3.6.28 \(\int \frac {x \sqrt {d+e x}}{a+b x+c x^2} \, dx\) [528]

Optimal. Leaf size=287 \[ \frac {2 \sqrt {d+e x}}{c}+\frac {\sqrt {2} \left (b c d-b^2 e+2 a c e-\sqrt {b^2-4 a c} (c d-b e)\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{3/2} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\sqrt {2} \left (b c d-b^2 e+2 a c e+\sqrt {b^2-4 a c} (c d-b e)\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{3/2} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]

[Out]

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

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Rubi [A]
time = 2.20, antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {838, 840, 1180, 214} \begin {gather*} \frac {\sqrt {2} \left (-\sqrt {b^2-4 a c} (c d-b e)+2 a c e+b^2 (-e)+b c d\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 )}{c^{3/2} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\sqrt {2} \left (\sqrt {b^2-4 a c} (c d-b e)+2 a c e+b^2 (-e)+b c d\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 )}{c^{3/2} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+\frac {2 \sqrt {d+e x}}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 838

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[g*
((d + e*x)^m/(c*m)), x] + Dist[1/c, Int[(d + e*x)^(m - 1)*(Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x]/
(a + b*x + c*x^2)), 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] && FractionQ[m] && GtQ[m, 0]

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 {x \sqrt {d+e x}}{a+b x+c x^2} \, dx &=\frac {2 \sqrt {d+e x}}{c}+\frac {\int \frac {-a e+(c d-b e) x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx}{c}\\ &=\frac {2 \sqrt {d+e x}}{c}+\frac {2 \text {Subst}\left (\int \frac {-a e^2-d (c d-b e)+(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 )}{c}\\ &=\frac {2 \sqrt {d+e x}}{c}-\frac {\left (b c d-b^2 e+2 a c e-\sqrt {b^2-4 a c} (c d-b e)\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 )}{c \sqrt {b^2-4 a c}}+\frac {\left (b c d-b^2 e+2 a c e+\sqrt {b^2-4 a c} (c d-b e)\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 )}{c \sqrt {b^2-4 a c}}\\ &=\frac {2 \sqrt {d+e x}}{c}+\frac {\sqrt {2} \left (b c d-b^2 e+2 a c e-\sqrt {b^2-4 a c} (c d-b e)\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 )}{c^{3/2} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\sqrt {2} \left (b c d-b^2 e+2 a c e+\sqrt {b^2-4 a c} (c d-b e)\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 )}{c^{3/2} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.96, size = 341, normalized size = 1.19 \begin {gather*} \frac {2 \sqrt {c} \sqrt {d+e x}-\frac {\left (-i b c d-c \sqrt {-b^2+4 a c} d+i b^2 e-2 i a c e+b \sqrt {-b^2+4 a c} e\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 {-\frac {b^2}{2}+2 a c} \sqrt {-2 c d+\left (b-i \sqrt {-b^2+4 a c}\right ) e}}-\frac {\left (i b c d-c \sqrt {-b^2+4 a c} d-i b^2 e+2 i a c e+b \sqrt {-b^2+4 a c} e\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 {-\frac {b^2}{2}+2 a c} \sqrt {-2 c d+\left (b+i \sqrt {-b^2+4 a c}\right ) e}}}{c^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[c]*Sqrt[d + e*x] - (((-I)*b*c*d - c*Sqrt[-b^2 + 4*a*c]*d + I*b^2*e - (2*I)*a*c*e + b*Sqrt[-b^2 + 4*a*c
]*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[-1/2*b^2 + 2*a
*c]*Sqrt[-2*c*d + (b - I*Sqrt[-b^2 + 4*a*c])*e]) - ((I*b*c*d - c*Sqrt[-b^2 + 4*a*c]*d - I*b^2*e + (2*I)*a*c*e
+ b*Sqrt[-b^2 + 4*a*c]*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[-1/2*b^2 + 2*a*c]*Sqrt[-2*c*d + (b + I*Sqrt[-b^2 + 4*a*c])*e]))/c^(3/2)

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Maple [A]
time = 0.15, size = 331, normalized size = 1.15

method	result	size
derivativedivides	\(\frac {2 \sqrt {e x +d}}{c}+\frac {\left (2 a c \,e^{2}-b^{2} e^{2}+b c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (e b -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (e b -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (-2 a c \,e^{2}+b^{2} e^{2}-b c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctanh \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-e b +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-e b +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\)	\(331\)
default	\(\frac {2 \sqrt {e x +d}}{c}+\frac {\left (2 a c \,e^{2}-b^{2} e^{2}+b c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (e b -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (e b -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (-2 a c \,e^{2}+b^{2} e^{2}-b c d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctanh \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-e b +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-e b +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\)	\(331\)
risch	\(\frac {2 \sqrt {e x +d}}{c}+\frac {2 \sqrt {2}\, \arctanh \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-e b +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right ) a \,e^{2}}{\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-e b +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\sqrt {2}\, \arctanh \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-e b +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right ) b^{2} e^{2}}{c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-e b +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\sqrt {2}\, \arctanh \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-e b +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right ) b d e}{\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-e b +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\sqrt {2}\, \arctanh \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-e b +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right ) b e}{c \sqrt {\left (-e b +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\sqrt {2}\, \arctanh \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-e b +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right ) d}{\sqrt {\left (-e b +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {2 \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (e b -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right ) a \,e^{2}}{\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (e b -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (e b -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right ) b^{2} e^{2}}{c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (e b -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (e b -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right ) b d e}{\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (e b -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (e b -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right ) b e}{c \sqrt {\left (e b -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (e b -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right ) d}{\sqrt {\left (e b -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\)	\(926\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(e*x+d)^(1/2)/(c*x^2+b*x+a),x,method=_RETURNVERBOSE)

[Out]

2*(e*x+d)^(1/2)/c+(2*a*c*e^2-b^2*e^2+b*c*d*e-(-e^2*(4*a*c-b^2))^(1/2)*b*e+(-e^2*(4*a*c-b^2))^(1/2)*c*d)/c/(-e^
2*(4*a*c-b^2))^(1/2)*2^(1/2)/((e*b-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)/((e
*b-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))-(-2*a*c*e^2+b^2*e^2-b*c*d*e-(-e^2*(4*a*c-b^2))^(1/2)*b*e+(-e^2*(4
*a*c-b^2))^(1/2)*c*d)/c/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((-e*b+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arcta
nh(c*(e*x+d)^(1/2)*2^(1/2)/((-e*b+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(x*(e*x+d)^(1/2)/(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

integrate(sqrt(x*e + d)*x/(c*x^2 + b*x + a), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1742 vs. \(2 (254) = 508\).
time = 2.38, size = 1742, normalized size = 6.07 \begin {gather*} \frac {\sqrt {2} c \sqrt {\frac {{\left (b^{2} c - 2 \, a c^{2}\right )} d - {\left (b^{3} - 3 \, a b c\right )} e + {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt {\frac {b^{2} c^{2} d^{2} - 2 \, {\left (b^{3} c - a b c^{2}\right )} d e + {\left (b^{4} - 2 \, a b^{2} c + a^{2} c^{2}\right )} e^{2}}{b^{2} c^{6} - 4 \, a c^{7}}}}{b^{2} c^{3} - 4 \, a c^{4}}} \log \left (\sqrt {2} {\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} d - {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e - {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} \sqrt {\frac {b^{2} c^{2} d^{2} - 2 \, {\left (b^{3} c - a b c^{2}\right )} d e + {\left (b^{4} - 2 \, a b^{2} c + a^{2} c^{2}\right )} e^{2}}{b^{2} c^{6} - 4 \, a c^{7}}}\right )} \sqrt {\frac {{\left (b^{2} c - 2 \, a c^{2}\right )} d - {\left (b^{3} - 3 \, a b c\right )} e + {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt {\frac {b^{2} c^{2} d^{2} - 2 \, {\left (b^{3} c - a b c^{2}\right )} d e + {\left (b^{4} - 2 \, a b^{2} c + a^{2} c^{2}\right )} e^{2}}{b^{2} c^{6} - 4 \, a c^{7}}}}{b^{2} c^{3} - 4 \, a c^{4}}} + 4 \, {\left (a b c d - {\left (a b^{2} - a^{2} c\right )} e\right )} \sqrt {x e + d}\right ) - \sqrt {2} c \sqrt {\frac {{\left (b^{2} c - 2 \, a c^{2}\right )} d - {\left (b^{3} - 3 \, a b c\right )} e + {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt {\frac {b^{2} c^{2} d^{2} - 2 \, {\left (b^{3} c - a b c^{2}\right )} d e + {\left (b^{4} - 2 \, a b^{2} c + a^{2} c^{2}\right )} e^{2}}{b^{2} c^{6} - 4 \, a c^{7}}}}{b^{2} c^{3} - 4 \, a c^{4}}} \log \left (-\sqrt {2} {\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} d - {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e - {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} \sqrt {\frac {b^{2} c^{2} d^{2} - 2 \, {\left (b^{3} c - a b c^{2}\right )} d e + {\left (b^{4} - 2 \, a b^{2} c + a^{2} c^{2}\right )} e^{2}}{b^{2} c^{6} - 4 \, a c^{7}}}\right )} \sqrt {\frac {{\left (b^{2} c - 2 \, a c^{2}\right )} d - {\left (b^{3} - 3 \, a b c\right )} e + {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt {\frac {b^{2} c^{2} d^{2} - 2 \, {\left (b^{3} c - a b c^{2}\right )} d e + {\left (b^{4} - 2 \, a b^{2} c + a^{2} c^{2}\right )} e^{2}}{b^{2} c^{6} - 4 \, a c^{7}}}}{b^{2} c^{3} - 4 \, a c^{4}}} + 4 \, {\left (a b c d - {\left (a b^{2} - a^{2} c\right )} e\right )} \sqrt {x e + d}\right ) + \sqrt {2} c \sqrt {\frac {{\left (b^{2} c - 2 \, a c^{2}\right )} d - {\left (b^{3} - 3 \, a b c\right )} e - {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt {\frac {b^{2} c^{2} d^{2} - 2 \, {\left (b^{3} c - a b c^{2}\right )} d e + {\left (b^{4} - 2 \, a b^{2} c + a^{2} c^{2}\right )} e^{2}}{b^{2} c^{6} - 4 \, a c^{7}}}}{b^{2} c^{3} - 4 \, a c^{4}}} \log \left (\sqrt {2} {\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} d - {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e + {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} \sqrt {\frac {b^{2} c^{2} d^{2} - 2 \, {\left (b^{3} c - a b c^{2}\right )} d e + {\left (b^{4} - 2 \, a b^{2} c + a^{2} c^{2}\right )} e^{2}}{b^{2} c^{6} - 4 \, a c^{7}}}\right )} \sqrt {\frac {{\left (b^{2} c - 2 \, a c^{2}\right )} d - {\left (b^{3} - 3 \, a b c\right )} e - {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt {\frac {b^{2} c^{2} d^{2} - 2 \, {\left (b^{3} c - a b c^{2}\right )} d e + {\left (b^{4} - 2 \, a b^{2} c + a^{2} c^{2}\right )} e^{2}}{b^{2} c^{6} - 4 \, a c^{7}}}}{b^{2} c^{3} - 4 \, a c^{4}}} + 4 \, {\left (a b c d - {\left (a b^{2} - a^{2} c\right )} e\right )} \sqrt {x e + d}\right ) - \sqrt {2} c \sqrt {\frac {{\left (b^{2} c - 2 \, a c^{2}\right )} d - {\left (b^{3} - 3 \, a b c\right )} e - {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt {\frac {b^{2} c^{2} d^{2} - 2 \, {\left (b^{3} c - a b c^{2}\right )} d e + {\left (b^{4} - 2 \, a b^{2} c + a^{2} c^{2}\right )} e^{2}}{b^{2} c^{6} - 4 \, a c^{7}}}}{b^{2} c^{3} - 4 \, a c^{4}}} \log \left (-\sqrt {2} {\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} d - {\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} e + {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} \sqrt {\frac {b^{2} c^{2} d^{2} - 2 \, {\left (b^{3} c - a b c^{2}\right )} d e + {\left (b^{4} - 2 \, a b^{2} c + a^{2} c^{2}\right )} e^{2}}{b^{2} c^{6} - 4 \, a c^{7}}}\right )} \sqrt {\frac {{\left (b^{2} c - 2 \, a c^{2}\right )} d - {\left (b^{3} - 3 \, a b c\right )} e - {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt {\frac {b^{2} c^{2} d^{2} - 2 \, {\left (b^{3} c - a b c^{2}\right )} d e + {\left (b^{4} - 2 \, a b^{2} c + a^{2} c^{2}\right )} e^{2}}{b^{2} c^{6} - 4 \, a c^{7}}}}{b^{2} c^{3} - 4 \, a c^{4}}} + 4 \, {\left (a b c d - {\left (a b^{2} - a^{2} c\right )} e\right )} \sqrt {x e + d}\right ) + 4 \, \sqrt {x e + d}}{2 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 753 vs. \(2 (254) = 508\).
time = 1.33, size = 753, normalized size = 2.62 \begin {gather*} \frac {2 \, \sqrt {x e + d}}{c} + \frac {{\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d e - {\left (b^{3} - 4 \, a b c\right )} e^{2}\right )} c^{2} - 2 \, {\left (\sqrt {b^{2} - 4 \, a c} c^{3} d^{2} - \sqrt {b^{2} - 4 \, a c} b c^{2} d e + \sqrt {b^{2} - 4 \, a c} a c^{2} e^{2}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | c \right |} + {\left (2 \, b c^{4} d^{2} - {\left (3 \, b^{2} c^{3} - 4 \, a c^{4}\right )} d e + {\left (b^{3} c^{2} - 2 \, a b c^{3}\right )} e^{2}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {2 \, c^{2} d - b c e + \sqrt {-4 \, {\left (c^{2} d^{2} - b c d e + a c e^{2}\right )} c^{2} + {\left (2 \, c^{2} d - b c e\right )}^{2}}}{c^{2}}}}\right )}{4 \, {\left (\sqrt {b^{2} - 4 \, a c} c^{4} d^{2} - \sqrt {b^{2} - 4 \, a c} b c^{3} d e + \sqrt {b^{2} - 4 \, a c} a c^{3} e^{2}\right )} c^{2}} - \frac {{\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d e - {\left (b^{3} - 4 \, a b c\right )} e^{2}\right )} c^{2} + 2 \, {\left (\sqrt {b^{2} - 4 \, a c} c^{3} d^{2} - \sqrt {b^{2} - 4 \, a c} b c^{2} d e + \sqrt {b^{2} - 4 \, a c} a c^{2} e^{2}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | c \right |} + {\left (2 \, b c^{4} d^{2} - {\left (3 \, b^{2} c^{3} - 4 \, a c^{4}\right )} d e + {\left (b^{3} c^{2} - 2 \, a b c^{3}\right )} e^{2}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x e + d}}{\sqrt {-\frac {2 \, c^{2} d - b c e - \sqrt {-4 \, {\left (c^{2} d^{2} - b c d e + a c e^{2}\right )} c^{2} + {\left (2 \, c^{2} d - b c e\right )}^{2}}}{c^{2}}}}\right )}{4 \, {\left (\sqrt {b^{2} - 4 \, a c} c^{4} d^{2} - \sqrt {b^{2} - 4 \, a c} b c^{3} d e + \sqrt {b^{2} - 4 \, a c} a c^{3} e^{2}\right )} c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x*e + d)/c + 1/4*(sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*((b^2*c - 4*a*c^2)*d*e - (b^3 - 4*a*
b*c)*e^2)*c^2 - 2*(sqrt(b^2 - 4*a*c)*c^3*d^2 - sqrt(b^2 - 4*a*c)*b*c^2*d*e + sqrt(b^2 - 4*a*c)*a*c^2*e^2)*sqrt
(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*abs(c) + (2*b*c^4*d^2 - (3*b^2*c^3 - 4*a*c^4)*d*e + (b^3*c^2 - 2*
a*b*c^3)*e^2)*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*c^2*
d - b*c*e + sqrt(-4*(c^2*d^2 - b*c*d*e + a*c*e^2)*c^2 + (2*c^2*d - b*c*e)^2))/c^2))/((sqrt(b^2 - 4*a*c)*c^4*d^
2 - sqrt(b^2 - 4*a*c)*b*c^3*d*e + sqrt(b^2 - 4*a*c)*a*c^3*e^2)*c^2) - 1/4*(sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 -
4*a*c)*c)*e)*((b^2*c - 4*a*c^2)*d*e - (b^3 - 4*a*b*c)*e^2)*c^2 + 2*(sqrt(b^2 - 4*a*c)*c^3*d^2 - sqrt(b^2 - 4*
a*c)*b*c^2*d*e + sqrt(b^2 - 4*a*c)*a*c^2*e^2)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*abs(c) + (2*b*c
^4*d^2 - (3*b^2*c^3 - 4*a*c^4)*d*e + (b^3*c^2 - 2*a*b*c^3)*e^2)*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*c^2*d - b*c*e - sqrt(-4*(c^2*d^2 - b*c*d*e + a*c*e^2)*c^2 + (2*c
^2*d - b*c*e)^2))/c^2))/((sqrt(b^2 - 4*a*c)*c^4*d^2 - sqrt(b^2 - 4*a*c)*b*c^3*d*e + sqrt(b^2 - 4*a*c)*a*c^3*e^
2)*c^2)

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Mupad [B]
time = 3.82, size = 2500, normalized size = 8.71 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(d + e*x)^(1/2))/c - atan(((((8*(4*a^2*c^3*e^4 - a*b^2*c^2*e^4 + 4*a*c^4*d^2*e^2 + b^3*c^2*d*e^3 - b^2*c^3*
d^2*e^2 - 4*a*b*c^3*d*e^3))/c - (8*(d + e*x)^(1/2)*((8*a^2*c^3*d - b^5*e - b^2*e*(-(4*a*c - b^2)^3)^(1/2) + b^
4*c*d + 7*a*b^3*c*e + a*c*e*(-(4*a*c - b^2)^3)^(1/2) + b*c*d*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d - 12*a^2
*b*c^2*e)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*(b^3*c^3*e^3 - 2*b^2*c^4*d*e^2 - 4*a*b*c^4*e^3 + 8*a
*c^5*d*e^2))/c)*((8*a^2*c^3*d - b^5*e - b^2*e*(-(4*a*c - b^2)^3)^(1/2) + b^4*c*d + 7*a*b^3*c*e + a*c*e*(-(4*a*
c - b^2)^3)^(1/2) + b*c*d*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d - 12*a^2*b*c^2*e)/(2*(16*a^2*c^5 + b^4*c^3
- 8*a*b^2*c^4)))^(1/2) - (8*(d + e*x)^(1/2)*(b^4*e^4 + 2*a^2*c^2*e^4 - 2*a*c^3*d^2*e^2 + b^2*c^2*d^2*e^2 - 4*a
*b^2*c*e^4 - 2*b^3*c*d*e^3 + 6*a*b*c^2*d*e^3))/c)*((8*a^2*c^3*d - b^5*e - b^2*e*(-(4*a*c - b^2)^3)^(1/2) + b^4
*c*d + 7*a*b^3*c*e + a*c*e*(-(4*a*c - b^2)^3)^(1/2) + b*c*d*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d - 12*a^2*
b*c^2*e)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*1i - (((8*(4*a^2*c^3*e^4 - a*b^2*c^2*e^4 + 4*a*c^4*d^
2*e^2 + b^3*c^2*d*e^3 - b^2*c^3*d^2*e^2 - 4*a*b*c^3*d*e^3))/c + (8*(d + e*x)^(1/2)*((8*a^2*c^3*d - b^5*e - b^2
*e*(-(4*a*c - b^2)^3)^(1/2) + b^4*c*d + 7*a*b^3*c*e + a*c*e*(-(4*a*c - b^2)^3)^(1/2) + b*c*d*(-(4*a*c - b^2)^3
)^(1/2) - 6*a*b^2*c^2*d - 12*a^2*b*c^2*e)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*(b^3*c^3*e^3 - 2*b^2
*c^4*d*e^2 - 4*a*b*c^4*e^3 + 8*a*c^5*d*e^2))/c)*((8*a^2*c^3*d - b^5*e - b^2*e*(-(4*a*c - b^2)^3)^(1/2) + b^4*c
*d + 7*a*b^3*c*e + a*c*e*(-(4*a*c - b^2)^3)^(1/2) + b*c*d*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d - 12*a^2*b*
c^2*e)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2) + (8*(d + e*x)^(1/2)*(b^4*e^4 + 2*a^2*c^2*e^4 - 2*a*c^3
*d^2*e^2 + b^2*c^2*d^2*e^2 - 4*a*b^2*c*e^4 - 2*b^3*c*d*e^3 + 6*a*b*c^2*d*e^3))/c)*((8*a^2*c^3*d - b^5*e - b^2*
e*(-(4*a*c - b^2)^3)^(1/2) + b^4*c*d + 7*a*b^3*c*e + a*c*e*(-(4*a*c - b^2)^3)^(1/2) + b*c*d*(-(4*a*c - b^2)^3)
^(1/2) - 6*a*b^2*c^2*d - 12*a^2*b*c^2*e)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*1i)/((((8*(4*a^2*c^3*
e^4 - a*b^2*c^2*e^4 + 4*a*c^4*d^2*e^2 + b^3*c^2*d*e^3 - b^2*c^3*d^2*e^2 - 4*a*b*c^3*d*e^3))/c - (8*(d + e*x)^(
1/2)*((8*a^2*c^3*d - b^5*e - b^2*e*(-(4*a*c - b^2)^3)^(1/2) + b^4*c*d + 7*a*b^3*c*e + a*c*e*(-(4*a*c - b^2)^3)
^(1/2) + b*c*d*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d - 12*a^2*b*c^2*e)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c
^4)))^(1/2)*(b^3*c^3*e^3 - 2*b^2*c^4*d*e^2 - 4*a*b*c^4*e^3 + 8*a*c^5*d*e^2))/c)*((8*a^2*c^3*d - b^5*e - b^2*e*
(-(4*a*c - b^2)^3)^(1/2) + b^4*c*d + 7*a*b^3*c*e + a*c*e*(-(4*a*c - b^2)^3)^(1/2) + b*c*d*(-(4*a*c - b^2)^3)^(
1/2) - 6*a*b^2*c^2*d - 12*a^2*b*c^2*e)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2) - (8*(d + e*x)^(1/2)*(b
^4*e^4 + 2*a^2*c^2*e^4 - 2*a*c^3*d^2*e^2 + b^2*c^2*d^2*e^2 - 4*a*b^2*c*e^4 - 2*b^3*c*d*e^3 + 6*a*b*c^2*d*e^3))
/c)*((8*a^2*c^3*d - b^5*e - b^2*e*(-(4*a*c - b^2)^3)^(1/2) + b^4*c*d + 7*a*b^3*c*e + a*c*e*(-(4*a*c - b^2)^3)^
(1/2) + b*c*d*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d - 12*a^2*b*c^2*e)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^
4)))^(1/2) - (16*(a*c^2*d^3*e^2 - a^2*b*e^5 + a*b^2*d*e^4 + a^2*c*d*e^4 - 2*a*b*c*d^2*e^3))/c + (((8*(4*a^2*c^
3*e^4 - a*b^2*c^2*e^4 + 4*a*c^4*d^2*e^2 + b^3*c^2*d*e^3 - b^2*c^3*d^2*e^2 - 4*a*b*c^3*d*e^3))/c + (8*(d + e*x)
^(1/2)*((8*a^2*c^3*d - b^5*e - b^2*e*(-(4*a*c - b^2)^3)^(1/2) + b^4*c*d + 7*a*b^3*c*e + a*c*e*(-(4*a*c - b^2)^
3)^(1/2) + b*c*d*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d - 12*a^2*b*c^2*e)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2
*c^4)))^(1/2)*(b^3*c^3*e^3 - 2*b^2*c^4*d*e^2 - 4*a*b*c^4*e^3 + 8*a*c^5*d*e^2))/c)*((8*a^2*c^3*d - b^5*e - b^2*
e*(-(4*a*c - b^2)^3)^(1/2) + b^4*c*d + 7*a*b^3*c*e + a*c*e*(-(4*a*c - b^2)^3)^(1/2) + b*c*d*(-(4*a*c - b^2)^3)
^(1/2) - 6*a*b^2*c^2*d - 12*a^2*b*c^2*e)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2) + (8*(d + e*x)^(1/2)*
(b^4*e^4 + 2*a^2*c^2*e^4 - 2*a*c^3*d^2*e^2 + b^2*c^2*d^2*e^2 - 4*a*b^2*c*e^4 - 2*b^3*c*d*e^3 + 6*a*b*c^2*d*e^3
))/c)*((8*a^2*c^3*d - b^5*e - b^2*e*(-(4*a*c - b^2)^3)^(1/2) + b^4*c*d + 7*a*b^3*c*e + a*c*e*(-(4*a*c - b^2)^3
)^(1/2) + b*c*d*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d - 12*a^2*b*c^2*e)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*
c^4)))^(1/2)))*((8*a^2*c^3*d - b^5*e - b^2*e*(-(4*a*c - b^2)^3)^(1/2) + b^4*c*d + 7*a*b^3*c*e + a*c*e*(-(4*a*c
- b^2)^3)^(1/2) + b*c*d*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c^2*d - 12*a^2*b*c^2*e)/(2*(16*a^2*c^5 + b^4*c^3 -
8*a*b^2*c^4)))^(1/2)*2i - atan(((((8*(4*a^2*c^3*e^4 - a*b^2*c^2*e^4 + 4*a*c^4*d^2*e^2 + b^3*c^2*d*e^3 - b^2*c
^3*d^2*e^2 - 4*a*b*c^3*d*e^3))/c - (8*(d + e*x)^(1/2)*(-(b^5*e - 8*a^2*c^3*d - b^2*e*(-(4*a*c - b^2)^3)^(1/2)
- b^4*c*d - 7*a*b^3*c*e + a*c*e*(-(4*a*c - b^2)^3)^(1/2) + b*c*d*(-(4*a*c - b^2)^3)^(1/2) + 6*a*b^2*c^2*d + 12
*a^2*b*c^2*e)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2)*(b^3*c^3*e^3 - 2*b^2*c^4*d*e^2 - 4*a*b*c^4*e^3 +
8*a*c^5*d*e^2))/c)*(-(b^5*e - 8*a^2*c^3*d - b^2*e*(-(4*a*c - b^2)^3)^(1/2) - b^4*c*d - 7*a*b^3*c*e + a*c*e*(-
(4*a*c - b^2)^3)^(1/2) + b*c*d*(-(4*a*c - b^2)^...

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