3.23.99 \(\int \frac {1}{\sqrt {d+e x} (a+b x+c x^2)^2} \, dx\) [2299]
Optimal. Leaf size=428 \[ -\frac {\sqrt {d+e x} \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}+\frac {\sqrt {c} \left (8 c^2 d^2-b \left (b+\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (4 b d-\sqrt {b^2-4 a c} d-6 a e\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )}-\frac {\sqrt {c} \left (8 c^2 d^2-b \left (b-\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (4 b d+\sqrt {b^2-4 a c} d-6 a e\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )} \]
[Out]
-(b*c*d-b^2*e+2*a*c*e+c*(-b*e+2*c*d)*x)*(e*x+d)^(1/2)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)/(c*x^2+b*x+a)+1/2*arcta
nh(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2))*c^(1/2)*(8*c^2*d^2-b*e^2*(b+(-4*a*c+b
^2)^(1/2))-2*c*e*(4*b*d-6*a*e-d*(-4*a*c+b^2)^(1/2)))/(-4*a*c+b^2)^(3/2)/(a*e^2-b*d*e+c*d^2)*2^(1/2)/(2*c*d-e*(
b-(-4*a*c+b^2)^(1/2)))^(1/2)-1/2*arctanh(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2))
*c^(1/2)*(8*c^2*d^2-b*e^2*(b-(-4*a*c+b^2)^(1/2))-2*c*e*(4*b*d-6*a*e+d*(-4*a*c+b^2)^(1/2)))/(-4*a*c+b^2)^(3/2)/
(a*e^2-b*d*e+c*d^2)*2^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)
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Rubi [A]
time = 1.00, antiderivative size = 428, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {754, 840, 1180,
214} \begin {gather*} \frac {\sqrt {c} \left (-2 c e \left (-d \sqrt {b^2-4 a c}-6 a e+4 b d\right )-b e^2 \left (\sqrt {b^2-4 a c}+b\right )+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )} \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {c} \left (-2 c e \left (d \sqrt {b^2-4 a c}-6 a e+4 b d\right )-b e^2 \left (b-\sqrt {b^2-4 a c}\right )+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )} \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {d+e x} \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/(Sqrt[d + e*x]*(a + b*x + c*x^2)^2),x]
[Out]
-((Sqrt[d + e*x]*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*
x + c*x^2))) + (Sqrt[c]*(8*c^2*d^2 - b*(b + Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(4*b*d - Sqrt[b^2 - 4*a*c]*d - 6*a*
e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*(b^2 - 4*a*c)^(
3/2)*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*(c*d^2 - b*d*e + a*e^2)) - (Sqrt[c]*(8*c^2*d^2 - b*(b - Sqrt[b^2
- 4*a*c])*e^2 - 2*c*e*(4*b*d + Sqrt[b^2 - 4*a*c]*d - 6*a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*
d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*(b^2 - 4*a*c)^(3/2)*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*(c*d^2 -
b*d*e + a*e^2))
Rule 214
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]
Rule 754
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(b
*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e +
a*e^2))), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b
*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]
Rule 840
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
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]
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {d+e x} \left (a+b x+c x^2\right )^2} \, dx &=-\frac {\sqrt {d+e x} \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}-\frac {\int \frac {\frac {1}{2} \left (4 c^2 d^2-b^2 e^2-3 c e (b d-2 a e)\right )+\frac {1}{2} c e (2 c d-b e) x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {\sqrt {d+e x} \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}-\frac {2 \text {Subst}\left (\int \frac {-\frac {1}{2} c d e (2 c d-b e)+\frac {1}{2} e \left (4 c^2 d^2-b^2 e^2-3 c e (b d-2 a e)\right )+\frac {1}{2} c e (2 c d-b e) x^2}{c d^2-b d e+a e^2+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {\sqrt {d+e x} \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}-\frac {\left (c \left (8 c^2 d^2-b \left (b+\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (4 b d-\sqrt {b^2-4 a c} d-6 a e\right )\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{2 \left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )}+\frac {\left (c \left (8 c^2 d^2-b \left (b-\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (4 b d+\sqrt {b^2-4 a c} d-6 a e\right )\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{2 \left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {\sqrt {d+e x} \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}+\frac {\sqrt {c} \left (8 c^2 d^2-b \left (b+\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (4 b d-\sqrt {b^2-4 a c} d-6 a e\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )}-\frac {\sqrt {c} \left (8 c^2 d^2-b \left (b-\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (4 b d+\sqrt {b^2-4 a c} d-6 a e\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 2.91, size = 439, normalized size = 1.03 \begin {gather*} \frac {\frac {2 \sqrt {d+e x} \left (-b^2 e+2 c (a e+c d x)+b c (d-e x)\right )}{a+x (b+c x)}+\frac {\sqrt {2} \sqrt {c} \left (-8 i c^2 d^2-b \left (-i b+\sqrt {-b^2+4 a c}\right ) e^2+2 c e \left (4 i b d+\sqrt {-b^2+4 a c} d-6 i a e\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-b^2+4 a c} \sqrt {-2 c d+\left (b-i \sqrt {-b^2+4 a c}\right ) e}}+\frac {\sqrt {2} \sqrt {c} \left (8 i c^2 d^2-b \left (i b+\sqrt {-b^2+4 a c}\right ) e^2+2 c e \left (-4 i b d+\sqrt {-b^2+4 a c} d+6 i a e\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-b^2+4 a c} \sqrt {-2 c d+\left (b+i \sqrt {-b^2+4 a c}\right ) e}}}{2 \left (b^2-4 a c\right ) \left (-c d^2+e (b d-a e)\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/(Sqrt[d + e*x]*(a + b*x + c*x^2)^2),x]
[Out]
((2*Sqrt[d + e*x]*(-(b^2*e) + 2*c*(a*e + c*d*x) + b*c*(d - e*x)))/(a + x*(b + c*x)) + (Sqrt[2]*Sqrt[c]*((-8*I)
*c^2*d^2 - b*((-I)*b + Sqrt[-b^2 + 4*a*c])*e^2 + 2*c*e*((4*I)*b*d + Sqrt[-b^2 + 4*a*c]*d - (6*I)*a*e))*ArcTan[
(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e - I*Sqrt[-b^2 + 4*a*c]*e]])/(Sqrt[-b^2 + 4*a*c]*Sqrt[-2*c*d
+ (b - I*Sqrt[-b^2 + 4*a*c])*e]) + (Sqrt[2]*Sqrt[c]*((8*I)*c^2*d^2 - b*(I*b + Sqrt[-b^2 + 4*a*c])*e^2 + 2*c*e*
((-4*I)*b*d + Sqrt[-b^2 + 4*a*c]*d + (6*I)*a*e))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[-2*c*d + b*e + I*
Sqrt[-b^2 + 4*a*c]*e]])/(Sqrt[-b^2 + 4*a*c]*Sqrt[-2*c*d + (b + I*Sqrt[-b^2 + 4*a*c])*e]))/(2*(b^2 - 4*a*c)*(-(
c*d^2) + e*(b*d - a*e)))
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Maple [A]
time = 0.88, size = 532, normalized size = 1.24
| | |
method |
result |
size |
| | |
derivativedivides |
\(32 e^{3} c^{2} \left (-\frac {\frac {\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, \sqrt {e x +d}}{4 c \left (-b e +2 c d -\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \left (-e x -\frac {b e}{2 c}-\frac {\sqrt {e^{2} \left (-4 a c +b^{2}\right )}}{2 c}\right )}+\frac {\left (2 b e -4 c d +3 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 \left (2 b e -4 c d +2 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, e^{2} \left (4 a c -b^{2}\right )}-\frac {\frac {\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, \sqrt {e x +d}}{4 c \left (-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \left (-e x -\frac {b e}{2 c}+\frac {\sqrt {e^{2} \left (-4 a c +b^{2}\right )}}{2 c}\right )}+\frac {\left (-2 b e +4 c d +3 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {2}\, \arctanh \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 \left (-2 b e +4 c d +2 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, e^{2} \left (4 a c -b^{2}\right )}\right )\) |
\(532\) |
default |
\(32 e^{3} c^{2} \left (-\frac {\frac {\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, \sqrt {e x +d}}{4 c \left (-b e +2 c d -\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \left (-e x -\frac {b e}{2 c}-\frac {\sqrt {e^{2} \left (-4 a c +b^{2}\right )}}{2 c}\right )}+\frac {\left (2 b e -4 c d +3 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 \left (2 b e -4 c d +2 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, e^{2} \left (4 a c -b^{2}\right )}-\frac {\frac {\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, \sqrt {e x +d}}{4 c \left (-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \left (-e x -\frac {b e}{2 c}+\frac {\sqrt {e^{2} \left (-4 a c +b^{2}\right )}}{2 c}\right )}+\frac {\left (-2 b e +4 c d +3 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {2}\, \arctanh \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 \left (-2 b e +4 c d +2 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{4 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, e^{2} \left (4 a c -b^{2}\right )}\right )\) |
\(532\) |
| | |
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|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(e*x+d)^(1/2)/(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)
[Out]
32*e^3*c^2*(-1/4/(-e^2*(4*a*c-b^2))^(1/2)/e^2/(4*a*c-b^2)*(1/4/c/(-b*e+2*c*d-(-4*a*c*e^2+b^2*e^2)^(1/2))*(-4*a
*c*e^2+b^2*e^2)^(1/2)*(e*x+d)^(1/2)/(-e*x-1/2*b*e/c-1/2/c*(e^2*(-4*a*c+b^2))^(1/2))+1/2*(2*b*e-4*c*d+3*(-4*a*c
*e^2+b^2*e^2)^(1/2))/(2*b*e-4*c*d+2*(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*
c)^(1/2)*arctan(c*(e*x+d)^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)))-1/4/(-e^2*(4*a*c-b^2)
)^(1/2)/e^2/(4*a*c-b^2)*(1/4/c/(-b*e+2*c*d+(-4*a*c*e^2+b^2*e^2)^(1/2))*(-4*a*c*e^2+b^2*e^2)^(1/2)*(e*x+d)^(1/2
)/(-e*x-1/2*b*e/c+1/2/c*(e^2*(-4*a*c+b^2))^(1/2))+1/2*(-2*b*e+4*c*d+3*(-4*a*c*e^2+b^2*e^2)^(1/2))/(-2*b*e+4*c*
d+2*(-4*a*c*e^2+b^2*e^2)^(1/2))*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/2
)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))))
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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/(e*x+d)^(1/2)/(c*x^2+b*x+a)^2,x, algorithm="maxima")
[Out]
integrate(1/((c*x^2 + b*x + a)^2*sqrt(x*e + d)), x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 20083 vs.
\(2 (391) = 782\).
time = 14.39, size = 20083, normalized size = 46.92 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)^(1/2)/(c*x^2+b*x+a)^2,x, algorithm="fricas")
[Out]
1/2*(sqrt(1/2)*((b^2*c^2 - 4*a*c^3)*d^2*x^2 + (b^3*c - 4*a*b*c^2)*d^2*x + (a*b^2*c - 4*a^2*c^2)*d^2 + (a^2*b^2
- 4*a^3*c + (a*b^2*c - 4*a^2*c^2)*x^2 + (a*b^3 - 4*a^2*b*c)*x)*e^2 - ((b^3*c - 4*a*b*c^2)*d*x^2 + (b^4 - 4*a*
b^2*c)*d*x + (a*b^3 - 4*a^2*b*c)*d)*e)*sqrt((32*c^5*d^5 - 80*b*c^4*d^4*e + 10*(5*b^2*c^3 + 12*a*c^4)*d^3*e^2 +
5*(b^3*c^2 - 36*a*b*c^3)*d^2*e^3 - 5*(b^4*c - 6*a*b^2*c^2 - 24*a^2*c^3)*d*e^4 - (b^5 - 15*a*b^3*c + 60*a^2*b*
c^2)*e^5 + ((b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*d^6 - 3*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^
3*c^4 - 64*a^3*b*c^5)*d^5*e + 3*(b^8*c - 11*a*b^6*c^2 + 36*a^2*b^4*c^3 - 16*a^3*b^2*c^4 - 64*a^4*c^5)*d^4*e^2
- (b^9 - 6*a*b^7*c - 24*a^2*b^5*c^2 + 224*a^3*b^3*c^3 - 384*a^4*b*c^4)*d^3*e^3 + 3*(a*b^8 - 11*a^2*b^6*c + 36*
a^3*b^4*c^2 - 16*a^4*b^2*c^3 - 64*a^5*c^4)*d^2*e^4 - 3*(a^2*b^7 - 12*a^3*b^5*c + 48*a^4*b^3*c^2 - 64*a^5*b*c^3
)*d*e^5 + (a^3*b^6 - 12*a^4*b^4*c + 48*a^5*b^2*c^2 - 64*a^6*c^3)*e^6)*sqrt((25*c^4*d^4*e^6 - 50*b*c^3*d^3*e^7
+ 15*(b^2*c^2 + 6*a*c^3)*d^2*e^8 + 10*(b^3*c - 9*a*b*c^2)*d*e^9 + (b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^10)/((b^6*
c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)*d^12 - 6*(b^7*c^5 - 12*a*b^5*c^6 + 48*a^2*b^3*c^7 - 64*a^3*b
*c^8)*d^11*e + 3*(5*b^8*c^4 - 58*a*b^6*c^5 + 216*a^2*b^4*c^6 - 224*a^3*b^2*c^7 - 128*a^4*c^8)*d^10*e^2 - 10*(2
*b^9*c^3 - 21*a*b^7*c^4 + 60*a^2*b^5*c^5 + 16*a^3*b^3*c^6 - 192*a^4*b*c^7)*d^9*e^3 + 15*(b^10*c^2 - 8*a*b^8*c^
3 + a^2*b^6*c^4 + 116*a^3*b^4*c^5 - 208*a^4*b^2*c^6 - 64*a^5*c^7)*d^8*e^4 - 6*(b^11*c - 2*a*b^9*c^2 - 62*a^2*b
^7*c^3 + 296*a^3*b^5*c^4 - 160*a^4*b^3*c^5 - 640*a^5*b*c^6)*d^7*e^5 + (b^12 + 18*a*b^10*c - 222*a^2*b^8*c^2 +
316*a^3*b^6*c^3 + 2160*a^4*b^4*c^4 - 4800*a^5*b^2*c^5 - 1280*a^6*c^6)*d^6*e^6 - 6*(a*b^11 - 2*a^2*b^9*c - 62*a
^3*b^7*c^2 + 296*a^4*b^5*c^3 - 160*a^5*b^3*c^4 - 640*a^6*b*c^5)*d^5*e^7 + 15*(a^2*b^10 - 8*a^3*b^8*c + a^4*b^6
*c^2 + 116*a^5*b^4*c^3 - 208*a^6*b^2*c^4 - 64*a^7*c^5)*d^4*e^8 - 10*(2*a^3*b^9 - 21*a^4*b^7*c + 60*a^5*b^5*c^2
+ 16*a^6*b^3*c^3 - 192*a^7*b*c^4)*d^3*e^9 + 3*(5*a^4*b^8 - 58*a^5*b^6*c + 216*a^6*b^4*c^2 - 224*a^7*b^2*c^3 -
128*a^8*c^4)*d^2*e^10 - 6*(a^5*b^7 - 12*a^6*b^5*c + 48*a^7*b^3*c^2 - 64*a^8*b*c^3)*d*e^11 + (a^6*b^6 - 12*a^7
*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*e^12)))/((b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*d^6 - 3*
(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*d^5*e + 3*(b^8*c - 11*a*b^6*c^2 + 36*a^2*b^4*c^3 - 16
*a^3*b^2*c^4 - 64*a^4*c^5)*d^4*e^2 - (b^9 - 6*a*b^7*c - 24*a^2*b^5*c^2 + 224*a^3*b^3*c^3 - 384*a^4*b*c^4)*d^3*
e^3 + 3*(a*b^8 - 11*a^2*b^6*c + 36*a^3*b^4*c^2 - 16*a^4*b^2*c^3 - 64*a^5*c^4)*d^2*e^4 - 3*(a^2*b^7 - 12*a^3*b^
5*c + 48*a^4*b^3*c^2 - 64*a^5*b*c^3)*d*e^5 + (a^3*b^6 - 12*a^4*b^4*c + 48*a^5*b^2*c^2 - 64*a^6*c^3)*e^6))*log(
sqrt(1/2)*(10*(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*d^4*e^4 - 20*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*d^3*e^5
+ 3*(b^6*c^2 + 8*a*b^4*c^3 - 112*a^2*b^2*c^4 + 256*a^3*c^5)*d^2*e^6 + (7*b^7*c - 104*a*b^5*c^2 + 496*a^2*b^3*
c^3 - 768*a^3*b*c^4)*d*e^7 + (b^8 - 23*a*b^6*c + 190*a^2*b^4*c^2 - 672*a^3*b^2*c^3 + 864*a^4*c^4)*e^8 + (8*(b^
6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)*d^9 - 36*(b^7*c^5 - 12*a*b^5*c^6 + 48*a^2*b^3*c^7 - 64*a^3
*b*c^8)*d^8*e + 2*(31*b^8*c^4 - 352*a*b^6*c^5 + 1248*a^2*b^4*c^6 - 1024*a^3*b^2*c^7 - 1280*a^4*c^8)*d^7*e^2 -
7*(7*b^9*c^3 - 64*a*b^7*c^4 + 96*a^2*b^5*c^5 + 512*a^3*b^3*c^6 - 1280*a^4*b*c^7)*d^6*e^3 + 3*(5*b^10*c^2 - 2*a
*b^8*c^3 - 432*a^2*b^6*c^4 + 2176*a^3*b^4*c^5 - 2560*a^4*b^2*c^6 - 1536*a^5*c^7)*d^5*e^4 + (b^11*c - 97*a*b^9*
c^2 + 888*a^2*b^7*c^3 - 1984*a^3*b^5*c^4 - 3200*a^4*b^3*c^5 + 11520*a^5*b*c^6)*d^4*e^5 - (b^12 - 20*a*b^10*c +
6*a^2*b^8*c^2 + 1152*a^3*b^6*c^3 - 5440*a^4*b^4*c^4 + 6144*a^5*b^2*c^5 + 3584*a^6*c^6)*d^3*e^6 + 3*(a*b^11 -
21*a^2*b^9*c + 128*a^3*b^7*c^2 - 160*a^4*b^5*c^3 - 768*a^5*b^3*c^4 + 1792*a^6*b*c^5)*d^2*e^7 - (3*a^2*b^10 - 6
2*a^3*b^8*c + 440*a^4*b^6*c^2 - 1248*a^5*b^4*c^3 + 896*a^6*b^2*c^4 + 1024*a^7*c^5)*d*e^8 + (a^3*b^9 - 20*a^4*b
^7*c + 144*a^5*b^5*c^2 - 448*a^6*b^3*c^3 + 512*a^7*b*c^4)*e^9)*sqrt((25*c^4*d^4*e^6 - 50*b*c^3*d^3*e^7 + 15*(b
^2*c^2 + 6*a*c^3)*d^2*e^8 + 10*(b^3*c - 9*a*b*c^2)*d*e^9 + (b^4 - 18*a*b^2*c + 81*a^2*c^2)*e^10)/((b^6*c^6 - 1
2*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)*d^12 - 6*(b^7*c^5 - 12*a*b^5*c^6 + 48*a^2*b^3*c^7 - 64*a^3*b*c^8)*d
^11*e + 3*(5*b^8*c^4 - 58*a*b^6*c^5 + 216*a^2*b^4*c^6 - 224*a^3*b^2*c^7 - 128*a^4*c^8)*d^10*e^2 - 10*(2*b^9*c^
3 - 21*a*b^7*c^4 + 60*a^2*b^5*c^5 + 16*a^3*b^3*c^6 - 192*a^4*b*c^7)*d^9*e^3 + 15*(b^10*c^2 - 8*a*b^8*c^3 + a^2
*b^6*c^4 + 116*a^3*b^4*c^5 - 208*a^4*b^2*c^6 - 64*a^5*c^7)*d^8*e^4 - 6*(b^11*c - 2*a*b^9*c^2 - 62*a^2*b^7*c^3
+ 296*a^3*b^5*c^4 - 160*a^4*b^3*c^5 - 640*a^5*b*c^6)*d^7*e^5 + (b^12 + 18*a*b^10*c - 222*a^2*b^8*c^2 + 316*a^3
*b^6*c^3 + 2160*a^4*b^4*c^4 - 4800*a^5*b^2*c^5 - 1280*a^6*c^6)*d^6*e^6 - 6*(a*b^11 - 2*a^2*b^9*c - 62*a^3*b^7*
c^2 + 296*a^4*b^5*c^3 - 160*a^5*b^3*c^4 - 640*a^6*b*c^5)*d^5*e^7 + 15*(a^2*b^10 - 8*a^3*b^8*c + a^4*b^6*c^2 +
116*a^5*b^4*c^3 - 208*a^6*b^2*c^4 - 64*a^7*c^5)...
________________________________________________________________________________________
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/(e*x+d)**(1/2)/(c*x**2+b*x+a)**2,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2822 vs.
\(2 (391) = 782\).
time = 1.49, size = 2822, normalized size = 6.59 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)^(1/2)/(c*x^2+b*x+a)^2,x, algorithm="giac")
[Out]
-(2*(x*e + d)^(3/2)*c^2*d*e - 2*sqrt(x*e + d)*c^2*d^2*e - (x*e + d)^(3/2)*b*c*e^2 + 2*sqrt(x*e + d)*b*c*d*e^2
- sqrt(x*e + d)*b^2*e^3 + 2*sqrt(x*e + d)*a*c*e^3)/((b^2*c*d^2 - 4*a*c^2*d^2 - b^3*d*e + 4*a*b*c*d*e + a*b^2*e
^2 - 4*a^2*c*e^2)*((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)*b*e - b*d*e + a*e^2)) - 1/8*((b^2*c*d^2
*e - 4*a*c^2*d^2*e - b^3*d*e^2 + 4*a*b*c*d*e^2 + a*b^2*e^3 - 4*a^2*c*e^3)^2*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2
- 4*a*c)*c)*e)*(2*c*d*e - b*e^2) + 2*(2*sqrt(b^2 - 4*a*c)*c^3*d^4*e - 4*sqrt(b^2 - 4*a*c)*b*c^2*d^3*e^2 + (b^2
*c + 8*a*c^2)*sqrt(b^2 - 4*a*c)*d^2*e^3 + (b^3 - 8*a*b*c)*sqrt(b^2 - 4*a*c)*d*e^4 - (a*b^2 - 6*a^2*c)*sqrt(b^2
- 4*a*c)*e^5)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*abs(b^2*c*d^2*e - 4*a*c^2*d^2*e - b^3*d*e^2 +
4*a*b*c*d*e^2 + a*b^2*e^3 - 4*a^2*c*e^3) - (16*(b^2*c^5 - 4*a*c^6)*d^7*e - 56*(b^3*c^4 - 4*a*b*c^5)*d^6*e^2 +
14*(5*b^4*c^3 - 16*a*b^2*c^4 - 16*a^2*c^5)*d^5*e^3 - 35*(b^5*c^2 - 16*a^2*b*c^4)*d^4*e^4 + 4*(b^6*c + 23*a*b^4
*c^2 - 92*a^2*b^2*c^3 - 64*a^3*c^4)*d^3*e^5 + (b^7 - 26*a*b^5*c - 8*a^2*b^3*c^2 + 384*a^3*b*c^3)*d^2*e^6 - 2*(
a*b^6 - 19*a^2*b^4*c + 48*a^3*b^2*c^2 + 48*a^4*c^3)*d*e^7 + (a^2*b^5 - 16*a^3*b^3*c + 48*a^4*b*c^2)*e^8)*sqrt(
-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e))*arctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(2*b^2*c^2*d^3 - 8*a*c^3*d
^3 - 3*b^3*c*d^2*e + 12*a*b*c^2*d^2*e + b^4*d*e^2 - 2*a*b^2*c*d*e^2 - 8*a^2*c^2*d*e^2 - a*b^3*e^3 + 4*a^2*b*c*
e^3 + sqrt((2*b^2*c^2*d^3 - 8*a*c^3*d^3 - 3*b^3*c*d^2*e + 12*a*b*c^2*d^2*e + b^4*d*e^2 - 2*a*b^2*c*d*e^2 - 8*a
^2*c^2*d*e^2 - a*b^3*e^3 + 4*a^2*b*c*e^3)^2 - 4*(b^2*c^2*d^4 - 4*a*c^3*d^4 - 2*b^3*c*d^3*e + 8*a*b*c^2*d^3*e +
b^4*d^2*e^2 - 2*a*b^2*c*d^2*e^2 - 8*a^2*c^2*d^2*e^2 - 2*a*b^3*d*e^3 + 8*a^2*b*c*d*e^3 + a^2*b^2*e^4 - 4*a^3*c
*e^4)*(b^2*c^2*d^2 - 4*a*c^3*d^2 - b^3*c*d*e + 4*a*b*c^2*d*e + a*b^2*c*e^2 - 4*a^2*c^2*e^2)))/(b^2*c^2*d^2 - 4
*a*c^3*d^2 - b^3*c*d*e + 4*a*b*c^2*d*e + a*b^2*c*e^2 - 4*a^2*c^2*e^2)))/(((b^2*c^3 - 4*a*c^4)*sqrt(b^2 - 4*a*c
)*d^6 - 3*(b^3*c^2 - 4*a*b*c^3)*sqrt(b^2 - 4*a*c)*d^5*e + 3*(b^4*c - 3*a*b^2*c^2 - 4*a^2*c^3)*sqrt(b^2 - 4*a*c
)*d^4*e^2 - (b^5 + 2*a*b^3*c - 24*a^2*b*c^2)*sqrt(b^2 - 4*a*c)*d^3*e^3 + 3*(a*b^4 - 3*a^2*b^2*c - 4*a^3*c^2)*s
qrt(b^2 - 4*a*c)*d^2*e^4 - 3*(a^2*b^3 - 4*a^3*b*c)*sqrt(b^2 - 4*a*c)*d*e^5 + (a^3*b^2 - 4*a^4*c)*sqrt(b^2 - 4*
a*c)*e^6)*abs(b^2*c*d^2*e - 4*a*c^2*d^2*e - b^3*d*e^2 + 4*a*b*c*d*e^2 + a*b^2*e^3 - 4*a^2*c*e^3)*abs(c)) + 1/8
*((b^2*c*d^2*e - 4*a*c^2*d^2*e - b^3*d*e^2 + 4*a*b*c*d*e^2 + a*b^2*e^3 - 4*a^2*c*e^3)^2*sqrt(-4*c^2*d + 2*(b*c
+ sqrt(b^2 - 4*a*c)*c)*e)*(2*c*d*e - b*e^2) - 2*(2*sqrt(b^2 - 4*a*c)*c^3*d^4*e - 4*sqrt(b^2 - 4*a*c)*b*c^2*d^
3*e^2 + (b^2*c + 8*a*c^2)*sqrt(b^2 - 4*a*c)*d^2*e^3 + (b^3 - 8*a*b*c)*sqrt(b^2 - 4*a*c)*d*e^4 - (a*b^2 - 6*a^2
*c)*sqrt(b^2 - 4*a*c)*e^5)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*abs(b^2*c*d^2*e - 4*a*c^2*d^2*e -
b^3*d*e^2 + 4*a*b*c*d*e^2 + a*b^2*e^3 - 4*a^2*c*e^3) - (16*(b^2*c^5 - 4*a*c^6)*d^7*e - 56*(b^3*c^4 - 4*a*b*c^5
)*d^6*e^2 + 14*(5*b^4*c^3 - 16*a*b^2*c^4 - 16*a^2*c^5)*d^5*e^3 - 35*(b^5*c^2 - 16*a^2*b*c^4)*d^4*e^4 + 4*(b^6*
c + 23*a*b^4*c^2 - 92*a^2*b^2*c^3 - 64*a^3*c^4)*d^3*e^5 + (b^7 - 26*a*b^5*c - 8*a^2*b^3*c^2 + 384*a^3*b*c^3)*d
^2*e^6 - 2*(a*b^6 - 19*a^2*b^4*c + 48*a^3*b^2*c^2 + 48*a^4*c^3)*d*e^7 + (a^2*b^5 - 16*a^3*b^3*c + 48*a^4*b*c^2
)*e^8)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e))*arctan(2*sqrt(1/2)*sqrt(x*e + d)/sqrt(-(2*b^2*c^2*d^3
- 8*a*c^3*d^3 - 3*b^3*c*d^2*e + 12*a*b*c^2*d^2*e + b^4*d*e^2 - 2*a*b^2*c*d*e^2 - 8*a^2*c^2*d*e^2 - a*b^3*e^3
+ 4*a^2*b*c*e^3 - sqrt((2*b^2*c^2*d^3 - 8*a*c^3*d^3 - 3*b^3*c*d^2*e + 12*a*b*c^2*d^2*e + b^4*d*e^2 - 2*a*b^2*c
*d*e^2 - 8*a^2*c^2*d*e^2 - a*b^3*e^3 + 4*a^2*b*c*e^3)^2 - 4*(b^2*c^2*d^4 - 4*a*c^3*d^4 - 2*b^3*c*d^3*e + 8*a*b
*c^2*d^3*e + b^4*d^2*e^2 - 2*a*b^2*c*d^2*e^2 - 8*a^2*c^2*d^2*e^2 - 2*a*b^3*d*e^3 + 8*a^2*b*c*d*e^3 + a^2*b^2*e
^4 - 4*a^3*c*e^4)*(b^2*c^2*d^2 - 4*a*c^3*d^2 - b^3*c*d*e + 4*a*b*c^2*d*e + a*b^2*c*e^2 - 4*a^2*c^2*e^2)))/(b^2
*c^2*d^2 - 4*a*c^3*d^2 - b^3*c*d*e + 4*a*b*c^2*d*e + a*b^2*c*e^2 - 4*a^2*c^2*e^2)))/(((b^2*c^3 - 4*a*c^4)*sqrt
(b^2 - 4*a*c)*d^6 - 3*(b^3*c^2 - 4*a*b*c^3)*sqrt(b^2 - 4*a*c)*d^5*e + 3*(b^4*c - 3*a*b^2*c^2 - 4*a^2*c^3)*sqrt
(b^2 - 4*a*c)*d^4*e^2 - (b^5 + 2*a*b^3*c - 24*a^2*b*c^2)*sqrt(b^2 - 4*a*c)*d^3*e^3 + 3*(a*b^4 - 3*a^2*b^2*c -
4*a^3*c^2)*sqrt(b^2 - 4*a*c)*d^2*e^4 - 3*(a^2*b^3 - 4*a^3*b*c)*sqrt(b^2 - 4*a*c)*d*e^5 + (a^3*b^2 - 4*a^4*c)*s
qrt(b^2 - 4*a*c)*e^6)*abs(b^2*c*d^2*e - 4*a*c^2*d^2*e - b^3*d*e^2 + 4*a*b*c*d*e^2 + a*b^2*e^3 - 4*a^2*c*e^3)*a
bs(c))
________________________________________________________________________________________
Mupad [B]
time = 7.89, size = 2500, normalized size = 5.84 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((d + e*x)^(1/2)*(a + b*x + c*x^2)^2),x)
[Out]
- (((d + e*x)^(1/2)*(b^2*e^3 + 2*c^2*d^2*e - 2*a*c*e^3 - 2*b*c*d*e^2))/((4*a*c - b^2)*(a*e^2 + c*d^2 - b*d*e))
+ (c*(b*e^2 - 2*c*d*e)*(d + e*x)^(3/2))/((4*a*c - b^2)*(a*e^2 + c*d^2 - b*d*e)))/((b*e - 2*c*d)*(d + e*x) + c
*(d + e*x)^2 + a*e^2 + c*d^2 - b*d*e) - atan(((((1536*a^5*c^6*e^7 + 4*a*b^8*c^2*e^7 - 4*b^9*c^2*d*e^6 - 72*a^2
*b^6*c^3*e^7 + 480*a^3*b^4*c^4*e^7 - 1408*a^4*b^2*c^5*e^7 + 512*a^3*c^8*d^4*e^3 + 2048*a^4*c^7*d^2*e^5 - 8*b^6
*c^5*d^4*e^3 + 16*b^7*c^4*d^3*e^4 - 4*b^8*c^3*d^2*e^5 - 384*a^2*b^2*c^7*d^4*e^3 + 768*a^2*b^3*c^6*d^3*e^4 + 19
2*a^2*b^4*c^5*d^2*e^5 - 1280*a^3*b^2*c^6*d^2*e^5 + 80*a*b^7*c^3*d*e^6 - 2048*a^4*b*c^6*d*e^6 + 96*a*b^4*c^6*d^
4*e^3 - 192*a*b^5*c^5*d^3*e^4 + 16*a*b^6*c^4*d^2*e^5 - 576*a^2*b^5*c^4*d*e^6 - 1024*a^3*b*c^7*d^3*e^4 + 1792*a
^3*b^3*c^5*d*e^6)/(a^2*b^6*e^4 - 64*a^3*c^5*d^4 - 64*a^5*c^3*e^4 + b^6*c^2*d^4 + b^8*d^2*e^2 - 12*a*b^4*c^3*d^
4 - 12*a^3*b^4*c*e^4 + 48*a^2*b^2*c^4*d^4 + 48*a^4*b^2*c^2*e^4 - 128*a^4*c^4*d^2*e^2 - 2*a*b^7*d*e^3 - 2*b^7*c
*d^3*e + 24*a^2*b^4*c^2*d^2*e^2 + 32*a^3*b^2*c^3*d^2*e^2 + 24*a*b^5*c^2*d^3*e - 10*a*b^6*c*d^2*e^2 + 24*a^2*b^
5*c*d*e^3 + 128*a^3*b*c^4*d^3*e + 128*a^4*b*c^3*d*e^3 - 96*a^2*b^3*c^3*d^3*e - 96*a^3*b^3*c^2*d*e^3) - (2*(d +
e*x)^(1/2)*(-(32*b^6*c^5*d^5 - 2048*a^3*c^8*d^5 - b^11*e^5 + b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6
*d^5 + 3840*a^5*b*c^5*e^5 - 7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5
+ 1504*a^3*b^5*c^3*e^5 - 3840*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3
- 5*c^2*d^2*e^3*(-(4*a*c - b^2)^9)^(1/2) + 27*a*b^9*c*e^5 - 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e
^4 + 5*b*c*d*e^4*(-(4*a*c - b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*
c^6*d^3*e^2 - 8960*a^3*b^3*c^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480
*a*b^6*c^4*d^3*e^2 - 240*a*b^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*
d*e^4 + 11520*a^4*b*c^6*d^2*e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(b^15*d^3*e^3 - 4096*a^6*c^9*d^6 - 4096*a^9*c^6*e
^6 - b^12*c^3*d^6 - a^3*b^12*e^6 + 24*a*b^10*c^4*d^6 + 24*a^4*b^10*c*e^6 - 3*a*b^14*d^2*e^4 + 3*a^2*b^13*d*e^5
+ 3*b^13*c^2*d^5*e - 3*b^14*c*d^4*e^2 - 240*a^2*b^8*c^5*d^6 + 1280*a^3*b^6*c^6*d^6 - 3840*a^4*b^4*c^7*d^6 + 6
144*a^5*b^2*c^8*d^6 - 240*a^5*b^8*c^2*e^6 + 1280*a^6*b^6*c^3*e^6 - 3840*a^7*b^4*c^4*e^6 + 6144*a^8*b^2*c^5*e^6
- 12288*a^7*c^8*d^4*e^2 - 12288*a^8*c^7*d^2*e^4 - 648*a^2*b^10*c^3*d^4*e^2 + 96*a^2*b^11*c^2*d^3*e^3 + 3120*a
^3*b^8*c^4*d^4*e^2 + 160*a^3*b^9*c^3*d^3*e^3 - 648*a^3*b^10*c^2*d^2*e^4 - 7680*a^4*b^6*c^5*d^4*e^2 - 3840*a^4*
b^7*c^4*d^3*e^3 + 3120*a^4*b^8*c^3*d^2*e^4 + 6912*a^5*b^4*c^6*d^4*e^2 + 16896*a^5*b^5*c^5*d^3*e^3 - 7680*a^5*b
^6*c^4*d^2*e^4 + 6144*a^6*b^2*c^7*d^4*e^2 - 32768*a^6*b^3*c^6*d^3*e^3 + 6912*a^6*b^4*c^5*d^2*e^4 + 6144*a^7*b^
2*c^6*d^2*e^4 - 72*a*b^11*c^3*d^5*e - 18*a*b^13*c*d^3*e^3 - 72*a^3*b^11*c*d*e^5 + 12288*a^6*b*c^8*d^5*e + 1228
8*a^8*b*c^6*d*e^5 + 69*a*b^12*c^2*d^4*e^2 + 720*a^2*b^9*c^4*d^5*e + 69*a^2*b^12*c*d^2*e^4 - 3840*a^3*b^7*c^5*d
^5*e + 11520*a^4*b^5*c^6*d^5*e + 720*a^4*b^9*c^2*d*e^5 - 18432*a^5*b^3*c^7*d^5*e - 3840*a^5*b^7*c^3*d*e^5 + 11
520*a^6*b^5*c^4*d*e^5 + 24576*a^7*b*c^7*d^3*e^3 - 18432*a^7*b^3*c^5*d*e^5)))^(1/2)*(512*a^5*c^6*d*e^6 - 256*a^
5*b*c^5*e^7 + 4*a^2*b^7*c^2*e^7 - 48*a^3*b^5*c^3*e^7 + 192*a^4*b^3*c^4*e^7 + 512*a^3*c^8*d^5*e^2 + 1024*a^4*c^
7*d^3*e^4 - 8*b^6*c^5*d^5*e^2 + 20*b^7*c^4*d^4*e^3 - 16*b^8*c^3*d^3*e^4 + 4*b^9*c^2*d^2*e^5 - 384*a^2*b^2*c^7*
d^5*e^2 + 960*a^2*b^3*c^6*d^4*e^3 - 576*a^2*b^4*c^5*d^3*e^4 - 96*a^2*b^5*c^4*d^2*e^5 + 256*a^3*b^2*c^6*d^3*e^4
+ 896*a^3*b^3*c^5*d^2*e^5 - 8*a*b^8*c^2*d*e^6 + 96*a*b^4*c^6*d^5*e^2 - 240*a*b^5*c^5*d^4*e^3 + 176*a*b^6*c^4*
d^3*e^4 - 24*a*b^7*c^3*d^2*e^5 + 88*a^2*b^6*c^3*d*e^6 - 1280*a^3*b*c^7*d^4*e^3 - 288*a^3*b^4*c^4*d*e^6 - 1536*
a^4*b*c^6*d^2*e^5 + 128*a^4*b^2*c^5*d*e^6))/(a^2*b^4*e^4 + 16*a^2*c^4*d^4 + 16*a^4*c^2*e^4 + b^4*c^2*d^4 + b^6
*d^2*e^2 - 8*a*b^2*c^3*d^4 - 8*a^3*b^2*c*e^4 + 32*a^3*c^3*d^2*e^2 - 2*a*b^5*d*e^3 - 2*b^5*c*d^3*e + 16*a*b^3*c
^2*d^3*e - 6*a*b^4*c*d^2*e^2 - 32*a^2*b*c^3*d^3*e + 16*a^2*b^3*c*d*e^3 - 32*a^3*b*c^2*d*e^3))*(-(32*b^6*c^5*d^
5 - 2048*a^3*c^8*d^5 - b^11*e^5 + b^2*e^5*(-(4*a*c - b^2)^9)^(1/2) - 384*a*b^4*c^6*d^5 + 3840*a^5*b*c^5*e^5 -
7680*a^5*c^6*d*e^4 - 80*b^7*c^4*d^4*e + 1536*a^2*b^2*c^7*d^5 - 288*a^2*b^7*c^2*e^5 + 1504*a^3*b^5*c^3*e^5 - 38
40*a^4*b^3*c^4*e^5 - 7680*a^4*c^7*d^3*e^2 + 50*b^8*c^3*d^3*e^2 + 5*b^9*c^2*d^2*e^3 - 5*c^2*d^2*e^3*(-(4*a*c -
b^2)^9)^(1/2) + 27*a*b^9*c*e^5 - 9*a*c*e^5*(-(4*a*c - b^2)^9)^(1/2) - 5*b^10*c*d*e^4 + 5*b*c*d*e^4*(-(4*a*c -
b^2)^9)^(1/2) + 960*a^2*b^4*c^5*d^3*e^2 + 2400*a^2*b^5*c^4*d^2*e^3 + 2560*a^3*b^2*c^6*d^3*e^2 - 8960*a^3*b^3*c
^5*d^2*e^3 + 960*a*b^5*c^5*d^4*e + 90*a*b^8*c^2*d*e^4 + 5120*a^3*b*c^7*d^4*e - 480*a*b^6*c^4*d^3*e^2 - 240*a*b
^7*c^3*d^2*e^3 - 3840*a^2*b^3*c^6*d^4*e - 480*a^2*b^6*c^3*d*e^4 + 320*a^3*b^4*c^4*d*e^4 + 11520*a^4*b*c^6*d^2*
e^3 + 3840*a^4*b^2*c^5*d*e^4)/(8*(b^15*d^3*e^3 ...
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