3.4.8 \(\int \frac {1}{x^2 (d+e x^2) (a+b x^2+c x^4)} \, dx\) [308]
Optimal. Leaf size=298 \[ -\frac {1}{a d x}-\frac {\sqrt {c} \left (c d-b e+\frac {b c d-b^2 e+2 a c e}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a \sqrt {b-\sqrt {b^2-4 a c}} \left (c d^2-b d e+a e^2\right )}-\frac {\sqrt {c} \left (c d-b e-\frac {b c d-b^2 e+2 a c e}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a \sqrt {b+\sqrt {b^2-4 a c}} \left (c d^2-b d e+a e^2\right )}-\frac {e^{5/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{3/2} \left (c d^2-b d e+a e^2\right )} \]
[Out]
-1/a/d/x-e^(5/2)*arctan(x*e^(1/2)/d^(1/2))/d^(3/2)/(a*e^2-b*d*e+c*d^2)-1/2*arctan(x*2^(1/2)*c^(1/2)/(b-(-4*a*c
+b^2)^(1/2))^(1/2))*c^(1/2)*(c*d-b*e+(2*a*c*e-b^2*e+b*c*d)/(-4*a*c+b^2)^(1/2))/a/(a*e^2-b*d*e+c*d^2)*2^(1/2)/(
b-(-4*a*c+b^2)^(1/2))^(1/2)-1/2*arctan(x*2^(1/2)*c^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2))*c^(1/2)*(c*d-b*e+(-2*a*
c*e+b^2*e-b*c*d)/(-4*a*c+b^2)^(1/2))/a/(a*e^2-b*d*e+c*d^2)*2^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)
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Rubi [A]
time = 0.62, antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1301, 211,
1180} \begin {gather*} -\frac {\sqrt {c} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (\frac {2 a c e+b^2 (-e)+b c d}{\sqrt {b^2-4 a c}}-b e+c d\right )}{\sqrt {2} a \sqrt {b-\sqrt {b^2-4 a c}} \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {c} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (-\frac {2 a c e+b^2 (-e)+b c d}{\sqrt {b^2-4 a c}}-b e+c d\right )}{\sqrt {2} a \sqrt {\sqrt {b^2-4 a c}+b} \left (a e^2-b d e+c d^2\right )}-\frac {e^{5/2} \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac {1}{a d x} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/(x^2*(d + e*x^2)*(a + b*x^2 + c*x^4)),x]
[Out]
-(1/(a*d*x)) - (Sqrt[c]*(c*d - b*e + (b*c*d - b^2*e + 2*a*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/S
qrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*a*Sqrt[b - Sqrt[b^2 - 4*a*c]]*(c*d^2 - b*d*e + a*e^2)) - (Sqrt[c]*(c*d -
b*e - (b*c*d - b^2*e + 2*a*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(
Sqrt[2]*a*Sqrt[b + Sqrt[b^2 - 4*a*c]]*(c*d^2 - b*d*e + a*e^2)) - (e^(5/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(d^(3/2
)*(c*d^2 - b*d*e + a*e^2))
Rule 211
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]
Rule 1180
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]
Rule 1301
Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[Ex
pandIntegrand[(f*x)^m*((d + e*x^2)^q/(a + b*x^2 + c*x^4)), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^
2 - 4*a*c, 0] && IntegerQ[q] && IntegerQ[m]
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx &=\int \left (\frac {1}{a d x^2}-\frac {e^3}{d \left (c d^2-b d e+a e^2\right ) \left (d+e x^2\right )}+\frac {-b c d+b^2 e-a c e-c (c d-b e) x^2}{a \left (c d^2-b d e+a e^2\right ) \left (a+b x^2+c x^4\right )}\right ) \, dx\\ &=-\frac {1}{a d x}+\frac {\int \frac {-b c d+b^2 e-a c e-c (c d-b e) x^2}{a+b x^2+c x^4} \, dx}{a \left (c d^2-b d e+a e^2\right )}-\frac {e^3 \int \frac {1}{d+e x^2} \, dx}{d \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {1}{a d x}-\frac {e^{5/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{3/2} \left (c d^2-b d e+a e^2\right )}-\frac {\left (c \left (c d-b e-\frac {b c d-b^2 e+2 a c e}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx}{2 a \left (c d^2-b d e+a e^2\right )}-\frac {\left (c \left (c d-b e+\frac {b c d-b^2 e+2 a c e}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx}{2 a \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {1}{a d x}-\frac {\sqrt {c} \left (c d-b e+\frac {b c d-b^2 e+2 a c e}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a \sqrt {b-\sqrt {b^2-4 a c}} \left (c d^2-b d e+a e^2\right )}-\frac {\sqrt {c} \left (c d-b e-\frac {b c d-b^2 e+2 a c e}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a \sqrt {b+\sqrt {b^2-4 a c}} \left (c d^2-b d e+a e^2\right )}-\frac {e^{5/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{3/2} \left (c d^2-b d e+a e^2\right )}\\ \end {align*}
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Mathematica [A]
time = 0.26, size = 340, normalized size = 1.14 \begin {gather*} -\frac {1}{a d x}-\frac {\sqrt {c} \left (b c d+c \sqrt {b^2-4 a c} d-b^2 e+2 a c e-b \sqrt {b^2-4 a c} e\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a \sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}} \left (c d^2+e (-b d+a e)\right )}+\frac {\sqrt {c} \left (b c d-c \sqrt {b^2-4 a c} d-b^2 e+2 a c e+b \sqrt {b^2-4 a c} e\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a \sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}} \left (c d^2+e (-b d+a e)\right )}-\frac {e^{5/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{3/2} \left (c d^2-b d e+a e^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/(x^2*(d + e*x^2)*(a + b*x^2 + c*x^4)),x]
[Out]
-(1/(a*d*x)) - (Sqrt[c]*(b*c*d + c*Sqrt[b^2 - 4*a*c]*d - b^2*e + 2*a*c*e - b*Sqrt[b^2 - 4*a*c]*e)*ArcTan[(Sqrt
[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*a*Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]*(c*d^2 +
e*(-(b*d) + a*e))) + (Sqrt[c]*(b*c*d - c*Sqrt[b^2 - 4*a*c]*d - b^2*e + 2*a*c*e + b*Sqrt[b^2 - 4*a*c]*e)*ArcTa
n[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*a*Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]*(
c*d^2 + e*(-(b*d) + a*e))) - (e^(5/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(d^(3/2)*(c*d^2 - b*d*e + a*e^2))
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Maple [A]
time = 0.21, size = 276, normalized size = 0.93
| | |
| method |
result |
size |
| | |
| default |
\(\frac {4 c \left (-\frac {\left (e b \sqrt {-4 a c +b^{2}}-c d \sqrt {-4 a c +b^{2}}-2 a c e +b^{2} e -b c d \right ) \sqrt {2}\, \arctanh \left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (e b \sqrt {-4 a c +b^{2}}-c d \sqrt {-4 a c +b^{2}}+2 a c e -b^{2} e +b c d \right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{\left (a \,e^{2}-d e b +c \,d^{2}\right ) a}-\frac {e^{3} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{d \left (a \,e^{2}-d e b +c \,d^{2}\right ) \sqrt {d e}}-\frac {1}{a d x}\) |
\(276\) |
| risch |
\(\text {Expression too large to display}\) |
\(3695\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/x^2/(e*x^2+d)/(c*x^4+b*x^2+a),x,method=_RETURNVERBOSE)
[Out]
4/(a*e^2-b*d*e+c*d^2)/a*c*(-1/8*(e*b*(-4*a*c+b^2)^(1/2)-c*d*(-4*a*c+b^2)^(1/2)-2*a*c*e+b^2*e-b*c*d)/(-4*a*c+b^
2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))+1/8*
(e*b*(-4*a*c+b^2)^(1/2)-c*d*(-4*a*c+b^2)^(1/2)+2*a*c*e-b^2*e+b*c*d)/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2
)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)))-1/d*e^3/(a*e^2-b*d*e+c*d^2)/(d*e)^(1/2
)*arctan(e*x/(d*e)^(1/2))-1/a/d/x
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^2/(e*x^2+d)/(c*x^4+b*x^2+a),x, algorithm="maxima")
[Out]
-arctan(x*e^(1/2)/sqrt(d))*e^(5/2)/((c*d^3 - b*d^2*e + a*d*e^2)*sqrt(d)) - integrate((b*c*d + (c^2*d - b*c*e)*
x^2 - b^2*e + a*c*e)/(c*x^4 + b*x^2 + a), x)/(a*c*d^2 - a*b*d*e + a^2*e^2) - 1/(a*d*x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 10147 vs.
\(2 (258) = 516\).
time = 163.30, size = 20327, normalized size = 68.21 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^2/(e*x^2+d)/(c*x^4+b*x^2+a),x, algorithm="fricas")
[Out]
[1/2*(a*x*sqrt(-e/d)*e^2*log((x^2*e - 2*d*x*sqrt(-e/d) - d)/(x^2*e + d)) - 2*c*d^2 + 2*b*d*e + sqrt(1/2)*(a*c*
d^3*x - a*b*d^2*x*e + a^2*d*x*e^2)*sqrt(-((b^3*c^2 - 3*a*b*c^3)*d^2 - 2*(b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d*e
+ (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e^2 + ((a^3*b^2*c^2 - 4*a^4*c^3)*d^4 - 2*(a^3*b^3*c - 4*a^4*b*c^2)*d^3*e + (
a^3*b^4 - 2*a^4*b^2*c - 8*a^5*c^2)*d^2*e^2 - 2*(a^4*b^3 - 4*a^5*b*c)*d*e^3 + (a^5*b^2 - 4*a^6*c)*e^4)*sqrt(((b
^4*c^4 - 2*a*b^2*c^5 + a^2*c^6)*d^4 - 4*(b^5*c^3 - 3*a*b^3*c^4 + 2*a^2*b*c^5)*d^3*e + 2*(3*b^6*c^2 - 12*a*b^4*
c^3 + 12*a^2*b^2*c^4 - a^3*c^5)*d^2*e^2 - 4*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^3*c^3 - 2*a^3*b*c^4)*d*e^3 + (b^8 -
6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^4)/((a^6*b^2*c^4 - 4*a^7*c^5)*d^8 - 4*(a^6*b^3*c^3 -
4*a^7*b*c^4)*d^7*e + 2*(3*a^6*b^4*c^2 - 10*a^7*b^2*c^3 - 8*a^8*c^4)*d^6*e^2 - 4*(a^6*b^5*c - a^7*b^3*c^2 - 12*
a^8*b*c^3)*d^5*e^3 + (a^6*b^6 + 8*a^7*b^4*c - 42*a^8*b^2*c^2 - 24*a^9*c^3)*d^4*e^4 - 4*(a^7*b^5 - a^8*b^3*c -
12*a^9*b*c^2)*d^3*e^5 + 2*(3*a^8*b^4 - 10*a^9*b^2*c - 8*a^10*c^2)*d^2*e^6 - 4*(a^9*b^3 - 4*a^10*b*c)*d*e^7 + (
a^10*b^2 - 4*a^11*c)*e^8)))/((a^3*b^2*c^2 - 4*a^4*c^3)*d^4 - 2*(a^3*b^3*c - 4*a^4*b*c^2)*d^3*e + (a^3*b^4 - 2*
a^4*b^2*c - 8*a^5*c^2)*d^2*e^2 - 2*(a^4*b^3 - 4*a^5*b*c)*d*e^3 + (a^5*b^2 - 4*a^6*c)*e^4))*log(2*(b^2*c^5 - a*
c^6)*d^2*x - 4*(b^3*c^4 - 2*a*b*c^5)*d*x*e + 2*(b^4*c^3 - 3*a*b^2*c^4 + a^2*c^5)*x*e^2 + sqrt(1/2)*((b^5*c^3 -
5*a*b^3*c^4 + 4*a^2*b*c^5)*d^3 - (3*b^6*c^2 - 18*a*b^4*c^3 + 25*a^2*b^2*c^4 - 4*a^3*c^5)*d^2*e + (3*b^7*c - 2
1*a*b^5*c^2 + 41*a^2*b^3*c^3 - 20*a^3*b*c^4)*d*e^2 - (b^8 - 8*a*b^6*c + 20*a^2*b^4*c^2 - 17*a^3*b^2*c^3 + 4*a^
4*c^4)*e^3 - ((a^3*b^4*c^3 - 6*a^4*b^2*c^4 + 8*a^5*c^5)*d^5 - (3*a^3*b^5*c^2 - 19*a^4*b^3*c^3 + 28*a^5*b*c^4)*
d^4*e + (3*a^3*b^6*c - 18*a^4*b^4*c^2 + 20*a^5*b^2*c^3 + 16*a^6*c^4)*d^3*e^2 - (a^3*b^7 - 3*a^4*b^5*c - 14*a^5
*b^3*c^2 + 40*a^6*b*c^3)*d^2*e^3 + (2*a^4*b^6 - 13*a^5*b^4*c + 18*a^6*b^2*c^2 + 8*a^7*c^3)*d*e^4 - (a^5*b^5 -
7*a^6*b^3*c + 12*a^7*b*c^2)*e^5)*sqrt(((b^4*c^4 - 2*a*b^2*c^5 + a^2*c^6)*d^4 - 4*(b^5*c^3 - 3*a*b^3*c^4 + 2*a^
2*b*c^5)*d^3*e + 2*(3*b^6*c^2 - 12*a*b^4*c^3 + 12*a^2*b^2*c^4 - a^3*c^5)*d^2*e^2 - 4*(b^7*c - 5*a*b^5*c^2 + 7*
a^2*b^3*c^3 - 2*a^3*b*c^4)*d*e^3 + (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^4)/((a^6*b^2
*c^4 - 4*a^7*c^5)*d^8 - 4*(a^6*b^3*c^3 - 4*a^7*b*c^4)*d^7*e + 2*(3*a^6*b^4*c^2 - 10*a^7*b^2*c^3 - 8*a^8*c^4)*d
^6*e^2 - 4*(a^6*b^5*c - a^7*b^3*c^2 - 12*a^8*b*c^3)*d^5*e^3 + (a^6*b^6 + 8*a^7*b^4*c - 42*a^8*b^2*c^2 - 24*a^9
*c^3)*d^4*e^4 - 4*(a^7*b^5 - a^8*b^3*c - 12*a^9*b*c^2)*d^3*e^5 + 2*(3*a^8*b^4 - 10*a^9*b^2*c - 8*a^10*c^2)*d^2
*e^6 - 4*(a^9*b^3 - 4*a^10*b*c)*d*e^7 + (a^10*b^2 - 4*a^11*c)*e^8)))*sqrt(-((b^3*c^2 - 3*a*b*c^3)*d^2 - 2*(b^4
*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d*e + (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e^2 + ((a^3*b^2*c^2 - 4*a^4*c^3)*d^4 - 2*(
a^3*b^3*c - 4*a^4*b*c^2)*d^3*e + (a^3*b^4 - 2*a^4*b^2*c - 8*a^5*c^2)*d^2*e^2 - 2*(a^4*b^3 - 4*a^5*b*c)*d*e^3 +
(a^5*b^2 - 4*a^6*c)*e^4)*sqrt(((b^4*c^4 - 2*a*b^2*c^5 + a^2*c^6)*d^4 - 4*(b^5*c^3 - 3*a*b^3*c^4 + 2*a^2*b*c^5
)*d^3*e + 2*(3*b^6*c^2 - 12*a*b^4*c^3 + 12*a^2*b^2*c^4 - a^3*c^5)*d^2*e^2 - 4*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^3
*c^3 - 2*a^3*b*c^4)*d*e^3 + (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^4)/((a^6*b^2*c^4 -
4*a^7*c^5)*d^8 - 4*(a^6*b^3*c^3 - 4*a^7*b*c^4)*d^7*e + 2*(3*a^6*b^4*c^2 - 10*a^7*b^2*c^3 - 8*a^8*c^4)*d^6*e^2
- 4*(a^6*b^5*c - a^7*b^3*c^2 - 12*a^8*b*c^3)*d^5*e^3 + (a^6*b^6 + 8*a^7*b^4*c - 42*a^8*b^2*c^2 - 24*a^9*c^3)*d
^4*e^4 - 4*(a^7*b^5 - a^8*b^3*c - 12*a^9*b*c^2)*d^3*e^5 + 2*(3*a^8*b^4 - 10*a^9*b^2*c - 8*a^10*c^2)*d^2*e^6 -
4*(a^9*b^3 - 4*a^10*b*c)*d*e^7 + (a^10*b^2 - 4*a^11*c)*e^8)))/((a^3*b^2*c^2 - 4*a^4*c^3)*d^4 - 2*(a^3*b^3*c -
4*a^4*b*c^2)*d^3*e + (a^3*b^4 - 2*a^4*b^2*c - 8*a^5*c^2)*d^2*e^2 - 2*(a^4*b^3 - 4*a^5*b*c)*d*e^3 + (a^5*b^2 -
4*a^6*c)*e^4))) - sqrt(1/2)*(a*c*d^3*x - a*b*d^2*x*e + a^2*d*x*e^2)*sqrt(-((b^3*c^2 - 3*a*b*c^3)*d^2 - 2*(b^4*
c - 4*a*b^2*c^2 + 2*a^2*c^3)*d*e + (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e^2 + ((a^3*b^2*c^2 - 4*a^4*c^3)*d^4 - 2*(a
^3*b^3*c - 4*a^4*b*c^2)*d^3*e + (a^3*b^4 - 2*a^4*b^2*c - 8*a^5*c^2)*d^2*e^2 - 2*(a^4*b^3 - 4*a^5*b*c)*d*e^3 +
(a^5*b^2 - 4*a^6*c)*e^4)*sqrt(((b^4*c^4 - 2*a*b^2*c^5 + a^2*c^6)*d^4 - 4*(b^5*c^3 - 3*a*b^3*c^4 + 2*a^2*b*c^5)
*d^3*e + 2*(3*b^6*c^2 - 12*a*b^4*c^3 + 12*a^2*b^2*c^4 - a^3*c^5)*d^2*e^2 - 4*(b^7*c - 5*a*b^5*c^2 + 7*a^2*b^3*
c^3 - 2*a^3*b*c^4)*d*e^3 + (b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)*e^4)/((a^6*b^2*c^4 - 4
*a^7*c^5)*d^8 - 4*(a^6*b^3*c^3 - 4*a^7*b*c^4)*d^7*e + 2*(3*a^6*b^4*c^2 - 10*a^7*b^2*c^3 - 8*a^8*c^4)*d^6*e^2 -
4*(a^6*b^5*c - a^7*b^3*c^2 - 12*a^8*b*c^3)*d^5*e^3 + (a^6*b^6 + 8*a^7*b^4*c - 42*a^8*b^2*c^2 - 24*a^9*c^3)*d^
4*e^4 - 4*(a^7*b^5 - a^8*b^3*c - 12*a^9*b*c^2)*d^3*e^5 + 2*(3*a^8*b^4 - 10*a^9*b^2*c - 8*a^10*c^2)*d^2*e^6 - 4
*(a^9*b^3 - 4*a^10*b*c)*d*e^7 + (a^10*b^2 - 4*a^11*c)*e^8)))/((a^3*b^2*c^2 - 4*a^4*c^3)*d^4 - 2*(a^3*b^3*c - 4
*a^4*b*c^2)*d^3*e + (a^3*b^4 - 2*a^4*b^2*c - 8*...
________________________________________________________________________________________
Sympy [F(-1)] Timed out
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/x**2/(e*x**2+d)/(c*x**4+b*x**2+a),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 10058 vs.
\(2 (258) = 516\).
time = 7.54, size = 10058, normalized size = 33.75 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/x^2/(e*x^2+d)/(c*x^4+b*x^2+a),x, algorithm="giac")
[Out]
-1/8*((2*a^2*b^4*c^5 - 8*a^3*b^2*c^6 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^3 +
4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^2*c^4 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*
c + sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^5 -
2*(b^2 - 4*a*c)*a^2*b^2*c^5)*d^5 - (6*a^2*b^5*c^4 - 28*a^3*b^3*c^5 + 16*a^4*b*c^6 - 3*sqrt(2)*sqrt(b^2 - 4*a*
c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^5*c^2 + 14*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*
a^3*b^3*c^3 + 6*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^3 - 8*sqrt(2)*sqrt(b^2 - 4
*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b*c^4 - 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*
a^3*b^2*c^4 - 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c^4 + 2*sqrt(2)*sqrt(b^2 - 4
*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b*c^5 - 6*(b^2 - 4*a*c)*a^2*b^3*c^4 + 4*(b^2 - 4*a*c)*a^3*b*c^5)*d^4
*e + 2*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^5*c^2 - 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^3*
c^3 - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c^3 - 2*a*b^5*c^3 + 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a
*c)*c)*a^3*b*c^4 + 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^4 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c
)*c)*a*b^3*c^4 + 16*a^2*b^3*c^4 - 4*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^5 - 32*a^3*b*c^5 + 2*(b^2
- 4*a*c)*a*b^3*c^3 - 8*(b^2 - 4*a*c)*a^2*b*c^4)*d^3*abs(a*c*d^2 - a*b*d*e + a^2*e^2) + (6*a^2*b^6*c^3 - 28*a^3
*b^4*c^4 + 16*a^4*b^2*c^5 - 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^6*c + 14*sqrt(2)
*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^4*c^2 + 6*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b
^2 - 4*a*c)*c)*a^2*b^5*c^2 - 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^2*c^3 - 4*sqrt(
2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^3*c^3 - 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt
(b^2 - 4*a*c)*c)*a^2*b^4*c^3 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^2*c^4 - 6*(b^
2 - 4*a*c)*a^2*b^4*c^3 + 4*(b^2 - 4*a*c)*a^3*b^2*c^4)*d^3*e^2 - 2*(2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a
*b^6*c - 17*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^2 - 4*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*
b^5*c^2 - 4*a*b^6*c^2 + 40*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^2*c^3 + 18*sqrt(2)*sqrt(b*c + sqrt(b^
2 - 4*a*c)*c)*a^2*b^3*c^3 + 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c^3 + 34*a^2*b^4*c^3 - 16*sqrt(2)*
sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*c^4 - 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b*c^4 - 9*sqrt(2)*sqrt
(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^4 - 80*a^3*b^2*c^4 + 4*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*c^5 +
32*a^4*c^5 + 4*(b^2 - 4*a*c)*a*b^4*c^2 - 18*(b^2 - 4*a*c)*a^2*b^2*c^3 + 8*(b^2 - 4*a*c)*a^3*c^4)*d^2*abs(a*c*
d^2 - a*b*d*e + a^2*e^2)*e + (2*b^4*c^3 - 16*a*b^2*c^4 + 32*a^2*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqr
t(b^2 - 4*a*c)*c)*b^4*c + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^2 + 2*sqrt(2)*sq
rt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^2 - 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4
*a*c)*c)*a^2*c^3 - 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^3 - sqrt(2)*sqrt(b^2 - 4*
a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c^3 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*c
^4 - 2*(b^2 - 4*a*c)*b^2*c^3 + 8*(b^2 - 4*a*c)*a*c^4)*(a*c*d^2 - a*b*d*e + a^2*e^2)^2*d - (2*a^2*b^7*c^2 - 4*a
^3*b^5*c^3 - 24*a^4*b^3*c^4 + 32*a^5*b*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^7
+ 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^5*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*
c + sqrt(b^2 - 4*a*c)*c)*a^2*b^6*c + 12*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^3*c^2
+ 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^4*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c
+ sqrt(b^2 - 4*a*c)*c)*a^2*b^5*c^2 - 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b*c^3 -
8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^2*c^3 - 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*
c + sqrt(b^2 - 4*a*c)*c)*a^3*b^3*c^3 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b*c^4 -
2*(b^2 - 4*a*c)*a^2*b^5*c^2 - 4*(b^2 - 4*a*c)*a^3*b^3*c^3 + 8*(b^2 - 4*a*c)*a^4*b*c^4)*d^2*e^3 + 2*(sqrt(2)*s
qrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^7 - 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^5*c - 2*sqrt(2)*sqrt(b*
c + sqrt(b^2 - 4*a*c)*c)*a*b^6*c - 2*a*b^7*c + 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^3*c^2 + 8*sqrt
(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^4*c^2 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^5*c^2 + 16*a^2*b
^5*c^2 - 4*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c^3 - 32*a^3*b^3*c^3 + 2*(b^2 - 4*a*c)*a*b^5*c - 8*
(b^2 - 4*a*c)*a^2*b^3*c^2)*d*abs(a*c*d^2 - a*b*d*e + a^2*e^2)*e^2 - (2*b^5*c^2 - 16*a*b^3*c^3 + 32*a^2*b*c^4 -
sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 ...
________________________________________________________________________________________
Mupad [B]
time = 5.89, size = 2500, normalized size = 8.39 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(x^2*(d + e*x^2)*(a + b*x^2 + c*x^4)),x)
[Out]
atan((((-(b^7*e^2 + b^5*c^2*d^2 + b^4*e^2*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c^3*d^2 + 12*a^2*b*c^4*d^2 - a*c^
3*d^2*(-(4*a*c - b^2)^3)^(1/2) - 20*a^3*b*c^3*e^2 - 2*b^6*c*d*e + 25*a^2*b^3*c^2*e^2 + a^2*c^2*e^2*(-(4*a*c -
b^2)^3)^(1/2) + b^2*c^2*d^2*(-(4*a*c - b^2)^3)^(1/2) - 9*a*b^5*c*e^2 + 16*a^3*c^4*d*e + 16*a*b^4*c^2*d*e - 2*b
^3*c*d*e*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*c*e^2*(-(4*a*c - b^2)^3)^(1/2) - 36*a^2*b^2*c^3*d*e + 4*a*b*c^2*d*
e*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^5*b^4*e^4 + 16*a^5*c^4*d^4 + 16*a^7*c^2*e^4 - 8*a^6*b^2*c*e^4 - 2*a^4*b^5*d*
e^3 + a^3*b^4*c^2*d^4 - 8*a^4*b^2*c^3*d^4 + a^3*b^6*d^2*e^2 + 32*a^6*c^3*d^2*e^2 - 2*a^3*b^5*c*d^3*e - 32*a^5*
b*c^3*d^3*e + 16*a^5*b^3*c*d*e^3 - 32*a^6*b*c^2*d*e^3 + 16*a^4*b^3*c^2*d^3*e - 6*a^4*b^4*c*d^2*e^2)))^(1/2)*((
(-(b^7*e^2 + b^5*c^2*d^2 + b^4*e^2*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c^3*d^2 + 12*a^2*b*c^4*d^2 - a*c^3*d^2*(
-(4*a*c - b^2)^3)^(1/2) - 20*a^3*b*c^3*e^2 - 2*b^6*c*d*e + 25*a^2*b^3*c^2*e^2 + a^2*c^2*e^2*(-(4*a*c - b^2)^3)
^(1/2) + b^2*c^2*d^2*(-(4*a*c - b^2)^3)^(1/2) - 9*a*b^5*c*e^2 + 16*a^3*c^4*d*e + 16*a*b^4*c^2*d*e - 2*b^3*c*d*
e*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*c*e^2*(-(4*a*c - b^2)^3)^(1/2) - 36*a^2*b^2*c^3*d*e + 4*a*b*c^2*d*e*(-(4*
a*c - b^2)^3)^(1/2))/(8*(a^5*b^4*e^4 + 16*a^5*c^4*d^4 + 16*a^7*c^2*e^4 - 8*a^6*b^2*c*e^4 - 2*a^4*b^5*d*e^3 + a
^3*b^4*c^2*d^4 - 8*a^4*b^2*c^3*d^4 + a^3*b^6*d^2*e^2 + 32*a^6*c^3*d^2*e^2 - 2*a^3*b^5*c*d^3*e - 32*a^5*b*c^3*d
^3*e + 16*a^5*b^3*c*d*e^3 - 32*a^6*b*c^2*d*e^3 + 16*a^4*b^3*c^2*d^3*e - 6*a^4*b^4*c*d^2*e^2)))^(1/2)*(192*a^10
*c^7*d^14*e^3 - x*(-(b^7*e^2 + b^5*c^2*d^2 + b^4*e^2*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c^3*d^2 + 12*a^2*b*c^4
*d^2 - a*c^3*d^2*(-(4*a*c - b^2)^3)^(1/2) - 20*a^3*b*c^3*e^2 - 2*b^6*c*d*e + 25*a^2*b^3*c^2*e^2 + a^2*c^2*e^2*
(-(4*a*c - b^2)^3)^(1/2) + b^2*c^2*d^2*(-(4*a*c - b^2)^3)^(1/2) - 9*a*b^5*c*e^2 + 16*a^3*c^4*d*e + 16*a*b^4*c^
2*d*e - 2*b^3*c*d*e*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*c*e^2*(-(4*a*c - b^2)^3)^(1/2) - 36*a^2*b^2*c^3*d*e + 4
*a*b*c^2*d*e*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^5*b^4*e^4 + 16*a^5*c^4*d^4 + 16*a^7*c^2*e^4 - 8*a^6*b^2*c*e^4 - 2
*a^4*b^5*d*e^3 + a^3*b^4*c^2*d^4 - 8*a^4*b^2*c^3*d^4 + a^3*b^6*d^2*e^2 + 32*a^6*c^3*d^2*e^2 - 2*a^3*b^5*c*d^3*
e - 32*a^5*b*c^3*d^3*e + 16*a^5*b^3*c*d*e^3 - 32*a^6*b*c^2*d*e^3 + 16*a^4*b^3*c^2*d^3*e - 6*a^4*b^4*c*d^2*e^2)
))^(1/2)*(512*a^11*c^7*d^15*e^3 + 512*a^12*c^6*d^13*e^5 - 512*a^13*c^5*d^11*e^7 - 512*a^14*c^4*d^9*e^9 - 32*a^
9*b^3*c^6*d^16*e^2 + 128*a^9*b^4*c^5*d^15*e^3 - 192*a^9*b^5*c^4*d^14*e^4 + 128*a^9*b^6*c^3*d^13*e^5 - 32*a^9*b
^7*c^2*d^12*e^6 - 640*a^10*b^2*c^6*d^15*e^3 + 1056*a^10*b^3*c^5*d^14*e^4 - 672*a^10*b^4*c^4*d^13*e^5 + 96*a^10
*b^5*c^3*d^12*e^6 + 32*a^10*b^6*c^2*d^11*e^7 + 512*a^11*b^2*c^5*d^13*e^5 + 288*a^11*b^3*c^4*d^12*e^6 - 192*a^1
1*b^4*c^3*d^11*e^7 + 32*a^11*b^5*c^2*d^10*e^8 + 384*a^12*b^2*c^4*d^11*e^7 - 288*a^12*b^3*c^3*d^10*e^8 - 32*a^1
2*b^4*c^2*d^9*e^9 + 256*a^13*b^2*c^3*d^9*e^9 + 128*a^10*b*c^7*d^16*e^2 - 1152*a^11*b*c^6*d^14*e^4 - 640*a^12*b
*c^5*d^12*e^6 + 640*a^13*b*c^4*d^10*e^8) + 128*a^11*c^6*d^12*e^5 - 320*a^12*c^5*d^10*e^7 - 256*a^13*c^4*d^8*e^
9 - 16*a^8*b^3*c^6*d^15*e^2 + 64*a^8*b^4*c^5*d^14*e^3 - 96*a^8*b^5*c^4*d^13*e^4 + 64*a^8*b^6*c^3*d^12*e^5 - 16
*a^8*b^7*c^2*d^11*e^6 - 304*a^9*b^2*c^6*d^14*e^3 + 512*a^9*b^3*c^5*d^13*e^4 - 352*a^9*b^4*c^4*d^12*e^5 + 64*a^
9*b^5*c^3*d^11*e^6 + 16*a^9*b^6*c^2*d^10*e^7 + 352*a^10*b^2*c^5*d^12*e^5 + 80*a^10*b^3*c^4*d^11*e^6 - 128*a^10
*b^4*c^3*d^10*e^7 + 16*a^10*b^5*c^2*d^9*e^8 + 336*a^11*b^2*c^4*d^10*e^7 - 128*a^11*b^3*c^3*d^9*e^8 - 16*a^11*b
^4*c^2*d^8*e^9 + 128*a^12*b^2*c^3*d^8*e^9 + 64*a^9*b*c^7*d^15*e^2 - 512*a^10*b*c^6*d^13*e^4 - 320*a^11*b*c^5*d
^11*e^6 + 256*a^12*b*c^4*d^9*e^8) + x*(112*a^10*c^6*d^10*e^6 - 32*a^9*c^7*d^12*e^4 - 16*a^8*c^8*d^14*e^2 - 128
*a^11*c^5*d^8*e^8 + 8*a^7*b^2*c^7*d^14*e^2 - 16*a^7*b^3*c^6*d^13*e^3 + 8*a^7*b^4*c^5*d^12*e^4 + 8*a^7*b^5*c^4*
d^11*e^5 - 16*a^7*b^6*c^3*d^10*e^6 + 8*a^7*b^7*c^2*d^9*e^7 - 72*a^8*b^3*c^5*d^11*e^5 + 128*a^8*b^4*c^4*d^10*e^
6 - 72*a^8*b^5*c^3*d^9*e^7 - 280*a^9*b^2*c^5*d^10*e^6 + 208*a^9*b^3*c^4*d^9*e^7 - 16*a^9*b^4*c^3*d^8*e^8 + 8*a
^9*b^5*c^2*d^7*e^9 + 96*a^10*b^2*c^4*d^8*e^8 - 56*a^10*b^3*c^3*d^7*e^9 + 32*a^8*b*c^7*d^13*e^3 + 128*a^9*b*c^6
*d^11*e^5 - 192*a^10*b*c^5*d^9*e^7 + 96*a^11*b*c^4*d^7*e^9))*(-(b^7*e^2 + b^5*c^2*d^2 + b^4*e^2*(-(4*a*c - b^2
)^3)^(1/2) - 7*a*b^3*c^3*d^2 + 12*a^2*b*c^4*d^2 - a*c^3*d^2*(-(4*a*c - b^2)^3)^(1/2) - 20*a^3*b*c^3*e^2 - 2*b^
6*c*d*e + 25*a^2*b^3*c^2*e^2 + a^2*c^2*e^2*(-(4*a*c - b^2)^3)^(1/2) + b^2*c^2*d^2*(-(4*a*c - b^2)^3)^(1/2) - 9
*a*b^5*c*e^2 + 16*a^3*c^4*d*e + 16*a*b^4*c^2*d*e - 2*b^3*c*d*e*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*c*e^2*(-(4*a
*c - b^2)^3)^(1/2) - 36*a^2*b^2*c^3*d*e + 4*a*b*c^2*d*e*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^5*b^4*e^4 + 16*a^5*c^4
*d^4 + 16*a^7*c^2*e^4 - 8*a^6*b^2*c*e^4 - 2*a^4*b^5*d*e^3 + a^3*b^4*c^2*d^4 - 8*a^4*b^2*c^3*d^4 + a^3*b^6*d^2*
e^2 + 32*a^6*c^3*d^2*e^2 - 2*a^3*b^5*c*d^3*e - 32*a^5*b*c^3*d^3*e + 16*a^5*b^3*c*d*e^3 - 32*a^6*b*c^2*d*e^3 +
16*a^4*b^3*c^2*d^3*e - 6*a^4*b^4*c*d^2*e^2)))^(...
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