3.2304 \(\int \frac{(d+e x)^{3/2}}{(a+b x+c x^2)^3} \, dx\)

Optimal. Leaf size=441 \[ -\frac{3 \sqrt{c} \left (-4 c e \left (-d \sqrt{b^2-4 a c}-a e+4 b d\right )+b e^2 \left (3 b-2 \sqrt{b^2-4 a c}\right )+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{2 \sqrt{2} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}+\frac{3 \sqrt{c} \left (-4 c e \left (d \sqrt{b^2-4 a c}-a e+4 b d\right )+b e^2 \left (2 \sqrt{b^2-4 a c}+3 b\right )+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{2 \sqrt{2} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{\sqrt{d+e x} (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{\sqrt{d+e x} \left (4 a c e-7 b^2 e+12 c x (2 c d-b e)+12 b c d\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )} \]

[Out]

-(Sqrt[d + e*x]*(b*d - 2*a*e + (2*c*d - b*e)*x))/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) + (Sqrt[d + e*x]*(12*b*
c*d - 7*b^2*e + 4*a*c*e + 12*c*(2*c*d - b*e)*x))/(4*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)) - (3*Sqrt[c]*(16*c^2*d^
2 + b*(3*b - 2*Sqrt[b^2 - 4*a*c])*e^2 - 4*c*e*(4*b*d - Sqrt[b^2 - 4*a*c]*d - a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sq
rt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(2*Sqrt[2]*(b^2 - 4*a*c)^(5/2)*Sqrt[2*c*d - (b - Sqrt[b
^2 - 4*a*c])*e]) + (3*Sqrt[c]*(16*c^2*d^2 + b*(3*b + 2*Sqrt[b^2 - 4*a*c])*e^2 - 4*c*e*(4*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]])/(2*Sqrt[2]*(b^2
- 4*a*c)^(5/2)*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

________________________________________________________________________________________

Rubi [A]  time = 1.94249, antiderivative size = 441, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {738, 822, 826, 1166, 208} \[ -\frac{3 \sqrt{c} \left (-4 c e \left (-d \sqrt{b^2-4 a c}-a e+4 b d\right )+b e^2 \left (3 b-2 \sqrt{b^2-4 a c}\right )+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{2 \sqrt{2} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}+\frac{3 \sqrt{c} \left (-4 c e \left (d \sqrt{b^2-4 a c}-a e+4 b d\right )+b e^2 \left (2 \sqrt{b^2-4 a c}+3 b\right )+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{2 \sqrt{2} \left (b^2-4 a c\right )^{5/2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{\sqrt{d+e x} (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{\sqrt{d+e x} \left (4 a c e-7 b^2 e+12 c x (2 c d-b e)+12 b c d\right )}{4 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[d + e*x]*(b*d - 2*a*e + (2*c*d - b*e)*x))/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) + (Sqrt[d + e*x]*(12*b*
c*d - 7*b^2*e + 4*a*c*e + 12*c*(2*c*d - b*e)*x))/(4*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)) - (3*Sqrt[c]*(16*c^2*d^
2 + b*(3*b - 2*Sqrt[b^2 - 4*a*c])*e^2 - 4*c*e*(4*b*d - Sqrt[b^2 - 4*a*c]*d - a*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sq
rt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(2*Sqrt[2]*(b^2 - 4*a*c)^(5/2)*Sqrt[2*c*d - (b - Sqrt[b
^2 - 4*a*c])*e]) + (3*Sqrt[c]*(16*c^2*d^2 + b*(3*b + 2*Sqrt[b^2 - 4*a*c])*e^2 - 4*c*e*(4*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]])/(2*Sqrt[2]*(b^2
- 4*a*c)^(5/2)*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])

Rule 738

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

Rule 822

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 826

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 1166

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]

Rule 208

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

Rubi steps

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

Mathematica [B]  time = 6.26712, size = 4707, normalized size = 10.67 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((d + e*x)^(5/2)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/(2*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a +
b*x + c*x^2)^2) - (-(((d + e*x)^(5/2)*(-(a*c*e*(2*c*d - b*e)^2)/2 + ((b*c*d - b^2*e + 2*a*c*e)*(12*c^2*d^2 +
b^2*e^2 - c*e*(11*b*d - 6*a*e)))/2 + c*(-(c*e*(b*d - 2*a*e)*(2*c*d - b*e))/2 + ((2*c*d - b*e)*(12*c^2*d^2 + b^
2*e^2 - c*e*(11*b*d - 6*a*e)))/2)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2))) - (-(e*(2*c*d
- b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e))*(d + e*x)^(3/2))/2 + (2*((3*((-3*c^2*d*e*(2*c*d - b*e)*
(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*b*c*e^2*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*
b*d - 2*a*e)))/4 + (3*c*e*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) - 4*c^3*d^2*e*(11*b*d - a*e) + c^2
*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4)*Sqrt[d + e*x])/c + (4*((Sqrt[2*c*d - b*e - Sqrt[b^2 - 4*a*c]*e
]*(((3*c*e*((3*a*c*e^2*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*d*(16*c^4*d^4 +
b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) - 4*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^
2)))/4))/2 + (3*c*d*((-3*c^2*d*e*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*b*c*e^2*
(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*e*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(
2*b*d - a*e) - 4*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4))/2 - (3*b*e*((-
3*c^2*d*e*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*b*c*e^2*(2*c*d - b*e)*(12*c^2*d
^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*e*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) - 4*c^3*d^
2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4))/2)/2 - (-((-2*c*d + b*e)*((3*c*e*((3*
a*c*e^2*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*d*(16*c^4*d^4 + b^4*e^4 - 7*b^2
*c*e^3*(2*b*d - a*e) - 4*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4))/2 + (3
*c*d*((-3*c^2*d*e*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*b*c*e^2*(2*c*d - b*e)*(
12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*e*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) -
4*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4))/2 - (3*b*e*((-3*c^2*d*e*(2*c*
d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*b*c*e^2*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 -
4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*e*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) - 4*c^3*d^2*e*(11*b*d - a
*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4))/2))/2 + 2*c*((e*((3*c*d*((3*a*c*e^2*(2*c*d - b*e)*(1
2*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*d*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) - 4
*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4))/2 - (3*a*e*((-3*c^2*d*e*(2*c*d
- b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*b*c*e^2*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4
*c*e*(3*b*d - 2*a*e)))/4 + (3*c*e*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) - 4*c^3*d^2*e*(11*b*d - a*
e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4))/2))/2 - (d*((3*c*e*((3*a*c*e^2*(2*c*d - b*e)*(12*c^2*
d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*d*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) - 4*c^3*d
^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4))/2 + (3*c*d*((-3*c^2*d*e*(2*c*d - b*e
)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*b*c*e^2*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(
3*b*d - 2*a*e)))/4 + (3*c*e*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) - 4*c^3*d^2*e*(11*b*d - a*e) + c
^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4))/2 - (3*b*e*((-3*c^2*d*e*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2
- 4*c*e*(3*b*d - 2*a*e)))/4 + (3*b*c*e^2*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3
*c*e*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) - 4*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12
*a*b*d*e - 4*a^2*e^2)))/4))/2))/2))/(Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d
- b*e - Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[2]*Sqrt[c]*(-2*c*d + b*e + Sqrt[b^2 - 4*a*c]*e)) + (Sqrt[2*c*d - b*e + Sq
rt[b^2 - 4*a*c]*e]*(((3*c*e*((3*a*c*e^2*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c
*d*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) - 4*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a
*b*d*e - 4*a^2*e^2)))/4))/2 + (3*c*d*((-3*c^2*d*e*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e))
)/4 + (3*b*c*e^2*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*e*(16*c^4*d^4 + b^4*e^
4 - 7*b^2*c*e^3*(2*b*d - a*e) - 4*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4
))/2 - (3*b*e*((-3*c^2*d*e*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*b*c*e^2*(2*c*d
- b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*e*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d
- a*e) - 4*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4))/2)/2 + (-((-2*c*d +
b*e)*((3*c*e*((3*a*c*e^2*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*d*(16*c^4*d^4
+ b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) - 4*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*
e^2)))/4))/2 + (3*c*d*((-3*c^2*d*e*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*b*c*e^
2*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*e*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3
*(2*b*d - a*e) - 4*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4))/2 - (3*b*e*(
(-3*c^2*d*e*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*b*c*e^2*(2*c*d - b*e)*(12*c^2
*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*e*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) - 4*c^3*
d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4))/2))/2 + 2*c*((e*((3*c*d*((3*a*c*e^2
*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*d*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*
(2*b*d - a*e) - 4*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4))/2 - (3*a*e*((
-3*c^2*d*e*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*b*c*e^2*(2*c*d - b*e)*(12*c^2*
d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*e*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) - 4*c^3*d
^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4))/2))/2 - (d*((3*c*e*((3*a*c*e^2*(2*c*
d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*d*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d
- a*e) - 4*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4))/2 + (3*c*d*((-3*c^2
*d*e*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*b*c*e^2*(2*c*d - b*e)*(12*c^2*d^2 +
b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*c*e*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) - 4*c^3*d^2*e*(
11*b*d - a*e) + c^2*e^2*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4))/2 - (3*b*e*((-3*c^2*d*e*(2*c*d - b*e)*(12*
c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - 2*a*e)))/4 + (3*b*c*e^2*(2*c*d - b*e)*(12*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d
- 2*a*e)))/4 + (3*c*e*(16*c^4*d^4 + b^4*e^4 - 7*b^2*c*e^3*(2*b*d - a*e) - 4*c^3*d^2*e*(11*b*d - a*e) + c^2*e^2
*(41*b^2*d^2 - 12*a*b*d*e - 4*a^2*e^2)))/4))/2))/2))/(Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d +
e*x])/Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]])/(Sqrt[2]*Sqrt[c]*(-2*c*d + b*e - Sqrt[b^2 - 4*a*c]*e))))/c))/(
3*c))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)))/(2*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))

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Maple [B]  time = 0.277, size = 2582, normalized size = 5.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27*e^2/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*(e*x+d)^(3/2)*b*c^2*d^2-9/4*e^3/(16*a^2*c^2-8*a
*b^2*c+b^4)*c/(-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((e*x+d)
^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b^2-12*e/(16*a^2*c^2-8*a*b^2*c+b^4)*c^3/(-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((e*x+d)^(1/2)*c*2^(1/2)/(
(-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*d^2+9*e^4/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(16*a^2*c^2-8*a*b^2*c+b^
4)*(e*x+d)^(1/2)*a*b*c*d-3*e^3/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2/(-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((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))
*a+12*e^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2/(-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((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b*d+12*e^2/(16*a^
2*c^2-8*a*b^2*c+b^4)*c^2/(-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)*arcta
n((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b*d-9*e^3/(c*e^2*x^2+b*e^2*x+a*e^2)^
2/(16*a^2*c^2-8*a*b^2*c+b^4)*(e*x+d)^(1/2)*a*c^2*d^2-4*e^4/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(16*a^2*c^2-8*a*b^2*c+b
^4)*(e*x+d)^(3/2)*a*b*c+8*e^3/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*(e*x+d)^(3/2)*c^2*a*d-5/4
*e^4/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*(e*x+d)^(3/2)*b^3+18*e^2/(c*e^2*x^2+b*e^2*x+a*e^2)
^2*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*(e*x+d)^(5/2)*b*d+3/2*e^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c*2^(1/2)/((-b*e+2*c*d+
(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(
1/2))*b+23/2*e^3/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*(e*x+d)^(3/2)*b^2*c*d-9/4*e^3/(16*a^2*
c^2-8*a*b^2*c+b^4)*c/(-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((e
*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b^2-12*e/(16*a^2*c^2-8*a*b^2*c+b^4)*c^3/
(-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((e*x+d)^(1/2)*c*2^(1/2)
/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*d^2-3*e^3/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2/(-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((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^
2*(4*a*c-b^2))^(1/2))*c)^(1/2))*a+12*e^2/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*(e*x+d)^(1/2)*
b*c^2*d^3-3/2*e^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan((
e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*b+3*e/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2*2^
(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b
^2))^(1/2))*c)^(1/2))*d-3*e/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(
1/2)*arctanh((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))*d-27/4*e^3/(c*e^2*x^2+b*
e^2*x+a*e^2)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*(e*x+d)^(1/2)*b^2*c*d^2+e^3/(c*e^2*x^2+b*e^2*x+a*e^2)^2*c^2/(16*a^2*
c^2-8*a*b^2*c+b^4)*(e*x+d)^(5/2)*a-19/4*e^3/(c*e^2*x^2+b*e^2*x+a*e^2)^2*c/(16*a^2*c^2-8*a*b^2*c+b^4)*(e*x+d)^(
5/2)*b^2-18*e/(c*e^2*x^2+b*e^2*x+a*e^2)^2*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*(e*x+d)^(5/2)*d^2+18*e/(c*e^2*x^2+b*e
^2*x+a*e^2)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*(e*x+d)^(3/2)*c^3*d^3+3/4*e^4/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(16*a^2*c^2
-8*a*b^2*c+b^4)*(e*x+d)^(1/2)*b^3*d-6*e/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*(e*x+d)^(1/2)*c
^3*d^4-3*e^5/(c*e^2*x^2+b*e^2*x+a*e^2)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*(e*x+d)^(1/2)*a^2*c-3/4*e^5/(c*e^2*x^2+b*e
^2*x+a*e^2)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*(e*x+d)^(1/2)*a*b^2-3*e^2/(c*e^2*x^2+b*e^2*x+a*e^2)^2*c^2/(16*a^2*c^2
-8*a*b^2*c+b^4)*(e*x+d)^(7/2)*b+6*e/(c*e^2*x^2+b*e^2*x+a*e^2)^2*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*(e*x+d)^(7/2)*d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{\frac{3}{2}}}{{\left (c x^{2} + b x + a\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 6.19812, size = 24935, normalized size = 56.54 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(3*sqrt(1/2)*(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^4 + 2*(b^5*c -
8*a*b^3*c^2 + 16*a^2*b*c^3)*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*
x)*sqrt((512*c^5*d^5 - 1280*b*c^4*d^4*e + 160*(7*b^2*c^3 + 4*a*c^4)*d^3*e^2 - 80*(5*b^3*c^2 + 12*a*b*c^3)*d^2*
e^3 + 10*(5*b^4*c + 40*a*b^2*c^2 + 16*a^2*c^3)*d*e^4 - (b^5 + 40*a*b^3*c + 80*a^2*b*c^2)*e^5 + sqrt(e^10/((b^1
0*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a^5*c^7)*d^4 - 2*(b^11*c -
20*a*b^9*c^2 + 160*a^2*b^7*c^3 - 640*a^3*b^5*c^4 + 1280*a^4*b^3*c^5 - 1024*a^5*b*c^6)*d^3*e + (b^12 - 18*a*b^1
0*c + 120*a^2*b^8*c^2 - 320*a^3*b^6*c^3 + 1536*a^5*b^2*c^5 - 2048*a^6*c^6)*d^2*e^2 - 2*(a*b^11 - 20*a^2*b^9*c
+ 160*a^3*b^7*c^2 - 640*a^4*b^5*c^3 + 1280*a^5*b^3*c^4 - 1024*a^6*b*c^5)*d*e^3 + (a^2*b^10 - 20*a^3*b^8*c + 16
0*a^4*b^6*c^2 - 640*a^5*b^4*c^3 + 1280*a^6*b^2*c^4 - 1024*a^7*c^5)*e^4))*((b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6
*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6)*d^2 - (b^11 - 20*a*b^9*c + 160*a^2*b^7*c^2 - 640*a^3
*b^5*c^3 + 1280*a^4*b^3*c^4 - 1024*a^5*b*c^5)*d*e + (a*b^10 - 20*a^2*b^8*c + 160*a^3*b^6*c^2 - 640*a^4*b^4*c^3
+ 1280*a^5*b^2*c^4 - 1024*a^6*c^5)*e^2))/((b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a
^4*b^2*c^5 - 1024*a^5*c^6)*d^2 - (b^11 - 20*a*b^9*c + 160*a^2*b^7*c^2 - 640*a^3*b^5*c^3 + 1280*a^4*b^3*c^4 - 1
024*a^5*b*c^5)*d*e + (a*b^10 - 20*a^2*b^8*c + 160*a^3*b^6*c^2 - 640*a^4*b^4*c^3 + 1280*a^5*b^2*c^4 - 1024*a^6*
c^5)*e^2))*log(27/2*sqrt(1/2)*(8*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^2*e^6 - 8*(b^7*c - 1
2*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d*e^7 + (b^8 - 8*a*b^6*c + 128*a^3*b^2*c^3 - 256*a^4*c^4)*e^8 - s
qrt(e^10/((b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a^5*c^7)*d^4
- 2*(b^11*c - 20*a*b^9*c^2 + 160*a^2*b^7*c^3 - 640*a^3*b^5*c^4 + 1280*a^4*b^3*c^5 - 1024*a^5*b*c^6)*d^3*e + (b
^12 - 18*a*b^10*c + 120*a^2*b^8*c^2 - 320*a^3*b^6*c^3 + 1536*a^5*b^2*c^5 - 2048*a^6*c^6)*d^2*e^2 - 2*(a*b^11 -
20*a^2*b^9*c + 160*a^3*b^7*c^2 - 640*a^4*b^5*c^3 + 1280*a^5*b^3*c^4 - 1024*a^6*b*c^5)*d*e^3 + (a^2*b^10 - 20*
a^3*b^8*c + 160*a^4*b^6*c^2 - 640*a^5*b^4*c^3 + 1280*a^6*b^2*c^4 - 1024*a^7*c^5)*e^4))*(32*(b^10*c^4 - 20*a*b^
8*c^5 + 160*a^2*b^6*c^6 - 640*a^3*b^4*c^7 + 1280*a^4*b^2*c^8 - 1024*a^5*c^9)*d^5 - 80*(b^11*c^3 - 20*a*b^9*c^4
+ 160*a^2*b^7*c^5 - 640*a^3*b^5*c^6 + 1280*a^4*b^3*c^7 - 1024*a^5*b*c^8)*d^4*e + 2*(33*b^12*c^2 - 632*a*b^10*
c^3 + 4720*a^2*b^8*c^4 - 16640*a^3*b^6*c^5 + 24320*a^4*b^4*c^6 + 2048*a^5*b^2*c^7 - 28672*a^6*c^8)*d^3*e^2 - (
19*b^13*c - 296*a*b^11*c^2 + 1360*a^2*b^9*c^3 + 1280*a^3*b^7*c^4 - 29440*a^4*b^5*c^5 + 88064*a^5*b^3*c^6 - 860
16*a^6*b*c^7)*d^2*e^3 + (b^14 + 10*a*b^12*c - 416*a^2*b^10*c^2 + 3680*a^3*b^8*c^3 - 14080*a^4*b^6*c^4 + 22016*
a^5*b^4*c^5 - 24576*a^7*c^7)*d*e^4 - (a*b^13 - 8*a^2*b^11*c - 80*a^3*b^9*c^2 + 1280*a^4*b^7*c^3 - 6400*a^5*b^5
*c^4 + 14336*a^6*b^3*c^5 - 12288*a^7*b*c^6)*e^5))*sqrt((512*c^5*d^5 - 1280*b*c^4*d^4*e + 160*(7*b^2*c^3 + 4*a*
c^4)*d^3*e^2 - 80*(5*b^3*c^2 + 12*a*b*c^3)*d^2*e^3 + 10*(5*b^4*c + 40*a*b^2*c^2 + 16*a^2*c^3)*d*e^4 - (b^5 + 4
0*a*b^3*c + 80*a^2*b*c^2)*e^5 + sqrt(e^10/((b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^3*b^4*c^5 + 1280
*a^4*b^2*c^6 - 1024*a^5*c^7)*d^4 - 2*(b^11*c - 20*a*b^9*c^2 + 160*a^2*b^7*c^3 - 640*a^3*b^5*c^4 + 1280*a^4*b^3
*c^5 - 1024*a^5*b*c^6)*d^3*e + (b^12 - 18*a*b^10*c + 120*a^2*b^8*c^2 - 320*a^3*b^6*c^3 + 1536*a^5*b^2*c^5 - 20
48*a^6*c^6)*d^2*e^2 - 2*(a*b^11 - 20*a^2*b^9*c + 160*a^3*b^7*c^2 - 640*a^4*b^5*c^3 + 1280*a^5*b^3*c^4 - 1024*a
^6*b*c^5)*d*e^3 + (a^2*b^10 - 20*a^3*b^8*c + 160*a^4*b^6*c^2 - 640*a^5*b^4*c^3 + 1280*a^6*b^2*c^4 - 1024*a^7*c
^5)*e^4))*((b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6)*d^2 -
(b^11 - 20*a*b^9*c + 160*a^2*b^7*c^2 - 640*a^3*b^5*c^3 + 1280*a^4*b^3*c^4 - 1024*a^5*b*c^5)*d*e + (a*b^10 - 2
0*a^2*b^8*c + 160*a^3*b^6*c^2 - 640*a^4*b^4*c^3 + 1280*a^5*b^2*c^4 - 1024*a^6*c^5)*e^2))/((b^10*c - 20*a*b^8*c
^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6)*d^2 - (b^11 - 20*a*b^9*c + 160*a^2*b
^7*c^2 - 640*a^3*b^5*c^3 + 1280*a^4*b^3*c^4 - 1024*a^5*b*c^5)*d*e + (a*b^10 - 20*a^2*b^8*c + 160*a^3*b^6*c^2 -
640*a^4*b^4*c^3 + 1280*a^5*b^2*c^4 - 1024*a^6*c^5)*e^2)) + 27*(256*c^5*d^4*e^5 - 512*b*c^4*d^3*e^6 + 48*(7*b^
2*c^3 + 4*a*c^4)*d^2*e^7 - 16*(5*b^3*c^2 + 12*a*b*c^3)*d*e^8 + (5*b^4*c + 40*a*b^2*c^2 + 16*a^2*c^3)*e^9)*sqrt
(e*x + d)) - 3*sqrt(1/2)*(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^4 + 2*(b
^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*
b*c^2)*x)*sqrt((512*c^5*d^5 - 1280*b*c^4*d^4*e + 160*(7*b^2*c^3 + 4*a*c^4)*d^3*e^2 - 80*(5*b^3*c^2 + 12*a*b*c^
3)*d^2*e^3 + 10*(5*b^4*c + 40*a*b^2*c^2 + 16*a^2*c^3)*d*e^4 - (b^5 + 40*a*b^3*c + 80*a^2*b*c^2)*e^5 + sqrt(e^1
0/((b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a^5*c^7)*d^4 - 2*(b^
11*c - 20*a*b^9*c^2 + 160*a^2*b^7*c^3 - 640*a^3*b^5*c^4 + 1280*a^4*b^3*c^5 - 1024*a^5*b*c^6)*d^3*e + (b^12 - 1
8*a*b^10*c + 120*a^2*b^8*c^2 - 320*a^3*b^6*c^3 + 1536*a^5*b^2*c^5 - 2048*a^6*c^6)*d^2*e^2 - 2*(a*b^11 - 20*a^2
*b^9*c + 160*a^3*b^7*c^2 - 640*a^4*b^5*c^3 + 1280*a^5*b^3*c^4 - 1024*a^6*b*c^5)*d*e^3 + (a^2*b^10 - 20*a^3*b^8
*c + 160*a^4*b^6*c^2 - 640*a^5*b^4*c^3 + 1280*a^6*b^2*c^4 - 1024*a^7*c^5)*e^4))*((b^10*c - 20*a*b^8*c^2 + 160*
a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6)*d^2 - (b^11 - 20*a*b^9*c + 160*a^2*b^7*c^2 -
640*a^3*b^5*c^3 + 1280*a^4*b^3*c^4 - 1024*a^5*b*c^5)*d*e + (a*b^10 - 20*a^2*b^8*c + 160*a^3*b^6*c^2 - 640*a^4*
b^4*c^3 + 1280*a^5*b^2*c^4 - 1024*a^6*c^5)*e^2))/((b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 +
1280*a^4*b^2*c^5 - 1024*a^5*c^6)*d^2 - (b^11 - 20*a*b^9*c + 160*a^2*b^7*c^2 - 640*a^3*b^5*c^3 + 1280*a^4*b^3*
c^4 - 1024*a^5*b*c^5)*d*e + (a*b^10 - 20*a^2*b^8*c + 160*a^3*b^6*c^2 - 640*a^4*b^4*c^3 + 1280*a^5*b^2*c^4 - 10
24*a^6*c^5)*e^2))*log(-27/2*sqrt(1/2)*(8*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^2*e^6 - 8*(b
^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d*e^7 + (b^8 - 8*a*b^6*c + 128*a^3*b^2*c^3 - 256*a^4*c^4)
*e^8 - sqrt(e^10/((b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a^5*c
^7)*d^4 - 2*(b^11*c - 20*a*b^9*c^2 + 160*a^2*b^7*c^3 - 640*a^3*b^5*c^4 + 1280*a^4*b^3*c^5 - 1024*a^5*b*c^6)*d^
3*e + (b^12 - 18*a*b^10*c + 120*a^2*b^8*c^2 - 320*a^3*b^6*c^3 + 1536*a^5*b^2*c^5 - 2048*a^6*c^6)*d^2*e^2 - 2*(
a*b^11 - 20*a^2*b^9*c + 160*a^3*b^7*c^2 - 640*a^4*b^5*c^3 + 1280*a^5*b^3*c^4 - 1024*a^6*b*c^5)*d*e^3 + (a^2*b^
10 - 20*a^3*b^8*c + 160*a^4*b^6*c^2 - 640*a^5*b^4*c^3 + 1280*a^6*b^2*c^4 - 1024*a^7*c^5)*e^4))*(32*(b^10*c^4 -
20*a*b^8*c^5 + 160*a^2*b^6*c^6 - 640*a^3*b^4*c^7 + 1280*a^4*b^2*c^8 - 1024*a^5*c^9)*d^5 - 80*(b^11*c^3 - 20*a
*b^9*c^4 + 160*a^2*b^7*c^5 - 640*a^3*b^5*c^6 + 1280*a^4*b^3*c^7 - 1024*a^5*b*c^8)*d^4*e + 2*(33*b^12*c^2 - 632
*a*b^10*c^3 + 4720*a^2*b^8*c^4 - 16640*a^3*b^6*c^5 + 24320*a^4*b^4*c^6 + 2048*a^5*b^2*c^7 - 28672*a^6*c^8)*d^3
*e^2 - (19*b^13*c - 296*a*b^11*c^2 + 1360*a^2*b^9*c^3 + 1280*a^3*b^7*c^4 - 29440*a^4*b^5*c^5 + 88064*a^5*b^3*c
^6 - 86016*a^6*b*c^7)*d^2*e^3 + (b^14 + 10*a*b^12*c - 416*a^2*b^10*c^2 + 3680*a^3*b^8*c^3 - 14080*a^4*b^6*c^4
+ 22016*a^5*b^4*c^5 - 24576*a^7*c^7)*d*e^4 - (a*b^13 - 8*a^2*b^11*c - 80*a^3*b^9*c^2 + 1280*a^4*b^7*c^3 - 6400
*a^5*b^5*c^4 + 14336*a^6*b^3*c^5 - 12288*a^7*b*c^6)*e^5))*sqrt((512*c^5*d^5 - 1280*b*c^4*d^4*e + 160*(7*b^2*c^
3 + 4*a*c^4)*d^3*e^2 - 80*(5*b^3*c^2 + 12*a*b*c^3)*d^2*e^3 + 10*(5*b^4*c + 40*a*b^2*c^2 + 16*a^2*c^3)*d*e^4 -
(b^5 + 40*a*b^3*c + 80*a^2*b*c^2)*e^5 + sqrt(e^10/((b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^3*b^4*c^
5 + 1280*a^4*b^2*c^6 - 1024*a^5*c^7)*d^4 - 2*(b^11*c - 20*a*b^9*c^2 + 160*a^2*b^7*c^3 - 640*a^3*b^5*c^4 + 1280
*a^4*b^3*c^5 - 1024*a^5*b*c^6)*d^3*e + (b^12 - 18*a*b^10*c + 120*a^2*b^8*c^2 - 320*a^3*b^6*c^3 + 1536*a^5*b^2*
c^5 - 2048*a^6*c^6)*d^2*e^2 - 2*(a*b^11 - 20*a^2*b^9*c + 160*a^3*b^7*c^2 - 640*a^4*b^5*c^3 + 1280*a^5*b^3*c^4
- 1024*a^6*b*c^5)*d*e^3 + (a^2*b^10 - 20*a^3*b^8*c + 160*a^4*b^6*c^2 - 640*a^5*b^4*c^3 + 1280*a^6*b^2*c^4 - 10
24*a^7*c^5)*e^4))*((b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^
6)*d^2 - (b^11 - 20*a*b^9*c + 160*a^2*b^7*c^2 - 640*a^3*b^5*c^3 + 1280*a^4*b^3*c^4 - 1024*a^5*b*c^5)*d*e + (a*
b^10 - 20*a^2*b^8*c + 160*a^3*b^6*c^2 - 640*a^4*b^4*c^3 + 1280*a^5*b^2*c^4 - 1024*a^6*c^5)*e^2))/((b^10*c - 20
*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6)*d^2 - (b^11 - 20*a*b^9*c + 1
60*a^2*b^7*c^2 - 640*a^3*b^5*c^3 + 1280*a^4*b^3*c^4 - 1024*a^5*b*c^5)*d*e + (a*b^10 - 20*a^2*b^8*c + 160*a^3*b
^6*c^2 - 640*a^4*b^4*c^3 + 1280*a^5*b^2*c^4 - 1024*a^6*c^5)*e^2)) + 27*(256*c^5*d^4*e^5 - 512*b*c^4*d^3*e^6 +
48*(7*b^2*c^3 + 4*a*c^4)*d^2*e^7 - 16*(5*b^3*c^2 + 12*a*b*c^3)*d*e^8 + (5*b^4*c + 40*a*b^2*c^2 + 16*a^2*c^3)*e
^9)*sqrt(e*x + d)) + 3*sqrt(1/2)*(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^
4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^2 + 2*(a*b^5 - 8*a^2*b^3*c +
16*a^3*b*c^2)*x)*sqrt((512*c^5*d^5 - 1280*b*c^4*d^4*e + 160*(7*b^2*c^3 + 4*a*c^4)*d^3*e^2 - 80*(5*b^3*c^2 + 1
2*a*b*c^3)*d^2*e^3 + 10*(5*b^4*c + 40*a*b^2*c^2 + 16*a^2*c^3)*d*e^4 - (b^5 + 40*a*b^3*c + 80*a^2*b*c^2)*e^5 -
sqrt(e^10/((b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a^5*c^7)*d^4
- 2*(b^11*c - 20*a*b^9*c^2 + 160*a^2*b^7*c^3 - 640*a^3*b^5*c^4 + 1280*a^4*b^3*c^5 - 1024*a^5*b*c^6)*d^3*e + (
b^12 - 18*a*b^10*c + 120*a^2*b^8*c^2 - 320*a^3*b^6*c^3 + 1536*a^5*b^2*c^5 - 2048*a^6*c^6)*d^2*e^2 - 2*(a*b^11
- 20*a^2*b^9*c + 160*a^3*b^7*c^2 - 640*a^4*b^5*c^3 + 1280*a^5*b^3*c^4 - 1024*a^6*b*c^5)*d*e^3 + (a^2*b^10 - 20
*a^3*b^8*c + 160*a^4*b^6*c^2 - 640*a^5*b^4*c^3 + 1280*a^6*b^2*c^4 - 1024*a^7*c^5)*e^4))*((b^10*c - 20*a*b^8*c^
2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6)*d^2 - (b^11 - 20*a*b^9*c + 160*a^2*b^
7*c^2 - 640*a^3*b^5*c^3 + 1280*a^4*b^3*c^4 - 1024*a^5*b*c^5)*d*e + (a*b^10 - 20*a^2*b^8*c + 160*a^3*b^6*c^2 -
640*a^4*b^4*c^3 + 1280*a^5*b^2*c^4 - 1024*a^6*c^5)*e^2))/((b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b
^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6)*d^2 - (b^11 - 20*a*b^9*c + 160*a^2*b^7*c^2 - 640*a^3*b^5*c^3 + 1280*
a^4*b^3*c^4 - 1024*a^5*b*c^5)*d*e + (a*b^10 - 20*a^2*b^8*c + 160*a^3*b^6*c^2 - 640*a^4*b^4*c^3 + 1280*a^5*b^2*
c^4 - 1024*a^6*c^5)*e^2))*log(27/2*sqrt(1/2)*(8*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^2*e^6
- 8*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d*e^7 + (b^8 - 8*a*b^6*c + 128*a^3*b^2*c^3 - 256*a
^4*c^4)*e^8 + sqrt(e^10/((b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 - 102
4*a^5*c^7)*d^4 - 2*(b^11*c - 20*a*b^9*c^2 + 160*a^2*b^7*c^3 - 640*a^3*b^5*c^4 + 1280*a^4*b^3*c^5 - 1024*a^5*b*
c^6)*d^3*e + (b^12 - 18*a*b^10*c + 120*a^2*b^8*c^2 - 320*a^3*b^6*c^3 + 1536*a^5*b^2*c^5 - 2048*a^6*c^6)*d^2*e^
2 - 2*(a*b^11 - 20*a^2*b^9*c + 160*a^3*b^7*c^2 - 640*a^4*b^5*c^3 + 1280*a^5*b^3*c^4 - 1024*a^6*b*c^5)*d*e^3 +
(a^2*b^10 - 20*a^3*b^8*c + 160*a^4*b^6*c^2 - 640*a^5*b^4*c^3 + 1280*a^6*b^2*c^4 - 1024*a^7*c^5)*e^4))*(32*(b^1
0*c^4 - 20*a*b^8*c^5 + 160*a^2*b^6*c^6 - 640*a^3*b^4*c^7 + 1280*a^4*b^2*c^8 - 1024*a^5*c^9)*d^5 - 80*(b^11*c^3
- 20*a*b^9*c^4 + 160*a^2*b^7*c^5 - 640*a^3*b^5*c^6 + 1280*a^4*b^3*c^7 - 1024*a^5*b*c^8)*d^4*e + 2*(33*b^12*c^
2 - 632*a*b^10*c^3 + 4720*a^2*b^8*c^4 - 16640*a^3*b^6*c^5 + 24320*a^4*b^4*c^6 + 2048*a^5*b^2*c^7 - 28672*a^6*c
^8)*d^3*e^2 - (19*b^13*c - 296*a*b^11*c^2 + 1360*a^2*b^9*c^3 + 1280*a^3*b^7*c^4 - 29440*a^4*b^5*c^5 + 88064*a^
5*b^3*c^6 - 86016*a^6*b*c^7)*d^2*e^3 + (b^14 + 10*a*b^12*c - 416*a^2*b^10*c^2 + 3680*a^3*b^8*c^3 - 14080*a^4*b
^6*c^4 + 22016*a^5*b^4*c^5 - 24576*a^7*c^7)*d*e^4 - (a*b^13 - 8*a^2*b^11*c - 80*a^3*b^9*c^2 + 1280*a^4*b^7*c^3
- 6400*a^5*b^5*c^4 + 14336*a^6*b^3*c^5 - 12288*a^7*b*c^6)*e^5))*sqrt((512*c^5*d^5 - 1280*b*c^4*d^4*e + 160*(7
*b^2*c^3 + 4*a*c^4)*d^3*e^2 - 80*(5*b^3*c^2 + 12*a*b*c^3)*d^2*e^3 + 10*(5*b^4*c + 40*a*b^2*c^2 + 16*a^2*c^3)*d
*e^4 - (b^5 + 40*a*b^3*c + 80*a^2*b*c^2)*e^5 - sqrt(e^10/((b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^3
*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a^5*c^7)*d^4 - 2*(b^11*c - 20*a*b^9*c^2 + 160*a^2*b^7*c^3 - 640*a^3*b^5*c^4
+ 1280*a^4*b^3*c^5 - 1024*a^5*b*c^6)*d^3*e + (b^12 - 18*a*b^10*c + 120*a^2*b^8*c^2 - 320*a^3*b^6*c^3 + 1536*a
^5*b^2*c^5 - 2048*a^6*c^6)*d^2*e^2 - 2*(a*b^11 - 20*a^2*b^9*c + 160*a^3*b^7*c^2 - 640*a^4*b^5*c^3 + 1280*a^5*b
^3*c^4 - 1024*a^6*b*c^5)*d*e^3 + (a^2*b^10 - 20*a^3*b^8*c + 160*a^4*b^6*c^2 - 640*a^5*b^4*c^3 + 1280*a^6*b^2*c
^4 - 1024*a^7*c^5)*e^4))*((b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024
*a^5*c^6)*d^2 - (b^11 - 20*a*b^9*c + 160*a^2*b^7*c^2 - 640*a^3*b^5*c^3 + 1280*a^4*b^3*c^4 - 1024*a^5*b*c^5)*d*
e + (a*b^10 - 20*a^2*b^8*c + 160*a^3*b^6*c^2 - 640*a^4*b^4*c^3 + 1280*a^5*b^2*c^4 - 1024*a^6*c^5)*e^2))/((b^10
*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6)*d^2 - (b^11 - 20*a*b^
9*c + 160*a^2*b^7*c^2 - 640*a^3*b^5*c^3 + 1280*a^4*b^3*c^4 - 1024*a^5*b*c^5)*d*e + (a*b^10 - 20*a^2*b^8*c + 16
0*a^3*b^6*c^2 - 640*a^4*b^4*c^3 + 1280*a^5*b^2*c^4 - 1024*a^6*c^5)*e^2)) + 27*(256*c^5*d^4*e^5 - 512*b*c^4*d^3
*e^6 + 48*(7*b^2*c^3 + 4*a*c^4)*d^2*e^7 - 16*(5*b^3*c^2 + 12*a*b*c^3)*d*e^8 + (5*b^4*c + 40*a*b^2*c^2 + 16*a^2
*c^3)*e^9)*sqrt(e*x + d)) - 3*sqrt(1/2)*(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*
c^4)*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^2 + 2*(a*b^5 - 8*a^2*
b^3*c + 16*a^3*b*c^2)*x)*sqrt((512*c^5*d^5 - 1280*b*c^4*d^4*e + 160*(7*b^2*c^3 + 4*a*c^4)*d^3*e^2 - 80*(5*b^3*
c^2 + 12*a*b*c^3)*d^2*e^3 + 10*(5*b^4*c + 40*a*b^2*c^2 + 16*a^2*c^3)*d*e^4 - (b^5 + 40*a*b^3*c + 80*a^2*b*c^2)
*e^5 - sqrt(e^10/((b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a^5*c
^7)*d^4 - 2*(b^11*c - 20*a*b^9*c^2 + 160*a^2*b^7*c^3 - 640*a^3*b^5*c^4 + 1280*a^4*b^3*c^5 - 1024*a^5*b*c^6)*d^
3*e + (b^12 - 18*a*b^10*c + 120*a^2*b^8*c^2 - 320*a^3*b^6*c^3 + 1536*a^5*b^2*c^5 - 2048*a^6*c^6)*d^2*e^2 - 2*(
a*b^11 - 20*a^2*b^9*c + 160*a^3*b^7*c^2 - 640*a^4*b^5*c^3 + 1280*a^5*b^3*c^4 - 1024*a^6*b*c^5)*d*e^3 + (a^2*b^
10 - 20*a^3*b^8*c + 160*a^4*b^6*c^2 - 640*a^5*b^4*c^3 + 1280*a^6*b^2*c^4 - 1024*a^7*c^5)*e^4))*((b^10*c - 20*a
*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6)*d^2 - (b^11 - 20*a*b^9*c + 160
*a^2*b^7*c^2 - 640*a^3*b^5*c^3 + 1280*a^4*b^3*c^4 - 1024*a^5*b*c^5)*d*e + (a*b^10 - 20*a^2*b^8*c + 160*a^3*b^6
*c^2 - 640*a^4*b^4*c^3 + 1280*a^5*b^2*c^4 - 1024*a^6*c^5)*e^2))/((b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 64
0*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6)*d^2 - (b^11 - 20*a*b^9*c + 160*a^2*b^7*c^2 - 640*a^3*b^5*c^3
+ 1280*a^4*b^3*c^4 - 1024*a^5*b*c^5)*d*e + (a*b^10 - 20*a^2*b^8*c + 160*a^3*b^6*c^2 - 640*a^4*b^4*c^3 + 1280*a
^5*b^2*c^4 - 1024*a^6*c^5)*e^2))*log(-27/2*sqrt(1/2)*(8*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)
*d^2*e^6 - 8*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d*e^7 + (b^8 - 8*a*b^6*c + 128*a^3*b^2*c^3
- 256*a^4*c^4)*e^8 + sqrt(e^10/((b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^3*b^4*c^5 + 1280*a^4*b^2*c
^6 - 1024*a^5*c^7)*d^4 - 2*(b^11*c - 20*a*b^9*c^2 + 160*a^2*b^7*c^3 - 640*a^3*b^5*c^4 + 1280*a^4*b^3*c^5 - 102
4*a^5*b*c^6)*d^3*e + (b^12 - 18*a*b^10*c + 120*a^2*b^8*c^2 - 320*a^3*b^6*c^3 + 1536*a^5*b^2*c^5 - 2048*a^6*c^6
)*d^2*e^2 - 2*(a*b^11 - 20*a^2*b^9*c + 160*a^3*b^7*c^2 - 640*a^4*b^5*c^3 + 1280*a^5*b^3*c^4 - 1024*a^6*b*c^5)*
d*e^3 + (a^2*b^10 - 20*a^3*b^8*c + 160*a^4*b^6*c^2 - 640*a^5*b^4*c^3 + 1280*a^6*b^2*c^4 - 1024*a^7*c^5)*e^4))*
(32*(b^10*c^4 - 20*a*b^8*c^5 + 160*a^2*b^6*c^6 - 640*a^3*b^4*c^7 + 1280*a^4*b^2*c^8 - 1024*a^5*c^9)*d^5 - 80*(
b^11*c^3 - 20*a*b^9*c^4 + 160*a^2*b^7*c^5 - 640*a^3*b^5*c^6 + 1280*a^4*b^3*c^7 - 1024*a^5*b*c^8)*d^4*e + 2*(33
*b^12*c^2 - 632*a*b^10*c^3 + 4720*a^2*b^8*c^4 - 16640*a^3*b^6*c^5 + 24320*a^4*b^4*c^6 + 2048*a^5*b^2*c^7 - 286
72*a^6*c^8)*d^3*e^2 - (19*b^13*c - 296*a*b^11*c^2 + 1360*a^2*b^9*c^3 + 1280*a^3*b^7*c^4 - 29440*a^4*b^5*c^5 +
88064*a^5*b^3*c^6 - 86016*a^6*b*c^7)*d^2*e^3 + (b^14 + 10*a*b^12*c - 416*a^2*b^10*c^2 + 3680*a^3*b^8*c^3 - 140
80*a^4*b^6*c^4 + 22016*a^5*b^4*c^5 - 24576*a^7*c^7)*d*e^4 - (a*b^13 - 8*a^2*b^11*c - 80*a^3*b^9*c^2 + 1280*a^4
*b^7*c^3 - 6400*a^5*b^5*c^4 + 14336*a^6*b^3*c^5 - 12288*a^7*b*c^6)*e^5))*sqrt((512*c^5*d^5 - 1280*b*c^4*d^4*e
+ 160*(7*b^2*c^3 + 4*a*c^4)*d^3*e^2 - 80*(5*b^3*c^2 + 12*a*b*c^3)*d^2*e^3 + 10*(5*b^4*c + 40*a*b^2*c^2 + 16*a^
2*c^3)*d*e^4 - (b^5 + 40*a*b^3*c + 80*a^2*b*c^2)*e^5 - sqrt(e^10/((b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 -
640*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a^5*c^7)*d^4 - 2*(b^11*c - 20*a*b^9*c^2 + 160*a^2*b^7*c^3 - 640*a^3
*b^5*c^4 + 1280*a^4*b^3*c^5 - 1024*a^5*b*c^6)*d^3*e + (b^12 - 18*a*b^10*c + 120*a^2*b^8*c^2 - 320*a^3*b^6*c^3
+ 1536*a^5*b^2*c^5 - 2048*a^6*c^6)*d^2*e^2 - 2*(a*b^11 - 20*a^2*b^9*c + 160*a^3*b^7*c^2 - 640*a^4*b^5*c^3 + 12
80*a^5*b^3*c^4 - 1024*a^6*b*c^5)*d*e^3 + (a^2*b^10 - 20*a^3*b^8*c + 160*a^4*b^6*c^2 - 640*a^5*b^4*c^3 + 1280*a
^6*b^2*c^4 - 1024*a^7*c^5)*e^4))*((b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^
5 - 1024*a^5*c^6)*d^2 - (b^11 - 20*a*b^9*c + 160*a^2*b^7*c^2 - 640*a^3*b^5*c^3 + 1280*a^4*b^3*c^4 - 1024*a^5*b
*c^5)*d*e + (a*b^10 - 20*a^2*b^8*c + 160*a^3*b^6*c^2 - 640*a^4*b^4*c^3 + 1280*a^5*b^2*c^4 - 1024*a^6*c^5)*e^2)
)/((b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6)*d^2 - (b^11 -
20*a*b^9*c + 160*a^2*b^7*c^2 - 640*a^3*b^5*c^3 + 1280*a^4*b^3*c^4 - 1024*a^5*b*c^5)*d*e + (a*b^10 - 20*a^2*b^
8*c + 160*a^3*b^6*c^2 - 640*a^4*b^4*c^3 + 1280*a^5*b^2*c^4 - 1024*a^6*c^5)*e^2)) + 27*(256*c^5*d^4*e^5 - 512*b
*c^4*d^3*e^6 + 48*(7*b^2*c^3 + 4*a*c^4)*d^2*e^7 - 16*(5*b^3*c^2 + 12*a*b*c^3)*d*e^8 + (5*b^4*c + 40*a*b^2*c^2
+ 16*a^2*c^3)*e^9)*sqrt(e*x + d)) - 2*(12*(2*c^3*d - b*c^2*e)*x^3 + (36*b*c^2*d - (19*b^2*c - 4*a*c^2)*e)*x^2
- 2*(b^3 - 10*a*b*c)*d - 3*(a*b^2 + 4*a^2*c)*e + (8*(b^2*c + 5*a*c^2)*d - (5*b^3 + 16*a*b*c)*e)*x)*sqrt(e*x +
d))/(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^4 + 2*(b^5*c - 8*a*b^3*c^2 +
16*a^2*b*c^3)*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out