3.2.61 \(\int \frac {1}{(d+e x^2)^2 (-c d^2+b d e+b e^2 x^2+c e^2 x^4)} \, dx\)

Optimal. Leaf size=187 \[ -\frac {\left (3 b^2 e^2-16 b c d e+28 c^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e} (2 c d-b e)^3}-\frac {c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right )}{\sqrt {e} \sqrt {c d-b e} (2 c d-b e)^3}-\frac {x (10 c d-3 b e)}{8 d^2 \left (d+e x^2\right ) (2 c d-b e)^2}-\frac {x}{4 d \left (d+e x^2\right )^2 (2 c d-b e)} \]

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Rubi [A]  time = 0.28, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1149, 414, 527, 522, 205, 208} \begin {gather*} -\frac {\left (3 b^2 e^2-16 b c d e+28 c^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e} (2 c d-b e)^3}-\frac {c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right )}{\sqrt {e} \sqrt {c d-b e} (2 c d-b e)^3}-\frac {x (10 c d-3 b e)}{8 d^2 \left (d+e x^2\right ) (2 c d-b e)^2}-\frac {x}{4 d \left (d+e x^2\right )^2 (2 c d-b e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x^2)^2*(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4)),x]

[Out]

-x/(4*d*(2*c*d - b*e)*(d + e*x^2)^2) - ((10*c*d - 3*b*e)*x)/(8*d^2*(2*c*d - b*e)^2*(d + e*x^2)) - ((28*c^2*d^2
- 16*b*c*d*e + 3*b^2*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(5/2)*Sqrt[e]*(2*c*d - b*e)^3) - (c^(5/2)*ArcTanh
[(Sqrt[c]*Sqrt[e]*x)/Sqrt[c*d - b*e]])/(Sqrt[e]*Sqrt[c*d - b*e]*(2*c*d - b*e)^3)

Rule 205

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

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 414

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 522

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 527

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> -Simp[
((b*e - a*f)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d
)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)
*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 1149

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[(d + e*x^2)^(p +
q)*(a/d + (c*x^2)/e)^p, x] /; FreeQ[{a, b, c, d, e, q}, 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}{\left (d+e x^2\right )^2 \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx &=\int \frac {1}{\left (d+e x^2\right )^3 \left (\frac {-c d^2+b d e}{d}+c e x^2\right )} \, dx\\ &=-\frac {x}{4 d (2 c d-b e) \left (d+e x^2\right )^2}+\frac {\int \frac {e (7 c d-3 b e)-3 c e^2 x^2}{\left (d+e x^2\right )^2 \left (\frac {-c d^2+b d e}{d}+c e x^2\right )} \, dx}{4 d e (2 c d-b e)}\\ &=-\frac {x}{4 d (2 c d-b e) \left (d+e x^2\right )^2}-\frac {(10 c d-3 b e) x}{8 d^2 (2 c d-b e)^2 \left (d+e x^2\right )}+\frac {\int \frac {e^2 \left (18 c^2 d^2-13 b c d e+3 b^2 e^2\right )-c e^3 (10 c d-3 b e) x^2}{\left (d+e x^2\right ) \left (\frac {-c d^2+b d e}{d}+c e x^2\right )} \, dx}{8 d^2 e^2 (2 c d-b e)^2}\\ &=-\frac {x}{4 d (2 c d-b e) \left (d+e x^2\right )^2}-\frac {(10 c d-3 b e) x}{8 d^2 (2 c d-b e)^2 \left (d+e x^2\right )}+\frac {c^3 \int \frac {1}{\frac {-c d^2+b d e}{d}+c e x^2} \, dx}{(2 c d-b e)^3}-\frac {\left (28 c^2 d^2-16 b c d e+3 b^2 e^2\right ) \int \frac {1}{d+e x^2} \, dx}{8 d^2 (2 c d-b e)^3}\\ &=-\frac {x}{4 d (2 c d-b e) \left (d+e x^2\right )^2}-\frac {(10 c d-3 b e) x}{8 d^2 (2 c d-b e)^2 \left (d+e x^2\right )}-\frac {\left (28 c^2 d^2-16 b c d e+3 b^2 e^2\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e} (2 c d-b e)^3}-\frac {c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right )}{\sqrt {e} \sqrt {c d-b e} (2 c d-b e)^3}\\ \end {align*}

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Mathematica [A]  time = 0.41, size = 177, normalized size = 0.95 \begin {gather*} \frac {1}{8} \left (-\frac {\left (3 b^2 e^2-16 b c d e+28 c^2 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{5/2} \sqrt {e} (2 c d-b e)^3}-\frac {\frac {8 c^{5/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {b e-c d}}\right )}{\sqrt {e} \sqrt {b e-c d}}+\frac {x (b e-2 c d) \left (2 c d \left (7 d+5 e x^2\right )-b e \left (5 d+3 e x^2\right )\right )}{d^2 \left (d+e x^2\right )^2}}{(b e-2 c d)^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x^2)^2*(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4)),x]

[Out]

(-(((28*c^2*d^2 - 16*b*c*d*e + 3*b^2*e^2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(d^(5/2)*Sqrt[e]*(2*c*d - b*e)^3)) - ((
(-2*c*d + b*e)*x*(-(b*e*(5*d + 3*e*x^2)) + 2*c*d*(7*d + 5*e*x^2)))/(d^2*(d + e*x^2)^2) + (8*c^(5/2)*ArcTan[(Sq
rt[c]*Sqrt[e]*x)/Sqrt[-(c*d) + b*e]])/(Sqrt[e]*Sqrt[-(c*d) + b*e]))/(-2*c*d + b*e)^3)/8

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (d+e x^2\right )^2 \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[1/((d + e*x^2)^2*(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4)),x]

[Out]

IntegrateAlgebraic[1/((d + e*x^2)^2*(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4)), x]

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fricas [B]  time = 3.71, size = 1765, normalized size = 9.44

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x^2+d)^2/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x, algorithm="fricas")

[Out]

[-1/16*(2*(20*c^2*d^3*e^2 - 16*b*c*d^2*e^3 + 3*b^2*d*e^4)*x^3 + 8*(c^2*d^3*e^3*x^4 + 2*c^2*d^4*e^2*x^2 + c^2*d
^5*e)*sqrt(c/(c*d*e - b*e^2))*log((c*e*x^2 + 2*(c*d*e - b*e^2)*x*sqrt(c/(c*d*e - b*e^2)) + c*d - b*e)/(c*e*x^2
- c*d + b*e)) - (28*c^2*d^4 - 16*b*c*d^3*e + 3*b^2*d^2*e^2 + (28*c^2*d^2*e^2 - 16*b*c*d*e^3 + 3*b^2*e^4)*x^4
+ 2*(28*c^2*d^3*e - 16*b*c*d^2*e^2 + 3*b^2*d*e^3)*x^2)*sqrt(-d*e)*log((e*x^2 - 2*sqrt(-d*e)*x - d)/(e*x^2 + d)
) + 2*(28*c^2*d^4*e - 24*b*c*d^3*e^2 + 5*b^2*d^2*e^3)*x)/(8*c^3*d^8*e - 12*b*c^2*d^7*e^2 + 6*b^2*c*d^6*e^3 - b
^3*d^5*e^4 + (8*c^3*d^6*e^3 - 12*b*c^2*d^5*e^4 + 6*b^2*c*d^4*e^5 - b^3*d^3*e^6)*x^4 + 2*(8*c^3*d^7*e^2 - 12*b*
c^2*d^6*e^3 + 6*b^2*c*d^5*e^4 - b^3*d^4*e^5)*x^2), -1/8*((20*c^2*d^3*e^2 - 16*b*c*d^2*e^3 + 3*b^2*d*e^4)*x^3 +
(28*c^2*d^4 - 16*b*c*d^3*e + 3*b^2*d^2*e^2 + (28*c^2*d^2*e^2 - 16*b*c*d*e^3 + 3*b^2*e^4)*x^4 + 2*(28*c^2*d^3*
e - 16*b*c*d^2*e^2 + 3*b^2*d*e^3)*x^2)*sqrt(d*e)*arctan(sqrt(d*e)*x/d) + 4*(c^2*d^3*e^3*x^4 + 2*c^2*d^4*e^2*x^
2 + c^2*d^5*e)*sqrt(c/(c*d*e - b*e^2))*log((c*e*x^2 + 2*(c*d*e - b*e^2)*x*sqrt(c/(c*d*e - b*e^2)) + c*d - b*e)
/(c*e*x^2 - c*d + b*e)) + (28*c^2*d^4*e - 24*b*c*d^3*e^2 + 5*b^2*d^2*e^3)*x)/(8*c^3*d^8*e - 12*b*c^2*d^7*e^2 +
6*b^2*c*d^6*e^3 - b^3*d^5*e^4 + (8*c^3*d^6*e^3 - 12*b*c^2*d^5*e^4 + 6*b^2*c*d^4*e^5 - b^3*d^3*e^6)*x^4 + 2*(8
*c^3*d^7*e^2 - 12*b*c^2*d^6*e^3 + 6*b^2*c*d^5*e^4 - b^3*d^4*e^5)*x^2), -1/16*(2*(20*c^2*d^3*e^2 - 16*b*c*d^2*e
^3 + 3*b^2*d*e^4)*x^3 - 16*(c^2*d^3*e^3*x^4 + 2*c^2*d^4*e^2*x^2 + c^2*d^5*e)*sqrt(-c/(c*d*e - b*e^2))*arctan(e
*x*sqrt(-c/(c*d*e - b*e^2))) - (28*c^2*d^4 - 16*b*c*d^3*e + 3*b^2*d^2*e^2 + (28*c^2*d^2*e^2 - 16*b*c*d*e^3 + 3
*b^2*e^4)*x^4 + 2*(28*c^2*d^3*e - 16*b*c*d^2*e^2 + 3*b^2*d*e^3)*x^2)*sqrt(-d*e)*log((e*x^2 - 2*sqrt(-d*e)*x -
d)/(e*x^2 + d)) + 2*(28*c^2*d^4*e - 24*b*c*d^3*e^2 + 5*b^2*d^2*e^3)*x)/(8*c^3*d^8*e - 12*b*c^2*d^7*e^2 + 6*b^2
*c*d^6*e^3 - b^3*d^5*e^4 + (8*c^3*d^6*e^3 - 12*b*c^2*d^5*e^4 + 6*b^2*c*d^4*e^5 - b^3*d^3*e^6)*x^4 + 2*(8*c^3*d
^7*e^2 - 12*b*c^2*d^6*e^3 + 6*b^2*c*d^5*e^4 - b^3*d^4*e^5)*x^2), -1/8*((20*c^2*d^3*e^2 - 16*b*c*d^2*e^3 + 3*b^
2*d*e^4)*x^3 - 8*(c^2*d^3*e^3*x^4 + 2*c^2*d^4*e^2*x^2 + c^2*d^5*e)*sqrt(-c/(c*d*e - b*e^2))*arctan(e*x*sqrt(-c
/(c*d*e - b*e^2))) + (28*c^2*d^4 - 16*b*c*d^3*e + 3*b^2*d^2*e^2 + (28*c^2*d^2*e^2 - 16*b*c*d*e^3 + 3*b^2*e^4)*
x^4 + 2*(28*c^2*d^3*e - 16*b*c*d^2*e^2 + 3*b^2*d*e^3)*x^2)*sqrt(d*e)*arctan(sqrt(d*e)*x/d) + (28*c^2*d^4*e - 2
4*b*c*d^3*e^2 + 5*b^2*d^2*e^3)*x)/(8*c^3*d^8*e - 12*b*c^2*d^7*e^2 + 6*b^2*c*d^6*e^3 - b^3*d^5*e^4 + (8*c^3*d^6
*e^3 - 12*b*c^2*d^5*e^4 + 6*b^2*c*d^4*e^5 - b^3*d^3*e^6)*x^4 + 2*(8*c^3*d^7*e^2 - 12*b*c^2*d^6*e^3 + 6*b^2*c*d
^5*e^4 - b^3*d^4*e^5)*x^2)]

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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^2+d)^2/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: (4*b^5*c*exp(1)*exp(2)^5+2*b^5*sqrt(2)*s
qrt(b*c*exp(2)^2-c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(1)*exp(2)^4-36*b^4*c^
2*d*exp(1)^2*exp(2)^4-4*b^4*c^2*d*exp(2)^5-4*b^4*c^2*exp(1)*exp(2)^5-18*b^4*c*d*sqrt(2)*sqrt(b*c*exp(2)^2-c*sq
rt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(1)^2*exp(2)^3-2*b^4*c*d*sqrt(2)*sqrt(b*c*e
xp(2)^2-c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(2)^4-4*b^4*c*sqrt(2)*sqrt(b*c*
exp(2)^2-c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(1)*exp(2)^4+2*b^4*sqrt(2)*sqr
t(b*c*exp(2)^2-c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*sqrt(b^2*exp(2)^2+4*c^2*d^2
*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(1)*exp(2)^3+96*b^3*c^3*d^2*exp(1)^3*exp(2)^3+64*b^3*c^3*d^2*exp(1)*exp(2)^4
+28*b^3*c^3*d*exp(1)^2*exp(2)^4+4*b^3*c^3*d*exp(2)^5+48*b^3*c^2*d^2*sqrt(2)*sqrt(b*c*exp(2)^2-c*sqrt(b^2*exp(2
)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(1)^3*exp(2)^2+32*b^3*c^2*d^2*sqrt(2)*sqrt(b*c*exp(2)^2
-c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(1)*exp(2)^3+20*b^3*c^2*d*sqrt(2)*sqrt
(b*c*exp(2)^2-c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(1)^2*exp(2)^3+4*b^3*c^2*
d*sqrt(2)*sqrt(b*c*exp(2)^2-c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(2)^4+2*b^3
*c^2*sqrt(2)*sqrt(b*c*exp(2)^2-c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(1)*exp(
2)^4-14*b^3*c*d*sqrt(2)*sqrt(b*c*exp(2)^2-c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*
sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(1)^2*exp(2)^2-2*b^3*c*d*sqrt(2)*sqrt(b*c*exp(2)^
2-c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c
*d*exp(1)*exp(2))*exp(2)^3-4*b^3*c*sqrt(2)*sqrt(b*c*exp(2)^2-c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(
1)*exp(2))*exp(2))*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(1)*exp(2)^3-4*b^3*c*(b^2*exp(
2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(1)*exp(2)^3-64*b^2*c^4*d^3*exp(1)^4*exp(2)^2-224*b^2*c^4*d^3*
exp(1)^2*exp(2)^3-32*b^2*c^4*d^3*exp(2)^4-48*b^2*c^4*d^2*exp(1)^3*exp(2)^3-48*b^2*c^4*d^2*exp(1)*exp(2)^4-32*b
^2*c^3*d^3*sqrt(2)*sqrt(b*c*exp(2)^2-c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(1
)^4*exp(2)-112*b^2*c^3*d^3*sqrt(2)*sqrt(b*c*exp(2)^2-c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2
))*exp(2))*exp(1)^2*exp(2)^2-16*b^2*c^3*d^3*sqrt(2)*sqrt(b*c*exp(2)^2-c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b
*c*d*exp(1)*exp(2))*exp(2))*exp(2)^3-16*b^2*c^3*d^2*sqrt(2)*sqrt(b*c*exp(2)^2-c*sqrt(b^2*exp(2)^2+4*c^2*d^2*ex
p(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(1)^3*exp(2)^2-32*b^2*c^3*d^2*sqrt(2)*sqrt(b*c*exp(2)^2-c*sqrt(b^2*exp(
2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(1)*exp(2)^3-10*b^2*c^3*d*sqrt(2)*sqrt(b*c*exp(2)^2-c*
sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(1)^2*exp(2)^3-2*b^2*c^3*d*sqrt(2)*sqrt(b
*c*exp(2)^2-c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(2)^4+24*b^2*c^2*d^2*sqrt(2
)*sqrt(b*c*exp(2)^2-c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*sqrt(b^2*exp(2)^2+4*c^
2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(1)^3*exp(2)+24*b^2*c^2*d^2*sqrt(2)*sqrt(b*c*exp(2)^2-c*sqrt(b^2*exp(2)
^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*e
xp(1)*exp(2)^2+12*b^2*c^2*d*sqrt(2)*sqrt(b*c*exp(2)^2-c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(
2))*exp(2))*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(1)^2*exp(2)^2+4*b^2*c^2*d*sqrt(2)*sq
rt(b*c*exp(2)^2-c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*sqrt(b^2*exp(2)^2+4*c^2*d^
2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2)^3+20*b^2*c^2*d*(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*ex
p(1)^2*exp(2)^2+4*b^2*c^2*d*(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2)^3+2*b^2*c^2*sqrt(2)*s
qrt(b*c*exp(2)^2-c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*sqrt(b^2*exp(2)^2+4*c^2*d
^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(1)*exp(2)^3+4*b^2*c^2*(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2
))*exp(1)*exp(2)^3+128*b*c^5*d^4*exp(1)^3*exp(2)^2+192*b*c^5*d^4*exp(1)*exp(2)^3+112*b*c^5*d^3*exp(1)^2*exp(2)
^3+16*b*c^5*d^3*exp(2)^4+64*b*c^4*d^4*sqrt(2)*sqrt(b*c*exp(2)^2-c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*e
xp(1)*exp(2))*exp(2))*exp(1)^3*exp(2)+96*b*c^4*d^4*sqrt(2)*sqrt(b*c*exp(2)^2-c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp
(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(1)*exp(2)^2+16*b*c^4*d^3*sqrt(2)*sqrt(b*c*exp(2)^2-c*sqrt(b^2*exp(2)^2+
4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(1)^2*exp(2)^2+16*b*c^4*d^3*sqrt(2)*sqrt(b*c*exp(2)^2-c*sqr
t(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(2)^3+8*b*c^4*d^2*sqrt(2)*sqrt(b*c*exp(2)^2-
c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(1)^3*exp(2)^2+16*b*c^4*d^2*sqrt(2)*sqr
t(b*c*exp(2)^2-c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(1)*exp(2)^3-56*b*c^3*d^
3*sqrt(2)*sqrt(b*c*exp(2)^2-c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*sqrt(b^2*exp(2
)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(1)^2*exp(2)-8*b*c^3*d^3*sqrt(2)*sqrt(b*c*exp(2)^2-c*sqrt(b^2*e
xp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(
2))*exp(2)^2-16*b*c^3*d^2*sqrt(2)*sqrt(b*c*exp(2)^2-c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2)
)*exp(2))*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(1)*exp(2)^2-16*b*c^3*d^2*(b^2*exp(2)^2
+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(1)^3*exp(2)-32*b*c^3*d^2*(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*e
xp(1)*exp(2))*exp(1)*exp(2)^2-6*b*c^3*d*sqrt(2)*sqrt(b*c*exp(2)^2-c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d
*exp(1)*exp(2))*exp(2))*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(1)^2*exp(2)^2-2*b*c^3*d*
sqrt(2)*sqrt(b*c*exp(2)^2-c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*sqrt(b^2*exp(2)^
2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2)^3-12*b*c^3*d*(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*ex
p(2))*exp(1)^2*exp(2)^2-4*b*c^3*d*(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2)^3-64*c^6*d^5*ex
p(1)^2*exp(2)^2-64*c^6*d^5*exp(2)^3-64*c^6*d^4*exp(1)*exp(2)^3-32*c^5*d^5*sqrt(2)*sqrt(b*c*exp(2)^2-c*sqrt(b^2
*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(1)^2*exp(2)-32*c^5*d^5*sqrt(2)*sqrt(b*c*exp(2)^2
-c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(2)^2-8*c^5*d^3*sqrt(2)*sqrt(b*c*exp(2
)^2-c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(1)^2*exp(2)^2-8*c^5*d^3*sqrt(2)*sq
rt(b*c*exp(2)^2-c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(2)^3+32*c^4*d^4*sqrt(2
)*sqrt(b*c*exp(2)^2-c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*sqrt(b^2*exp(2)^2+4*c^
2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(1)*exp(2)+16*c^4*d^3*(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp
(2))*exp(1)^2*exp(2)+16*c^4*d^3*(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2)^2+8*c^4*d^2*sqrt(
2)*sqrt(b*c*exp(2)^2-c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*sqrt(b^2*exp(2)^2+4*c
^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(1)*exp(2)^2+16*c^4*d^2*(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*
exp(2))*exp(1)*exp(2)^2)/(8*b^6*d^3*exp(1)^6*exp(2)^3-16*b^6*d^3*exp(1)^4*exp(2)^4+8*b^6*d^3*exp(1)^2*exp(2)^5
-64*b^5*c*d^4*exp(1)^7*exp(2)^2+112*b^5*c*d^4*exp(1)^5*exp(2)^3-32*b^5*c*d^4*exp(1)^3*exp(2)^4-16*b^5*c*d^4*ex
p(1)*exp(2)^5-16*b^5*c*d^3*exp(1)^6*exp(2)^3+32*b^5*c*d^3*exp(1)^4*exp(2)^4-16*b^5*c*d^3*exp(1)^2*exp(2)^5+128
*b^4*c^2*d^5*exp(1)^8*exp(2)-64*b^4*c^2*d^5*exp(1)^6*exp(2)^2-248*b^4*c^2*d^5*exp(1)^4*exp(2)^3+176*b^4*c^2*d^
5*exp(1)^2*exp(2)^4+8*b^4*c^2*d^5*exp(2)^5+64*b^4*c^2*d^4*exp(1)^7*exp(2)^2-96*b^4*c^2*d^4*exp(1)^5*exp(2)^3+3
2*b^4*c^2*d^4*exp(1)*exp(2)^5+8*b^4*c^2*d^3*exp(1)^6*exp(2)^3-16*b^4*c^2*d^3*exp(1)^4*exp(2)^4+8*b^4*c^2*d^3*e
xp(1)^2*exp(2)^5-512*b^3*c^3*d^6*exp(1)^7*exp(2)+832*b^3*c^3*d^6*exp(1)^5*exp(2)^2-128*b^3*c^3*d^6*exp(1)^3*ex
p(2)^3-192*b^3*c^3*d^6*exp(1)*exp(2)^4-192*b^3*c^3*d^5*exp(1)^6*exp(2)^2+368*b^3*c^3*d^5*exp(1)^4*exp(2)^3-160
*b^3*c^3*d^5*exp(1)^2*exp(2)^4-16*b^3*c^3*d^5*exp(2)^5-32*b^3*c^3*d^4*exp(1)^7*exp(2)^2+48*b^3*c^3*d^4*exp(1)^
5*exp(2)^3-16*b^3*c^3*d^4*exp(1)*exp(2)^5+768*b^2*c^4*d^7*exp(1)^6*exp(2)-1472*b^2*c^4*d^7*exp(1)^4*exp(2)^2+6
40*b^2*c^4*d^7*exp(1)^2*exp(2)^3+64*b^2*c^4*d^7*exp(2)^4+192*b^2*c^4*d^6*exp(1)^5*exp(2)^2-384*b^2*c^4*d^6*exp
(1)^3*exp(2)^3+192*b^2*c^4*d^6*exp(1)*exp(2)^4+96*b^2*c^4*d^5*exp(1)^6*exp(2)^2-184*b^2*c^4*d^5*exp(1)^4*exp(2
)^3+80*b^2*c^4*d^5*exp(1)^2*exp(2)^4+8*b^2*c^4*d^5*exp(2)^5-512*b*c^5*d^8*exp(1)^5*exp(2)+1024*b*c^5*d^8*exp(1
)^3*exp(2)^2-512*b*c^5*d^8*exp(1)*exp(2)^3-64*b*c^5*d^7*exp(1)^4*exp(2)^2+128*b*c^5*d^7*exp(1)^2*exp(2)^3-64*b
*c^5*d^7*exp(2)^4-96*b*c^5*d^6*exp(1)^5*exp(2)^2+192*b*c^5*d^6*exp(1)^3*exp(2)^3-96*b*c^5*d^6*exp(1)*exp(2)^4+
128*c^6*d^9*exp(1)^4*exp(2)-256*c^6*d^9*exp(1)^2*exp(2)^2+128*c^6*d^9*exp(2)^3+32*c^6*d^7*exp(1)^4*exp(2)^2-64
*c^6*d^7*exp(1)^2*exp(2)^3+32*c^6*d^7*exp(2)^4)/abs(c)*atan(x/sqrt(-(c^2*exp(2)^3*b*d^4-2*c^2*exp(2)^2*b*d^4*e
xp(1)^2+c^2*exp(2)*b*d^4*exp(1)^4-2*c*exp(2)^3*b^2*d^3*exp(1)+4*c*exp(2)^2*b^2*d^3*exp(1)^3-2*c*exp(2)*b^2*d^3
*exp(1)^5+exp(2)^3*b^3*d^2*exp(1)^2-2*exp(2)^2*b^3*d^2*exp(1)^4+exp(2)*b^3*d^2*exp(1)^6+sqrt((-c^2*exp(2)^3*b*
d^4+2*c^2*exp(2)^2*b*d^4*exp(1)^2-c^2*exp(2)*b*d^4*exp(1)^4+2*c*exp(2)^3*b^2*d^3*exp(1)-4*c*exp(2)^2*b^2*d^3*e
xp(1)^3+2*c*exp(2)*b^2*d^3*exp(1)^5-exp(2)^3*b^3*d^2*exp(1)^2+2*exp(2)^2*b^3*d^2*exp(1)^4-exp(2)*b^3*d^2*exp(1
)^6)*(-c^2*exp(2)^3*b*d^4+2*c^2*exp(2)^2*b*d^4*exp(1)^2-c^2*exp(2)*b*d^4*exp(1)^4+2*c*exp(2)^3*b^2*d^3*exp(1)-
4*c*exp(2)^2*b^2*d^3*exp(1)^3+2*c*exp(2)*b^2*d^3*exp(1)^5-exp(2)^3*b^3*d^2*exp(1)^2+2*exp(2)^2*b^3*d^2*exp(1)^
4-exp(2)*b^3*d^2*exp(1)^6)-4*(-c^3*exp(2)^3*d^4+2*c^3*exp(2)^2*d^4*exp(1)^2-c^3*exp(2)*d^4*exp(1)^4+2*c^2*exp(
2)^3*b*d^3*exp(1)-4*c^2*exp(2)^2*b*d^3*exp(1)^3+2*c^2*exp(2)*b*d^3*exp(1)^5-c*exp(2)^3*b^2*d^2*exp(1)^2+2*c*ex
p(2)^2*b^2*d^2*exp(1)^4-c*exp(2)*b^2*d^2*exp(1)^6)*(c^3*exp(2)^2*d^6-2*c^3*exp(2)*d^6*exp(1)^2+c^3*d^6*exp(1)^
4-3*c^2*exp(2)^2*b*d^5*exp(1)+6*c^2*exp(2)*b*d^5*exp(1)^3-3*c^2*b*d^5*exp(1)^5+3*c*exp(2)^2*b^2*d^4*exp(1)^2-6
*c*exp(2)*b^2*d^4*exp(1)^4+3*c*b^2*d^4*exp(1)^6-exp(2)^2*b^3*d^3*exp(1)^3+2*exp(2)*b^3*d^3*exp(1)^5-b^3*d^3*ex
p(1)^7)))/2/(-c^3*exp(2)^3*d^4+2*c^3*exp(2)^2*d^4*exp(1)^2-c^3*exp(2)*d^4*exp(1)^4+2*c^2*exp(2)^3*b*d^3*exp(1)
-4*c^2*exp(2)^2*b*d^3*exp(1)^3+2*c^2*exp(2)*b*d^3*exp(1)^5-c*exp(2)^3*b^2*d^2*exp(1)^2+2*c*exp(2)^2*b^2*d^2*ex
p(1)^4-c*exp(2)*b^2*d^2*exp(1)^6)))-(4*b^5*c*exp(1)*exp(2)^5-2*b^5*sqrt(2)*sqrt(b*c*exp(2)^2+c*sqrt(b^2*exp(2)
^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(1)*exp(2)^4-36*b^4*c^2*d*exp(1)^2*exp(2)^4-4*b^4*c^2*d*
exp(2)^5-4*b^4*c^2*exp(1)*exp(2)^5+18*b^4*c*d*sqrt(2)*sqrt(b*c*exp(2)^2+c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4
*b*c*d*exp(1)*exp(2))*exp(2))*exp(1)^2*exp(2)^3+2*b^4*c*d*sqrt(2)*sqrt(b*c*exp(2)^2+c*sqrt(b^2*exp(2)^2+4*c^2*
d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(2)^4+4*b^4*c*sqrt(2)*sqrt(b*c*exp(2)^2+c*sqrt(b^2*exp(2)^2+4*c^2
*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(1)*exp(2)^4+2*b^4*sqrt(2)*sqrt(b*c*exp(2)^2+c*sqrt(b^2*exp(2)^2
+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp
(1)*exp(2)^3+96*b^3*c^3*d^2*exp(1)^3*exp(2)^3+64*b^3*c^3*d^2*exp(1)*exp(2)^4+28*b^3*c^3*d*exp(1)^2*exp(2)^4+4*
b^3*c^3*d*exp(2)^5-48*b^3*c^2*d^2*sqrt(2)*sqrt(b*c*exp(2)^2+c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1
)*exp(2))*exp(2))*exp(1)^3*exp(2)^2-32*b^3*c^2*d^2*sqrt(2)*sqrt(b*c*exp(2)^2+c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp
(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(1)*exp(2)^3-20*b^3*c^2*d*sqrt(2)*sqrt(b*c*exp(2)^2+c*sqrt(b^2*exp(2)^2+
4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(1)^2*exp(2)^3-4*b^3*c^2*d*sqrt(2)*sqrt(b*c*exp(2)^2+c*sqrt
(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(2)^4-2*b^3*c^2*sqrt(2)*sqrt(b*c*exp(2)^2+c*s
qrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(1)*exp(2)^4-14*b^3*c*d*sqrt(2)*sqrt(b*c*e
xp(2)^2+c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)
-4*b*c*d*exp(1)*exp(2))*exp(1)^2*exp(2)^2-2*b^3*c*d*sqrt(2)*sqrt(b*c*exp(2)^2+c*sqrt(b^2*exp(2)^2+4*c^2*d^2*ex
p(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2)^3-4*b^3*c
*sqrt(2)*sqrt(b*c*exp(2)^2+c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*sqrt(b^2*exp(2)
^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(1)*exp(2)^3-4*b^3*c*(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(
1)*exp(2))*exp(1)*exp(2)^3-64*b^2*c^4*d^3*exp(1)^4*exp(2)^2-224*b^2*c^4*d^3*exp(1)^2*exp(2)^3-32*b^2*c^4*d^3*e
xp(2)^4-48*b^2*c^4*d^2*exp(1)^3*exp(2)^3-48*b^2*c^4*d^2*exp(1)*exp(2)^4+32*b^2*c^3*d^3*sqrt(2)*sqrt(b*c*exp(2)
^2+c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(1)^4*exp(2)+112*b^2*c^3*d^3*sqrt(2)
*sqrt(b*c*exp(2)^2+c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(1)^2*exp(2)^2+16*b^
2*c^3*d^3*sqrt(2)*sqrt(b*c*exp(2)^2+c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(2)
^3+16*b^2*c^3*d^2*sqrt(2)*sqrt(b*c*exp(2)^2+c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2)
)*exp(1)^3*exp(2)^2+32*b^2*c^3*d^2*sqrt(2)*sqrt(b*c*exp(2)^2+c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(
1)*exp(2))*exp(2))*exp(1)*exp(2)^3+10*b^2*c^3*d*sqrt(2)*sqrt(b*c*exp(2)^2+c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)
-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(1)^2*exp(2)^3+2*b^2*c^3*d*sqrt(2)*sqrt(b*c*exp(2)^2+c*sqrt(b^2*exp(2)^2+4*
c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(2)^4+24*b^2*c^2*d^2*sqrt(2)*sqrt(b*c*exp(2)^2+c*sqrt(b^2*exp
(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2)
)*exp(1)^3*exp(2)+24*b^2*c^2*d^2*sqrt(2)*sqrt(b*c*exp(2)^2+c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)
*exp(2))*exp(2))*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(1)*exp(2)^2+12*b^2*c^2*d*sqrt(2
)*sqrt(b*c*exp(2)^2+c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*sqrt(b^2*exp(2)^2+4*c^
2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(1)^2*exp(2)^2+4*b^2*c^2*d*sqrt(2)*sqrt(b*c*exp(2)^2+c*sqrt(b^2*exp(2)^
2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*ex
p(2)^3+20*b^2*c^2*d*(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(1)^2*exp(2)^2+4*b^2*c^2*d*(b^2*e
xp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2)^3+2*b^2*c^2*sqrt(2)*sqrt(b*c*exp(2)^2+c*sqrt(b^2*exp(2)
^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*e
xp(1)*exp(2)^3+4*b^2*c^2*(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(1)*exp(2)^3+128*b*c^5*d^4*e
xp(1)^3*exp(2)^2+192*b*c^5*d^4*exp(1)*exp(2)^3+112*b*c^5*d^3*exp(1)^2*exp(2)^3+16*b*c^5*d^3*exp(2)^4-64*b*c^4*
d^4*sqrt(2)*sqrt(b*c*exp(2)^2+c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(1)^3*exp
(2)-96*b*c^4*d^4*sqrt(2)*sqrt(b*c*exp(2)^2+c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))
*exp(1)*exp(2)^2-16*b*c^4*d^3*sqrt(2)*sqrt(b*c*exp(2)^2+c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*ex
p(2))*exp(2))*exp(1)^2*exp(2)^2-16*b*c^4*d^3*sqrt(2)*sqrt(b*c*exp(2)^2+c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*
b*c*d*exp(1)*exp(2))*exp(2))*exp(2)^3-8*b*c^4*d^2*sqrt(2)*sqrt(b*c*exp(2)^2+c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(
2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(1)^3*exp(2)^2-16*b*c^4*d^2*sqrt(2)*sqrt(b*c*exp(2)^2+c*sqrt(b^2*exp(2)^2
+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(1)*exp(2)^3-56*b*c^3*d^3*sqrt(2)*sqrt(b*c*exp(2)^2+c*sqrt
(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1
)*exp(2))*exp(1)^2*exp(2)-8*b*c^3*d^3*sqrt(2)*sqrt(b*c*exp(2)^2+c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*e
xp(1)*exp(2))*exp(2))*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2)^2-16*b*c^3*d^2*sqrt(2)*
sqrt(b*c*exp(2)^2+c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*sqrt(b^2*exp(2)^2+4*c^2*
d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(1)*exp(2)^2-16*b*c^3*d^2*(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*e
xp(2))*exp(1)^3*exp(2)-32*b*c^3*d^2*(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(1)*exp(2)^2-6*b*
c^3*d*sqrt(2)*sqrt(b*c*exp(2)^2+c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*sqrt(b^2*e
xp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(1)^2*exp(2)^2-2*b*c^3*d*sqrt(2)*sqrt(b*c*exp(2)^2+c*sqrt(b
^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*
exp(2))*exp(2)^3-12*b*c^3*d*(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(1)^2*exp(2)^2-4*b*c^3*d*
(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2)^3-64*c^6*d^5*exp(1)^2*exp(2)^2-64*c^6*d^5*exp(2)^
3-64*c^6*d^4*exp(1)*exp(2)^3+32*c^5*d^5*sqrt(2)*sqrt(b*c*exp(2)^2+c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d
*exp(1)*exp(2))*exp(2))*exp(1)^2*exp(2)+32*c^5*d^5*sqrt(2)*sqrt(b*c*exp(2)^2+c*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp
(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(2)^2+8*c^5*d^3*sqrt(2)*sqrt(b*c*exp(2)^2+c*sqrt(b^2*exp(2)^2+4*c^2*d^2*
exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(1)^2*exp(2)^2+8*c^5*d^3*sqrt(2)*sqrt(b*c*exp(2)^2+c*sqrt(b^2*exp(2)^
2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*exp(2)^3+32*c^4*d^4*sqrt(2)*sqrt(b*c*exp(2)^2+c*sqrt(b^2*exp
(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2)
)*exp(1)*exp(2)+16*c^4*d^3*(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(1)^2*exp(2)+16*c^4*d^3*(b
^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2)^2+8*c^4*d^2*sqrt(2)*sqrt(b*c*exp(2)^2+c*sqrt(b^2*ex
p(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(2))*sqrt(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2
))*exp(1)*exp(2)^2+16*c^4*d^2*(b^2*exp(2)^2+4*c^2*d^2*exp(2)-4*b*c*d*exp(1)*exp(2))*exp(1)*exp(2)^2)/(8*b^6*d^
3*exp(1)^6*exp(2)^3-16*b^6*d^3*exp(1)^4*exp(2)^4+8*b^6*d^3*exp(1)^2*exp(2)^5-64*b^5*c*d^4*exp(1)^7*exp(2)^2+11
2*b^5*c*d^4*exp(1)^5*exp(2)^3-32*b^5*c*d^4*exp(1)^3*exp(2)^4-16*b^5*c*d^4*exp(1)*exp(2)^5-16*b^5*c*d^3*exp(1)^
6*exp(2)^3+32*b^5*c*d^3*exp(1)^4*exp(2)^4-16*b^5*c*d^3*exp(1)^2*exp(2)^5+128*b^4*c^2*d^5*exp(1)^8*exp(2)-64*b^
4*c^2*d^5*exp(1)^6*exp(2)^2-248*b^4*c^2*d^5*exp(1)^4*exp(2)^3+176*b^4*c^2*d^5*exp(1)^2*exp(2)^4+8*b^4*c^2*d^5*
exp(2)^5+64*b^4*c^2*d^4*exp(1)^7*exp(2)^2-96*b^4*c^2*d^4*exp(1)^5*exp(2)^3+32*b^4*c^2*d^4*exp(1)*exp(2)^5+8*b^
4*c^2*d^3*exp(1)^6*exp(2)^3-16*b^4*c^2*d^3*exp(1)^4*exp(2)^4+8*b^4*c^2*d^3*exp(1)^2*exp(2)^5-512*b^3*c^3*d^6*e
xp(1)^7*exp(2)+832*b^3*c^3*d^6*exp(1)^5*exp(2)^2-128*b^3*c^3*d^6*exp(1)^3*exp(2)^3-192*b^3*c^3*d^6*exp(1)*exp(
2)^4-192*b^3*c^3*d^5*exp(1)^6*exp(2)^2+368*b^3*c^3*d^5*exp(1)^4*exp(2)^3-160*b^3*c^3*d^5*exp(1)^2*exp(2)^4-16*
b^3*c^3*d^5*exp(2)^5-32*b^3*c^3*d^4*exp(1)^7*exp(2)^2+48*b^3*c^3*d^4*exp(1)^5*exp(2)^3-16*b^3*c^3*d^4*exp(1)*e
xp(2)^5+768*b^2*c^4*d^7*exp(1)^6*exp(2)-1472*b^2*c^4*d^7*exp(1)^4*exp(2)^2+640*b^2*c^4*d^7*exp(1)^2*exp(2)^3+6
4*b^2*c^4*d^7*exp(2)^4+192*b^2*c^4*d^6*exp(1)^5*exp(2)^2-384*b^2*c^4*d^6*exp(1)^3*exp(2)^3+192*b^2*c^4*d^6*exp
(1)*exp(2)^4+96*b^2*c^4*d^5*exp(1)^6*exp(2)^2-184*b^2*c^4*d^5*exp(1)^4*exp(2)^3+80*b^2*c^4*d^5*exp(1)^2*exp(2)
^4+8*b^2*c^4*d^5*exp(2)^5-512*b*c^5*d^8*exp(1)^5*exp(2)+1024*b*c^5*d^8*exp(1)^3*exp(2)^2-512*b*c^5*d^8*exp(1)*
exp(2)^3-64*b*c^5*d^7*exp(1)^4*exp(2)^2+128*b*c^5*d^7*exp(1)^2*exp(2)^3-64*b*c^5*d^7*exp(2)^4-96*b*c^5*d^6*exp
(1)^5*exp(2)^2+192*b*c^5*d^6*exp(1)^3*exp(2)^3-96*b*c^5*d^6*exp(1)*exp(2)^4+128*c^6*d^9*exp(1)^4*exp(2)-256*c^
6*d^9*exp(1)^2*exp(2)^2+128*c^6*d^9*exp(2)^3+32*c^6*d^7*exp(1)^4*exp(2)^2-64*c^6*d^7*exp(1)^2*exp(2)^3+32*c^6*
d^7*exp(2)^4)/abs(c)*atan(x/sqrt(-(c^2*exp(2)^3*b*d^4-2*c^2*exp(2)^2*b*d^4*exp(1)^2+c^2*exp(2)*b*d^4*exp(1)^4-
2*c*exp(2)^3*b^2*d^3*exp(1)+4*c*exp(2)^2*b^2*d^3*exp(1)^3-2*c*exp(2)*b^2*d^3*exp(1)^5+exp(2)^3*b^3*d^2*exp(1)^
2-2*exp(2)^2*b^3*d^2*exp(1)^4+exp(2)*b^3*d^2*exp(1)^6-sqrt((-c^2*exp(2)^3*b*d^4+2*c^2*exp(2)^2*b*d^4*exp(1)^2-
c^2*exp(2)*b*d^4*exp(1)^4+2*c*exp(2)^3*b^2*d^3*exp(1)-4*c*exp(2)^2*b^2*d^3*exp(1)^3+2*c*exp(2)*b^2*d^3*exp(1)^
5-exp(2)^3*b^3*d^2*exp(1)^2+2*exp(2)^2*b^3*d^2*exp(1)^4-exp(2)*b^3*d^2*exp(1)^6)*(-c^2*exp(2)^3*b*d^4+2*c^2*ex
p(2)^2*b*d^4*exp(1)^2-c^2*exp(2)*b*d^4*exp(1)^4+2*c*exp(2)^3*b^2*d^3*exp(1)-4*c*exp(2)^2*b^2*d^3*exp(1)^3+2*c*
exp(2)*b^2*d^3*exp(1)^5-exp(2)^3*b^3*d^2*exp(1)^2+2*exp(2)^2*b^3*d^2*exp(1)^4-exp(2)*b^3*d^2*exp(1)^6)-4*(-c^3
*exp(2)^3*d^4+2*c^3*exp(2)^2*d^4*exp(1)^2-c^3*exp(2)*d^4*exp(1)^4+2*c^2*exp(2)^3*b*d^3*exp(1)-4*c^2*exp(2)^2*b
*d^3*exp(1)^3+2*c^2*exp(2)*b*d^3*exp(1)^5-c*exp(2)^3*b^2*d^2*exp(1)^2+2*c*exp(2)^2*b^2*d^2*exp(1)^4-c*exp(2)*b
^2*d^2*exp(1)^6)*(c^3*exp(2)^2*d^6-2*c^3*exp(2)*d^6*exp(1)^2+c^3*d^6*exp(1)^4-3*c^2*exp(2)^2*b*d^5*exp(1)+6*c^
2*exp(2)*b*d^5*exp(1)^3-3*c^2*b*d^5*exp(1)^5+3*c*exp(2)^2*b^2*d^4*exp(1)^2-6*c*exp(2)*b^2*d^4*exp(1)^4+3*c*b^2
*d^4*exp(1)^6-exp(2)^2*b^3*d^3*exp(1)^3+2*exp(2)*b^3*d^3*exp(1)^5-b^3*d^3*exp(1)^7)))/2/(-c^3*exp(2)^3*d^4+2*c
^3*exp(2)^2*d^4*exp(1)^2-c^3*exp(2)*d^4*exp(1)^4+2*c^2*exp(2)^3*b*d^3*exp(1)-4*c^2*exp(2)^2*b*d^3*exp(1)^3+2*c
^2*exp(2)*b*d^3*exp(1)^5-c*exp(2)^3*b^2*d^2*exp(1)^2+2*c*exp(2)^2*b^2*d^2*exp(1)^4-c*exp(2)*b^2*d^2*exp(1)^6))
)+(-5*c*exp(2)*d*exp(1)^2+c*d*exp(1)^4+3*exp(2)*b*exp(1)^3-b*exp(1)^5)*1/2/(-c^2*exp(2)^2*d^4+2*c^2*exp(2)*d^4
*exp(1)^2-c^2*d^4*exp(1)^4+2*c*exp(2)^2*b*d^3*exp(1)-4*c*exp(2)*b*d^3*exp(1)^3+2*c*b*d^3*exp(1)^5-exp(2)^2*b^2
*d^2*exp(1)^2+2*exp(2)*b^2*d^2*exp(1)^4-b^2*d^2*exp(1)^6)/sqrt(d*exp(1))*atan(x*exp(1)/sqrt(d*exp(1)))-x*exp(1
)^2/(-2*c*exp(2)*d^3+2*c*d^3*exp(1)^2+2*exp(2)*b*d^2*exp(1)-2*b*d^2*exp(1)^3)/(x^2*exp(1)+d)

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maple [A]  time = 0.01, size = 319, normalized size = 1.71 \begin {gather*} \frac {3 b^{2} e^{3} x^{3}}{8 \left (b e -2 c d \right )^{3} \left (e \,x^{2}+d \right )^{2} d^{2}}-\frac {2 b c \,e^{2} x^{3}}{\left (b e -2 c d \right )^{3} \left (e \,x^{2}+d \right )^{2} d}+\frac {5 c^{2} e \,x^{3}}{2 \left (b e -2 c d \right )^{3} \left (e \,x^{2}+d \right )^{2}}+\frac {5 b^{2} e^{2} x}{8 \left (b e -2 c d \right )^{3} \left (e \,x^{2}+d \right )^{2} d}-\frac {3 b c e x}{\left (b e -2 c d \right )^{3} \left (e \,x^{2}+d \right )^{2}}-\frac {c^{3} \arctan \left (\frac {c e x}{\sqrt {\left (b e -c d \right ) c e}}\right )}{\left (b e -2 c d \right )^{3} \sqrt {\left (b e -c d \right ) c e}}+\frac {7 c^{2} d x}{2 \left (b e -2 c d \right )^{3} \left (e \,x^{2}+d \right )^{2}}+\frac {3 b^{2} e^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{8 \left (b e -2 c d \right )^{3} \sqrt {d e}\, d^{2}}-\frac {2 b c e \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\left (b e -2 c d \right )^{3} \sqrt {d e}\, d}+\frac {7 c^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \left (b e -2 c d \right )^{3} \sqrt {d e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x^2+d)^2/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x)

[Out]

-c^3/(b*e-2*c*d)^3/((b*e-c*d)*c*e)^(1/2)*arctan(1/((b*e-c*d)*c*e)^(1/2)*c*e*x)+3/8/(b*e-2*c*d)^3/(e*x^2+d)^2*e
^3/d^2*x^3*b^2-2/(b*e-2*c*d)^3/(e*x^2+d)^2*e^2/d*x^3*b*c+5/2/(b*e-2*c*d)^3/(e*x^2+d)^2*e*x^3*c^2+5/8/(b*e-2*c*
d)^3/(e*x^2+d)^2/d*x*b^2*e^2-3/(b*e-2*c*d)^3/(e*x^2+d)^2*x*b*c*e+7/2/(b*e-2*c*d)^3/(e*x^2+d)^2*d*x*c^2+3/8/(b*
e-2*c*d)^3/d^2/(d*e)^(1/2)*arctan(1/(d*e)^(1/2)*e*x)*b^2*e^2-2/(b*e-2*c*d)^3/d/(d*e)^(1/2)*arctan(1/(d*e)^(1/2
)*e*x)*b*c*e+7/2/(b*e-2*c*d)^3/(d*e)^(1/2)*arctan(1/(d*e)^(1/2)*e*x)*c^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^2+d)^2/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^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(b*e-c*d>0)', see `assume?` for
more details)Is b*e-c*d positive or negative?

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mupad [B]  time = 6.45, size = 6267, normalized size = 33.51

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d + e*x^2)^2*(b*e^2*x^2 - c*d^2 + c*e^2*x^4 + b*d*e)),x)

[Out]

((x*(5*b*e - 14*c*d))/(8*d*(b^2*e^2 + 4*c^2*d^2 - 4*b*c*d*e)) + (e*x^3*(3*b*e - 10*c*d))/(8*d^2*(b^2*e^2 + 4*c
^2*d^2 - 4*b*c*d*e)))/(d^2 + e^2*x^4 + 2*d*e*x^2) - (atan(((((x*(9*b^4*c^3*e^10 + 848*c^7*d^4*e^6 - 896*b*c^6*
d^3*e^7 - 96*b^3*c^4*d*e^9 + 424*b^2*c^5*d^2*e^8))/(64*(16*c^4*d^8 + b^4*d^4*e^4 - 8*b^3*c*d^5*e^3 + 24*b^2*c^
2*d^6*e^2 - 32*b*c^3*d^7*e)) - (((576*c^10*d^10*e^6 - 2144*b*c^9*d^9*e^7 + 3504*b^2*c^8*d^8*e^8 - 3288*b^3*c^7
*d^7*e^9 + 1940*b^4*c^6*d^6*e^10 - 738*b^5*c^5*d^5*e^11 + 177*b^6*c^4*d^4*e^12 - (49*b^7*c^3*d^3*e^13)/2 + (3*
b^8*c^2*d^2*e^14)/2)/(2*(64*c^6*d^10 + b^6*d^4*e^6 - 12*b^5*c*d^5*e^5 + 240*b^2*c^4*d^8*e^2 - 160*b^3*c^3*d^7*
e^3 + 60*b^4*c^2*d^6*e^4 - 192*b*c^5*d^9*e)) - (x*(-c^5*e*(b*e - c*d))^(1/2)*(16384*b*c^8*d^10*e^8 - 49152*b^2
*c^7*d^9*e^9 + 61440*b^3*c^6*d^8*e^10 - 40960*b^4*c^5*d^7*e^11 + 15360*b^5*c^4*d^6*e^12 - 3072*b^6*c^3*d^5*e^1
3 + 256*b^7*c^2*d^4*e^14))/(128*(16*c^4*d^8 + b^4*d^4*e^4 - 8*b^3*c*d^5*e^3 + 24*b^2*c^2*d^6*e^2 - 32*b*c^3*d^
7*e)*(b^4*e^5 + 8*c^4*d^4*e - 20*b*c^3*d^3*e^2 + 18*b^2*c^2*d^2*e^3 - 7*b^3*c*d*e^4)))*(-c^5*e*(b*e - c*d))^(1
/2))/(2*(b^4*e^5 + 8*c^4*d^4*e - 20*b*c^3*d^3*e^2 + 18*b^2*c^2*d^2*e^3 - 7*b^3*c*d*e^4)))*(-c^5*e*(b*e - c*d))
^(1/2)*1i)/(b^4*e^5 + 8*c^4*d^4*e - 20*b*c^3*d^3*e^2 + 18*b^2*c^2*d^2*e^3 - 7*b^3*c*d*e^4) + (((x*(9*b^4*c^3*e
^10 + 848*c^7*d^4*e^6 - 896*b*c^6*d^3*e^7 - 96*b^3*c^4*d*e^9 + 424*b^2*c^5*d^2*e^8))/(64*(16*c^4*d^8 + b^4*d^4
*e^4 - 8*b^3*c*d^5*e^3 + 24*b^2*c^2*d^6*e^2 - 32*b*c^3*d^7*e)) + (((576*c^10*d^10*e^6 - 2144*b*c^9*d^9*e^7 + 3
504*b^2*c^8*d^8*e^8 - 3288*b^3*c^7*d^7*e^9 + 1940*b^4*c^6*d^6*e^10 - 738*b^5*c^5*d^5*e^11 + 177*b^6*c^4*d^4*e^
12 - (49*b^7*c^3*d^3*e^13)/2 + (3*b^8*c^2*d^2*e^14)/2)/(2*(64*c^6*d^10 + b^6*d^4*e^6 - 12*b^5*c*d^5*e^5 + 240*
b^2*c^4*d^8*e^2 - 160*b^3*c^3*d^7*e^3 + 60*b^4*c^2*d^6*e^4 - 192*b*c^5*d^9*e)) + (x*(-c^5*e*(b*e - c*d))^(1/2)
*(16384*b*c^8*d^10*e^8 - 49152*b^2*c^7*d^9*e^9 + 61440*b^3*c^6*d^8*e^10 - 40960*b^4*c^5*d^7*e^11 + 15360*b^5*c
^4*d^6*e^12 - 3072*b^6*c^3*d^5*e^13 + 256*b^7*c^2*d^4*e^14))/(128*(16*c^4*d^8 + b^4*d^4*e^4 - 8*b^3*c*d^5*e^3
+ 24*b^2*c^2*d^6*e^2 - 32*b*c^3*d^7*e)*(b^4*e^5 + 8*c^4*d^4*e - 20*b*c^3*d^3*e^2 + 18*b^2*c^2*d^2*e^3 - 7*b^3*
c*d*e^4)))*(-c^5*e*(b*e - c*d))^(1/2))/(2*(b^4*e^5 + 8*c^4*d^4*e - 20*b*c^3*d^3*e^2 + 18*b^2*c^2*d^2*e^3 - 7*b
^3*c*d*e^4)))*(-c^5*e*(b*e - c*d))^(1/2)*1i)/(b^4*e^5 + 8*c^4*d^4*e - 20*b*c^3*d^3*e^2 + 18*b^2*c^2*d^2*e^3 -
7*b^3*c*d*e^4))/(((9*b^3*c^5*e^8)/32 - (35*c^8*d^3*e^5)/4 + (61*b*c^7*d^2*e^6)/8 - (39*b^2*c^6*d*e^7)/16)/(64*
c^6*d^10 + b^6*d^4*e^6 - 12*b^5*c*d^5*e^5 + 240*b^2*c^4*d^8*e^2 - 160*b^3*c^3*d^7*e^3 + 60*b^4*c^2*d^6*e^4 - 1
92*b*c^5*d^9*e) + (((x*(9*b^4*c^3*e^10 + 848*c^7*d^4*e^6 - 896*b*c^6*d^3*e^7 - 96*b^3*c^4*d*e^9 + 424*b^2*c^5*
d^2*e^8))/(64*(16*c^4*d^8 + b^4*d^4*e^4 - 8*b^3*c*d^5*e^3 + 24*b^2*c^2*d^6*e^2 - 32*b*c^3*d^7*e)) - (((576*c^1
0*d^10*e^6 - 2144*b*c^9*d^9*e^7 + 3504*b^2*c^8*d^8*e^8 - 3288*b^3*c^7*d^7*e^9 + 1940*b^4*c^6*d^6*e^10 - 738*b^
5*c^5*d^5*e^11 + 177*b^6*c^4*d^4*e^12 - (49*b^7*c^3*d^3*e^13)/2 + (3*b^8*c^2*d^2*e^14)/2)/(2*(64*c^6*d^10 + b^
6*d^4*e^6 - 12*b^5*c*d^5*e^5 + 240*b^2*c^4*d^8*e^2 - 160*b^3*c^3*d^7*e^3 + 60*b^4*c^2*d^6*e^4 - 192*b*c^5*d^9*
e)) - (x*(-c^5*e*(b*e - c*d))^(1/2)*(16384*b*c^8*d^10*e^8 - 49152*b^2*c^7*d^9*e^9 + 61440*b^3*c^6*d^8*e^10 - 4
0960*b^4*c^5*d^7*e^11 + 15360*b^5*c^4*d^6*e^12 - 3072*b^6*c^3*d^5*e^13 + 256*b^7*c^2*d^4*e^14))/(128*(16*c^4*d
^8 + b^4*d^4*e^4 - 8*b^3*c*d^5*e^3 + 24*b^2*c^2*d^6*e^2 - 32*b*c^3*d^7*e)*(b^4*e^5 + 8*c^4*d^4*e - 20*b*c^3*d^
3*e^2 + 18*b^2*c^2*d^2*e^3 - 7*b^3*c*d*e^4)))*(-c^5*e*(b*e - c*d))^(1/2))/(2*(b^4*e^5 + 8*c^4*d^4*e - 20*b*c^3
*d^3*e^2 + 18*b^2*c^2*d^2*e^3 - 7*b^3*c*d*e^4)))*(-c^5*e*(b*e - c*d))^(1/2))/(b^4*e^5 + 8*c^4*d^4*e - 20*b*c^3
*d^3*e^2 + 18*b^2*c^2*d^2*e^3 - 7*b^3*c*d*e^4) - (((x*(9*b^4*c^3*e^10 + 848*c^7*d^4*e^6 - 896*b*c^6*d^3*e^7 -
96*b^3*c^4*d*e^9 + 424*b^2*c^5*d^2*e^8))/(64*(16*c^4*d^8 + b^4*d^4*e^4 - 8*b^3*c*d^5*e^3 + 24*b^2*c^2*d^6*e^2
- 32*b*c^3*d^7*e)) + (((576*c^10*d^10*e^6 - 2144*b*c^9*d^9*e^7 + 3504*b^2*c^8*d^8*e^8 - 3288*b^3*c^7*d^7*e^9 +
1940*b^4*c^6*d^6*e^10 - 738*b^5*c^5*d^5*e^11 + 177*b^6*c^4*d^4*e^12 - (49*b^7*c^3*d^3*e^13)/2 + (3*b^8*c^2*d^
2*e^14)/2)/(2*(64*c^6*d^10 + b^6*d^4*e^6 - 12*b^5*c*d^5*e^5 + 240*b^2*c^4*d^8*e^2 - 160*b^3*c^3*d^7*e^3 + 60*b
^4*c^2*d^6*e^4 - 192*b*c^5*d^9*e)) + (x*(-c^5*e*(b*e - c*d))^(1/2)*(16384*b*c^8*d^10*e^8 - 49152*b^2*c^7*d^9*e
^9 + 61440*b^3*c^6*d^8*e^10 - 40960*b^4*c^5*d^7*e^11 + 15360*b^5*c^4*d^6*e^12 - 3072*b^6*c^3*d^5*e^13 + 256*b^
7*c^2*d^4*e^14))/(128*(16*c^4*d^8 + b^4*d^4*e^4 - 8*b^3*c*d^5*e^3 + 24*b^2*c^2*d^6*e^2 - 32*b*c^3*d^7*e)*(b^4*
e^5 + 8*c^4*d^4*e - 20*b*c^3*d^3*e^2 + 18*b^2*c^2*d^2*e^3 - 7*b^3*c*d*e^4)))*(-c^5*e*(b*e - c*d))^(1/2))/(2*(b
^4*e^5 + 8*c^4*d^4*e - 20*b*c^3*d^3*e^2 + 18*b^2*c^2*d^2*e^3 - 7*b^3*c*d*e^4)))*(-c^5*e*(b*e - c*d))^(1/2))/(b
^4*e^5 + 8*c^4*d^4*e - 20*b*c^3*d^3*e^2 + 18*b^2*c^2*d^2*e^3 - 7*b^3*c*d*e^4)))*(-c^5*e*(b*e - c*d))^(1/2)*1i)
/(b^4*e^5 + 8*c^4*d^4*e - 20*b*c^3*d^3*e^2 + 18*b^2*c^2*d^2*e^3 - 7*b^3*c*d*e^4) - (atan(((((x*(9*b^4*c^3*e^10
+ 848*c^7*d^4*e^6 - 896*b*c^6*d^3*e^7 - 96*b^3*c^4*d*e^9 + 424*b^2*c^5*d^2*e^8))/(32*(16*c^4*d^8 + b^4*d^4*e^
4 - 8*b^3*c*d^5*e^3 + 24*b^2*c^2*d^6*e^2 - 32*b*c^3*d^7*e)) - (((576*c^10*d^10*e^6 - 2144*b*c^9*d^9*e^7 + 3504
*b^2*c^8*d^8*e^8 - 3288*b^3*c^7*d^7*e^9 + 1940*b^4*c^6*d^6*e^10 - 738*b^5*c^5*d^5*e^11 + 177*b^6*c^4*d^4*e^12
- (49*b^7*c^3*d^3*e^13)/2 + (3*b^8*c^2*d^2*e^14)/2)/(64*c^6*d^10 + b^6*d^4*e^6 - 12*b^5*c*d^5*e^5 + 240*b^2*c^
4*d^8*e^2 - 160*b^3*c^3*d^7*e^3 + 60*b^4*c^2*d^6*e^4 - 192*b*c^5*d^9*e) - (x*(-d^5*e)^(1/2)*(3*b^2*e^2 + 28*c^
2*d^2 - 16*b*c*d*e)*(16384*b*c^8*d^10*e^8 - 49152*b^2*c^7*d^9*e^9 + 61440*b^3*c^6*d^8*e^10 - 40960*b^4*c^5*d^7
*e^11 + 15360*b^5*c^4*d^6*e^12 - 3072*b^6*c^3*d^5*e^13 + 256*b^7*c^2*d^4*e^14))/(512*(8*c^3*d^8*e - b^3*d^5*e^
4 - 12*b*c^2*d^7*e^2 + 6*b^2*c*d^6*e^3)*(16*c^4*d^8 + b^4*d^4*e^4 - 8*b^3*c*d^5*e^3 + 24*b^2*c^2*d^6*e^2 - 32*
b*c^3*d^7*e)))*(-d^5*e)^(1/2)*(3*b^2*e^2 + 28*c^2*d^2 - 16*b*c*d*e))/(16*(8*c^3*d^8*e - b^3*d^5*e^4 - 12*b*c^2
*d^7*e^2 + 6*b^2*c*d^6*e^3)))*(-d^5*e)^(1/2)*(3*b^2*e^2 + 28*c^2*d^2 - 16*b*c*d*e)*1i)/(16*(8*c^3*d^8*e - b^3*
d^5*e^4 - 12*b*c^2*d^7*e^2 + 6*b^2*c*d^6*e^3)) + (((x*(9*b^4*c^3*e^10 + 848*c^7*d^4*e^6 - 896*b*c^6*d^3*e^7 -
96*b^3*c^4*d*e^9 + 424*b^2*c^5*d^2*e^8))/(32*(16*c^4*d^8 + b^4*d^4*e^4 - 8*b^3*c*d^5*e^3 + 24*b^2*c^2*d^6*e^2
- 32*b*c^3*d^7*e)) + (((576*c^10*d^10*e^6 - 2144*b*c^9*d^9*e^7 + 3504*b^2*c^8*d^8*e^8 - 3288*b^3*c^7*d^7*e^9 +
1940*b^4*c^6*d^6*e^10 - 738*b^5*c^5*d^5*e^11 + 177*b^6*c^4*d^4*e^12 - (49*b^7*c^3*d^3*e^13)/2 + (3*b^8*c^2*d^
2*e^14)/2)/(64*c^6*d^10 + b^6*d^4*e^6 - 12*b^5*c*d^5*e^5 + 240*b^2*c^4*d^8*e^2 - 160*b^3*c^3*d^7*e^3 + 60*b^4*
c^2*d^6*e^4 - 192*b*c^5*d^9*e) + (x*(-d^5*e)^(1/2)*(3*b^2*e^2 + 28*c^2*d^2 - 16*b*c*d*e)*(16384*b*c^8*d^10*e^8
- 49152*b^2*c^7*d^9*e^9 + 61440*b^3*c^6*d^8*e^10 - 40960*b^4*c^5*d^7*e^11 + 15360*b^5*c^4*d^6*e^12 - 3072*b^6
*c^3*d^5*e^13 + 256*b^7*c^2*d^4*e^14))/(512*(8*c^3*d^8*e - b^3*d^5*e^4 - 12*b*c^2*d^7*e^2 + 6*b^2*c*d^6*e^3)*(
16*c^4*d^8 + b^4*d^4*e^4 - 8*b^3*c*d^5*e^3 + 24*b^2*c^2*d^6*e^2 - 32*b*c^3*d^7*e)))*(-d^5*e)^(1/2)*(3*b^2*e^2
+ 28*c^2*d^2 - 16*b*c*d*e))/(16*(8*c^3*d^8*e - b^3*d^5*e^4 - 12*b*c^2*d^7*e^2 + 6*b^2*c*d^6*e^3)))*(-d^5*e)^(1
/2)*(3*b^2*e^2 + 28*c^2*d^2 - 16*b*c*d*e)*1i)/(16*(8*c^3*d^8*e - b^3*d^5*e^4 - 12*b*c^2*d^7*e^2 + 6*b^2*c*d^6*
e^3)))/(((9*b^3*c^5*e^8)/32 - (35*c^8*d^3*e^5)/4 + (61*b*c^7*d^2*e^6)/8 - (39*b^2*c^6*d*e^7)/16)/(64*c^6*d^10
+ b^6*d^4*e^6 - 12*b^5*c*d^5*e^5 + 240*b^2*c^4*d^8*e^2 - 160*b^3*c^3*d^7*e^3 + 60*b^4*c^2*d^6*e^4 - 192*b*c^5*
d^9*e) + (((x*(9*b^4*c^3*e^10 + 848*c^7*d^4*e^6 - 896*b*c^6*d^3*e^7 - 96*b^3*c^4*d*e^9 + 424*b^2*c^5*d^2*e^8))
/(32*(16*c^4*d^8 + b^4*d^4*e^4 - 8*b^3*c*d^5*e^3 + 24*b^2*c^2*d^6*e^2 - 32*b*c^3*d^7*e)) - (((576*c^10*d^10*e^
6 - 2144*b*c^9*d^9*e^7 + 3504*b^2*c^8*d^8*e^8 - 3288*b^3*c^7*d^7*e^9 + 1940*b^4*c^6*d^6*e^10 - 738*b^5*c^5*d^5
*e^11 + 177*b^6*c^4*d^4*e^12 - (49*b^7*c^3*d^3*e^13)/2 + (3*b^8*c^2*d^2*e^14)/2)/(64*c^6*d^10 + b^6*d^4*e^6 -
12*b^5*c*d^5*e^5 + 240*b^2*c^4*d^8*e^2 - 160*b^3*c^3*d^7*e^3 + 60*b^4*c^2*d^6*e^4 - 192*b*c^5*d^9*e) - (x*(-d^
5*e)^(1/2)*(3*b^2*e^2 + 28*c^2*d^2 - 16*b*c*d*e)*(16384*b*c^8*d^10*e^8 - 49152*b^2*c^7*d^9*e^9 + 61440*b^3*c^6
*d^8*e^10 - 40960*b^4*c^5*d^7*e^11 + 15360*b^5*c^4*d^6*e^12 - 3072*b^6*c^3*d^5*e^13 + 256*b^7*c^2*d^4*e^14))/(
512*(8*c^3*d^8*e - b^3*d^5*e^4 - 12*b*c^2*d^7*e^2 + 6*b^2*c*d^6*e^3)*(16*c^4*d^8 + b^4*d^4*e^4 - 8*b^3*c*d^5*e
^3 + 24*b^2*c^2*d^6*e^2 - 32*b*c^3*d^7*e)))*(-d^5*e)^(1/2)*(3*b^2*e^2 + 28*c^2*d^2 - 16*b*c*d*e))/(16*(8*c^3*d
^8*e - b^3*d^5*e^4 - 12*b*c^2*d^7*e^2 + 6*b^2*c*d^6*e^3)))*(-d^5*e)^(1/2)*(3*b^2*e^2 + 28*c^2*d^2 - 16*b*c*d*e
))/(16*(8*c^3*d^8*e - b^3*d^5*e^4 - 12*b*c^2*d^7*e^2 + 6*b^2*c*d^6*e^3)) - (((x*(9*b^4*c^3*e^10 + 848*c^7*d^4*
e^6 - 896*b*c^6*d^3*e^7 - 96*b^3*c^4*d*e^9 + 424*b^2*c^5*d^2*e^8))/(32*(16*c^4*d^8 + b^4*d^4*e^4 - 8*b^3*c*d^5
*e^3 + 24*b^2*c^2*d^6*e^2 - 32*b*c^3*d^7*e)) + (((576*c^10*d^10*e^6 - 2144*b*c^9*d^9*e^7 + 3504*b^2*c^8*d^8*e^
8 - 3288*b^3*c^7*d^7*e^9 + 1940*b^4*c^6*d^6*e^10 - 738*b^5*c^5*d^5*e^11 + 177*b^6*c^4*d^4*e^12 - (49*b^7*c^3*d
^3*e^13)/2 + (3*b^8*c^2*d^2*e^14)/2)/(64*c^6*d^10 + b^6*d^4*e^6 - 12*b^5*c*d^5*e^5 + 240*b^2*c^4*d^8*e^2 - 160
*b^3*c^3*d^7*e^3 + 60*b^4*c^2*d^6*e^4 - 192*b*c^5*d^9*e) + (x*(-d^5*e)^(1/2)*(3*b^2*e^2 + 28*c^2*d^2 - 16*b*c*
d*e)*(16384*b*c^8*d^10*e^8 - 49152*b^2*c^7*d^9*e^9 + 61440*b^3*c^6*d^8*e^10 - 40960*b^4*c^5*d^7*e^11 + 15360*b
^5*c^4*d^6*e^12 - 3072*b^6*c^3*d^5*e^13 + 256*b^7*c^2*d^4*e^14))/(512*(8*c^3*d^8*e - b^3*d^5*e^4 - 12*b*c^2*d^
7*e^2 + 6*b^2*c*d^6*e^3)*(16*c^4*d^8 + b^4*d^4*e^4 - 8*b^3*c*d^5*e^3 + 24*b^2*c^2*d^6*e^2 - 32*b*c^3*d^7*e)))*
(-d^5*e)^(1/2)*(3*b^2*e^2 + 28*c^2*d^2 - 16*b*c*d*e))/(16*(8*c^3*d^8*e - b^3*d^5*e^4 - 12*b*c^2*d^7*e^2 + 6*b^
2*c*d^6*e^3)))*(-d^5*e)^(1/2)*(3*b^2*e^2 + 28*c^2*d^2 - 16*b*c*d*e))/(16*(8*c^3*d^8*e - b^3*d^5*e^4 - 12*b*c^2
*d^7*e^2 + 6*b^2*c*d^6*e^3))))*(-d^5*e)^(1/2)*(3*b^2*e^2 + 28*c^2*d^2 - 16*b*c*d*e)*1i)/(8*(8*c^3*d^8*e - b^3*
d^5*e^4 - 12*b*c^2*d^7*e^2 + 6*b^2*c*d^6*e^3))

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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(1/(e*x**2+d)**2/(c*e**2*x**4+b*e**2*x**2+b*d*e-c*d**2),x)

[Out]

Timed out

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