\(\int \frac {(d+e x)^{3/2}}{a+b x+c x^2} \, dx\) [2290]

Optimal result

Integrand size = 22, antiderivative size = 322 \[ \int \frac {(d+e x)^{3/2}}{a+b x+c x^2} \, dx=\frac {2 e \sqrt {d+e x}}{c}-\frac {\sqrt {2} \left (2 c^2 d^2+b \left (b-\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (b d-\sqrt {b^2-4 a c} d+a e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{3/2} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {2} \left (2 c^2 d^2+b \left (b+\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (b d+\sqrt {b^2-4 a c} d+a e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{c^{3/2} \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {717, 840, 1180, 214} \[ \int \frac {(d+e x)^{3/2}}{a+b x+c x^2} \, dx=-\frac {\sqrt {2} \left (-2 c e \left (-d \sqrt {b^2-4 a c}+a e+b d\right )+b e^2 \left (b-\sqrt {b^2-4 a c}\right )+2 c^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{c^{3/2} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}+\frac {\sqrt {2} \left (-2 c e \left (d \sqrt {b^2-4 a c}+a e+b d\right )+b e^2 \left (\sqrt {b^2-4 a c}+b\right )+2 c^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{c^{3/2} \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}+\frac {2 e \sqrt {d+e x}}{c} \]

[In]

Int[(d + e*x)^(3/2)/(a + b*x + c*x^2),x]

[Out]

(2*e*Sqrt[d + e*x])/c - (Sqrt[2]*(2*c^2*d^2 + b*(b - Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(b*d - Sqrt[b^2 - 4*a*c]*d
+ a*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]*(2*c^2*d^2 + b*(b + Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e
*(b*d + Sqrt[b^2 - 4*a*c]*d + a*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 717

Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[e*((d + e*x)^(m - 1)/(c*(
m - 1))), x] + Dist[1/c, Int[(d + e*x)^(m - 2)*(Simp[c*d^2 - a*e^2 + e*(2*c*d - b*e)*x, x]/(a + b*x + c*x^2)),
x], x] /; FreeQ[{a, b, c, d, e}, 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] && GtQ[m, 1]

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.14 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.13 \[ \int \frac {(d+e x)^{3/2}}{a+b x+c x^2} \, dx=\frac {2 \sqrt {c} e \sqrt {d+e x}+\frac {\left (-2 i c^2 d^2-b \left (i b+\sqrt {-b^2+4 a c}\right ) e^2+2 c e \left (i b d+\sqrt {-b^2+4 a c} d+i a e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-\frac {b^2}{2}+2 a c} \sqrt {-2 c d+\left (b-i \sqrt {-b^2+4 a c}\right ) e}}+\frac {\left (2 i c^2 d^2-b \left (-i b+\sqrt {-b^2+4 a c}\right ) e^2+2 c e \left (-i b d+\sqrt {-b^2+4 a c} d-i a e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-\frac {b^2}{2}+2 a c} \sqrt {-2 c d+\left (b+i \sqrt {-b^2+4 a c}\right ) e}}}{c^{3/2}} \]

[In]

Integrate[(d + e*x)^(3/2)/(a + b*x + c*x^2),x]

[Out]

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

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.07

method	result	size
pseudoelliptic	\(\frac {e \left (2 \sqrt {e x +d}+\frac {\left (2 a c \,e^{2}-b^{2} e^{2}+2 b c d e -2 c^{2} d^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (-2 a c \,e^{2}+b^{2} e^{2}-2 b c d e +2 c^{2} d^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{c}\)	\(344\)
derivativedivides	\(2 e \left (\frac {\sqrt {e x +d}}{c}+\frac {\left (2 a c \,e^{2}-b^{2} e^{2}+2 b c d e -2 c^{2} d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (-2 a c \,e^{2}+b^{2} e^{2}-2 b c d e +2 c^{2} d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )\)	\(353\)
default	\(2 e \left (\frac {\sqrt {e x +d}}{c}+\frac {\left (2 a c \,e^{2}-b^{2} e^{2}+2 b c d e -2 c^{2} d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (-2 a c \,e^{2}+b^{2} e^{2}-2 b c d e +2 c^{2} d^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{2 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )\)	\(353\)
risch	\(\frac {2 e \sqrt {e x +d}}{c}-8 e \left (-\frac {\left (2 a c \,e^{2}-b^{2} e^{2}+2 b c d e -2 c^{2} d^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (-2 a c \,e^{2}+b^{2} e^{2}-2 b c d e +2 c^{2} d^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e -2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )\)	\(354\)

[In]

int((e*x+d)^(3/2)/(c*x^2+b*x+a),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2770 vs. \(2 (272) = 544\).

Time = 0.48 (sec) , antiderivative size = 2770, normalized size of antiderivative = 8.60 \[ \int \frac {(d+e x)^{3/2}}{a+b x+c x^2} \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)^(3/2)/(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

-1/2*(sqrt(2)*c*sqrt((2*c^3*d^3 - 3*b*c^2*d^2*e + 3*(b^2*c - 2*a*c^2)*d*e^2 - (b^3 - 3*a*b*c)*e^3 + (b^2*c^3 -
4*a*c^4)*sqrt((9*c^4*d^4*e^2 - 18*b*c^3*d^3*e^3 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^4 - 6*(b^3*c - a*b*c^2)*d*e^5
+ (b^4 - 2*a*b^2*c + a^2*c^2)*e^6)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4))*log(sqrt(2)*(3*(b^2*c^2 - 4*a*c
^3)*d^2*e^2 - 3*(b^3*c - 4*a*b*c^2)*d*e^3 + (b^4 - 5*a*b^2*c + 4*a^2*c^2)*e^4 - (2*(b^2*c^4 - 4*a*c^5)*d - (b^
3*c^3 - 4*a*b*c^4)*e)*sqrt((9*c^4*d^4*e^2 - 18*b*c^3*d^3*e^3 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^4 - 6*(b^3*c - a*
b*c^2)*d*e^5 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^6)/(b^2*c^6 - 4*a*c^7)))*sqrt((2*c^3*d^3 - 3*b*c^2*d^2*e + 3*(b^2
*c - 2*a*c^2)*d*e^2 - (b^3 - 3*a*b*c)*e^3 + (b^2*c^3 - 4*a*c^4)*sqrt((9*c^4*d^4*e^2 - 18*b*c^3*d^3*e^3 + 3*(5*
b^2*c^2 - 2*a*c^3)*d^2*e^4 - 6*(b^3*c - a*b*c^2)*d*e^5 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^6)/(b^2*c^6 - 4*a*c^7))
)/(b^2*c^3 - 4*a*c^4)) - 4*(3*c^3*d^4*e - 6*b*c^2*d^3*e^2 + 2*(2*b^2*c + a*c^2)*d^2*e^3 - (b^3 + 2*a*b*c)*d*e^
4 + (a*b^2 - a^2*c)*e^5)*sqrt(e*x + d)) - sqrt(2)*c*sqrt((2*c^3*d^3 - 3*b*c^2*d^2*e + 3*(b^2*c - 2*a*c^2)*d*e^
2 - (b^3 - 3*a*b*c)*e^3 + (b^2*c^3 - 4*a*c^4)*sqrt((9*c^4*d^4*e^2 - 18*b*c^3*d^3*e^3 + 3*(5*b^2*c^2 - 2*a*c^3)
*d^2*e^4 - 6*(b^3*c - a*b*c^2)*d*e^5 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^6)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c
^4))*log(-sqrt(2)*(3*(b^2*c^2 - 4*a*c^3)*d^2*e^2 - 3*(b^3*c - 4*a*b*c^2)*d*e^3 + (b^4 - 5*a*b^2*c + 4*a^2*c^2)
*e^4 - (2*(b^2*c^4 - 4*a*c^5)*d - (b^3*c^3 - 4*a*b*c^4)*e)*sqrt((9*c^4*d^4*e^2 - 18*b*c^3*d^3*e^3 + 3*(5*b^2*c
^2 - 2*a*c^3)*d^2*e^4 - 6*(b^3*c - a*b*c^2)*d*e^5 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^6)/(b^2*c^6 - 4*a*c^7)))*sqr
t((2*c^3*d^3 - 3*b*c^2*d^2*e + 3*(b^2*c - 2*a*c^2)*d*e^2 - (b^3 - 3*a*b*c)*e^3 + (b^2*c^3 - 4*a*c^4)*sqrt((9*c
^4*d^4*e^2 - 18*b*c^3*d^3*e^3 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^4 - 6*(b^3*c - a*b*c^2)*d*e^5 + (b^4 - 2*a*b^2*c
+ a^2*c^2)*e^6)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4)) - 4*(3*c^3*d^4*e - 6*b*c^2*d^3*e^2 + 2*(2*b^2*c +
a*c^2)*d^2*e^3 - (b^3 + 2*a*b*c)*d*e^4 + (a*b^2 - a^2*c)*e^5)*sqrt(e*x + d)) + sqrt(2)*c*sqrt((2*c^3*d^3 - 3*b
*c^2*d^2*e + 3*(b^2*c - 2*a*c^2)*d*e^2 - (b^3 - 3*a*b*c)*e^3 - (b^2*c^3 - 4*a*c^4)*sqrt((9*c^4*d^4*e^2 - 18*b*
c^3*d^3*e^3 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^4 - 6*(b^3*c - a*b*c^2)*d*e^5 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^6)/(
b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4))*log(sqrt(2)*(3*(b^2*c^2 - 4*a*c^3)*d^2*e^2 - 3*(b^3*c - 4*a*b*c^2)*d
*e^3 + (b^4 - 5*a*b^2*c + 4*a^2*c^2)*e^4 + (2*(b^2*c^4 - 4*a*c^5)*d - (b^3*c^3 - 4*a*b*c^4)*e)*sqrt((9*c^4*d^4
*e^2 - 18*b*c^3*d^3*e^3 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^4 - 6*(b^3*c - a*b*c^2)*d*e^5 + (b^4 - 2*a*b^2*c + a^2
*c^2)*e^6)/(b^2*c^6 - 4*a*c^7)))*sqrt((2*c^3*d^3 - 3*b*c^2*d^2*e + 3*(b^2*c - 2*a*c^2)*d*e^2 - (b^3 - 3*a*b*c)
*e^3 - (b^2*c^3 - 4*a*c^4)*sqrt((9*c^4*d^4*e^2 - 18*b*c^3*d^3*e^3 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^4 - 6*(b^3*c
- a*b*c^2)*d*e^5 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^6)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4)) - 4*(3*c^3*d^4
*e - 6*b*c^2*d^3*e^2 + 2*(2*b^2*c + a*c^2)*d^2*e^3 - (b^3 + 2*a*b*c)*d*e^4 + (a*b^2 - a^2*c)*e^5)*sqrt(e*x + d
)) - sqrt(2)*c*sqrt((2*c^3*d^3 - 3*b*c^2*d^2*e + 3*(b^2*c - 2*a*c^2)*d*e^2 - (b^3 - 3*a*b*c)*e^3 - (b^2*c^3 -
4*a*c^4)*sqrt((9*c^4*d^4*e^2 - 18*b*c^3*d^3*e^3 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^4 - 6*(b^3*c - a*b*c^2)*d*e^5
+ (b^4 - 2*a*b^2*c + a^2*c^2)*e^6)/(b^2*c^6 - 4*a*c^7)))/(b^2*c^3 - 4*a*c^4))*log(-sqrt(2)*(3*(b^2*c^2 - 4*a*c
^3)*d^2*e^2 - 3*(b^3*c - 4*a*b*c^2)*d*e^3 + (b^4 - 5*a*b^2*c + 4*a^2*c^2)*e^4 + (2*(b^2*c^4 - 4*a*c^5)*d - (b^
3*c^3 - 4*a*b*c^4)*e)*sqrt((9*c^4*d^4*e^2 - 18*b*c^3*d^3*e^3 + 3*(5*b^2*c^2 - 2*a*c^3)*d^2*e^4 - 6*(b^3*c - a*
b*c^2)*d*e^5 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^6)/(b^2*c^6 - 4*a*c^7)))*sqrt((2*c^3*d^3 - 3*b*c^2*d^2*e + 3*(b^2
*c - 2*a*c^2)*d*e^2 - (b^3 - 3*a*b*c)*e^3 - (b^2*c^3 - 4*a*c^4)*sqrt((9*c^4*d^4*e^2 - 18*b*c^3*d^3*e^3 + 3*(5*
b^2*c^2 - 2*a*c^3)*d^2*e^4 - 6*(b^3*c - a*b*c^2)*d*e^5 + (b^4 - 2*a*b^2*c + a^2*c^2)*e^6)/(b^2*c^6 - 4*a*c^7))
)/(b^2*c^3 - 4*a*c^4)) - 4*(3*c^3*d^4*e - 6*b*c^2*d^3*e^2 + 2*(2*b^2*c + a*c^2)*d^2*e^3 - (b^3 + 2*a*b*c)*d*e^
4 + (a*b^2 - a^2*c)*e^5)*sqrt(e*x + d)) - 4*sqrt(e*x + d)*e)/c

Sympy [F]

\[ \int \frac {(d+e x)^{3/2}}{a+b x+c x^2} \, dx=\int \frac {\left (d + e x\right )^{\frac {3}{2}}}{a + b x + c x^{2}}\, dx \]

[In]

integrate((e*x+d)**(3/2)/(c*x**2+b*x+a),x)

[Out]

Integral((d + e*x)**(3/2)/(a + b*x + c*x**2), x)

Maxima [F]

\[ \int \frac {(d+e x)^{3/2}}{a+b x+c x^2} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{c x^{2} + b x + a} \,d x } \]

[In]

integrate((e*x+d)^(3/2)/(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

integrate((e*x + d)^(3/2)/(c*x^2 + b*x + a), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 797 vs. \(2 (272) = 544\).

Time = 0.36 (sec) , antiderivative size = 797, normalized size of antiderivative = 2.48 \[ \int \frac {(d+e x)^{3/2}}{a+b x+c x^2} \, dx=\frac {2 \, \sqrt {e x + d} e}{c} + \frac {{\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left (2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d e - {\left (b^{3} - 4 \, a b c\right )} e^{2}\right )} c^{2} e^{2} - 2 \, {\left (\sqrt {b^{2} - 4 \, a c} c^{3} d^{2} e - \sqrt {b^{2} - 4 \, a c} b c^{2} d e^{2} + \sqrt {b^{2} - 4 \, a c} a c^{2} e^{3}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | c \right |} {\left | e \right |} - {\left (4 \, c^{5} d^{3} e - 6 \, b c^{4} d^{2} e^{2} + 4 \, {\left (b^{2} c^{3} - a c^{4}\right )} d e^{3} - {\left (b^{3} c^{2} - 2 \, a b c^{3}\right )} e^{4}\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 {e x + 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} {\left | e \right |}} - \frac {{\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left (2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d e - {\left (b^{3} - 4 \, a b c\right )} e^{2}\right )} c^{2} e^{2} + 2 \, {\left (\sqrt {b^{2} - 4 \, a c} c^{3} d^{2} e - \sqrt {b^{2} - 4 \, a c} b c^{2} d e^{2} + \sqrt {b^{2} - 4 \, a c} a c^{2} e^{3}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | c \right |} {\left | e \right |} - {\left (4 \, c^{5} d^{3} e - 6 \, b c^{4} d^{2} e^{2} + 4 \, {\left (b^{2} c^{3} - a c^{4}\right )} d e^{3} - {\left (b^{3} c^{2} - 2 \, a b c^{3}\right )} e^{4}\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 {e x + 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} {\left | e \right |}} \]

[In]

integrate((e*x+d)^(3/2)/(c*x^2+b*x+a),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 11.63 (sec) , antiderivative size = 8334, normalized size of antiderivative = 25.88 \[ \int \frac {(d+e x)^{3/2}}{a+b x+c x^2} \, dx=\text {Too large to display} \]

[In]

int((d + e*x)^(3/2)/(a + b*x + c*x^2),x)

[Out]

(2*e*(d + e*x)^(1/2))/c - atan(((((8*(4*a^2*c^3*e^5 - a*b^2*c^2*e^5 + 4*a*c^4*d^2*e^3 + b^3*c^2*d*e^4 - b^2*c^
3*d^2*e^3 - 4*a*b*c^3*d*e^4))/c - (8*(d + e*x)^(1/2)*(-(b^5*e^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 - b^2*e^3*(-(4*a
*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e - 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1
/2) - 7*a*b^3*c*e^3 + a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e + 3*b*c*d*e^2*(-(4*a
*c - b^2)^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(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^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 - b^2*e^3*(-(4*a*c - b
^2)^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e - 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) -
7*a*b^3*c*e^3 + a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e + 3*b*c*d*e^2*(-(4*a*c - b
^2)^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(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
^6 + 2*a^2*c^2*e^6 + 2*c^4*d^4*e^2 - 12*a*c^3*d^2*e^4 - 4*b*c^3*d^3*e^3 + 6*b^2*c^2*d^2*e^4 - 4*a*b^2*c*e^6 -
4*b^3*c*d*e^5 + 12*a*b*c^2*d*e^5))/c)*(-(b^5*e^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 - b^2*e^3*(-(4*a*c - b^2)^3)^(1
/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e - 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c
*e^3 + a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e + 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1
/2) + 18*a*b^2*c^2*d*e^2)/(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^5 - a*b^2*c^2
*e^5 + 4*a*c^4*d^2*e^3 + b^3*c^2*d*e^4 - b^2*c^3*d^2*e^3 - 4*a*b*c^3*d*e^4))/c + (8*(d + e*x)^(1/2)*(-(b^5*e^3
+ 8*a*c^4*d^3 - 2*b^2*c^3*d^3 - b^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^
3*c^2*d^2*e - 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^3 + a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*
c*d*e^2 - 12*a*b*c^3*d^2*e + 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(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^3 + 8*a
*c^4*d^3 - 2*b^2*c^3*d^3 - b^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*
d^2*e - 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^3 + a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^
2 - 12*a*b*c^3*d^2*e + 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(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^6 + 2*a^2*c^2*e^6 + 2*c^4*d^4*e^2 - 12*a*c^3*d^2*e^4 - 4*b*c^3
*d^3*e^3 + 6*b^2*c^2*d^2*e^4 - 4*a*b^2*c*e^6 - 4*b^3*c*d*e^5 + 12*a*b*c^2*d*e^5))/c)*(-(b^5*e^3 + 8*a*c^4*d^3
- 2*b^2*c^3*d^3 - b^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e - 3
*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^3 + a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*
b*c^3*d^2*e + 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(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^5 - a*b^2*c^2*e^5 + 4*a*c^4*d^2*e^3 + b^3*c^2*d*e^4 - b^2*c^3*d^2*e^3 - 4*a*
b*c^3*d*e^4))/c - (8*(d + e*x)^(1/2)*(-(b^5*e^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 - b^2*e^3*(-(4*a*c - b^2)^3)^(1/
2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e - 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*
e^3 + a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e + 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1/
2) + 18*a*b^2*c^2*d*e^2)/(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^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 - b^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 1
2*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e - 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^3 +
a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e + 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 1
8*a*b^2*c^2*d*e^2)/(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^6 + 2*a^2*c^2*e
^6 + 2*c^4*d^4*e^2 - 12*a*c^3*d^2*e^4 - 4*b*c^3*d^3*e^3 + 6*b^2*c^2*d^2*e^4 - 4*a*b^2*c*e^6 - 4*b^3*c*d*e^5 +
12*a*b*c^2*d*e^5))/c)*(-(b^5*e^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 - b^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c
^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e - 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^3 + a*c*e^3*(
-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e + 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^2*c
^2*d*e^2)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2) + (((8*(4*a^2*c^3*e^5 - a*b^2*c^2*e^5 + 4*a*c^4*d^2*
e^3 + b^3*c^2*d*e^4 - b^2*c^3*d^2*e^3 - 4*a*b*c^3*d*e^4))/c + (8*(d + e*x)^(1/2)*(-(b^5*e^3 + 8*a*c^4*d^3 - 2*
b^2*c^3*d^3 - b^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e - 3*c^2
*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^3 + a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*b*c^
3*d^2*e + 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(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^3 + 8*a*c^4*d^3 - 2*b^2*c^
3*d^3 - b^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e - 3*c^2*d^2*e
*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^3 + a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*
e + 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(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^6 + 2*a^2*c^2*e^6 + 2*c^4*d^4*e^2 - 12*a*c^3*d^2*e^4 - 4*b*c^3*d^3*e^3 + 6*b^2*c^
2*d^2*e^4 - 4*a*b^2*c*e^6 - 4*b^3*c*d*e^5 + 12*a*b*c^2*d*e^5))/c)*(-(b^5*e^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 - b
^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e - 3*c^2*d^2*e*(-(4*a*c
- b^2)^3)^(1/2) - 7*a*b^3*c*e^3 + a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e + 3*b*c
*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2) - (16*(2
*c^3*d^5*e^3 - b^3*d^2*e^6 - a^2*b*e^8 + 4*a*c^2*d^3*e^5 - 5*b*c^2*d^4*e^4 + 4*b^2*c*d^3*e^5 + 2*a*b^2*d*e^7 +
2*a^2*c*d*e^7 - 6*a*b*c*d^2*e^6))/c))*(-(b^5*e^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 - b^2*e^3*(-(4*a*c - b^2)^3)^(
1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e - 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*
c*e^3 + a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e + 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(
1/2) + 18*a*b^2*c^2*d*e^2)/(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^5 - a*
b^2*c^2*e^5 + 4*a*c^4*d^2*e^3 + b^3*c^2*d*e^4 - b^2*c^3*d^2*e^3 - 4*a*b*c^3*d*e^4))/c - (8*(d + e*x)^(1/2)*(-(
b^5*e^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 + b^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2
+ 3*b^3*c^2*d^2*e + 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^3 - a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) -
3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e - 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(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^
3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 + b^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b
^3*c^2*d^2*e + 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^3 - a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4
*c*d*e^2 - 12*a*b*c^3*d^2*e - 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(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^6 + 2*a^2*c^2*e^6 + 2*c^4*d^4*e^2 - 12*a*c^3*d^2*e^4 -
4*b*c^3*d^3*e^3 + 6*b^2*c^2*d^2*e^4 - 4*a*b^2*c*e^6 - 4*b^3*c*d*e^5 + 12*a*b*c^2*d*e^5))/c)*(-(b^5*e^3 + 8*a*c
^4*d^3 - 2*b^2*c^3*d^3 + b^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^
2*e + 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^3 - a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2
- 12*a*b*c^3*d^2*e - 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(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^5 - a*b^2*c^2*e^5 + 4*a*c^4*d^2*e^3 + b^3*c^2*d*e^4 - b^2*c^3*d^2*e^3
- 4*a*b*c^3*d*e^4))/c + (8*(d + e*x)^(1/2)*(-(b^5*e^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 + b^2*e^3*(-(4*a*c - b^2)
^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e + 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a
*b^3*c*e^3 - a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e - 3*b*c*d*e^2*(-(4*a*c - b^2)
^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(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^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 + b^2*e^3*(-(4*a*c - b^2)^3)^(1
/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e + 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c
*e^3 - a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e - 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1
/2) + 18*a*b^2*c^2*d*e^2)/(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^6 + 2*a^
2*c^2*e^6 + 2*c^4*d^4*e^2 - 12*a*c^3*d^2*e^4 - 4*b*c^3*d^3*e^3 + 6*b^2*c^2*d^2*e^4 - 4*a*b^2*c*e^6 - 4*b^3*c*d
*e^5 + 12*a*b*c^2*d*e^5))/c)*(-(b^5*e^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 + b^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 12*
a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e + 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^3 - a*
c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e - 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*
a*b^2*c^2*d*e^2)/(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^5 - a*b^2*c^2*e^5 + 4*
a*c^4*d^2*e^3 + b^3*c^2*d*e^4 - b^2*c^3*d^2*e^3 - 4*a*b*c^3*d*e^4))/c - (8*(d + e*x)^(1/2)*(-(b^5*e^3 + 8*a*c^
4*d^3 - 2*b^2*c^3*d^3 + b^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2
*e + 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^3 - a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 -
12*a*b*c^3*d^2*e - 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(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^3 + 8*a*c^4*d^3
- 2*b^2*c^3*d^3 + b^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e + 3
*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^3 - a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*
b*c^3*d^2*e - 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(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^6 + 2*a^2*c^2*e^6 + 2*c^4*d^4*e^2 - 12*a*c^3*d^2*e^4 - 4*b*c^3*d^3*e^3
+ 6*b^2*c^2*d^2*e^4 - 4*a*b^2*c*e^6 - 4*b^3*c*d*e^5 + 12*a*b*c^2*d*e^5))/c)*(-(b^5*e^3 + 8*a*c^4*d^3 - 2*b^2*c
^3*d^3 + b^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e + 3*c^2*d^2*
e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^3 - a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2
*e - 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(2*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2
) + (((8*(4*a^2*c^3*e^5 - a*b^2*c^2*e^5 + 4*a*c^4*d^2*e^3 + b^3*c^2*d*e^4 - b^2*c^3*d^2*e^3 - 4*a*b*c^3*d*e^4)
)/c + (8*(d + e*x)^(1/2)*(-(b^5*e^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 + b^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*
b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e + 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^3 - a*c*e^
3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e - 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^
2*c^2*d*e^2)/(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^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 + b^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*
e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e + 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^3 - a*c*e^3*(-(4
*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e - 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^2*c^2*
d*e^2)/(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^6 + 2*a^2*c^2*e^6 + 2*c^4*d
^4*e^2 - 12*a*c^3*d^2*e^4 - 4*b*c^3*d^3*e^3 + 6*b^2*c^2*d^2*e^4 - 4*a*b^2*c*e^6 - 4*b^3*c*d*e^5 + 12*a*b*c^2*d
*e^5))/c)*(-(b^5*e^3 + 8*a*c^4*d^3 - 2*b^2*c^3*d^3 + b^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*
a^2*c^3*d*e^2 + 3*b^3*c^2*d^2*e + 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^3 - a*c*e^3*(-(4*a*c - b^
2)^3)^(1/2) - 3*b^4*c*d*e^2 - 12*a*b*c^3*d^2*e - 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(2
*(16*a^2*c^5 + b^4*c^3 - 8*a*b^2*c^4)))^(1/2) - (16*(2*c^3*d^5*e^3 - b^3*d^2*e^6 - a^2*b*e^8 + 4*a*c^2*d^3*e^5
- 5*b*c^2*d^4*e^4 + 4*b^2*c*d^3*e^5 + 2*a*b^2*d*e^7 + 2*a^2*c*d*e^7 - 6*a*b*c*d^2*e^6))/c))*(-(b^5*e^3 + 8*a*
c^4*d^3 - 2*b^2*c^3*d^3 + b^2*e^3*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2*e^3 - 24*a^2*c^3*d*e^2 + 3*b^3*c^2*d
^2*e + 3*c^2*d^2*e*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c*e^3 - a*c*e^3*(-(4*a*c - b^2)^3)^(1/2) - 3*b^4*c*d*e^2
- 12*a*b*c^3*d^2*e - 3*b*c*d*e^2*(-(4*a*c - b^2)^3)^(1/2) + 18*a*b^2*c^2*d*e^2)/(2*(16*a^2*c^5 + b^4*c^3 - 8*
a*b^2*c^4)))^(1/2)*2i