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

Optimal. Leaf size=121 \[ \frac {x \left (b^2 e^2-5 b c d e+7 c^2 d^2\right )}{c^3}-\frac {(2 c d-b e)^3 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right )}{c^{7/2} \sqrt {e} \sqrt {c d-b e}}+\frac {e x^3 (4 c d-b e)}{3 c^2}+\frac {e^2 x^5}{5 c} \]

[Out]

(b^2*e^2-5*b*c*d*e+7*c^2*d^2)*x/c^3+1/3*e*(-b*e+4*c*d)*x^3/c^2+1/5*e^2*x^5/c-(-b*e+2*c*d)^3*arctanh(x*c^(1/2)*
e^(1/2)/(-b*e+c*d)^(1/2))/c^(7/2)/e^(1/2)/(-b*e+c*d)^(1/2)

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Rubi [A]  time = 0.16, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1149, 390, 208} \[ \frac {x \left (b^2 e^2-5 b c d e+7 c^2 d^2\right )}{c^3}+\frac {e x^3 (4 c d-b e)}{3 c^2}-\frac {(2 c d-b e)^3 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right )}{c^{7/2} \sqrt {e} \sqrt {c d-b e}}+\frac {e^2 x^5}{5 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

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 {\left (d+e x^2\right )^4}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx &=\int \frac {\left (d+e x^2\right )^3}{\frac {-c d^2+b d e}{d}+c e x^2} \, dx\\ &=\int \left (\frac {7 c^2 d^2-5 b c d e+b^2 e^2}{c^3}+\frac {e (4 c d-b e) x^2}{c^2}+\frac {e^2 x^4}{c}+\frac {8 c^3 d^3-12 b c^2 d^2 e+6 b^2 c d e^2-b^3 e^3}{c^3 \left (-c d+b e+c e x^2\right )}\right ) \, dx\\ &=\frac {\left (7 c^2 d^2-5 b c d e+b^2 e^2\right ) x}{c^3}+\frac {e (4 c d-b e) x^3}{3 c^2}+\frac {e^2 x^5}{5 c}+\frac {(2 c d-b e)^3 \int \frac {1}{-c d+b e+c e x^2} \, dx}{c^3}\\ &=\frac {\left (7 c^2 d^2-5 b c d e+b^2 e^2\right ) x}{c^3}+\frac {e (4 c d-b e) x^3}{3 c^2}+\frac {e^2 x^5}{5 c}-\frac {(2 c d-b e)^3 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right )}{c^{7/2} \sqrt {e} \sqrt {c d-b e}}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 121, normalized size = 1.00 \[ -\frac {x \left (-b^2 e^2+5 b c d e-7 c^2 d^2\right )}{c^3}-\frac {(b e-2 c d)^3 \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {b e-c d}}\right )}{c^{7/2} \sqrt {e} \sqrt {b e-c d}}-\frac {e x^3 (b e-4 c d)}{3 c^2}+\frac {e^2 x^5}{5 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.75, size = 446, normalized size = 3.69 \[ \left [\frac {6 \, {\left (c^{4} d e^{3} - b c^{3} e^{4}\right )} x^{5} + 10 \, {\left (4 \, c^{4} d^{2} e^{2} - 5 \, b c^{3} d e^{3} + b^{2} c^{2} e^{4}\right )} x^{3} - 15 \, {\left (8 \, c^{3} d^{3} - 12 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \sqrt {c^{2} d e - b c e^{2}} \log \left (\frac {c e x^{2} + c d - b e + 2 \, \sqrt {c^{2} d e - b c e^{2}} x}{c e x^{2} - c d + b e}\right ) + 30 \, {\left (7 \, c^{4} d^{3} e - 12 \, b c^{3} d^{2} e^{2} + 6 \, b^{2} c^{2} d e^{3} - b^{3} c e^{4}\right )} x}{30 \, {\left (c^{5} d e - b c^{4} e^{2}\right )}}, \frac {3 \, {\left (c^{4} d e^{3} - b c^{3} e^{4}\right )} x^{5} + 5 \, {\left (4 \, c^{4} d^{2} e^{2} - 5 \, b c^{3} d e^{3} + b^{2} c^{2} e^{4}\right )} x^{3} - 15 \, {\left (8 \, c^{3} d^{3} - 12 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \sqrt {-c^{2} d e + b c e^{2}} \arctan \left (-\frac {\sqrt {-c^{2} d e + b c e^{2}} x}{c d - b e}\right ) + 15 \, {\left (7 \, c^{4} d^{3} e - 12 \, b c^{3} d^{2} e^{2} + 6 \, b^{2} c^{2} d e^{3} - b^{3} c e^{4}\right )} x}{15 \, {\left (c^{5} d e - b c^{4} e^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/30*(6*(c^4*d*e^3 - b*c^3*e^4)*x^5 + 10*(4*c^4*d^2*e^2 - 5*b*c^3*d*e^3 + b^2*c^2*e^4)*x^3 - 15*(8*c^3*d^3 -
12*b*c^2*d^2*e + 6*b^2*c*d*e^2 - b^3*e^3)*sqrt(c^2*d*e - b*c*e^2)*log((c*e*x^2 + c*d - b*e + 2*sqrt(c^2*d*e -
b*c*e^2)*x)/(c*e*x^2 - c*d + b*e)) + 30*(7*c^4*d^3*e - 12*b*c^3*d^2*e^2 + 6*b^2*c^2*d*e^3 - b^3*c*e^4)*x)/(c^5
*d*e - b*c^4*e^2), 1/15*(3*(c^4*d*e^3 - b*c^3*e^4)*x^5 + 5*(4*c^4*d^2*e^2 - 5*b*c^3*d*e^3 + b^2*c^2*e^4)*x^3 -
15*(8*c^3*d^3 - 12*b*c^2*d^2*e + 6*b^2*c*d*e^2 - b^3*e^3)*sqrt(-c^2*d*e + b*c*e^2)*arctan(-sqrt(-c^2*d*e + b*
c*e^2)*x/(c*d - b*e)) + 15*(7*c^4*d^3*e - 12*b*c^3*d^2*e^2 + 6*b^2*c^2*d*e^3 - b^3*c*e^4)*x)/(c^5*d*e - b*c^4*
e^2)]

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giac [B]  time = 5.85, size = 10312, normalized size = 85.22 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(128*b*c^10*d^6*e^10 - 64*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d
^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b*c^8*d^6*e^6 - 384*b^2*c^9*d^5*e^11 + 192*sqrt(2)*sqrt(4*c^2*d^2*e^2 -
4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^2*c^7*d^5*e^7 + 48
0*b^3*c^8*d^4*e^12 - 240*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2
- 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^3*c^6*d^4*e^8 + 32*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt
(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^2*c^7*d^4*e^8 - 16*sqrt(2)*sqrt(4*c^2*d^2*e^2
- 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b*c^8*d^4*e^8 - 320
*b^4*c^7*d^3*e^13 - 32*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b*c^8*d^4*e^8 + 160*sqrt(2)*sqrt(4*c^2*d^2*e^2
- 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^4*c^5*d^3*e^9 - 6
4*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^
4)*c*e^2)*b^3*c^6*d^3*e^9 + 32*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d
^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^2*c^7*d^3*e^9 + 120*b^5*c^6*d^2*e^14 + 64*(4*c^2*d^2*e^2 - 4*b*c*d*e^
3 + b^2*e^4)*b^2*c^7*d^3*e^9 - 60*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^
2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^5*c^4*d^2*e^10 + 48*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2
*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^4*c^5*d^2*e^10 - 24*sqrt(2)*sqrt(4*c
^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^3*c^6*
d^2*e^10 - 24*b^6*c^5*d*e^15 - 48*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b^3*c^6*d^2*e^10 + 12*sqrt(2)*sqrt(4
*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^6*c^
3*d*e^11 - 16*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*
e^3 + b^2*e^4)*c*e^2)*b^5*c^4*d*e^11 + 8*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sq
rt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^4*c^5*d*e^11 + 2*b^7*c^4*e^16 + 16*(4*c^2*d^2*e^2 - 4*b*c*d
*e^3 + b^2*e^4)*b^4*c^5*d*e^11 - sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2
*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^7*c^2*e^12 + 2*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*
sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^6*c^3*e^12 - sqrt(2)*sqrt(4*c^2*d^2*e^2 -
4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^5*c^4*e^12 - 2*(4*c
^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b^5*c^4*e^12 + (256*c^9*d^7*e^9 - 128*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d
*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*c^7*d^7*e^5 - 896*b*c^8*d^6*
e^10 + 448*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3
+ b^2*e^4)*c*e^2)*b*c^6*d^6*e^6 + 1344*b^2*c^7*d^5*e^11 - 672*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*
e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^2*c^5*d^5*e^7 + 64*sqrt(2)*sqrt(4*c^2
*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b*c^6*d^5*
e^7 - 32*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 +
b^2*e^4)*c*e^2)*c^7*d^5*e^7 - 1120*b^3*c^6*d^4*e^12 - 64*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c^7*d^5*e^7
+ 560*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^
2*e^4)*c*e^2)*b^3*c^4*d^4*e^8 - 160*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*
c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^2*c^5*d^4*e^8 + 80*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^
2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b*c^6*d^4*e^8 + 560*b^4*c^5*d^3*e^13
+ 160*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b*c^6*d^4*e^8 - 280*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b
^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^4*c^3*d^3*e^9 + 160*sqrt(2)*sqrt(4
*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^3*c^
4*d^3*e^9 - 80*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d
*e^3 + b^2*e^4)*c*e^2)*b^2*c^5*d^3*e^9 - 168*b^5*c^4*d^2*e^14 - 160*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b^
2*c^5*d^3*e^9 + 84*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b
*c*d*e^3 + b^2*e^4)*c*e^2)*b^5*c^2*d^2*e^10 - 80*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*
e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^4*c^3*d^2*e^10 + 40*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*
b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^3*c^4*d^2*e^10 + 28*b
^6*c^3*d*e^15 + 80*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b^3*c^4*d^2*e^10 - 14*sqrt(2)*sqrt(4*c^2*d^2*e^2 -
4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^6*c*d*e^11 + 20*sqr
t(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*
e^2)*b^5*c^2*d*e^11 - 10*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2
- 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^4*c^3*d*e^11 - 2*b^7*c^2*e^16 - 20*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*
b^4*c^3*d*e^11 + sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c
*d*e^3 + b^2*e^4)*c*e^2)*b^7*e^12 - 2*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 + sqrt(
4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^6*c*e^12 + sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)
*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^5*c^2*e^12 + 2*(4*c^2*d^2*e^2 - 4*b*c*d*e
^3 + b^2*e^4)*b^5*c^2*e^12)*c^2 - 2*(256*c^10*d^8*e^8 - 128*sqrt(2)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*
d*e^3 + b^2*e^4)*c*e^2)*c^9*d^8*e^6 - 896*b*c^9*d^7*e^9 + 448*sqrt(2)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*
c*d*e^3 + b^2*e^4)*c*e^2)*b*c^8*d^7*e^7 + 1344*b^2*c^8*d^6*e^10 - 672*sqrt(2)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^
2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^2*c^7*d^6*e^8 + 64*sqrt(2)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3
+ b^2*e^4)*c*e^2)*b*c^8*d^6*e^8 - 32*sqrt(2)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2
)*c^9*d^6*e^8 - 1120*b^3*c^7*d^5*e^11 + 560*sqrt(2)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)
*c*e^2)*b^3*c^6*d^5*e^9 - 160*sqrt(2)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^2*c^
7*d^5*e^9 + 80*sqrt(2)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b*c^8*d^5*e^9 - 64*(4
*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c^8*d^6*e^6 + 560*b^4*c^6*d^4*e^12 - 280*sqrt(2)*sqrt(b*c*e^4 + sqrt(4*c
^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^4*c^5*d^4*e^10 + 160*sqrt(2)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 -
4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^3*c^6*d^4*e^10 - 80*sqrt(2)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 +
b^2*e^4)*c*e^2)*b^2*c^7*d^4*e^10 + 160*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b*c^7*d^5*e^7 - 168*b^5*c^5*d^3
*e^13 + 84*sqrt(2)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^5*c^4*d^3*e^11 - 80*sqr
t(2)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^4*c^5*d^3*e^11 + 40*sqrt(2)*sqrt(b*c*
e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^3*c^6*d^3*e^11 - 160*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 +
b^2*e^4)*b^2*c^6*d^4*e^8 + 28*b^6*c^4*d^2*e^14 - 14*sqrt(2)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 +
b^2*e^4)*c*e^2)*b^6*c^3*d^2*e^12 + 20*sqrt(2)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^
2)*b^5*c^4*d^2*e^12 - 10*sqrt(2)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^4*c^5*d^2
*e^12 + 80*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b^3*c^5*d^3*e^9 - 2*b^7*c^3*d*e^15 + sqrt(2)*sqrt(b*c*e^4 +
sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^7*c^2*d*e^13 - 2*sqrt(2)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e
^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^6*c^3*d*e^13 + sqrt(2)*sqrt(b*c*e^4 + sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 +
b^2*e^4)*c*e^2)*b^5*c^4*d*e^13 - 20*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b^4*c^4*d^2*e^10 + 2*(4*c^2*d^2*e^
2 - 4*b*c*d*e^3 + b^2*e^4)*b^5*c^3*d*e^11)*abs(c))*arctan(2*sqrt(1/2)*x*e^6/sqrt((b*c^5*e^12 + sqrt(b^2*c^10*e
^24 + 4*(c^6*d^2*e^10 - b*c^5*d*e^11)*c^6*e^12))/c^6))/((16*c^10*d^6*e^8 - 48*b*c^9*d^5*e^9 + 56*b^2*c^8*d^4*e
^10 - 8*b*c^9*d^4*e^10 + 4*c^10*d^4*e^10 - 32*b^3*c^7*d^3*e^11 + 16*b^2*c^8*d^3*e^11 - 8*b*c^9*d^3*e^11 + 9*b^
4*c^6*d^2*e^12 - 10*b^3*c^7*d^2*e^12 + 5*b^2*c^8*d^2*e^12 - b^5*c^5*d*e^13 + 2*b^4*c^6*d*e^13 - b^3*c^7*d*e^13
)*c^2) + 1/8*(128*b*c^10*d^6*e^10 - 64*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt
(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b*c^8*d^6*e^6 - 384*b^2*c^9*d^5*e^11 + 192*sqrt(2)*sqrt(4*c^2*d
^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^2*c^7*d^5*
e^7 + 480*b^3*c^8*d^4*e^12 - 240*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2
*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^3*c^6*d^4*e^8 + 32*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e
^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^2*c^7*d^4*e^8 - 16*sqrt(2)*sqrt(4*c^2*
d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b*c^8*d^4*e
^8 - 320*b^4*c^7*d^3*e^13 - 32*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b*c^8*d^4*e^8 + 160*sqrt(2)*sqrt(4*c^2*
d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^4*c^5*d^3
*e^9 - 64*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3
+ b^2*e^4)*c*e^2)*b^3*c^6*d^3*e^9 + 32*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt
(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^2*c^7*d^3*e^9 + 120*b^5*c^6*d^2*e^14 + 64*(4*c^2*d^2*e^2 - 4*
b*c*d*e^3 + b^2*e^4)*b^2*c^7*d^3*e^9 - 60*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - s
qrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^5*c^4*d^2*e^10 + 48*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e
^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^4*c^5*d^2*e^10 - 24*sqrt(2)*
sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*
b^3*c^6*d^2*e^10 - 24*b^6*c^5*d*e^15 - 48*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b^3*c^6*d^2*e^10 + 12*sqrt(2
)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2
)*b^6*c^3*d*e^11 - 16*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 -
4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^5*c^4*d*e^11 + 8*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*
e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^4*c^5*d*e^11 + 2*b^7*c^4*e^16 + 16*(4*c^2*d^2*e^2 -
4*b*c*d*e^3 + b^2*e^4)*b^4*c^5*d*e^11 - sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sq
rt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^7*c^2*e^12 + 2*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b
^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^6*c^3*e^12 - sqrt(2)*sqrt(4*c^2*d^
2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^5*c^4*e^12
- 2*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b^5*c^4*e^12 + (256*c^9*d^7*e^9 - 128*sqrt(2)*sqrt(4*c^2*d^2*e^2 -
4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*c^7*d^7*e^5 - 896*b*
c^8*d^6*e^10 + 448*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b
*c*d*e^3 + b^2*e^4)*c*e^2)*b*c^6*d^6*e^6 + 1344*b^2*c^7*d^5*e^11 - 672*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^
3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^2*c^5*d^5*e^7 + 64*sqrt(2)*sq
rt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b*
c^6*d^5*e^7 - 32*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c
*d*e^3 + b^2*e^4)*c*e^2)*c^7*d^5*e^7 - 1120*b^3*c^6*d^4*e^12 - 64*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c^7*
d^5*e^7 + 560*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*
e^3 + b^2*e^4)*c*e^2)*b^3*c^4*d^4*e^8 - 160*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 -
sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^2*c^5*d^4*e^8 + 80*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*
e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b*c^6*d^4*e^8 + 560*b^4*c^5*d
^3*e^13 + 160*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b*c^6*d^4*e^8 - 280*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d
*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^4*c^3*d^3*e^9 + 160*sqrt(2
)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2
)*b^3*c^4*d^3*e^9 - 80*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 -
4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^2*c^5*d^3*e^9 - 168*b^5*c^4*d^2*e^14 - 160*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2
*e^4)*b^2*c^5*d^3*e^9 + 84*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e
^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^5*c^2*d^2*e^10 - 80*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*s
qrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^4*c^3*d^2*e^10 + 40*sqrt(2)*sqrt(4*c^2*d^2*
e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^3*c^4*d^2*e^1
0 + 28*b^6*c^3*d*e^15 + 80*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b^3*c^4*d^2*e^10 - 14*sqrt(2)*sqrt(4*c^2*d^
2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^6*c*d*e^11
+ 20*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2
*e^4)*c*e^2)*b^5*c^2*d*e^11 - 10*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2
*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^4*c^3*d*e^11 - 2*b^7*c^2*e^16 - 20*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b
^2*e^4)*b^4*c^3*d*e^11 + sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2
- 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^7*e^12 - 2*sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*sqrt(b*c*e^4
- sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^6*c*e^12 + sqrt(2)*sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 +
b^2*e^4)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^5*c^2*e^12 + 2*(4*c^2*d^2*e^2 - 4
*b*c*d*e^3 + b^2*e^4)*b^5*c^2*e^12)*c^2 - 2*(256*c^10*d^8*e^8 + 128*sqrt(2)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2
- 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*c^9*d^8*e^6 - 896*b*c^9*d^7*e^9 - 448*sqrt(2)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^
2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b*c^8*d^7*e^7 + 1344*b^2*c^8*d^6*e^10 + 672*sqrt(2)*sqrt(b*c*e^4 - sqrt(4*c^
2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^2*c^7*d^6*e^8 - 64*sqrt(2)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b
*c*d*e^3 + b^2*e^4)*c*e^2)*b*c^8*d^6*e^8 + 32*sqrt(2)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^
4)*c*e^2)*c^9*d^6*e^8 - 1120*b^3*c^7*d^5*e^11 - 560*sqrt(2)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 +
b^2*e^4)*c*e^2)*b^3*c^6*d^5*e^9 + 160*sqrt(2)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2
)*b^2*c^7*d^5*e^9 - 80*sqrt(2)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b*c^8*d^5*e^9
- 64*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c^8*d^6*e^6 + 560*b^4*c^6*d^4*e^12 + 280*sqrt(2)*sqrt(b*c*e^4 -
sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^4*c^5*d^4*e^10 - 160*sqrt(2)*sqrt(b*c*e^4 - sqrt(4*c^2*d^
2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^3*c^6*d^4*e^10 + 80*sqrt(2)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*
d*e^3 + b^2*e^4)*c*e^2)*b^2*c^7*d^4*e^10 + 160*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b*c^7*d^5*e^7 - 168*b^5
*c^5*d^3*e^13 - 84*sqrt(2)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^5*c^4*d^3*e^11
+ 80*sqrt(2)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^4*c^5*d^3*e^11 - 40*sqrt(2)*s
qrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^3*c^6*d^3*e^11 - 160*(4*c^2*d^2*e^2 - 4*b*c
*d*e^3 + b^2*e^4)*b^2*c^6*d^4*e^8 + 28*b^6*c^4*d^2*e^14 + 14*sqrt(2)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c
*d*e^3 + b^2*e^4)*c*e^2)*b^6*c^3*d^2*e^12 - 20*sqrt(2)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e
^4)*c*e^2)*b^5*c^4*d^2*e^12 + 10*sqrt(2)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^4
*c^5*d^2*e^12 + 80*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b^3*c^5*d^3*e^9 - 2*b^7*c^3*d*e^15 - sqrt(2)*sqrt(b
*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^7*c^2*d*e^13 + 2*sqrt(2)*sqrt(b*c*e^4 - sqrt(4*c
^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*c*e^2)*b^6*c^3*d*e^13 - sqrt(2)*sqrt(b*c*e^4 - sqrt(4*c^2*d^2*e^2 - 4*b*c*
d*e^3 + b^2*e^4)*c*e^2)*b^5*c^4*d*e^13 - 20*(4*c^2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b^4*c^4*d^2*e^10 + 2*(4*c^
2*d^2*e^2 - 4*b*c*d*e^3 + b^2*e^4)*b^5*c^3*d*e^11)*abs(c))*arctan(2*sqrt(1/2)*x*e^6/sqrt((b*c^5*e^12 - sqrt(b^
2*c^10*e^24 + 4*(c^6*d^2*e^10 - b*c^5*d*e^11)*c^6*e^12))/c^6))/((16*c^10*d^6*e^8 - 48*b*c^9*d^5*e^9 + 56*b^2*c
^8*d^4*e^10 - 8*b*c^9*d^4*e^10 + 4*c^10*d^4*e^10 - 32*b^3*c^7*d^3*e^11 + 16*b^2*c^8*d^3*e^11 - 8*b*c^9*d^3*e^1
1 + 9*b^4*c^6*d^2*e^12 - 10*b^3*c^7*d^2*e^12 + 5*b^2*c^8*d^2*e^12 - b^5*c^5*d*e^13 + 2*b^4*c^6*d*e^13 - b^3*c^
7*d*e^13)*c^2) + 1/15*(3*c^4*x^5*e^12 + 20*c^4*d*x^3*e^11 - 5*b*c^3*x^3*e^12 + 105*c^4*d^2*x*e^10 - 75*b*c^3*d
*x*e^11 + 15*b^2*c^2*x*e^12)*e^(-10)/c^5

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maple [B]  time = 0.01, size = 226, normalized size = 1.87 \[ \frac {e^{2} x^{5}}{5 c}-\frac {b^{3} e^{3} \arctan \left (\frac {c e x}{\sqrt {\left (b e -c d \right ) c e}}\right )}{\sqrt {\left (b e -c d \right ) c e}\, c^{3}}+\frac {6 b^{2} d \,e^{2} \arctan \left (\frac {c e x}{\sqrt {\left (b e -c d \right ) c e}}\right )}{\sqrt {\left (b e -c d \right ) c e}\, c^{2}}-\frac {12 b \,d^{2} e \arctan \left (\frac {c e x}{\sqrt {\left (b e -c d \right ) c e}}\right )}{\sqrt {\left (b e -c d \right ) c e}\, c}-\frac {b \,e^{2} x^{3}}{3 c^{2}}+\frac {4 d e \,x^{3}}{3 c}+\frac {8 d^{3} \arctan \left (\frac {c e x}{\sqrt {\left (b e -c d \right ) c e}}\right )}{\sqrt {\left (b e -c d \right ) c e}}+\frac {b^{2} e^{2} x}{c^{3}}-\frac {5 b d e x}{c^{2}}+\frac {7 d^{2} x}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*e^2*x^5/c-1/3/c^2*x^3*b*e^2+4/3/c*x^3*d*e+1/c^3*b^2*e^2*x-5/c^2*b*d*e*x+7/c*d^2*x-1/c^3/((b*e-c*d)*c*e)^(1
/2)*arctan(c*e*x/((b*e-c*d)*c*e)^(1/2))*b^3*e^3+6/c^2/((b*e-c*d)*c*e)^(1/2)*arctan(c*e*x/((b*e-c*d)*c*e)^(1/2)
)*b^2*d*e^2-12/c/((b*e-c*d)*c*e)^(1/2)*arctan(c*e*x/((b*e-c*d)*c*e)^(1/2))*b*d^2*e+8/((b*e-c*d)*c*e)^(1/2)*arc
tan(c*e*x/((b*e-c*d)*c*e)^(1/2))*d^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^4/(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 = 4.53, size = 182, normalized size = 1.50 \[ x\,\left (\frac {3\,d^2}{c}+\frac {\left (\frac {e\,\left (b\,e-c\,d\right )}{c^2}-\frac {3\,d\,e}{c}\right )\,\left (b\,e-c\,d\right )}{c\,e}\right )-x^3\,\left (\frac {e\,\left (b\,e-c\,d\right )}{3\,c^2}-\frac {d\,e}{c}\right )+\frac {e^2\,x^5}{5\,c}-\frac {\mathrm {atan}\left (\frac {\sqrt {c}\,e\,x\,{\left (b\,e-2\,c\,d\right )}^3}{\sqrt {b\,e^2-c\,d\,e}\,\left (b^3\,e^3-6\,b^2\,c\,d\,e^2+12\,b\,c^2\,d^2\,e-8\,c^3\,d^3\right )}\right )\,{\left (b\,e-2\,c\,d\right )}^3}{c^{7/2}\,\sqrt {b\,e^2-c\,d\,e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*((3*d^2)/c + (((e*(b*e - c*d))/c^2 - (3*d*e)/c)*(b*e - c*d))/(c*e)) - x^3*((e*(b*e - c*d))/(3*c^2) - (d*e)/c
) + (e^2*x^5)/(5*c) - (atan((c^(1/2)*e*x*(b*e - 2*c*d)^3)/((b*e^2 - c*d*e)^(1/2)*(b^3*e^3 - 8*c^3*d^3 + 12*b*c
^2*d^2*e - 6*b^2*c*d*e^2)))*(b*e - 2*c*d)^3)/(c^(7/2)*(b*e^2 - c*d*e)^(1/2))

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sympy [B]  time = 1.00, size = 345, normalized size = 2.85 \[ x^{3} \left (- \frac {b e^{2}}{3 c^{2}} + \frac {4 d e}{3 c}\right ) + x \left (\frac {b^{2} e^{2}}{c^{3}} - \frac {5 b d e}{c^{2}} + \frac {7 d^{2}}{c}\right ) + \frac {\sqrt {- \frac {1}{c^{7} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{3} \log {\left (x + \frac {- b c^{3} e \sqrt {- \frac {1}{c^{7} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{3} + c^{4} d \sqrt {- \frac {1}{c^{7} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{3}}{b^{3} e^{3} - 6 b^{2} c d e^{2} + 12 b c^{2} d^{2} e - 8 c^{3} d^{3}} \right )}}{2} - \frac {\sqrt {- \frac {1}{c^{7} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{3} \log {\left (x + \frac {b c^{3} e \sqrt {- \frac {1}{c^{7} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{3} - c^{4} d \sqrt {- \frac {1}{c^{7} e \left (b e - c d\right )}} \left (b e - 2 c d\right )^{3}}{b^{3} e^{3} - 6 b^{2} c d e^{2} + 12 b c^{2} d^{2} e - 8 c^{3} d^{3}} \right )}}{2} + \frac {e^{2} x^{5}}{5 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**3*(-b*e**2/(3*c**2) + 4*d*e/(3*c)) + x*(b**2*e**2/c**3 - 5*b*d*e/c**2 + 7*d**2/c) + sqrt(-1/(c**7*e*(b*e -
c*d)))*(b*e - 2*c*d)**3*log(x + (-b*c**3*e*sqrt(-1/(c**7*e*(b*e - c*d)))*(b*e - 2*c*d)**3 + c**4*d*sqrt(-1/(c*
*7*e*(b*e - c*d)))*(b*e - 2*c*d)**3)/(b**3*e**3 - 6*b**2*c*d*e**2 + 12*b*c**2*d**2*e - 8*c**3*d**3))/2 - sqrt(
-1/(c**7*e*(b*e - c*d)))*(b*e - 2*c*d)**3*log(x + (b*c**3*e*sqrt(-1/(c**7*e*(b*e - c*d)))*(b*e - 2*c*d)**3 - c
**4*d*sqrt(-1/(c**7*e*(b*e - c*d)))*(b*e - 2*c*d)**3)/(b**3*e**3 - 6*b**2*c*d*e**2 + 12*b*c**2*d**2*e - 8*c**3
*d**3))/2 + e**2*x**5/(5*c)

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