3.17.91 \(\int \frac {1}{(d+e x)^{5/2} (a d e+(c d^2+a e^2) x+c d e x^2)} \, dx\)

Optimal. Leaf size=153 \[ -\frac {2 c^{5/2} d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{7/2}}+\frac {2 c^2 d^2}{\sqrt {d+e x} \left (c d^2-a e^2\right )^3}+\frac {2 c d}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}+\frac {2}{5 (d+e x)^{5/2} \left (c d^2-a e^2\right )} \]

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Rubi [A]  time = 0.13, antiderivative size = 153, 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 {2 c^2 d^2}{\sqrt {d+e x} \left (c d^2-a e^2\right )^3}-\frac {2 c^{5/2} d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{7/2}}+\frac {2 c d}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}+\frac {2}{5 (d+e x)^{5/2} \left (c d^2-a e^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)),x]

[Out]

2/(5*(c*d^2 - a*e^2)*(d + e*x)^(5/2)) + (2*c*d)/(3*(c*d^2 - a*e^2)^2*(d + e*x)^(3/2)) + (2*c^2*d^2)/((c*d^2 -
a*e^2)^3*Sqrt[d + e*x]) - (2*c^(5/2)*d^(5/2)*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(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 {1}{(d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx &=\int \frac {1}{(a e+c d x) (d+e x)^{7/2}} \, dx\\ &=\frac {2}{5 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {(c d) \int \frac {1}{(a e+c d x) (d+e x)^{5/2}} \, dx}{c d^2-a e^2}\\ &=\frac {2}{5 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {2 c d}{3 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}+\frac {\left (c^2 d^2\right ) \int \frac {1}{(a e+c d x) (d+e x)^{3/2}} \, dx}{\left (c d^2-a e^2\right )^2}\\ &=\frac {2}{5 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {2 c d}{3 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}+\frac {2 c^2 d^2}{\left (c d^2-a e^2\right )^3 \sqrt {d+e x}}+\frac {\left (c^3 d^3\right ) \int \frac {1}{(a e+c d x) \sqrt {d+e x}} \, dx}{\left (c d^2-a e^2\right )^3}\\ &=\frac {2}{5 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {2 c d}{3 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}+\frac {2 c^2 d^2}{\left (c d^2-a e^2\right )^3 \sqrt {d+e x}}+\frac {\left (2 c^3 d^3\right ) \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 )}{e \left (c d^2-a e^2\right )^3}\\ &=\frac {2}{5 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {2 c d}{3 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}+\frac {2 c^2 d^2}{\left (c d^2-a e^2\right )^3 \sqrt {d+e x}}-\frac {2 c^{5/2} d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{7/2}}\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 57, normalized size = 0.37 \begin {gather*} \frac {2 \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};\frac {c d (d+e x)}{c d^2-a e^2}\right )}{5 (d+e x)^{5/2} \left (c d^2-a e^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)),x]

[Out]

(2*Hypergeometric2F1[-5/2, 1, -3/2, (c*d*(d + e*x))/(c*d^2 - a*e^2)])/(5*(c*d^2 - a*e^2)*(d + e*x)^(5/2))

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IntegrateAlgebraic [A]  time = 0.28, size = 175, normalized size = 1.14 \begin {gather*} \frac {2 \left (3 a^2 e^4-6 a c d^2 e^2-5 a c d e^2 (d+e x)+3 c^2 d^4+5 c^2 d^3 (d+e x)+15 c^2 d^2 (d+e x)^2\right )}{15 (d+e x)^{5/2} \left (c d^2-a e^2\right )^3}+\frac {2 c^{5/2} d^{5/2} \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 )}{\left (a e^2-c d^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((d + e*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)),x]

[Out]

(2*(3*c^2*d^4 - 6*a*c*d^2*e^2 + 3*a^2*e^4 + 5*c^2*d^3*(d + e*x) - 5*a*c*d*e^2*(d + e*x) + 15*c^2*d^2*(d + e*x)
^2))/(15*(c*d^2 - a*e^2)^3*(d + e*x)^(5/2)) + (2*c^(5/2)*d^(5/2)*ArcTan[(Sqrt[c]*Sqrt[d]*Sqrt[-(c*d^2) + a*e^2
]*Sqrt[d + e*x])/(c*d^2 - a*e^2)])/(-(c*d^2) + a*e^2)^(7/2)

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fricas [B]  time = 0.44, size = 783, normalized size = 5.12 \begin {gather*} \left [-\frac {15 \, {\left (c^{2} d^{2} e^{3} x^{3} + 3 \, c^{2} d^{3} e^{2} x^{2} + 3 \, c^{2} d^{4} e x + c^{2} d^{5}\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} + 23 \, c^{2} d^{4} - 11 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 5 \, {\left (7 \, c^{2} d^{3} e - a c d e^{3}\right )} x\right )} \sqrt {e x + d}}{15 \, {\left (c^{3} d^{9} - 3 \, a c^{2} d^{7} e^{2} + 3 \, a^{2} c d^{5} e^{4} - a^{3} d^{3} e^{6} + {\left (c^{3} d^{6} e^{3} - 3 \, a c^{2} d^{4} e^{5} + 3 \, a^{2} c d^{2} e^{7} - a^{3} e^{9}\right )} x^{3} + 3 \, {\left (c^{3} d^{7} e^{2} - 3 \, a c^{2} d^{5} e^{4} + 3 \, a^{2} c d^{3} e^{6} - a^{3} d e^{8}\right )} x^{2} + 3 \, {\left (c^{3} d^{8} e - 3 \, a c^{2} d^{6} e^{3} + 3 \, a^{2} c d^{4} e^{5} - a^{3} d^{2} e^{7}\right )} x\right )}}, -\frac {2 \, {\left (15 \, {\left (c^{2} d^{2} e^{3} x^{3} + 3 \, c^{2} d^{3} e^{2} x^{2} + 3 \, c^{2} d^{4} e x + c^{2} d^{5}\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} + 23 \, c^{2} d^{4} - 11 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 5 \, {\left (7 \, c^{2} d^{3} e - a c d e^{3}\right )} x\right )} \sqrt {e x + d}\right )}}{15 \, {\left (c^{3} d^{9} - 3 \, a c^{2} d^{7} e^{2} + 3 \, a^{2} c d^{5} e^{4} - a^{3} d^{3} e^{6} + {\left (c^{3} d^{6} e^{3} - 3 \, a c^{2} d^{4} e^{5} + 3 \, a^{2} c d^{2} e^{7} - a^{3} e^{9}\right )} x^{3} + 3 \, {\left (c^{3} d^{7} e^{2} - 3 \, a c^{2} d^{5} e^{4} + 3 \, a^{2} c d^{3} e^{6} - a^{3} d e^{8}\right )} x^{2} + 3 \, {\left (c^{3} d^{8} e - 3 \, a c^{2} d^{6} e^{3} + 3 \, a^{2} c d^{4} e^{5} - a^{3} d^{2} e^{7}\right )} x\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="fricas")

[Out]

[-1/15*(15*(c^2*d^2*e^3*x^3 + 3*c^2*d^3*e^2*x^2 + 3*c^2*d^4*e*x + c^2*d^5)*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 + 23*c^2*d^4 - 11*a*c*d^2*e^2 + 3*a^2*e^4 + 5*(7*c^2*d^3*e - a*c*d*e^3)*x)*sqrt(e*x + d))/(c^3*d^9
- 3*a*c^2*d^7*e^2 + 3*a^2*c*d^5*e^4 - a^3*d^3*e^6 + (c^3*d^6*e^3 - 3*a*c^2*d^4*e^5 + 3*a^2*c*d^2*e^7 - a^3*e^
9)*x^3 + 3*(c^3*d^7*e^2 - 3*a*c^2*d^5*e^4 + 3*a^2*c*d^3*e^6 - a^3*d*e^8)*x^2 + 3*(c^3*d^8*e - 3*a*c^2*d^6*e^3
+ 3*a^2*c*d^4*e^5 - a^3*d^2*e^7)*x), -2/15*(15*(c^2*d^2*e^3*x^3 + 3*c^2*d^3*e^2*x^2 + 3*c^2*d^4*e*x + c^2*d^5)
*sqrt(-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 + 23*c^2*d^4 - 11*a*c*d^2*e^2 + 3*a^2*e^4 + 5*(7*c^2*d^3*e - a*c*d*e^3)*x)*sqrt(e*x +
d))/(c^3*d^9 - 3*a*c^2*d^7*e^2 + 3*a^2*c*d^5*e^4 - a^3*d^3*e^6 + (c^3*d^6*e^3 - 3*a*c^2*d^4*e^5 + 3*a^2*c*d^2*
e^7 - a^3*e^9)*x^3 + 3*(c^3*d^7*e^2 - 3*a*c^2*d^5*e^4 + 3*a^2*c*d^3*e^6 - a^3*d*e^8)*x^2 + 3*(c^3*d^8*e - 3*a*
c^2*d^6*e^3 + 3*a^2*c*d^4*e^5 - a^3*d^2*e^7)*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(1/(e*x+d)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: 2*((4*a^6*c*d*exp(2)^6+2*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(2)^6-36*a^5*c^
2*d^3*exp(1)^2*exp(2)^4+12*a^5*c^2*d^3*exp(2)^5-4*a^5*c^2*d^2*exp(2)^5-18*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)^4+6*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(2)^5-4
*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(2)^5+2*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(2)^5+96*a^4*c^3*d^5*exp(1)^4*exp(
2)^2-48*a^4*c^3*d^5*exp(1)^2*exp(2)^3+12*a^4*c^3*d^5*exp(2)^4+28*a^4*c^3*d^4*exp(1)^2*exp(2)^3-8*a^4*c^3*d^4*e
xp(2)^4+48*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)^2-24*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)^3+6*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(2)^4+20*a^4*c^2*d^3*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)^2*exp(2)^3+2*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)*e
xp(2)^4-14*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*ex
p(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+4*a^4*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)*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(2)^4-4*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(2)^4-4*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(2)^4-64*a^
3*c^4*d^7*exp(1)^6-24*a^3*c^4*d^7*exp(1)^2*exp(2)^2+8*a^3*c^4*d^7*exp(2)^3-48*a^3*c^4*d^6*exp(1)^4*exp(2)+12*a
^3*c^4*d^6*exp(1)^2*exp(2)^2-4*a^3*c^4*d^6*exp(2)^3-32*a^3*c^3*d^6*sqrt(-c^2*d^3-c*d*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))+a*c*d*exp(2))*sqrt(2)*exp(1)^6-12*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)^2+4*a^3*c^3*d^6*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(2)^3
-16*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)-28*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)^2+4*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(2)^3-10*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)^2+2*a^3*c^3*d^4*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(2)^3+2
4*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))*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)^4*exp(2)-6*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)^2+2*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(2)^3+12*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)^2+4*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(2)^3+20*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)^2-4*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(2)^3+2*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(2)^3+4*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(2)^3+96*a^2*c^5*d^9
*exp(1)^4-48*a^2*c^5*d^9*exp(1)^2*exp(2)+12*a^2*c^5*d^9*exp(2)^2+48*a^2*c^5*d^8*exp(1)^4-12*a^2*c^5*d^8*exp(1)
^2*exp(2)+4*a^2*c^5*d^8*exp(2)^2+48*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)^4-24*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)+6*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(2)^2+16*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+28*
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))*sqr
t(2)*exp(1)^2*exp(2)-4*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*ex
p(2))+a*c*d*exp(2))*sqrt(2)*exp(2)^2+8*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+4*a^2*c^4*d^6*sqrt(-c^2*d^3-c*d*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))+a*c*d*exp(2))*sqrt(2)*exp(1)^2*exp(2)-24*a^2*c^3*d^6*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)*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+6*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*ex
p(2))*exp(1)^2*exp(2)-2*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*e
xp(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(2)^2-24*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)*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))*exp(1)^2*exp(2)-16*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)^4-8*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)-6*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*e
xp(2)-12*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)-36*a*c^6*d^11*
exp(1)^2+12*a*c^6*d^11*exp(2)-28*a*c^6*d^10*exp(1)^2+8*a*c^6*d^10*exp(2)-18*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+6*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(2)-20*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)*ex
p(1)^2-10*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+2*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(2)+14*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)^
2-4*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))*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(2)+12*a*c^4*d^7*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)*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+4*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(2)+20*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)^2-4*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(2)+6*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+12*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+4*c
^7*d^13+4*c^7*d^12+2*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)+4*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)+2*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)-2*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))-4*c^5*d^9*sq
rt(-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))-4*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))-2*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))-4*c^5*d^8*(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)))/(8*a^7*d^3*exp(1)^6*exp(2)^4-24*a^7*d^3*exp(1)^4*exp(2)^5+24*a^7*d^3*ex
p(1)^2*exp(2)^6-8*a^7*d^3*exp(2)^7-64*a^6*c*d^5*exp(1)^8*exp(2)^2+224*a^6*c*d^5*exp(1)^6*exp(2)^3-288*a^6*c*d^
5*exp(1)^4*exp(2)^4+160*a^6*c*d^5*exp(1)^2*exp(2)^5-32*a^6*c*d^5*exp(2)^6-16*a^6*c*d^4*exp(1)^6*exp(2)^3+48*a^
6*c*d^4*exp(1)^4*exp(2)^4-48*a^6*c*d^4*exp(1)^2*exp(2)^5+16*a^6*c*d^4*exp(2)^6+128*a^5*c^2*d^7*exp(1)^10-512*a
^5*c^2*d^7*exp(1)^8*exp(2)+816*a^5*c^2*d^7*exp(1)^6*exp(2)^2-656*a^5*c^2*d^7*exp(1)^4*exp(2)^3+272*a^5*c^2*d^7
*exp(1)^2*exp(2)^4-48*a^5*c^2*d^7*exp(2)^5+64*a^5*c^2*d^6*exp(1)^8*exp(2)-208*a^5*c^2*d^6*exp(1)^6*exp(2)^2+24
0*a^5*c^2*d^6*exp(1)^4*exp(2)^3-112*a^5*c^2*d^6*exp(1)^2*exp(2)^4+16*a^5*c^2*d^6*exp(2)^5+8*a^5*c^2*d^5*exp(1)
^6*exp(2)^2-24*a^5*c^2*d^5*exp(1)^4*exp(2)^3+24*a^5*c^2*d^5*exp(1)^2*exp(2)^4-8*a^5*c^2*d^5*exp(2)^5-64*a^4*c^
3*d^9*exp(1)^8+224*a^4*c^3*d^9*exp(1)^6*exp(2)-288*a^4*c^3*d^9*exp(1)^4*exp(2)^2+160*a^4*c^3*d^9*exp(1)^2*exp(
2)^3-32*a^4*c^3*d^9*exp(2)^4-64*a^4*c^3*d^8*exp(1)^8+208*a^4*c^3*d^8*exp(1)^6*exp(2)-240*a^4*c^3*d^8*exp(1)^4*
exp(2)^2+112*a^4*c^3*d^8*exp(1)^2*exp(2)^3-16*a^4*c^3*d^8*exp(2)^4-32*a^4*c^3*d^7*exp(1)^8+112*a^4*c^3*d^7*exp
(1)^6*exp(2)-144*a^4*c^3*d^7*exp(1)^4*exp(2)^2+80*a^4*c^3*d^7*exp(1)^2*exp(2)^3-16*a^4*c^3*d^7*exp(2)^4+8*a^3*
c^4*d^11*exp(1)^6-24*a^3*c^4*d^11*exp(1)^4*exp(2)+24*a^3*c^4*d^11*exp(1)^2*exp(2)^2-8*a^3*c^4*d^11*exp(2)^3+16
*a^3*c^4*d^10*exp(1)^6-48*a^3*c^4*d^10*exp(1)^4*exp(2)+48*a^3*c^4*d^10*exp(1)^2*exp(2)^2-16*a^3*c^4*d^10*exp(2
)^3+8*a^3*c^4*d^9*exp(1)^6-24*a^3*c^4*d^9*exp(1)^4*exp(2)+24*a^3*c^4*d^9*exp(1)^2*exp(2)^2-8*a^3*c^4*d^9*exp(2
)^3)/abs(c)/abs(d)*atan(sqrt(d+x*exp(1))/sqrt(-(-c*d^4*exp(1)^4*a^2+2*c*d^4*exp(1)^2*a^2*exp(2)-c*d^4*a^2*exp(
2)^2+d^2*exp(1)^4*a^3*exp(2)-2*d^2*exp(1)^2*a^3*exp(2)^2+d^2*a^3*exp(2)^3+sqrt((c*d^4*exp(1)^4*a^2-2*c*d^4*exp
(1)^2*a^2*exp(2)+c*d^4*a^2*exp(2)^2-d^2*exp(1)^4*a^3*exp(2)+2*d^2*exp(1)^2*a^3*exp(2)^2-d^2*a^3*exp(2)^3)*(c*d
^4*exp(1)^4*a^2-2*c*d^4*exp(1)^2*a^2*exp(2)+c*d^4*a^2*exp(2)^2-d^2*exp(1)^4*a^3*exp(2)+2*d^2*exp(1)^2*a^3*exp(
2)^2-d^2*a^3*exp(2)^3)-4*(-c*d^3*exp(1)^4*a^2+2*c*d^3*exp(1)^2*a^2*exp(2)-c*d^3*a^2*exp(2)^2)*(-d^3*exp(1)^6*a
^3+3*d^3*exp(1)^4*a^3*exp(2)-3*d^3*exp(1)^2*a^3*exp(2)^2+d^3*a^3*exp(2)^3)))/2/(-c*d^3*exp(1)^4*a^2+2*c*d^3*ex
p(1)^2*a^2*exp(2)-c*d^3*a^2*exp(2)^2)))-(4*a^6*c*d*exp(2)^6-2*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(2)^6-36*a^5*c^2*d^3*exp(1)^2*exp(2)^4+12*a^5*c^
2*d^3*exp(2)^5-4*a^5*c^2*d^2*exp(2)^5+18*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)^4-6*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(2)^5+4*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(2)^5+2*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(2)^5+96*a^4*c^3*d^5*exp(1)^4*exp(2)^2-48*a^4*c^3*d^5*exp(1)^2*exp(
2)^3+12*a^4*c^3*d^5*exp(2)^4+28*a^4*c^3*d^4*exp(1)^2*exp(2)^3-8*a^4*c^3*d^4*exp(2)^4-48*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)^2
+24*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)^3-6*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(2)^4-20*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)^3-2*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(2)^4-14*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)^3+4*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(2)^4-4*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(2)^4-4*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(2)^4-64*a^3*c^4*d^7*exp(1)^6-24*a^3*c^4*d^7
*exp(1)^2*exp(2)^2+8*a^3*c^4*d^7*exp(2)^3-48*a^3*c^4*d^6*exp(1)^4*exp(2)+12*a^3*c^4*d^6*exp(1)^2*exp(2)^2-4*a^
3*c^4*d^6*exp(2)^3+32*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+12*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)^2-4*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(2)^3+16*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)+28*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)^2-4*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(2)^3+10*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)^2-2*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(2)^3+24*a^3*c^2*d^4*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)*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)-6*a^3*c^2*d^4*sqrt(-c^2*d^3+c*d*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))+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+2*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(2)^3+
12*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)^2+4*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(2)^3+20*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)^2-4*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(2)^3+2*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(2)^3+4*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(2)^3+96*a^2*c^5*d^9*exp(1)^4-48*a^2*c^5*d^9*exp(1)^2
*exp(2)+12*a^2*c^5*d^9*exp(2)^2+48*a^2*c^5*d^8*exp(1)^4-12*a^2*c^5*d^8*exp(1)^2*exp(2)+4*a^2*c^5*d^8*exp(2)^2-
48*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+24*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)-6*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*e
xp(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(2)^2-16*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)^4-28*a^2*c^4*d^7*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)^2*exp(2)+4*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(2
)^2-8*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)^4-4*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)-24*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+6*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)^2*exp(2)-2*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(2)^2-24*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)-16*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)^4-8*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)-6*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)-12*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)-36*a*c^6*d^11*exp(1)^2+12*a*c^6*d^11*exp(2)-28*
a*c^6*d^10*exp(1)^2+8*a*c^6*d^10*exp(2)+18*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*ex
p(2)^2+2*a*c*d^2*exp(2))+a*c*d*exp(2))*sqrt(2)*exp(1)^2-6*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(2)+20*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+10*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)^2-2*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(2)
+14*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))*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)^2-4*a*c^4*d^8*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)*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(2)+12*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+4*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(2)+20*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)^2-4*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(2)+6*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+12*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+4*c^7*d^13+4*c^7*d^12-2*c^6*d^12*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)-4*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)-2*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)-2*
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))-4*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))-4*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))-2*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))-4*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)))/(8*a^7*d^3*exp(1)^6*exp(2)^4-24*a^7*d^3*exp(1)^4*exp(2)^5+24*a^7*d^3*exp(1)^2*exp(2)^6-8*a^7*d^3*exp(2)^
7-64*a^6*c*d^5*exp(1)^8*exp(2)^2+224*a^6*c*d^5*exp(1)^6*exp(2)^3-288*a^6*c*d^5*exp(1)^4*exp(2)^4+160*a^6*c*d^5
*exp(1)^2*exp(2)^5-32*a^6*c*d^5*exp(2)^6-16*a^6*c*d^4*exp(1)^6*exp(2)^3+48*a^6*c*d^4*exp(1)^4*exp(2)^4-48*a^6*
c*d^4*exp(1)^2*exp(2)^5+16*a^6*c*d^4*exp(2)^6+128*a^5*c^2*d^7*exp(1)^10-512*a^5*c^2*d^7*exp(1)^8*exp(2)+816*a^
5*c^2*d^7*exp(1)^6*exp(2)^2-656*a^5*c^2*d^7*exp(1)^4*exp(2)^3+272*a^5*c^2*d^7*exp(1)^2*exp(2)^4-48*a^5*c^2*d^7
*exp(2)^5+64*a^5*c^2*d^6*exp(1)^8*exp(2)-208*a^5*c^2*d^6*exp(1)^6*exp(2)^2+240*a^5*c^2*d^6*exp(1)^4*exp(2)^3-1
12*a^5*c^2*d^6*exp(1)^2*exp(2)^4+16*a^5*c^2*d^6*exp(2)^5+8*a^5*c^2*d^5*exp(1)^6*exp(2)^2-24*a^5*c^2*d^5*exp(1)
^4*exp(2)^3+24*a^5*c^2*d^5*exp(1)^2*exp(2)^4-8*a^5*c^2*d^5*exp(2)^5-64*a^4*c^3*d^9*exp(1)^8+224*a^4*c^3*d^9*ex
p(1)^6*exp(2)-288*a^4*c^3*d^9*exp(1)^4*exp(2)^2+160*a^4*c^3*d^9*exp(1)^2*exp(2)^3-32*a^4*c^3*d^9*exp(2)^4-64*a
^4*c^3*d^8*exp(1)^8+208*a^4*c^3*d^8*exp(1)^6*exp(2)-240*a^4*c^3*d^8*exp(1)^4*exp(2)^2+112*a^4*c^3*d^8*exp(1)^2
*exp(2)^3-16*a^4*c^3*d^8*exp(2)^4-32*a^4*c^3*d^7*exp(1)^8+112*a^4*c^3*d^7*exp(1)^6*exp(2)-144*a^4*c^3*d^7*exp(
1)^4*exp(2)^2+80*a^4*c^3*d^7*exp(1)^2*exp(2)^3-16*a^4*c^3*d^7*exp(2)^4+8*a^3*c^4*d^11*exp(1)^6-24*a^3*c^4*d^11
*exp(1)^4*exp(2)+24*a^3*c^4*d^11*exp(1)^2*exp(2)^2-8*a^3*c^4*d^11*exp(2)^3+16*a^3*c^4*d^10*exp(1)^6-48*a^3*c^4
*d^10*exp(1)^4*exp(2)+48*a^3*c^4*d^10*exp(1)^2*exp(2)^2-16*a^3*c^4*d^10*exp(2)^3+8*a^3*c^4*d^9*exp(1)^6-24*a^3
*c^4*d^9*exp(1)^4*exp(2)+24*a^3*c^4*d^9*exp(1)^2*exp(2)^2-8*a^3*c^4*d^9*exp(2)^3)/abs(c)/abs(d)*atan(sqrt(d+x*
exp(1))/sqrt(-(-c*d^4*exp(1)^4*a^2+2*c*d^4*exp(1)^2*a^2*exp(2)-c*d^4*a^2*exp(2)^2+d^2*exp(1)^4*a^3*exp(2)-2*d^
2*exp(1)^2*a^3*exp(2)^2+d^2*a^3*exp(2)^3-sqrt((c*d^4*exp(1)^4*a^2-2*c*d^4*exp(1)^2*a^2*exp(2)+c*d^4*a^2*exp(2)
^2-d^2*exp(1)^4*a^3*exp(2)+2*d^2*exp(1)^2*a^3*exp(2)^2-d^2*a^3*exp(2)^3)*(c*d^4*exp(1)^4*a^2-2*c*d^4*exp(1)^2*
a^2*exp(2)+c*d^4*a^2*exp(2)^2-d^2*exp(1)^4*a^3*exp(2)+2*d^2*exp(1)^2*a^3*exp(2)^2-d^2*a^3*exp(2)^3)-4*(-c*d^3*
exp(1)^4*a^2+2*c*d^3*exp(1)^2*a^2*exp(2)-c*d^3*a^2*exp(2)^2)*(-d^3*exp(1)^6*a^3+3*d^3*exp(1)^4*a^3*exp(2)-3*d^
3*exp(1)^2*a^3*exp(2)^2+d^3*a^3*exp(2)^3)))/2/(-c*d^3*exp(1)^4*a^2+2*c*d^3*exp(1)^2*a^2*exp(2)-c*d^3*a^2*exp(2
)^2)))+(3*(d+x*exp(1))*c*d^2-3*(d+x*exp(1))*a*exp(2)+d*exp(1)^2*a-d*a*exp(2))/(-3*d^2*exp(1)^4*a^2+6*d^2*exp(1
)^2*a^2*exp(2)-3*d^2*a^2*exp(2)^2)/sqrt(d+x*exp(1))/(d+x*exp(1)))

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maple [A]  time = 0.06, size = 146, normalized size = 0.95 \begin {gather*} -\frac {2 c^{3} d^{3} \arctan \left (\frac {\sqrt {e x +d}\, c d}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) c d}}\right )}{\left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) c d}}-\frac {2 c^{2} d^{2}}{\left (a \,e^{2}-c \,d^{2}\right )^{3} \sqrt {e x +d}}+\frac {2 c d}{3 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2}{5 \left (a \,e^{2}-c \,d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(5/2)/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x),x)

[Out]

-2*c^3*d^3/(a*e^2-c*d^2)^3/((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)-2/5/(
a*e^2-c*d^2)/(e*x+d)^(5/2)-2*c^2*d^2/(a*e^2-c*d^2)^3/(e*x+d)^(1/2)+2/3*c*d/(a*e^2-c*d^2)^2/(e*x+d)^(3/2)

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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(1/(e*x+d)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),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 = 0.14, size = 171, normalized size = 1.12 \begin {gather*} -\frac {\frac {2}{5\,\left (a\,e^2-c\,d^2\right )}+\frac {2\,c^2\,d^2\,{\left (d+e\,x\right )}^2}{{\left (a\,e^2-c\,d^2\right )}^3}-\frac {2\,c\,d\,\left (d+e\,x\right )}{3\,{\left (a\,e^2-c\,d^2\right )}^2}}{{\left (d+e\,x\right )}^{5/2}}-\frac {2\,c^{5/2}\,d^{5/2}\,\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 )}{{\left (a\,e^2-c\,d^2\right )}^{7/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d + e*x)^(5/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)),x)

[Out]

- (2/(5*(a*e^2 - c*d^2)) + (2*c^2*d^2*(d + e*x)^2)/(a*e^2 - c*d^2)^3 - (2*c*d*(d + e*x))/(3*(a*e^2 - c*d^2)^2)
)/(d + e*x)^(5/2) - (2*c^(5/2)*d^(5/2)*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)))/(a*e^2 - c*d^2)^(7/2)

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sympy [A]  time = 34.85, size = 141, normalized size = 0.92 \begin {gather*} - \frac {2 c^{2} d^{2}}{\sqrt {d + e x} \left (a e^{2} - c d^{2}\right )^{3}} - \frac {2 c^{2} d^{2} \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {a e^{2} - c d^{2}}{c d}}} \right )}}{\sqrt {\frac {a e^{2} - c d^{2}}{c d}} \left (a e^{2} - c d^{2}\right )^{3}} + \frac {2 c d}{3 \left (d + e x\right )^{\frac {3}{2}} \left (a e^{2} - c d^{2}\right )^{2}} - \frac {2}{5 \left (d + e x\right )^{\frac {5}{2}} \left (a e^{2} - c d^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**(5/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)

[Out]

-2*c**2*d**2/(sqrt(d + e*x)*(a*e**2 - c*d**2)**3) - 2*c**2*d**2*atan(sqrt(d + e*x)/sqrt((a*e**2 - c*d**2)/(c*d
)))/(sqrt((a*e**2 - c*d**2)/(c*d))*(a*e**2 - c*d**2)**3) + 2*c*d/(3*(d + e*x)**(3/2)*(a*e**2 - c*d**2)**2) - 2
/(5*(d + e*x)**(5/2)*(a*e**2 - c*d**2))

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