3.17.16 \(\int \frac {b+2 c x}{(d+e x)^{3/2} (a+b x+c x^2)} \, dx\) [1616]

Optimal. Leaf size=354 \[ \frac {2 (2 c d-b e)}{\left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}+\frac {\sqrt {2} \sqrt {c} \left (b \left (b+\sqrt {b^2-4 a c}\right ) e-2 c \left (\sqrt {b^2-4 a c} d+2 a e\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )}-\frac {\sqrt {2} \sqrt {c} \left (b \left (b-\sqrt {b^2-4 a c}\right ) e+2 c \left (\sqrt {b^2-4 a c} d-2 a e\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.49, antiderivative size = 354, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {842, 840, 1180, 214} \begin {gather*} \frac {\sqrt {2} \sqrt {c} \left (b e \left (\sqrt {b^2-4 a c}+b\right )-2 c \left (d \sqrt {b^2-4 a c}+2 a e\right )\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 {b^2-4 a c} \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 {2} \sqrt {c} \left (2 c \left (d \sqrt {b^2-4 a c}-2 a e\right )+b e \left (b-\sqrt {b^2-4 a c}\right )\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 {b^2-4 a c} \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 {2 (2 c d-b e)}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(2*c*d - b*e))/((c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x]) + (Sqrt[2]*Sqrt[c]*(b*(b + Sqrt[b^2 - 4*a*c])*e - 2*
c*(Sqrt[b^2 - 4*a*c]*d + 2*a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*
e]])/(Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*(c*d^2 - b*d*e + a*e^2)) - (Sqrt[2]*Sqrt[c]*(b
*(b - Sqrt[b^2 - 4*a*c])*e + 2*c*(Sqrt[b^2 - 4*a*c]*d - 2*a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2
*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[b^2 - 4*a*c]*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 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 842

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e
*f - d*g)*((d + e*x)^(m + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[(d +
e*x)^(m + 1)*(Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x]/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c,
d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && FractionQ[m] && LtQ[m, -1]

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 {b+2 c x}{(d+e x)^{3/2} \left (a+b x+c x^2\right )} \, dx &=\frac {2 (2 c d-b e)}{\left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}+\frac {\int \frac {b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\sqrt {d+e x} \left (a+b x+c x^2\right )} \, dx}{c d^2-b d e+a e^2}\\ &=\frac {2 (2 c d-b e)}{\left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}+\frac {2 \text {Subst}\left (\int \frac {-c d (2 c d-b e)+e \left (b c d-b^2 e+2 a c e\right )+c (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 d^2-b d e+a e^2}\\ &=\frac {2 (2 c d-b e)}{\left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}+\frac {\left (2 \left (\frac {1}{2} c (2 c d-b e)-\frac {-c (2 c d-b e) (-2 c d+b e)+2 c \left (-c d (2 c d-b e)+e \left (b c d-b^2 e+2 a c e\right )\right )}{2 \sqrt {b^2-4 a c} 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 d^2-b d e+a e^2}+\frac {\left (2 \left (\frac {1}{2} c (2 c d-b e)+\frac {-c (2 c d-b e) (-2 c d+b e)+2 c \left (-c d (2 c d-b e)+e \left (b c d-b^2 e+2 a c e\right )\right )}{2 \sqrt {b^2-4 a c} 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 d^2-b d e+a e^2}\\ &=\frac {2 (2 c d-b e)}{\left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}+\frac {\sqrt {2} \sqrt {c} \left (b \left (b+\sqrt {b^2-4 a c}\right ) e-2 c \left (\sqrt {b^2-4 a c} d+2 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 {b^2-4 a c} \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 {2} \sqrt {c} \left (b \left (b-\sqrt {b^2-4 a c}\right ) e+c \left (2 \sqrt {b^2-4 a c} d-4 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 {b^2-4 a c} \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*}

________________________________________________________________________________________

Mathematica [A]
time = 1.10, size = 350, normalized size = 0.99 \begin {gather*} \frac {4 c d-2 b e}{\left (c d^2+e (-b d+a e)\right ) \sqrt {d+e x}}+\frac {\sqrt {2} \sqrt {c} \left (b \left (b+\sqrt {b^2-4 a c}\right ) e-2 c \left (\sqrt {b^2-4 a c} d+2 a e\right )\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-\sqrt {b^2-4 a c} e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e} \left (-c d^2+e (b d-a e)\right )}+\frac {\sqrt {2} \sqrt {c} \left (b \left (-b+\sqrt {b^2-4 a c}\right ) e+c \left (-2 \sqrt {b^2-4 a c} d+4 a e\right )\right ) \tan ^{-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 {b^2-4 a c} \sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e} \left (-c d^2+e (b d-a e)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 1.09, size = 362, normalized size = 1.02

method	result	size
derivativedivides	\(-\frac {2 \left (b e -2 c d \right )}{\left (e^{2} a -b d e +c \,d^{2}\right ) \sqrt {e x +d}}+\frac {8 c \left (\frac {\left (-4 a c \,e^{2}+b^{2} e^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (4 a c \,e^{2}-b^{2} e^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctanh \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{e^{2} a -b d e +c \,d^{2}}\)	\(362\)
default	\(-\frac {2 \left (b e -2 c d \right )}{\left (e^{2} a -b d e +c \,d^{2}\right ) \sqrt {e x +d}}+\frac {8 c \left (\frac {\left (-4 a c \,e^{2}+b^{2} e^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (4 a c \,e^{2}-b^{2} e^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b e +2 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, c d \right ) \sqrt {2}\, \arctanh \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{e^{2} a -b d e +c \,d^{2}}\)	\(362\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(b*e-2*c*d)/(a*e^2-b*d*e+c*d^2)/(e*x+d)^(1/2)+8/(a*e^2-b*d*e+c*d^2)*c*(1/8*(-4*a*c*e^2+b^2*e^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))-1/8*(4*a*c*e
^2-b^2*e^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)))

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 8283 vs. \(2 (317) = 634\).
time = 2.42, size = 8283, normalized size = 23.40 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(sqrt(2)*(c*d^3 + a*x*e^3 - (b*d*x - a*d)*e^2 + (c*d^2*x - b*d^2)*e)*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 + (c^3*d^6 - 3*b*c^2*d^5*e + 3*(b^2*c + a*c^2)*d^4*e^2 - 3*a^2*b*
d*e^5 - (b^3 + 6*a*b*c)*d^3*e^3 + a^3*e^6 + 3*(a*b^2 + a^2*c)*d^2*e^4)*sqrt((9*(b^2*c^4 - 4*a*c^5)*d^4*e^2 - 1
8*(b^3*c^3 - 4*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 22*a*b^2*c^3 + 8*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 5*a*b^3*c^2 +
4*a^2*b*c^3)*d*e^5 + (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 4*a^3*c^3)*e^6)/(c^6*d^12 - 6*b*c^5*d^11*e + 3*(5*b^2*
c^4 + 2*a*c^5)*d^10*e^2 - 10*(2*b^3*c^3 + 3*a*b*c^4)*d^9*e^3 + 15*(b^4*c^2 + 4*a*b^2*c^3 + a^2*c^4)*d^8*e^4 -
6*(b^5*c + 10*a*b^3*c^2 + 10*a^2*b*c^3)*d^7*e^5 - 6*a^5*b*d*e^11 + (b^6 + 30*a*b^4*c + 90*a^2*b^2*c^2 + 20*a^3
*c^3)*d^6*e^6 + a^6*e^12 - 6*(a*b^5 + 10*a^2*b^3*c + 10*a^3*b*c^2)*d^5*e^7 + 15*(a^2*b^4 + 4*a^3*b^2*c + a^4*c
^2)*d^4*e^8 - 10*(2*a^3*b^3 + 3*a^4*b*c)*d^3*e^9 + 3*(5*a^4*b^2 + 2*a^5*c)*d^2*e^10)))/(c^3*d^6 - 3*b*c^2*d^5*
e + 3*(b^2*c + a*c^2)*d^4*e^2 - 3*a^2*b*d*e^5 - (b^3 + 6*a*b*c)*d^3*e^3 + a^3*e^6 + 3*(a*b^2 + a^2*c)*d^2*e^4)
)*log(sqrt(2)*(6*c^4*d^4 - 12*b*c^3*d^3*e + (11*b^2*c^2 - 8*a*c^3)*d^2*e^2 - (5*b^3*c - 8*a*b*c^2)*d*e^3 + (b^
4 - 3*a*b^2*c + 2*a^2*c^2)*e^4 - (2*c^4*d^7 - 7*b*c^3*d^6*e + 3*(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 - 5*(b^3*c + 3*a
*b*c^2)*d^4*e^3 - a^3*b*e^7 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^4 - 3*(a*b^3 + 3*a^2*b*c)*d^2*e^5 + (3*a^2*
b^2 + 2*a^3*c)*d*e^6)*sqrt((9*(b^2*c^4 - 4*a*c^5)*d^4*e^2 - 18*(b^3*c^3 - 4*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 -
22*a*b^2*c^3 + 8*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 5*a*b^3*c^2 + 4*a^2*b*c^3)*d*e^5 + (b^6 - 6*a*b^4*c + 9*a^2*b^2
*c^2 - 4*a^3*c^3)*e^6)/(c^6*d^12 - 6*b*c^5*d^11*e + 3*(5*b^2*c^4 + 2*a*c^5)*d^10*e^2 - 10*(2*b^3*c^3 + 3*a*b*c
^4)*d^9*e^3 + 15*(b^4*c^2 + 4*a*b^2*c^3 + a^2*c^4)*d^8*e^4 - 6*(b^5*c + 10*a*b^3*c^2 + 10*a^2*b*c^3)*d^7*e^5 -
6*a^5*b*d*e^11 + (b^6 + 30*a*b^4*c + 90*a^2*b^2*c^2 + 20*a^3*c^3)*d^6*e^6 + a^6*e^12 - 6*(a*b^5 + 10*a^2*b^3*
c + 10*a^3*b*c^2)*d^5*e^7 + 15*(a^2*b^4 + 4*a^3*b^2*c + a^4*c^2)*d^4*e^8 - 10*(2*a^3*b^3 + 3*a^4*b*c)*d^3*e^9
+ 3*(5*a^4*b^2 + 2*a^5*c)*d^2*e^10)))*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 + (c^3*d^6 - 3*b*c^2*d^5*e + 3*(b^2*c + a*c^2)*d^4*e^2 - 3*a^2*b*d*e^5 - (b^3 + 6*a*b*c)*d^3*e^3 + a
^3*e^6 + 3*(a*b^2 + a^2*c)*d^2*e^4)*sqrt((9*(b^2*c^4 - 4*a*c^5)*d^4*e^2 - 18*(b^3*c^3 - 4*a*b*c^4)*d^3*e^3 + 3
*(5*b^4*c^2 - 22*a*b^2*c^3 + 8*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 5*a*b^3*c^2 + 4*a^2*b*c^3)*d*e^5 + (b^6 - 6*a*b^4
*c + 9*a^2*b^2*c^2 - 4*a^3*c^3)*e^6)/(c^6*d^12 - 6*b*c^5*d^11*e + 3*(5*b^2*c^4 + 2*a*c^5)*d^10*e^2 - 10*(2*b^3
*c^3 + 3*a*b*c^4)*d^9*e^3 + 15*(b^4*c^2 + 4*a*b^2*c^3 + a^2*c^4)*d^8*e^4 - 6*(b^5*c + 10*a*b^3*c^2 + 10*a^2*b*
c^3)*d^7*e^5 - 6*a^5*b*d*e^11 + (b^6 + 30*a*b^4*c + 90*a^2*b^2*c^2 + 20*a^3*c^3)*d^6*e^6 + a^6*e^12 - 6*(a*b^5
+ 10*a^2*b^3*c + 10*a^3*b*c^2)*d^5*e^7 + 15*(a^2*b^4 + 4*a^3*b^2*c + a^4*c^2)*d^4*e^8 - 10*(2*a^3*b^3 + 3*a^4
*b*c)*d^3*e^9 + 3*(5*a^4*b^2 + 2*a^5*c)*d^2*e^10)))/(c^3*d^6 - 3*b*c^2*d^5*e + 3*(b^2*c + a*c^2)*d^4*e^2 - 3*a
^2*b*d*e^5 - (b^3 + 6*a*b*c)*d^3*e^3 + a^3*e^6 + 3*(a*b^2 + a^2*c)*d^2*e^4)) - 4*(3*c^4*d^2 - 3*b*c^3*d*e + (b
^2*c^2 - a*c^3)*e^2)*sqrt(x*e + d)) - sqrt(2)*(c*d^3 + a*x*e^3 - (b*d*x - a*d)*e^2 + (c*d^2*x - b*d^2)*e)*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 + (c^3*d^6 - 3*b*c^2*d^5*e + 3*(
b^2*c + a*c^2)*d^4*e^2 - 3*a^2*b*d*e^5 - (b^3 + 6*a*b*c)*d^3*e^3 + a^3*e^6 + 3*(a*b^2 + a^2*c)*d^2*e^4)*sqrt((
9*(b^2*c^4 - 4*a*c^5)*d^4*e^2 - 18*(b^3*c^3 - 4*a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 22*a*b^2*c^3 + 8*a^2*c^4)*d^
2*e^4 - 6*(b^5*c - 5*a*b^3*c^2 + 4*a^2*b*c^3)*d*e^5 + (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 4*a^3*c^3)*e^6)/(c^6*
d^12 - 6*b*c^5*d^11*e + 3*(5*b^2*c^4 + 2*a*c^5)*d^10*e^2 - 10*(2*b^3*c^3 + 3*a*b*c^4)*d^9*e^3 + 15*(b^4*c^2 +
4*a*b^2*c^3 + a^2*c^4)*d^8*e^4 - 6*(b^5*c + 10*a*b^3*c^2 + 10*a^2*b*c^3)*d^7*e^5 - 6*a^5*b*d*e^11 + (b^6 + 30*
a*b^4*c + 90*a^2*b^2*c^2 + 20*a^3*c^3)*d^6*e^6 + a^6*e^12 - 6*(a*b^5 + 10*a^2*b^3*c + 10*a^3*b*c^2)*d^5*e^7 +
15*(a^2*b^4 + 4*a^3*b^2*c + a^4*c^2)*d^4*e^8 - 10*(2*a^3*b^3 + 3*a^4*b*c)*d^3*e^9 + 3*(5*a^4*b^2 + 2*a^5*c)*d^
2*e^10)))/(c^3*d^6 - 3*b*c^2*d^5*e + 3*(b^2*c + a*c^2)*d^4*e^2 - 3*a^2*b*d*e^5 - (b^3 + 6*a*b*c)*d^3*e^3 + a^3
*e^6 + 3*(a*b^2 + a^2*c)*d^2*e^4))*log(-sqrt(2)*(6*c^4*d^4 - 12*b*c^3*d^3*e + (11*b^2*c^2 - 8*a*c^3)*d^2*e^2 -
(5*b^3*c - 8*a*b*c^2)*d*e^3 + (b^4 - 3*a*b^2*c + 2*a^2*c^2)*e^4 - (2*c^4*d^7 - 7*b*c^3*d^6*e + 3*(3*b^2*c^2 +
2*a*c^3)*d^5*e^2 - 5*(b^3*c + 3*a*b*c^2)*d^4*e^3 - a^3*b*e^7 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^4 - 3*(a*
b^3 + 3*a^2*b*c)*d^2*e^5 + (3*a^2*b^2 + 2*a^3*c)*d*e^6)*sqrt((9*(b^2*c^4 - 4*a*c^5)*d^4*e^2 - 18*(b^3*c^3 - 4*
a*b*c^4)*d^3*e^3 + 3*(5*b^4*c^2 - 22*a*b^2*c^3 + 8*a^2*c^4)*d^2*e^4 - 6*(b^5*c - 5*a*b^3*c^2 + 4*a^2*b*c^3)*d*
e^5 + (b^6 - 6*a*b^4*c + 9*a^2*b^2*c^2 - 4*a^3*c^3)*e^6)/(c^6*d^12 - 6*b*c^5*d^11*e + 3*(5*b^2*c^4 + 2*a*c^5)*
d^10*e^2 - 10*(2*b^3*c^3 + 3*a*b*c^4)*d^9*e^3 +...

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1366 vs. \(2 (317) = 634\).
time = 3.07, size = 1366, normalized size = 3.86 \begin {gather*} \frac {2 \, {\left (2 \, c d - b e\right )}}{{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt {x e + d}} + \frac {{\left ({\left (c d^{2} e - b d e^{2} + a e^{3}\right )}^{2} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left (2 \, \sqrt {b^{2} - 4 \, a c} c d - \sqrt {b^{2} - 4 \, a c} b e\right )} - 2 \, {\left (2 \, c^{3} d^{4} - 4 \, b c^{2} d^{3} e + 3 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3} + {\left (a b^{2} - 2 \, a^{2} c\right )} e^{4}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | c d^{2} e - b d e^{2} + a e^{3} \right |} + {\left (2 \, \sqrt {b^{2} - 4 \, a c} c^{3} d^{5} e^{2} - 5 \, \sqrt {b^{2} - 4 \, a c} b c^{2} d^{4} e^{3} + 4 \, {\left (b^{2} c + a c^{2}\right )} \sqrt {b^{2} - 4 \, a c} d^{3} e^{4} - \sqrt {b^{2} - 4 \, a c} a^{2} b e^{7} - {\left (b^{3} + 6 \, a b c\right )} \sqrt {b^{2} - 4 \, a c} d^{2} e^{5} + 2 \, {\left (a b^{2} + a^{2} c\right )} \sqrt {b^{2} - 4 \, a c} d e^{6}\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^{3} - 3 \, b c d^{2} e + b^{2} d e^{2} + 2 \, a c d e^{2} - a b e^{3} + \sqrt {{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2} + 2 \, a c d e^{2} - a b e^{3}\right )}^{2} - 4 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} {\left (c^{2} d^{2} - b c d e + a c e^{2}\right )}}}{c^{2} d^{2} - b c d e + a c e^{2}}}}\right )}{4 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - 3 \, a^{2} b d e^{5} - {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + a^{3} e^{6} + 3 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4}\right )} {\left | c d^{2} e - b d e^{2} + a e^{3} \right |} {\left | c \right |}} - \frac {{\left ({\left (c d^{2} e - b d e^{2} + a e^{3}\right )}^{2} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left (2 \, \sqrt {b^{2} - 4 \, a c} c d - \sqrt {b^{2} - 4 \, a c} b e\right )} + 2 \, {\left (2 \, c^{3} d^{4} - 4 \, b c^{2} d^{3} e + 3 \, b^{2} c d^{2} e^{2} - b^{3} d e^{3} + {\left (a b^{2} - 2 \, a^{2} c\right )} e^{4}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | c d^{2} e - b d e^{2} + a e^{3} \right |} + {\left (2 \, \sqrt {b^{2} - 4 \, a c} c^{3} d^{5} e^{2} - 5 \, \sqrt {b^{2} - 4 \, a c} b c^{2} d^{4} e^{3} + 4 \, {\left (b^{2} c + a c^{2}\right )} \sqrt {b^{2} - 4 \, a c} d^{3} e^{4} - \sqrt {b^{2} - 4 \, a c} a^{2} b e^{7} - {\left (b^{3} + 6 \, a b c\right )} \sqrt {b^{2} - 4 \, a c} d^{2} e^{5} + 2 \, {\left (a b^{2} + a^{2} c\right )} \sqrt {b^{2} - 4 \, a c} d e^{6}\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^{3} - 3 \, b c d^{2} e + b^{2} d e^{2} + 2 \, a c d e^{2} - a b e^{3} - \sqrt {{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2} + 2 \, a c d e^{2} - a b e^{3}\right )}^{2} - 4 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} {\left (c^{2} d^{2} - b c d e + a c e^{2}\right )}}}{c^{2} d^{2} - b c d e + a c e^{2}}}}\right )}{4 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - 3 \, a^{2} b d e^{5} - {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + a^{3} e^{6} + 3 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4}\right )} {\left | c d^{2} e - b d e^{2} + a e^{3} \right |} {\left | c \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 9.40, size = 2500, normalized size = 7.06 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*c*d - 2*b*e + 2^(1/2)*a*e^2*atan(-((2^(1/2)*(-(b^3*e^3 - 2*c^3*d^3 + b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*
e^3 - a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 3*c^2*d^2*e*(b^2 - 4*a*c)^
(1/2) - 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2
+ 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2)*((d + e*x)^(1/2)
*(64*a*c^9*d^8*e^2 - 64*a^5*c^5*e^10 - 8*a^3*b^4*c^3*e^10 + 48*a^4*b^2*c^4*e^10 + 128*a^2*c^8*d^6*e^4 - 128*a^
4*c^6*d^2*e^8 - 16*b^2*c^8*d^8*e^2 + 64*b^3*c^7*d^7*e^3 - 104*b^4*c^6*d^6*e^4 + 88*b^5*c^5*d^5*e^5 - 40*b^6*c^
4*d^4*e^6 + 8*b^7*c^3*d^3*e^7 + 480*a^2*b^2*c^6*d^4*e^6 - 320*a^2*b^3*c^5*d^3*e^7 + 72*a^2*b^4*c^4*d^2*e^8 + 1
28*a^3*b^2*c^5*d^2*e^8 - 256*a*b*c^8*d^7*e^3 + 128*a^4*b*c^5*d*e^9 + 384*a*b^2*c^7*d^6*e^4 - 256*a*b^3*c^6*d^5
*e^5 + 40*a*b^4*c^5*d^4*e^6 + 48*a*b^5*c^4*d^3*e^7 - 24*a*b^6*c^3*d^2*e^8 - 384*a^2*b*c^7*d^5*e^5 + 24*a^2*b^5
*c^3*d*e^9 - 128*a^3*b^3*c^4*d*e^9) + (2^(1/2)*(-(b^3*e^3 - 2*c^3*d^3 + b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*
e^3 - a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 3*c^2*d^2*e*(b^2 - 4*a*c)^
(1/2) - 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2
+ 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2)*(64*a*c^9*d^10*e
^2 - 64*a^6*c^4*e^12 - 8*a^4*b^4*c^2*e^12 + 48*a^5*b^2*c^3*e^12 + 192*a^2*c^8*d^8*e^4 + 128*a^3*c^7*d^6*e^6 -
128*a^4*c^6*d^4*e^8 - 192*a^5*c^5*d^2*e^10 - 16*b^2*c^8*d^10*e^2 + 80*b^3*c^7*d^9*e^3 - 168*b^4*c^6*d^8*e^4 +
192*b^5*c^5*d^7*e^5 - 128*b^6*c^4*d^6*e^6 + 48*b^7*c^3*d^5*e^7 - 8*b^8*c^2*d^4*e^8 - (2^(1/2)*(-(b^3*e^3 - 2*c
^3*d^3 + b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 - a*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2
*e - 3*b^2*c*d*e^2 + 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) - 3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(a^3*e^6 + c^3*d^6 - b
^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d
^5*e - 6*a*b*c*d^3*e^3))^(1/2)*(d + e*x)^(1/2)*(64*a*c^9*d^11*e^2 - 32*a^6*b*c^3*e^13 + 64*a^6*c^4*d*e^12 + 8*
a^5*b^3*c^2*e^13 + 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 + 640*a^4*c^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10 - 16
*b^2*c^8*d^11*e^2 + 88*b^3*c^7*d^10*e^3 - 200*b^4*c^6*d^9*e^4 + 240*b^5*c^5*d^8*e^5 - 160*b^6*c^4*d^7*e^6 + 56
*b^7*c^3*d^6*e^7 - 8*b^8*c^2*d^5*e^8 + 2400*a^2*b^2*c^6*d^7*e^6 - 1680*a^2*b^3*c^5*d^6*e^7 + 240*a^2*b^4*c^4*d
^5*e^8 + 240*a^2*b^5*c^3*d^4*e^9 - 80*a^2*b^6*c^2*d^3*e^10 + 2720*a^3*b^2*c^5*d^5*e^8 - 1200*a^3*b^3*c^4*d^4*e
^9 + 80*a^3*b^5*c^2*d^2*e^11 + 1200*a^4*b^2*c^4*d^3*e^10 - 200*a^4*b^3*c^3*d^2*e^11 - 352*a*b*c^8*d^10*e^3 + 7
20*a*b^2*c^7*d^9*e^4 - 600*a*b^3*c^6*d^8*e^5 + 336*a*b^5*c^4*d^6*e^7 - 208*a*b^6*c^3*d^5*e^8 + 40*a*b^7*c^2*d^
4*e^9 - 1440*a^2*b*c^7*d^8*e^5 - 2240*a^3*b*c^6*d^6*e^7 - 1600*a^4*b*c^5*d^4*e^9 - 40*a^4*b^4*c^2*d*e^12 - 480
*a^5*b*c^4*d^2*e^11 + 144*a^5*b^2*c^3*d*e^12))/2 + 1248*a^2*b^2*c^6*d^6*e^6 - 1056*a^2*b^3*c^5*d^5*e^7 + 432*a
^2*b^4*c^4*d^4*e^8 - 48*a^2*b^6*c^2*d^2*e^10 + 608*a^3*b^2*c^5*d^4*e^8 - 576*a^3*b^3*c^4*d^3*e^9 + 192*a^3*b^4
*c^3*d^2*e^10 + 48*a^4*b^2*c^4*d^2*e^10 - 320*a*b*c^8*d^9*e^3 + 192*a^5*b*c^4*d*e^11 + 624*a*b^2*c^7*d^8*e^4 -
576*a*b^3*c^6*d^7*e^5 + 192*a*b^4*c^5*d^6*e^6 + 96*a*b^5*c^4*d^5*e^7 - 112*a*b^6*c^3*d^4*e^8 + 32*a*b^7*c^2*d
^3*e^9 - 768*a^2*b*c^7*d^7*e^5 - 384*a^3*b*c^6*d^5*e^7 + 32*a^3*b^5*c^2*d*e^11 + 256*a^4*b*c^5*d^3*e^9 - 176*a
^4*b^3*c^3*d*e^11))/2)*1i)/2 + (2^(1/2)*(-(b^3*e^3 - 2*c^3*d^3 + b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 - a
*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) -
3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2
*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2)*((d + e*x)^(1/2)*(64*a*
c^9*d^8*e^2 - 64*a^5*c^5*e^10 - 8*a^3*b^4*c^3*e^10 + 48*a^4*b^2*c^4*e^10 + 128*a^2*c^8*d^6*e^4 - 128*a^4*c^6*d
^2*e^8 - 16*b^2*c^8*d^8*e^2 + 64*b^3*c^7*d^7*e^3 - 104*b^4*c^6*d^6*e^4 + 88*b^5*c^5*d^5*e^5 - 40*b^6*c^4*d^4*e
^6 + 8*b^7*c^3*d^3*e^7 + 480*a^2*b^2*c^6*d^4*e^6 - 320*a^2*b^3*c^5*d^3*e^7 + 72*a^2*b^4*c^4*d^2*e^8 + 128*a^3*
b^2*c^5*d^2*e^8 - 256*a*b*c^8*d^7*e^3 + 128*a^4*b*c^5*d*e^9 + 384*a*b^2*c^7*d^6*e^4 - 256*a*b^3*c^6*d^5*e^5 +
40*a*b^4*c^5*d^4*e^6 + 48*a*b^5*c^4*d^3*e^7 - 24*a*b^6*c^3*d^2*e^8 - 384*a^2*b*c^7*d^5*e^5 + 24*a^2*b^5*c^3*d*
e^9 - 128*a^3*b^3*c^4*d*e^9) - (2^(1/2)*(-(b^3*e^3 - 2*c^3*d^3 + b^2*e^3*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e^3 - a
*c*e^3*(b^2 - 4*a*c)^(1/2) + 6*a*c^2*d*e^2 + 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 3*c^2*d^2*e*(b^2 - 4*a*c)^(1/2) -
3*b*c*d*e^2*(b^2 - 4*a*c)^(1/2))/(a^3*e^6 + c^3*d^6 - b^3*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a*c^2*d^4*e^2 + 3*a^2
*c*d^2*e^4 + 3*b^2*c*d^4*e^2 - 3*a^2*b*d*e^5 - 3*b*c^2*d^5*e - 6*a*b*c*d^3*e^3))^(1/2)*(64*a*c^9*d^10*e^2 - 64
*a^6*c^4*e^12 - 8*a^4*b^4*c^2*e^12 + 48*a^5*b^2...

________________________________________________________________________________________