3.4.83 \(\int \frac {1}{\sqrt {d+e x} (b x+c x^2)^3} \, dx\) [383]

Optimal. Leaf size=299 \[ -\frac {\sqrt {d+e x} (b (c d-b e)+c (2 c d-b e) x)}{2 b^2 d (c d-b e) \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b (c d-b e) \left (12 c^2 d^2-7 b c d e-3 b^2 e^2\right )+3 c (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{4 b^4 d^2 (c d-b e)^2 \left (b x+c x^2\right )}-\frac {3 \left (16 c^2 d^2+4 b c d e+b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 d^{5/2}}+\frac {3 c^{5/2} \left (16 c^2 d^2-36 b c d e+21 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 (c d-b e)^{5/2}} \]

[Out]

-3/4*(b^2*e^2+4*b*c*d*e+16*c^2*d^2)*arctanh((e*x+d)^(1/2)/d^(1/2))/b^5/d^(5/2)+3/4*c^(5/2)*(21*b^2*e^2-36*b*c*
d*e+16*c^2*d^2)*arctanh(c^(1/2)*(e*x+d)^(1/2)/(-b*e+c*d)^(1/2))/b^5/(-b*e+c*d)^(5/2)-1/2*(b*(-b*e+c*d)+c*(-b*e
+2*c*d)*x)*(e*x+d)^(1/2)/b^2/d/(-b*e+c*d)/(c*x^2+b*x)^2+1/4*(b*(-b*e+c*d)*(-3*b^2*e^2-7*b*c*d*e+12*c^2*d^2)+3*
c*(-b*e+2*c*d)*(-b^2*e^2-4*b*c*d*e+4*c^2*d^2)*x)*(e*x+d)^(1/2)/b^4/d^2/(-b*e+c*d)^2/(c*x^2+b*x)

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Rubi [A]
time = 0.29, antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {754, 836, 840, 1180, 214} \begin {gather*} -\frac {\sqrt {d+e x} (c x (2 c d-b e)+b (c d-b e))}{2 b^2 d \left (b x+c x^2\right )^2 (c d-b e)}-\frac {3 \left (b^2 e^2+4 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 d^{5/2}}+\frac {3 c^{5/2} \left (21 b^2 e^2-36 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 (c d-b e)^{5/2}}+\frac {\sqrt {d+e x} \left (3 c x (2 c d-b e) \left (-b^2 e^2-4 b c d e+4 c^2 d^2\right )+b (c d-b e) \left (-3 b^2 e^2-7 b c d e+12 c^2 d^2\right )\right )}{4 b^4 d^2 \left (b x+c x^2\right ) (c d-b e)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[d + e*x]*(b*x + c*x^2)^3),x]

[Out]

-1/2*(Sqrt[d + e*x]*(b*(c*d - b*e) + c*(2*c*d - b*e)*x))/(b^2*d*(c*d - b*e)*(b*x + c*x^2)^2) + (Sqrt[d + e*x]*
(b*(c*d - b*e)*(12*c^2*d^2 - 7*b*c*d*e - 3*b^2*e^2) + 3*c*(2*c*d - b*e)*(4*c^2*d^2 - 4*b*c*d*e - b^2*e^2)*x))/
(4*b^4*d^2*(c*d - b*e)^2*(b*x + c*x^2)) - (3*(16*c^2*d^2 + 4*b*c*d*e + b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]
)/(4*b^5*d^(5/2)) + (3*c^(5/2)*(16*c^2*d^2 - 36*b*c*d*e + 21*b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d
- b*e]])/(4*b^5*(c*d - b*e)^(5/2))

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 754

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(b
*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e +
a*e^2))), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b
*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 836

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)
*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], 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] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 840

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

Rule 1180

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {d+e x} \left (b x+c x^2\right )^3} \, dx &=-\frac {\sqrt {d+e x} (b (c d-b e)+c (2 c d-b e) x)}{2 b^2 d (c d-b e) \left (b x+c x^2\right )^2}-\frac {\int \frac {\frac {1}{2} \left (12 c^2 d^2-7 b c d e-3 b^2 e^2\right )+\frac {5}{2} c e (2 c d-b e) x}{\sqrt {d+e x} \left (b x+c x^2\right )^2} \, dx}{2 b^2 d (c d-b e)}\\ &=-\frac {\sqrt {d+e x} (b (c d-b e)+c (2 c d-b e) x)}{2 b^2 d (c d-b e) \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b (c d-b e) \left (12 c^2 d^2-7 b c d e-3 b^2 e^2\right )+3 c (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{4 b^4 d^2 (c d-b e)^2 \left (b x+c x^2\right )}+\frac {\int \frac {\frac {3}{4} (c d-b e)^2 \left (16 c^2 d^2+4 b c d e+b^2 e^2\right )+\frac {3}{4} c e (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{2 b^4 d^2 (c d-b e)^2}\\ &=-\frac {\sqrt {d+e x} (b (c d-b e)+c (2 c d-b e) x)}{2 b^2 d (c d-b e) \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b (c d-b e) \left (12 c^2 d^2-7 b c d e-3 b^2 e^2\right )+3 c (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{4 b^4 d^2 (c d-b e)^2 \left (b x+c x^2\right )}+\frac {\text {Subst}\left (\int \frac {-\frac {3}{4} c d e (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right )+\frac {3}{4} e (c d-b e)^2 \left (16 c^2 d^2+4 b c d e+b^2 e^2\right )+\frac {3}{4} c e (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x^2}{c d^2-b d e+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{b^4 d^2 (c d-b e)^2}\\ &=-\frac {\sqrt {d+e x} (b (c d-b e)+c (2 c d-b e) x)}{2 b^2 d (c d-b e) \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b (c d-b e) \left (12 c^2 d^2-7 b c d e-3 b^2 e^2\right )+3 c (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{4 b^4 d^2 (c d-b e)^2 \left (b x+c x^2\right )}+\frac {\left (3 c \left (16 c^2 d^2+4 b c d e+b^2 e^2\right )\right ) \text {Subst}\left (\int \frac {1}{-\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 b^5 d^2}-\frac {\left (3 c^3 \left (16 c^2 d^2-36 b c d e+21 b^2 e^2\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 b^5 (c d-b e)^2}\\ &=-\frac {\sqrt {d+e x} (b (c d-b e)+c (2 c d-b e) x)}{2 b^2 d (c d-b e) \left (b x+c x^2\right )^2}+\frac {\sqrt {d+e x} \left (b (c d-b e) \left (12 c^2 d^2-7 b c d e-3 b^2 e^2\right )+3 c (2 c d-b e) \left (4 c^2 d^2-4 b c d e-b^2 e^2\right ) x\right )}{4 b^4 d^2 (c d-b e)^2 \left (b x+c x^2\right )}-\frac {3 \left (16 c^2 d^2+4 b c d e+b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 d^{5/2}}+\frac {3 c^{5/2} \left (16 c^2 d^2-36 b c d e+21 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 (c d-b e)^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 1.82, size = 292, normalized size = 0.98 \begin {gather*} \frac {\frac {b \sqrt {d+e x} \left (24 c^5 d^3 x^3+36 b c^4 d^2 x^2 (d-e x)+b^5 e^2 (-2 d+3 e x)+2 b^4 c e \left (2 d^2+d e x+3 e^2 x^2\right )+b^2 c^3 d x \left (8 d^2-55 d e x+6 e^2 x^2\right )+b^3 c^2 \left (-2 d^3-13 d^2 e x+10 d e^2 x^2+3 e^3 x^3\right )\right )}{d^2 (c d-b e)^2 x^2 (b+c x)^2}-\frac {3 c^{5/2} \left (16 c^2 d^2-36 b c d e+21 b^2 e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{(-c d+b e)^{5/2}}-\frac {3 \left (16 c^2 d^2+4 b c d e+b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{5/2}}}{4 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[d + e*x]*(b*x + c*x^2)^3),x]

[Out]

((b*Sqrt[d + e*x]*(24*c^5*d^3*x^3 + 36*b*c^4*d^2*x^2*(d - e*x) + b^5*e^2*(-2*d + 3*e*x) + 2*b^4*c*e*(2*d^2 + d
*e*x + 3*e^2*x^2) + b^2*c^3*d*x*(8*d^2 - 55*d*e*x + 6*e^2*x^2) + b^3*c^2*(-2*d^3 - 13*d^2*e*x + 10*d*e^2*x^2 +
3*e^3*x^3)))/(d^2*(c*d - b*e)^2*x^2*(b + c*x)^2) - (3*c^(5/2)*(16*c^2*d^2 - 36*b*c*d*e + 21*b^2*e^2)*ArcTan[(
Sqrt[c]*Sqrt[d + e*x])/Sqrt[-(c*d) + b*e]])/(-(c*d) + b*e)^(5/2) - (3*(16*c^2*d^2 + 4*b*c*d*e + b^2*e^2)*ArcTa
nh[Sqrt[d + e*x]/Sqrt[d]])/d^(5/2))/(4*b^5)

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Maple [A]
time = 0.52, size = 295, normalized size = 0.99

method	result	size
derivativedivides	\(2 e^{5} \left (-\frac {c^{3} \left (\frac {\frac {3 c b e \left (5 b e -4 c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 \left (b^{2} e^{2}-2 b c d e +d^{2} c^{2}\right )}+\frac {\left (17 b e -12 c d \right ) b e \sqrt {e x +d}}{8 b e -8 c d}}{\left (c \left (e x +d \right )+b e -c d \right )^{2}}+\frac {3 \left (21 b^{2} e^{2}-36 b c d e +16 d^{2} c^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{8 \left (b^{2} e^{2}-2 b c d e +d^{2} c^{2}\right ) \sqrt {\left (b e -c d \right ) c}}\right )}{b^{5} e^{5}}-\frac {\frac {-\frac {3 b e \left (b e +4 c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 d^{2}}+\frac {b e \left (5 b e +12 c d \right ) \sqrt {e x +d}}{8 d}}{e^{2} x^{2}}+\frac {3 \left (b^{2} e^{2}+4 b c d e +16 d^{2} c^{2}\right ) \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 d^{\frac {5}{2}}}}{b^{5} e^{5}}\right )\)	\(295\)
default	\(2 e^{5} \left (-\frac {c^{3} \left (\frac {\frac {3 c b e \left (5 b e -4 c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 \left (b^{2} e^{2}-2 b c d e +d^{2} c^{2}\right )}+\frac {\left (17 b e -12 c d \right ) b e \sqrt {e x +d}}{8 b e -8 c d}}{\left (c \left (e x +d \right )+b e -c d \right )^{2}}+\frac {3 \left (21 b^{2} e^{2}-36 b c d e +16 d^{2} c^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{8 \left (b^{2} e^{2}-2 b c d e +d^{2} c^{2}\right ) \sqrt {\left (b e -c d \right ) c}}\right )}{b^{5} e^{5}}-\frac {\frac {-\frac {3 b e \left (b e +4 c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 d^{2}}+\frac {b e \left (5 b e +12 c d \right ) \sqrt {e x +d}}{8 d}}{e^{2} x^{2}}+\frac {3 \left (b^{2} e^{2}+4 b c d e +16 d^{2} c^{2}\right ) \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 d^{\frac {5}{2}}}}{b^{5} e^{5}}\right )\)	\(295\)
risch	\(-\frac {\sqrt {e x +d}\, \left (-3 b e x -12 c d x +2 b d \right )}{4 d^{2} b^{4} x^{2}}-\frac {15 e^{2} c^{4} \left (e x +d \right )^{\frac {3}{2}}}{4 b^{3} \left (c e x +b e \right )^{2} \left (b^{2} e^{2}-2 b c d e +d^{2} c^{2}\right )}+\frac {3 d e \,c^{5} \left (e x +d \right )^{\frac {3}{2}}}{b^{4} \left (c e x +b e \right )^{2} \left (b^{2} e^{2}-2 b c d e +d^{2} c^{2}\right )}-\frac {17 e^{2} c^{3} \sqrt {e x +d}}{4 b^{3} \left (c e x +b e \right )^{2} \left (b e -c d \right )}+\frac {3 d e \,c^{4} \sqrt {e x +d}}{b^{4} \left (c e x +b e \right )^{2} \left (b e -c d \right )}-\frac {63 e^{2} c^{3} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{4 b^{3} \left (b^{2} e^{2}-2 b c d e +d^{2} c^{2}\right ) \sqrt {\left (b e -c d \right ) c}}+\frac {27 d e \,c^{4} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{b^{4} \left (b^{2} e^{2}-2 b c d e +d^{2} c^{2}\right ) \sqrt {\left (b e -c d \right ) c}}-\frac {12 d^{2} c^{5} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{b^{5} \left (b^{2} e^{2}-2 b c d e +d^{2} c^{2}\right ) \sqrt {\left (b e -c d \right ) c}}-\frac {3 e^{2} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{4 b^{3} d^{\frac {5}{2}}}-\frac {3 e \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right ) c}{b^{4} d^{\frac {3}{2}}}-\frac {12 \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right ) c^{2}}{b^{5} \sqrt {d}}\)	\(482\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(1/2)/(c*x^2+b*x)^3,x,method=_RETURNVERBOSE)

[Out]

2*e^5*(-c^3/b^5/e^5*((3/8*c*b*e*(5*b*e-4*c*d)/(b^2*e^2-2*b*c*d*e+c^2*d^2)*(e*x+d)^(3/2)+1/8*(17*b*e-12*c*d)*b*
e/(b*e-c*d)*(e*x+d)^(1/2))/(c*(e*x+d)+b*e-c*d)^2+3/8*(21*b^2*e^2-36*b*c*d*e+16*c^2*d^2)/(b^2*e^2-2*b*c*d*e+c^2
*d^2)/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)))-1/b^5/e^5*((-3/8*b*e*(b*e+4*c*d)/d^2*(e
*x+d)^(3/2)+1/8*b*e*(5*b*e+12*c*d)/d*(e*x+d)^(1/2))/e^2/x^2+3/8*(b^2*e^2+4*b*c*d*e+16*c^2*d^2)/d^(5/2)*arctanh
((e*x+d)^(1/2)/d^(1/2))))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(1/2)/(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(c*d-%e*b>0)', see `assume?` fo
r more detai

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 732 vs. \(2 (283) = 566\).
time = 5.93, size = 2964, normalized size = 9.91 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(1/2)/(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

[1/8*(3*(16*c^6*d^5*x^4 + 32*b*c^5*d^5*x^3 + 16*b^2*c^4*d^5*x^2 + 21*(b^2*c^4*d^3*x^4 + 2*b^3*c^3*d^3*x^3 + b^
4*c^2*d^3*x^2)*e^2 - 36*(b*c^5*d^4*x^4 + 2*b^2*c^4*d^4*x^3 + b^3*c^3*d^4*x^2)*e)*sqrt(c/(c*d - b*e))*log((2*c*
d + 2*(c*d - b*e)*sqrt(x*e + d)*sqrt(c/(c*d - b*e)) + (c*x - b)*e)/(c*x + b)) + 3*(16*c^6*d^4*x^4 + 32*b*c^5*d
^4*x^3 + 16*b^2*c^4*d^4*x^2 + (b^4*c^2*x^4 + 2*b^5*c*x^3 + b^6*x^2)*e^4 + 2*(b^3*c^3*d*x^4 + 2*b^4*c^2*d*x^3 +
b^5*c*d*x^2)*e^3 + 9*(b^2*c^4*d^2*x^4 + 2*b^3*c^3*d^2*x^3 + b^4*c^2*d^2*x^2)*e^2 - 28*(b*c^5*d^3*x^4 + 2*b^2*
c^4*d^3*x^3 + b^3*c^3*d^3*x^2)*e)*sqrt(d)*log((x*e - 2*sqrt(x*e + d)*sqrt(d) + 2*d)/x) + 2*(24*b*c^5*d^4*x^3 +
36*b^2*c^4*d^4*x^2 + 8*b^3*c^3*d^4*x - 2*b^4*c^2*d^4 + 3*(b^4*c^2*d*x^3 + 2*b^5*c*d*x^2 + b^6*d*x)*e^3 + 2*(3
*b^3*c^3*d^2*x^3 + 5*b^4*c^2*d^2*x^2 + b^5*c*d^2*x - b^6*d^2)*e^2 - (36*b^2*c^4*d^3*x^3 + 55*b^3*c^3*d^3*x^2 +
13*b^4*c^2*d^3*x - 4*b^5*c*d^3)*e)*sqrt(x*e + d))/(b^5*c^4*d^5*x^4 + 2*b^6*c^3*d^5*x^3 + b^7*c^2*d^5*x^2 + (b
^7*c^2*d^3*x^4 + 2*b^8*c*d^3*x^3 + b^9*d^3*x^2)*e^2 - 2*(b^6*c^3*d^4*x^4 + 2*b^7*c^2*d^4*x^3 + b^8*c*d^4*x^2)*
e), 1/8*(6*(16*c^6*d^5*x^4 + 32*b*c^5*d^5*x^3 + 16*b^2*c^4*d^5*x^2 + 21*(b^2*c^4*d^3*x^4 + 2*b^3*c^3*d^3*x^3 +
b^4*c^2*d^3*x^2)*e^2 - 36*(b*c^5*d^4*x^4 + 2*b^2*c^4*d^4*x^3 + b^3*c^3*d^4*x^2)*e)*sqrt(-c/(c*d - b*e))*arcta
n(-(c*d - b*e)*sqrt(x*e + d)*sqrt(-c/(c*d - b*e))/(c*x*e + c*d)) + 3*(16*c^6*d^4*x^4 + 32*b*c^5*d^4*x^3 + 16*b
^2*c^4*d^4*x^2 + (b^4*c^2*x^4 + 2*b^5*c*x^3 + b^6*x^2)*e^4 + 2*(b^3*c^3*d*x^4 + 2*b^4*c^2*d*x^3 + b^5*c*d*x^2)
*e^3 + 9*(b^2*c^4*d^2*x^4 + 2*b^3*c^3*d^2*x^3 + b^4*c^2*d^2*x^2)*e^2 - 28*(b*c^5*d^3*x^4 + 2*b^2*c^4*d^3*x^3 +
b^3*c^3*d^3*x^2)*e)*sqrt(d)*log((x*e - 2*sqrt(x*e + d)*sqrt(d) + 2*d)/x) + 2*(24*b*c^5*d^4*x^3 + 36*b^2*c^4*d
^4*x^2 + 8*b^3*c^3*d^4*x - 2*b^4*c^2*d^4 + 3*(b^4*c^2*d*x^3 + 2*b^5*c*d*x^2 + b^6*d*x)*e^3 + 2*(3*b^3*c^3*d^2*
x^3 + 5*b^4*c^2*d^2*x^2 + b^5*c*d^2*x - b^6*d^2)*e^2 - (36*b^2*c^4*d^3*x^3 + 55*b^3*c^3*d^3*x^2 + 13*b^4*c^2*d
^3*x - 4*b^5*c*d^3)*e)*sqrt(x*e + d))/(b^5*c^4*d^5*x^4 + 2*b^6*c^3*d^5*x^3 + b^7*c^2*d^5*x^2 + (b^7*c^2*d^3*x^
4 + 2*b^8*c*d^3*x^3 + b^9*d^3*x^2)*e^2 - 2*(b^6*c^3*d^4*x^4 + 2*b^7*c^2*d^4*x^3 + b^8*c*d^4*x^2)*e), 1/8*(6*(1
6*c^6*d^4*x^4 + 32*b*c^5*d^4*x^3 + 16*b^2*c^4*d^4*x^2 + (b^4*c^2*x^4 + 2*b^5*c*x^3 + b^6*x^2)*e^4 + 2*(b^3*c^3
*d*x^4 + 2*b^4*c^2*d*x^3 + b^5*c*d*x^2)*e^3 + 9*(b^2*c^4*d^2*x^4 + 2*b^3*c^3*d^2*x^3 + b^4*c^2*d^2*x^2)*e^2 -
28*(b*c^5*d^3*x^4 + 2*b^2*c^4*d^3*x^3 + b^3*c^3*d^3*x^2)*e)*sqrt(-d)*arctan(sqrt(x*e + d)*sqrt(-d)/d) + 3*(16*
c^6*d^5*x^4 + 32*b*c^5*d^5*x^3 + 16*b^2*c^4*d^5*x^2 + 21*(b^2*c^4*d^3*x^4 + 2*b^3*c^3*d^3*x^3 + b^4*c^2*d^3*x^
2)*e^2 - 36*(b*c^5*d^4*x^4 + 2*b^2*c^4*d^4*x^3 + b^3*c^3*d^4*x^2)*e)*sqrt(c/(c*d - b*e))*log((2*c*d + 2*(c*d -
b*e)*sqrt(x*e + d)*sqrt(c/(c*d - b*e)) + (c*x - b)*e)/(c*x + b)) + 2*(24*b*c^5*d^4*x^3 + 36*b^2*c^4*d^4*x^2 +
8*b^3*c^3*d^4*x - 2*b^4*c^2*d^4 + 3*(b^4*c^2*d*x^3 + 2*b^5*c*d*x^2 + b^6*d*x)*e^3 + 2*(3*b^3*c^3*d^2*x^3 + 5*
b^4*c^2*d^2*x^2 + b^5*c*d^2*x - b^6*d^2)*e^2 - (36*b^2*c^4*d^3*x^3 + 55*b^3*c^3*d^3*x^2 + 13*b^4*c^2*d^3*x - 4
*b^5*c*d^3)*e)*sqrt(x*e + d))/(b^5*c^4*d^5*x^4 + 2*b^6*c^3*d^5*x^3 + b^7*c^2*d^5*x^2 + (b^7*c^2*d^3*x^4 + 2*b^
8*c*d^3*x^3 + b^9*d^3*x^2)*e^2 - 2*(b^6*c^3*d^4*x^4 + 2*b^7*c^2*d^4*x^3 + b^8*c*d^4*x^2)*e), 1/4*(3*(16*c^6*d^
5*x^4 + 32*b*c^5*d^5*x^3 + 16*b^2*c^4*d^5*x^2 + 21*(b^2*c^4*d^3*x^4 + 2*b^3*c^3*d^3*x^3 + b^4*c^2*d^3*x^2)*e^2
- 36*(b*c^5*d^4*x^4 + 2*b^2*c^4*d^4*x^3 + b^3*c^3*d^4*x^2)*e)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(x
*e + d)*sqrt(-c/(c*d - b*e))/(c*x*e + c*d)) + 3*(16*c^6*d^4*x^4 + 32*b*c^5*d^4*x^3 + 16*b^2*c^4*d^4*x^2 + (b^4
*c^2*x^4 + 2*b^5*c*x^3 + b^6*x^2)*e^4 + 2*(b^3*c^3*d*x^4 + 2*b^4*c^2*d*x^3 + b^5*c*d*x^2)*e^3 + 9*(b^2*c^4*d^2
*x^4 + 2*b^3*c^3*d^2*x^3 + b^4*c^2*d^2*x^2)*e^2 - 28*(b*c^5*d^3*x^4 + 2*b^2*c^4*d^3*x^3 + b^3*c^3*d^3*x^2)*e)*
sqrt(-d)*arctan(sqrt(x*e + d)*sqrt(-d)/d) + (24*b*c^5*d^4*x^3 + 36*b^2*c^4*d^4*x^2 + 8*b^3*c^3*d^4*x - 2*b^4*c
^2*d^4 + 3*(b^4*c^2*d*x^3 + 2*b^5*c*d*x^2 + b^6*d*x)*e^3 + 2*(3*b^3*c^3*d^2*x^3 + 5*b^4*c^2*d^2*x^2 + b^5*c*d^
2*x - b^6*d^2)*e^2 - (36*b^2*c^4*d^3*x^3 + 55*b^3*c^3*d^3*x^2 + 13*b^4*c^2*d^3*x - 4*b^5*c*d^3)*e)*sqrt(x*e +
d))/(b^5*c^4*d^5*x^4 + 2*b^6*c^3*d^5*x^3 + b^7*c^2*d^5*x^2 + (b^7*c^2*d^3*x^4 + 2*b^8*c*d^3*x^3 + b^9*d^3*x^2)
*e^2 - 2*(b^6*c^3*d^4*x^4 + 2*b^7*c^2*d^4*x^3 + b^8*c*d^4*x^2)*e)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{3} \left (b + c x\right )^{3} \sqrt {d + e x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**(1/2)/(c*x**2+b*x)**3,x)

[Out]

Integral(1/(x**3*(b + c*x)**3*sqrt(d + e*x)), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 619 vs. \(2 (283) = 566\).
time = 1.33, size = 619, normalized size = 2.07 \begin {gather*} -\frac {3 \, {\left (16 \, c^{5} d^{2} - 36 \, b c^{4} d e + 21 \, b^{2} c^{3} e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{4 \, {\left (b^{5} c^{2} d^{2} - 2 \, b^{6} c d e + b^{7} e^{2}\right )} \sqrt {-c^{2} d + b c e}} + \frac {24 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{5} d^{3} e - 72 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{5} d^{4} e + 72 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{5} d^{5} e - 24 \, \sqrt {x e + d} c^{5} d^{6} e - 36 \, {\left (x e + d\right )}^{\frac {7}{2}} b c^{4} d^{2} e^{2} + 144 \, {\left (x e + d\right )}^{\frac {5}{2}} b c^{4} d^{3} e^{2} - 180 \, {\left (x e + d\right )}^{\frac {3}{2}} b c^{4} d^{4} e^{2} + 72 \, \sqrt {x e + d} b c^{4} d^{5} e^{2} + 6 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{2} c^{3} d e^{3} - 73 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{2} c^{3} d^{2} e^{3} + 136 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} c^{3} d^{3} e^{3} - 69 \, \sqrt {x e + d} b^{2} c^{3} d^{4} e^{3} + 3 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{3} c^{2} e^{4} + {\left (x e + d\right )}^{\frac {5}{2}} b^{3} c^{2} d e^{4} - 24 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} c^{2} d^{2} e^{4} + 18 \, \sqrt {x e + d} b^{3} c^{2} d^{3} e^{4} + 6 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{4} c e^{5} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{4} c d e^{5} + 8 \, \sqrt {x e + d} b^{4} c d^{2} e^{5} + 3 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{5} e^{6} - 5 \, \sqrt {x e + d} b^{5} d e^{6}}{4 \, {\left (b^{4} c^{2} d^{4} - 2 \, b^{5} c d^{3} e + b^{6} d^{2} e^{2}\right )} {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + {\left (x e + d\right )} b e - b d e\right )}^{2}} + \frac {3 \, {\left (16 \, c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{4 \, b^{5} \sqrt {-d} d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(1/2)/(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

-3/4*(16*c^5*d^2 - 36*b*c^4*d*e + 21*b^2*c^3*e^2)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/((b^5*c^2*d^2 -
2*b^6*c*d*e + b^7*e^2)*sqrt(-c^2*d + b*c*e)) + 1/4*(24*(x*e + d)^(7/2)*c^5*d^3*e - 72*(x*e + d)^(5/2)*c^5*d^4
*e + 72*(x*e + d)^(3/2)*c^5*d^5*e - 24*sqrt(x*e + d)*c^5*d^6*e - 36*(x*e + d)^(7/2)*b*c^4*d^2*e^2 + 144*(x*e +
d)^(5/2)*b*c^4*d^3*e^2 - 180*(x*e + d)^(3/2)*b*c^4*d^4*e^2 + 72*sqrt(x*e + d)*b*c^4*d^5*e^2 + 6*(x*e + d)^(7/
2)*b^2*c^3*d*e^3 - 73*(x*e + d)^(5/2)*b^2*c^3*d^2*e^3 + 136*(x*e + d)^(3/2)*b^2*c^3*d^3*e^3 - 69*sqrt(x*e + d)
*b^2*c^3*d^4*e^3 + 3*(x*e + d)^(7/2)*b^3*c^2*e^4 + (x*e + d)^(5/2)*b^3*c^2*d*e^4 - 24*(x*e + d)^(3/2)*b^3*c^2*
d^2*e^4 + 18*sqrt(x*e + d)*b^3*c^2*d^3*e^4 + 6*(x*e + d)^(5/2)*b^4*c*e^5 - 10*(x*e + d)^(3/2)*b^4*c*d*e^5 + 8*
sqrt(x*e + d)*b^4*c*d^2*e^5 + 3*(x*e + d)^(3/2)*b^5*e^6 - 5*sqrt(x*e + d)*b^5*d*e^6)/((b^4*c^2*d^4 - 2*b^5*c*d
^3*e + b^6*d^2*e^2)*((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)*b*e - b*d*e)^2) + 3/4*(16*c^2*d^2 + 4
*b*c*d*e + b^2*e^2)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^5*sqrt(-d)*d^2)

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Mupad [B]
time = 2.88, size = 2500, normalized size = 8.36 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((b*x + c*x^2)^3*(d + e*x)^(1/2)),x)

[Out]

((e*(d + e*x)^(3/2)*(3*b^5*e^5 + 72*c^5*d^5 + 136*b^2*c^3*d^3*e^2 - 24*b^3*c^2*d^2*e^3 - 180*b*c^4*d^4*e - 10*
b^4*c*d*e^4))/(4*b^4*(c*d^2 - b*d*e)^2) - ((d + e*x)^(1/2)*(24*c^4*d^4*e - 5*b^4*e^5 - 48*b*c^3*d^3*e^2 + 21*b
^2*c^2*d^2*e^3 + 3*b^3*c*d*e^4))/(4*b^4*(c*d^2 - b*d*e)) + (e*(d + e*x)^(5/2)*(6*b^4*c*e^4 - 72*c^5*d^4 + b^3*
c^2*d*e^3 - 73*b^2*c^3*d^2*e^2 + 144*b*c^4*d^3*e))/(4*b^4*(c*d^2 - b*d*e)^2) + (3*c*e*(d + e*x)^(7/2)*(8*c^4*d
^3 + b^3*c*e^3 + 2*b^2*c^2*d*e^2 - 12*b*c^3*d^2*e))/(4*b^4*(c*d^2 - b*d*e)^2))/(c^2*(d + e*x)^4 - (d + e*x)*(4
*c^2*d^3 + 2*b^2*d*e^2 - 6*b*c*d^2*e) - (4*c^2*d - 2*b*c*e)*(d + e*x)^3 + (d + e*x)^2*(b^2*e^2 + 6*c^2*d^2 - 6
*b*c*d*e) + c^2*d^4 + b^2*d^2*e^2 - 2*b*c*d^3*e) - (atan((((-c^5*(b*e - c*d)^5)^(1/2)*(((d + e*x)^(1/2)*(9*b^8
*c^3*e^10 + 4608*c^11*d^8*e^2 - 18432*b*c^10*d^7*e^3 + 36*b^7*c^4*d*e^9 + 27360*b^2*c^9*d^6*e^4 - 17568*b^3*c^
8*d^5*e^5 + 3978*b^4*c^7*d^4*e^6 - 180*b^5*c^6*d^3*e^7 + 198*b^6*c^5*d^2*e^8))/(8*(b^8*c^4*d^8 + b^12*d^4*e^4
- 4*b^9*c^3*d^7*e - 4*b^11*c*d^5*e^3 + 6*b^10*c^2*d^6*e^2)) + (3*(-c^5*(b*e - c*d)^5)^(1/2)*((24*b^10*c^8*d^8*
e^3 - 96*b^11*c^7*d^7*e^4 + 141*b^12*c^6*d^6*e^5 - 87*b^13*c^5*d^5*e^6 + 18*b^14*c^4*d^4*e^7 - 3*b^15*c^3*d^3*
e^8 + 3*b^16*c^2*d^2*e^9)/(b^12*c^4*d^8 + b^16*d^4*e^4 - 4*b^13*c^3*d^7*e - 4*b^15*c*d^5*e^3 + 6*b^14*c^2*d^6*
e^2) - (3*(-c^5*(b*e - c*d)^5)^(1/2)*(d + e*x)^(1/2)*(21*b^2*e^2 + 16*c^2*d^2 - 36*b*c*d*e)*(128*b^10*c^7*d^9*
e^2 - 576*b^11*c^6*d^8*e^3 + 1024*b^12*c^5*d^7*e^4 - 896*b^13*c^4*d^6*e^5 + 384*b^14*c^3*d^5*e^6 - 64*b^15*c^2
*d^4*e^7))/(64*(b^8*c^4*d^8 + b^12*d^4*e^4 - 4*b^9*c^3*d^7*e - 4*b^11*c*d^5*e^3 + 6*b^10*c^2*d^6*e^2)*(b^10*e^
5 - b^5*c^5*d^5 + 5*b^6*c^4*d^4*e - 10*b^7*c^3*d^3*e^2 + 10*b^8*c^2*d^2*e^3 - 5*b^9*c*d*e^4)))*(21*b^2*e^2 + 1
6*c^2*d^2 - 36*b*c*d*e))/(8*(b^10*e^5 - b^5*c^5*d^5 + 5*b^6*c^4*d^4*e - 10*b^7*c^3*d^3*e^2 + 10*b^8*c^2*d^2*e^
3 - 5*b^9*c*d*e^4)))*(21*b^2*e^2 + 16*c^2*d^2 - 36*b*c*d*e)*3i)/(8*(b^10*e^5 - b^5*c^5*d^5 + 5*b^6*c^4*d^4*e -
10*b^7*c^3*d^3*e^2 + 10*b^8*c^2*d^2*e^3 - 5*b^9*c*d*e^4)) + ((-c^5*(b*e - c*d)^5)^(1/2)*(((d + e*x)^(1/2)*(9*
b^8*c^3*e^10 + 4608*c^11*d^8*e^2 - 18432*b*c^10*d^7*e^3 + 36*b^7*c^4*d*e^9 + 27360*b^2*c^9*d^6*e^4 - 17568*b^3
*c^8*d^5*e^5 + 3978*b^4*c^7*d^4*e^6 - 180*b^5*c^6*d^3*e^7 + 198*b^6*c^5*d^2*e^8))/(8*(b^8*c^4*d^8 + b^12*d^4*e
^4 - 4*b^9*c^3*d^7*e - 4*b^11*c*d^5*e^3 + 6*b^10*c^2*d^6*e^2)) - (3*(-c^5*(b*e - c*d)^5)^(1/2)*((24*b^10*c^8*d
^8*e^3 - 96*b^11*c^7*d^7*e^4 + 141*b^12*c^6*d^6*e^5 - 87*b^13*c^5*d^5*e^6 + 18*b^14*c^4*d^4*e^7 - 3*b^15*c^3*d
^3*e^8 + 3*b^16*c^2*d^2*e^9)/(b^12*c^4*d^8 + b^16*d^4*e^4 - 4*b^13*c^3*d^7*e - 4*b^15*c*d^5*e^3 + 6*b^14*c^2*d
^6*e^2) + (3*(-c^5*(b*e - c*d)^5)^(1/2)*(d + e*x)^(1/2)*(21*b^2*e^2 + 16*c^2*d^2 - 36*b*c*d*e)*(128*b^10*c^7*d
^9*e^2 - 576*b^11*c^6*d^8*e^3 + 1024*b^12*c^5*d^7*e^4 - 896*b^13*c^4*d^6*e^5 + 384*b^14*c^3*d^5*e^6 - 64*b^15*
c^2*d^4*e^7))/(64*(b^8*c^4*d^8 + b^12*d^4*e^4 - 4*b^9*c^3*d^7*e - 4*b^11*c*d^5*e^3 + 6*b^10*c^2*d^6*e^2)*(b^10
*e^5 - b^5*c^5*d^5 + 5*b^6*c^4*d^4*e - 10*b^7*c^3*d^3*e^2 + 10*b^8*c^2*d^2*e^3 - 5*b^9*c*d*e^4)))*(21*b^2*e^2
+ 16*c^2*d^2 - 36*b*c*d*e))/(8*(b^10*e^5 - b^5*c^5*d^5 + 5*b^6*c^4*d^4*e - 10*b^7*c^3*d^3*e^2 + 10*b^8*c^2*d^2
*e^3 - 5*b^9*c*d*e^4)))*(21*b^2*e^2 + 16*c^2*d^2 - 36*b*c*d*e)*3i)/(8*(b^10*e^5 - b^5*c^5*d^5 + 5*b^6*c^4*d^4*
e - 10*b^7*c^3*d^3*e^2 + 10*b^8*c^2*d^2*e^3 - 5*b^9*c*d*e^4)))/(((567*b^7*c^5*e^10)/32 + 1728*c^12*d^7*e^3 - 6
048*b*c^11*d^6*e^4 + (1215*b^6*c^6*d*e^9)/16 + 7020*b^2*c^10*d^5*e^5 - 2430*b^3*c^9*d^4*e^6 - (1701*b^4*c^8*d^
3*e^7)/4 + (351*b^5*c^7*d^2*e^8)/8)/(b^12*c^4*d^8 + b^16*d^4*e^4 - 4*b^13*c^3*d^7*e - 4*b^15*c*d^5*e^3 + 6*b^1
4*c^2*d^6*e^2) - (3*(-c^5*(b*e - c*d)^5)^(1/2)*(((d + e*x)^(1/2)*(9*b^8*c^3*e^10 + 4608*c^11*d^8*e^2 - 18432*b
*c^10*d^7*e^3 + 36*b^7*c^4*d*e^9 + 27360*b^2*c^9*d^6*e^4 - 17568*b^3*c^8*d^5*e^5 + 3978*b^4*c^7*d^4*e^6 - 180*
b^5*c^6*d^3*e^7 + 198*b^6*c^5*d^2*e^8))/(8*(b^8*c^4*d^8 + b^12*d^4*e^4 - 4*b^9*c^3*d^7*e - 4*b^11*c*d^5*e^3 +
6*b^10*c^2*d^6*e^2)) + (3*(-c^5*(b*e - c*d)^5)^(1/2)*((24*b^10*c^8*d^8*e^3 - 96*b^11*c^7*d^7*e^4 + 141*b^12*c^
6*d^6*e^5 - 87*b^13*c^5*d^5*e^6 + 18*b^14*c^4*d^4*e^7 - 3*b^15*c^3*d^3*e^8 + 3*b^16*c^2*d^2*e^9)/(b^12*c^4*d^8
+ b^16*d^4*e^4 - 4*b^13*c^3*d^7*e - 4*b^15*c*d^5*e^3 + 6*b^14*c^2*d^6*e^2) - (3*(-c^5*(b*e - c*d)^5)^(1/2)*(d
+ e*x)^(1/2)*(21*b^2*e^2 + 16*c^2*d^2 - 36*b*c*d*e)*(128*b^10*c^7*d^9*e^2 - 576*b^11*c^6*d^8*e^3 + 1024*b^12*
c^5*d^7*e^4 - 896*b^13*c^4*d^6*e^5 + 384*b^14*c^3*d^5*e^6 - 64*b^15*c^2*d^4*e^7))/(64*(b^8*c^4*d^8 + b^12*d^4*
e^4 - 4*b^9*c^3*d^7*e - 4*b^11*c*d^5*e^3 + 6*b^10*c^2*d^6*e^2)*(b^10*e^5 - b^5*c^5*d^5 + 5*b^6*c^4*d^4*e - 10*
b^7*c^3*d^3*e^2 + 10*b^8*c^2*d^2*e^3 - 5*b^9*c*d*e^4)))*(21*b^2*e^2 + 16*c^2*d^2 - 36*b*c*d*e))/(8*(b^10*e^5 -
b^5*c^5*d^5 + 5*b^6*c^4*d^4*e - 10*b^7*c^3*d^3*e^2 + 10*b^8*c^2*d^2*e^3 - 5*b^9*c*d*e^4)))*(21*b^2*e^2 + 16*c
^2*d^2 - 36*b*c*d*e))/(8*(b^10*e^5 - b^5*c^5*d^5 + 5*b^6*c^4*d^4*e - 10*b^7*c^3*d^3*e^2 + 10*b^8*c^2*d^2*e^3 -
5*b^9*c*d*e^4)) + (3*(-c^5*(b*e - c*d)^5)^(1/2...

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