1))*d^6*exp(1)^2*c^3-16*
sqrt(d+x*exp(1))*(d+x*exp(1))*d^4*exp(1)^4*a*c^2+31*sqrt(d+x*exp(1))*(d+x*exp(1))*d^4*exp(1)^2*a*exp(2)*c^2+16
*sqrt(d+x*exp(1))*(d+x*exp(1))*d^2*exp(1)^4*a^2*exp(2)*c-31*sqrt(d+x*exp(1))*(d+x*exp(1))*d^2*exp(1)^2*a^2*exp
(2)^2*c+5*sqrt(d+x*exp(1))*(d+x*exp(1))*exp(1)^2*a^3*exp(2)^3+3*sqrt(d+x*exp(1))*d^5*exp(1)^4*a*c^2-3*sqrt(d+x
*exp(1))*d^5*exp(1)^2*a*exp(2)*c^2+12*sqrt(d+x*exp(1))*d^3*exp(1)^6*a^2*c-30*sqrt(d+x*exp(1))*d^3*exp(1)^4*a^2
*exp(2)*c+18*sqrt(d+x*exp(1))*d^3*exp(1)^2*a^2*exp(2)^2*c+3*sqrt(d+x*exp(1))*d*exp(1)^4*a^3*exp(2)^2-3*sqrt(d+
x*exp(1))*d*exp(1)^2*a^3*exp(2)^3)/(-4*d^8*c^4+32*d^6*exp(1)^2*a*c^3-16*d^6*a*exp(2)*c^3-64*d^4*exp(1)^4*a^2*c
^2+64*d^4*exp(1)^2*a^2*exp(2)*c^2-24*d^4*a^2*exp(2)^2*c^2+32*d^2*exp(1)^2*a^3*exp(2)^2*c-16*d^2*a^3*exp(2)^3*c
-4*a^4*exp(2)^4)/((d+x*exp(1))^2*d*c-(d+x*exp(1))*d^2*c+(d+x*exp(1))*a*exp(2)+d*exp(1)^2*a-d*a*exp(2))^2

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maple [A]  time = 0.07, size = 226, normalized size = 1.28 \begin {gather*} -\frac {9 \sqrt {e x +d}\, a c d \,e^{4}}{4 \left (a \,e^{2}-c \,d^{2}\right )^{3} \left (c d e x +a \,e^{2}\right )^{2}}+\frac {9 \sqrt {e x +d}\, c^{2} d^{3} e^{2}}{4 \left (a \,e^{2}-c \,d^{2}\right )^{3} \left (c d e x +a \,e^{2}\right )^{2}}-\frac {7 \left (e x +d \right )^{\frac {3}{2}} c^{2} d^{2} e^{2}}{4 \left (a \,e^{2}-c \,d^{2}\right )^{3} \left (c d e x +a \,e^{2}\right )^{2}}-\frac {15 c d \,e^{2} \arctan \left (\frac {\sqrt {e x +d}\, c d}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) c d}}\right )}{4 \left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) c d}}-\frac {2 e^{2}}{\left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {e x +d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(3/2)/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^3,x)

[Out]

-7/4*e^2/(a*e^2-c*d^2)^3*c^2*d^2/(c*d*e*x+a*e^2)^2*(e*x+d)^(3/2)-9/4*e^4/(a*e^2-c*d^2)^3*c*d/(c*d*e*x+a*e^2)^2
*(e*x+d)^(1/2)*a+9/4*e^2/(a*e^2-c*d^2)^3*c^2*d^3/(c*d*e*x+a*e^2)^2*(e*x+d)^(1/2)-15/4*e^2/(a*e^2-c*d^2)^3*c*d/
((a*e^2-c*d^2)*c*d)^(1/2)*arctan((e*x+d)^(1/2)/((a*e^2-c*d^2)*c*d)^(1/2)*c*d)-2*e^2/(a*e^2-c*d^2)^3/(e*x+d)^(1
/2)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e^2-c*d^2>0)', see `assume?`
for more details)Is a*e^2-c*d^2 positive or negative?

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mupad [B]  time = 0.80, size = 251, normalized size = 1.43 \begin {gather*} -\frac {\frac {2\,e^2}{a\,e^2-c\,d^2}+\frac {25\,c\,d\,e^2\,\left (d+e\,x\right )}{4\,{\left (a\,e^2-c\,d^2\right )}^2}+\frac {15\,c^2\,d^2\,e^2\,{\left (d+e\,x\right )}^2}{4\,{\left (a\,e^2-c\,d^2\right )}^3}}{\sqrt {d+e\,x}\,\left (a^2\,e^4-2\,a\,c\,d^2\,e^2+c^2\,d^4\right )-\left (2\,c^2\,d^3-2\,a\,c\,d\,e^2\right )\,{\left (d+e\,x\right )}^{3/2}+c^2\,d^2\,{\left (d+e\,x\right )}^{5/2}}-\frac {15\,\sqrt {c}\,\sqrt {d}\,e^2\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {d}\,\sqrt {d+e\,x}\,\left (a^3\,e^6-3\,a^2\,c\,d^2\,e^4+3\,a\,c^2\,d^4\,e^2-c^3\,d^6\right )}{{\left (a\,e^2-c\,d^2\right )}^{7/2}}\right )}{4\,{\left (a\,e^2-c\,d^2\right )}^{7/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(3/2)/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^3,x)

[Out]

- ((2*e^2)/(a*e^2 - c*d^2) + (25*c*d*e^2*(d + e*x))/(4*(a*e^2 - c*d^2)^2) + (15*c^2*d^2*e^2*(d + e*x)^2)/(4*(a
*e^2 - c*d^2)^3))/((d + e*x)^(1/2)*(a^2*e^4 + c^2*d^4 - 2*a*c*d^2*e^2) - (2*c^2*d^3 - 2*a*c*d*e^2)*(d + e*x)^(
3/2) + c^2*d^2*(d + e*x)^(5/2)) - (15*c^(1/2)*d^(1/2)*e^2*atan((c^(1/2)*d^(1/2)*(d + e*x)^(1/2)*(a^3*e^6 - c^3
*d^6 + 3*a*c^2*d^4*e^2 - 3*a^2*c*d^2*e^4))/(a*e^2 - c*d^2)^(7/2)))/(4*(a*e^2 - c*d^2)^(7/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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