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3.2399 \(\int \frac {1}{(d+e x)^3 (a+b x+c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=621 \[ \frac {e \sqrt {a+b x+c x^2} (2 c d-b e) \left (-16 c^2 e^2 \left (81 a^2 e^2+28 a b d e+b^2 d^2\right )+40 b^2 c e^3 (19 a e+2 b d)-64 c^3 d^2 e (2 b d-7 a e)-105 b^4 e^4+64 c^4 d^4\right )}{12 \left (b^2-4 a c\right )^2 (d+e x) \left (a e^2-b d e+c d^2\right )^4}-\frac {2 \left (-c x (2 c d-b e) \left (-4 c e (2 b d-9 a e)-7 b^2 e^2+8 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (20 a c e^2-7 b^2 e^2+8 c^2 d^2\right )+8 a c e (2 c d-b e)^2\right )}{3 \left (b^2-4 a c\right )^2 (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}+\frac {5 e^4 \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \tanh ^{-1}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{9/2}}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{3 \left (b^2-4 a c\right ) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}+\frac {e \sqrt {a+b x+c x^2} \left (8 b^2 c e^3 (27 a e+8 b d)-128 c^3 d^2 e (b d-3 a e)-48 a c^2 e^3 (5 a e+8 b d)-35 b^4 e^4+64 c^4 d^4\right )}{6 \left (b^2-4 a c\right )^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^3} \]

[Out]

-2/3*(b*c*d-b^2*e+2*a*c*e+c*(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)/(e*x+d)^2/(c*x^2+b*x+a)^(3/2)+5/8
*e^4*(24*c^2*d^2+7*b^2*e^2-4*c*e*(a*e+6*b*d))*arctanh(1/2*(b*d-2*a*e+(-b*e+2*c*d)*x)/(a*e^2-b*d*e+c*d^2)^(1/2)
/(c*x^2+b*x+a)^(1/2))/(a*e^2-b*d*e+c*d^2)^(9/2)-2/3*(8*a*c*e*(-b*e+2*c*d)^2-(2*a*c*e-b^2*e+b*c*d)*(20*a*c*e^2-
7*b^2*e^2+8*c^2*d^2)-c*(-b*e+2*c*d)*(8*c^2*d^2-7*b^2*e^2-4*c*e*(-9*a*e+2*b*d))*x)/(-4*a*c+b^2)^2/(a*e^2-b*d*e+
c*d^2)^2/(e*x+d)^2/(c*x^2+b*x+a)^(1/2)+1/6*e*(64*c^4*d^4-35*b^4*e^4-128*c^3*d^2*e*(-3*a*e+b*d)-48*a*c^2*e^3*(5
*a*e+8*b*d)+8*b^2*c*e^3*(27*a*e+8*b*d))*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^2/(a*e^2-b*d*e+c*d^2)^3/(e*x+d)^2+1/1
2*e*(-b*e+2*c*d)*(64*c^4*d^4-105*b^4*e^4-64*c^3*d^2*e*(-7*a*e+2*b*d)+40*b^2*c*e^3*(19*a*e+2*b*d)-16*c^2*e^2*(8
1*a^2*e^2+28*a*b*d*e+b^2*d^2))*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^2/(a*e^2-b*d*e+c*d^2)^4/(e*x+d)

________________________________________________________________________________________

Rubi [A]  time = 1.04, antiderivative size = 621, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {740, 822, 834, 806, 724, 206} \[ \frac {e \sqrt {a+b x+c x^2} (2 c d-b e) \left (-16 c^2 e^2 \left (81 a^2 e^2+28 a b d e+b^2 d^2\right )+40 b^2 c e^3 (19 a e+2 b d)-64 c^3 d^2 e (2 b d-7 a e)-105 b^4 e^4+64 c^4 d^4\right )}{12 \left (b^2-4 a c\right )^2 (d+e x) \left (a e^2-b d e+c d^2\right )^4}+\frac {e \sqrt {a+b x+c x^2} \left (8 b^2 c e^3 (27 a e+8 b d)-128 c^3 d^2 e (b d-3 a e)-48 a c^2 e^3 (5 a e+8 b d)-35 b^4 e^4+64 c^4 d^4\right )}{6 \left (b^2-4 a c\right )^2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )^3}-\frac {2 \left (-c x (2 c d-b e) \left (-4 c e (2 b d-9 a e)-7 b^2 e^2+8 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (20 a c e^2-7 b^2 e^2+8 c^2 d^2\right )+8 a c e (2 c d-b e)^2\right )}{3 \left (b^2-4 a c\right )^2 (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}+\frac {5 e^4 \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \tanh ^{-1}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{9/2}}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{3 \left (b^2-4 a c\right ) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^3*(a + b*x + c*x^2)^(5/2)),x]

[Out]

(-2*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/(3*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^2*(a + b
*x + c*x^2)^(3/2)) - (2*(8*a*c*e*(2*c*d - b*e)^2 - (b*c*d - b^2*e + 2*a*c*e)*(8*c^2*d^2 - 7*b^2*e^2 + 20*a*c*e
^2) - c*(2*c*d - b*e)*(8*c^2*d^2 - 7*b^2*e^2 - 4*c*e*(2*b*d - 9*a*e))*x))/(3*(b^2 - 4*a*c)^2*(c*d^2 - b*d*e +
a*e^2)^2*(d + e*x)^2*Sqrt[a + b*x + c*x^2]) + (e*(64*c^4*d^4 - 35*b^4*e^4 - 128*c^3*d^2*e*(b*d - 3*a*e) - 48*a
*c^2*e^3*(8*b*d + 5*a*e) + 8*b^2*c*e^3*(8*b*d + 27*a*e))*Sqrt[a + b*x + c*x^2])/(6*(b^2 - 4*a*c)^2*(c*d^2 - b*
d*e + a*e^2)^3*(d + e*x)^2) + (e*(2*c*d - b*e)*(64*c^4*d^4 - 105*b^4*e^4 - 64*c^3*d^2*e*(2*b*d - 7*a*e) + 40*b
^2*c*e^3*(2*b*d + 19*a*e) - 16*c^2*e^2*(b^2*d^2 + 28*a*b*d*e + 81*a^2*e^2))*Sqrt[a + b*x + c*x^2])/(12*(b^2 -
4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^4*(d + e*x)) + (5*e^4*(24*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*e))*ArcTanh[
(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(8*(c*d^2 - b*d*e + a*
e^2)^(9/2))

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
 /; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 724

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

Rule 740

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 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x],
x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 0]

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 834

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

Rubi steps

\begin {align*} \int \frac {1}{(d+e x)^3 \left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \int \frac {\frac {1}{2} \left (8 c^2 d^2-7 b^2 e^2+20 a c e^2\right )+4 c e (2 c d-b e) x}{(d+e x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (8 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-7 b^2 e^2+20 a c e^2\right )-c (2 c d-b e) \left (8 c^2 d^2-7 b^2 e^2-4 c e (2 b d-9 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2 \sqrt {a+b x+c x^2}}+\frac {4 \int \frac {\frac {1}{4} e \left (8 c e (2 c d-b e) \left (b^2 d-12 a c d+4 a b e\right )+\left (4 b c d-5 b^2 e+12 a c e\right ) \left (8 c^2 d^2-7 b^2 e^2+20 a c e^2\right )\right )+c e (2 c d-b e) \left (8 c^2 d^2-7 b^2 e^2-4 c e (2 b d-9 a e)\right ) x}{(d+e x)^3 \sqrt {a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (8 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-7 b^2 e^2+20 a c e^2\right )-c (2 c d-b e) \left (8 c^2 d^2-7 b^2 e^2-4 c e (2 b d-9 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2 \sqrt {a+b x+c x^2}}+\frac {e \left (64 c^4 d^4-35 b^4 e^4-128 c^3 d^2 e (b d-3 a e)-48 a c^2 e^3 (8 b d+5 a e)+8 b^2 c e^3 (8 b d+27 a e)\right ) \sqrt {a+b x+c x^2}}{6 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}-\frac {2 \int \frac {\frac {1}{8} \left (-220 b^4 c d e^4+105 b^5 e^5+8 b^3 c e^3 \left (6 c d^2-95 a e^2\right )+64 a c^3 d e^2 \left (2 c d^2-33 a e^2\right )+96 b^2 c^2 d e^2 \left (c d^2+16 a e^2\right )-16 b c^2 e \left (4 c^2 d^4+36 a c d^2 e^2-81 a^2 e^4\right )\right )-\frac {1}{4} c e \left (64 c^4 d^4-35 b^4 e^4-128 c^3 d^2 e (b d-3 a e)-48 a c^2 e^3 (8 b d+5 a e)+8 b^2 c e^3 (8 b d+27 a e)\right ) x}{(d+e x)^2 \sqrt {a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (8 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-7 b^2 e^2+20 a c e^2\right )-c (2 c d-b e) \left (8 c^2 d^2-7 b^2 e^2-4 c e (2 b d-9 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2 \sqrt {a+b x+c x^2}}+\frac {e \left (64 c^4 d^4-35 b^4 e^4-128 c^3 d^2 e (b d-3 a e)-48 a c^2 e^3 (8 b d+5 a e)+8 b^2 c e^3 (8 b d+27 a e)\right ) \sqrt {a+b x+c x^2}}{6 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}+\frac {e (2 c d-b e) \left (64 c^4 d^4-105 b^4 e^4-64 c^3 d^2 e (2 b d-7 a e)+40 b^2 c e^3 (2 b d+19 a e)-16 c^2 e^2 \left (b^2 d^2+28 a b d e+81 a^2 e^2\right )\right ) \sqrt {a+b x+c x^2}}{12 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^4 (d+e x)}+\frac {\left (5 e^4 \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right )\right ) \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{8 \left (c d^2-b d e+a e^2\right )^4}\\ &=-\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (8 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-7 b^2 e^2+20 a c e^2\right )-c (2 c d-b e) \left (8 c^2 d^2-7 b^2 e^2-4 c e (2 b d-9 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2 \sqrt {a+b x+c x^2}}+\frac {e \left (64 c^4 d^4-35 b^4 e^4-128 c^3 d^2 e (b d-3 a e)-48 a c^2 e^3 (8 b d+5 a e)+8 b^2 c e^3 (8 b d+27 a e)\right ) \sqrt {a+b x+c x^2}}{6 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}+\frac {e (2 c d-b e) \left (64 c^4 d^4-105 b^4 e^4-64 c^3 d^2 e (2 b d-7 a e)+40 b^2 c e^3 (2 b d+19 a e)-16 c^2 e^2 \left (b^2 d^2+28 a b d e+81 a^2 e^2\right )\right ) \sqrt {a+b x+c x^2}}{12 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^4 (d+e x)}-\frac {\left (5 e^4 \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac {-b d+2 a e-(2 c d-b e) x}{\sqrt {a+b x+c x^2}}\right )}{4 \left (c d^2-b d e+a e^2\right )^4}\\ &=-\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (8 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-7 b^2 e^2+20 a c e^2\right )-c (2 c d-b e) \left (8 c^2 d^2-7 b^2 e^2-4 c e (2 b d-9 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2 \sqrt {a+b x+c x^2}}+\frac {e \left (64 c^4 d^4-35 b^4 e^4-128 c^3 d^2 e (b d-3 a e)-48 a c^2 e^3 (8 b d+5 a e)+8 b^2 c e^3 (8 b d+27 a e)\right ) \sqrt {a+b x+c x^2}}{6 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}+\frac {e (2 c d-b e) \left (64 c^4 d^4-105 b^4 e^4-64 c^3 d^2 e (2 b d-7 a e)+40 b^2 c e^3 (2 b d+19 a e)-16 c^2 e^2 \left (b^2 d^2+28 a b d e+81 a^2 e^2\right )\right ) \sqrt {a+b x+c x^2}}{12 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^4 (d+e x)}+\frac {5 e^4 \left (24 c^2 d^2+7 b^2 e^2-4 c e (6 b d+a e)\right ) \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{8 \left (c d^2-b d e+a e^2\right )^{9/2}}\\ \end {align*}

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Mathematica [A]  time = 2.76, size = 626, normalized size = 1.01 \[ \frac {2 \left (\frac {e \left (\frac {2 \sqrt {a+x (b+c x)} (2 c d-b e) \left (-16 c^2 e^2 \left (81 a^2 e^2+28 a b d e+b^2 d^2\right )+40 b^2 c e^3 (19 a e+2 b d)-64 c^3 d^2 e (2 b d-7 a e)-105 b^4 e^4+64 c^4 d^4\right )}{(d+e x) \left (e (a e-b d)+c d^2\right )}-\frac {15 e^3 \left (b^2-4 a c\right )^2 \left (-4 c e (a e+6 b d)+7 b^2 e^2+24 c^2 d^2\right ) \tanh ^{-1}\left (\frac {2 a e-b d+b e x-2 c d x}{2 \sqrt {a+x (b+c x)} \sqrt {e (a e-b d)+c d^2}}\right )}{\left (e (a e-b d)+c d^2\right )^{3/2}}+\frac {4 \sqrt {a+x (b+c x)} \left (8 b^2 c e^3 (27 a e+8 b d)-128 c^3 d^2 e (b d-3 a e)-48 a c^2 e^3 (5 a e+8 b d)-35 b^4 e^4+64 c^4 d^4\right )}{(d+e x)^2}\right )}{16 \left (b^2-4 a c\right ) \left (e (a e-b d)+c d^2\right )^2}+\frac {-8 c^2 \left (5 a^2 e^3+a c d e (9 e x-2 d)+2 c^2 d^3 x\right )+2 b^2 c e \left (21 a e^2+c d (4 d+3 e x)\right )-4 b c^2 \left (a e^2 (13 d-9 e x)+2 c d^2 (d-3 e x)\right )-7 b^4 e^3+7 b^3 c e^2 (d-e x)}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+x (b+c x)} \left (e (b d-a e)-c d^2\right )}+\frac {-2 c (a e+c d x)+b^2 e+b c (e x-d)}{(d+e x)^2 (a+x (b+c x))^{3/2}}\right )}{3 \left (b^2-4 a c\right ) \left (e (a e-b d)+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^3*(a + b*x + c*x^2)^(5/2)),x]

[Out]

(2*((b^2*e - 2*c*(a*e + c*d*x) + b*c*(-d + e*x))/((d + e*x)^2*(a + x*(b + c*x))^(3/2)) + (-7*b^4*e^3 + 7*b^3*c
*e^2*(d - e*x) - 4*b*c^2*(a*e^2*(13*d - 9*e*x) + 2*c*d^2*(d - 3*e*x)) + 2*b^2*c*e*(21*a*e^2 + c*d*(4*d + 3*e*x
)) - 8*c^2*(5*a^2*e^3 + 2*c^2*d^3*x + a*c*d*e*(-2*d + 9*e*x)))/((b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*(d +
e*x)^2*Sqrt[a + x*(b + c*x)]) + (e*((4*(64*c^4*d^4 - 35*b^4*e^4 - 128*c^3*d^2*e*(b*d - 3*a*e) - 48*a*c^2*e^3*(
8*b*d + 5*a*e) + 8*b^2*c*e^3*(8*b*d + 27*a*e))*Sqrt[a + x*(b + c*x)])/(d + e*x)^2 + (2*(2*c*d - b*e)*(64*c^4*d
^4 - 105*b^4*e^4 - 64*c^3*d^2*e*(2*b*d - 7*a*e) + 40*b^2*c*e^3*(2*b*d + 19*a*e) - 16*c^2*e^2*(b^2*d^2 + 28*a*b
*d*e + 81*a^2*e^2))*Sqrt[a + x*(b + c*x)])/((c*d^2 + e*(-(b*d) + a*e))*(d + e*x)) - (15*(b^2 - 4*a*c)^2*e^3*(2
4*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(6*b*d + a*e))*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b
*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(c*d^2 + e*(-(b*d) + a*e))^(3/2)))/(16*(b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) +
 a*e))^2)))/(3*(b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e)))

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fricas [B]  time = 124.96, size = 14440, normalized size = 23.25 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")

[Out]

[-1/48*(15*(24*(a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*d^4*e^4 - 24*(a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c
^3)*d^3*e^5 + (7*a^2*b^6 - 60*a^3*b^4*c + 144*a^4*b^2*c^2 - 64*a^5*c^3)*d^2*e^6 + (24*(b^4*c^4 - 8*a*b^2*c^5 +
 16*a^2*c^6)*d^2*e^6 - 24*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*d*e^7 + (7*b^6*c^2 - 60*a*b^4*c^3 + 144*a^2*b
^2*c^4 - 64*a^3*c^5)*e^8)*x^6 + 2*(24*(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*d^3*e^5 - (17*b^6*c^2 - 132*a*b^4*c
^3 + 240*a^2*b^2*c^4 + 64*a^3*c^5)*d*e^7 + (7*b^7*c - 60*a*b^5*c^2 + 144*a^2*b^3*c^3 - 64*a^3*b*c^4)*e^8)*x^5
+ (24*(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*d^4*e^4 + 72*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*d^3*e^5 - (65*b
^6*c^2 - 564*a*b^4*c^3 + 1392*a^2*b^2*c^4 - 704*a^3*c^5)*d^2*e^6 + 4*(b^7*c - 24*a*b^5*c^2 + 144*a^2*b^3*c^3 -
 256*a^3*b*c^4)*d*e^7 + (7*b^8 - 46*a*b^6*c + 24*a^2*b^4*c^2 + 224*a^3*b^2*c^3 - 128*a^4*c^4)*e^8)*x^4 + 2*(24
*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*d^4*e^4 + 48*(a*b^4*c^3 - 8*a^2*b^2*c^4 + 16*a^3*c^5)*d^3*e^5 - (17*b^
7*c - 108*a*b^5*c^2 + 48*a^2*b^3*c^3 + 448*a^3*b*c^4)*d^2*e^6 + (7*b^8 - 70*a*b^6*c + 216*a^2*b^4*c^2 - 160*a^
3*b^2*c^3 - 128*a^4*c^4)*d*e^7 + (7*a*b^7 - 60*a^2*b^5*c + 144*a^3*b^3*c^2 - 64*a^4*b*c^3)*e^8)*x^3 + (24*(b^6
*c^2 - 6*a*b^4*c^3 + 32*a^3*c^5)*d^4*e^4 - 24*(b^7*c - 10*a*b^5*c^2 + 32*a^2*b^3*c^3 - 32*a^3*b*c^4)*d^3*e^5 +
 (7*b^8 - 142*a*b^6*c + 816*a^2*b^4*c^2 - 1504*a^3*b^2*c^3 + 256*a^4*c^4)*d^2*e^6 + 4*(7*a*b^7 - 66*a^2*b^5*c
+ 192*a^3*b^3*c^2 - 160*a^4*b*c^3)*d*e^7 + (7*a^2*b^6 - 60*a^3*b^4*c + 144*a^4*b^2*c^2 - 64*a^5*c^3)*e^8)*x^2
+ 2*(24*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c^4)*d^4*e^4 - 24*(a*b^6*c - 9*a^2*b^4*c^2 + 24*a^3*b^2*c^3 - 16
*a^4*c^4)*d^3*e^5 + 7*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d^2*e^6 + (7*a^2*b^6 - 60*a^3*b^4
*c + 144*a^4*b^2*c^2 - 64*a^5*c^3)*d*e^7)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*log((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4
*a*c)*d^2 - (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 + 4*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x +
 a)*(b*d - 2*a*e + (2*c*d - b*e)*x) - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)/(e^2*x^2 + 2*d*e*x +
d^2)) + 4*(8*(b^3*c^5 - 12*a*b*c^6)*d^9 - 8*(5*b^4*c^4 - 54*a*b^2*c^5 + 24*a^2*c^6)*d^8*e + 8*(10*b^5*c^3 - 85
*a*b^3*c^4 + 12*a^2*b*c^5)*d^7*e^2 - 80*(b^6*c^2 - 4*a*b^4*c^3 - 21*a^2*b^2*c^4 + 28*a^3*c^5)*d^6*e^3 + 40*(b^
7*c + 6*a*b^5*c^2 - 85*a^2*b^3*c^3 + 124*a^3*b*c^4)*d^5*e^4 - 8*(b^8 + 38*a*b^6*c - 300*a^2*b^4*c^2 + 430*a^3*
b^2*c^3 + 56*a^4*c^4)*d^4*e^5 + (88*a*b^7 - 543*a^2*b^5*c + 480*a^3*b^3*c^2 + 688*a^4*b*c^3)*d^3*e^6 - (41*a^2
*b^6 - 390*a^3*b^4*c + 1296*a^4*b^2*c^2 - 1696*a^5*c^3)*d^2*e^7 - 45*(a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*d*
e^8 + 6*(a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2)*e^9 - (128*c^8*d^7*e^2 - 448*b*c^7*d^6*e^3 + 32*(13*b^2*c^6 + 32*
a*c^7)*d^5*e^4 + 80*(b^3*c^5 - 32*a*b*c^6)*d^4*e^5 - 2*(233*b^4*c^4 - 1704*a*b^2*c^5 + 848*a^2*c^6)*d^3*e^6 +
(395*b^5*c^3 - 2552*a*b^3*c^4 + 2544*a^2*b*c^5)*d^2*e^7 - (105*b^6*c^2 - 470*a*b^4*c^3 - 672*a^2*b^2*c^4 + 259
2*a^3*c^5)*d*e^8 + (105*a*b^5*c^2 - 760*a^2*b^3*c^3 + 1296*a^3*b*c^4)*e^9)*x^5 - (256*c^8*d^8*e - 704*b*c^7*d^
7*e^2 + 32*(5*b^2*c^6 + 64*a*c^7)*d^6*e^3 + 112*(7*b^3*c^5 - 32*a*b*c^6)*d^5*e^4 - 8*(79*b^4*c^4 - 192*a*b^2*c
^5 + 64*a^2*c^6)*d^4*e^5 - (269*b^5*c^3 - 2888*a*b^3*c^4 + 3216*a^2*b*c^5)*d^3*e^6 + (615*b^6*c^2 - 4598*a*b^4
*c^3 + 7680*a^2*b^2*c^4 - 2784*a^3*c^5)*d^2*e^7 - (210*b^7*c - 1185*a*b^5*c^2 + 152*a^2*b^3*c^3 + 3696*a^3*b*c
^4)*d*e^8 + 6*(35*a*b^6*c - 265*a^2*b^4*c^2 + 504*a^3*b^2*c^3 - 80*a^4*c^4)*e^9)*x^4 - (128*c^8*d^9 - 64*b*c^7
*d^8*e - 16*(55*b^2*c^6 - 76*a*c^7)*d^7*e^2 + 40*(29*b^3*c^5 - 4*a*b*c^6)*d^6*e^3 + 8*(25*b^4*c^4 - 678*a*b^2*
c^5 + 520*a^2*c^6)*d^5*e^4 - 56*(23*b^5*c^3 - 167*a*b^3*c^4 + 204*a^2*b*c^5)*d^4*e^5 + 4*(201*b^6*c^2 - 1393*a
*b^4*c^3 + 2340*a^2*b^2*c^4 - 384*a^3*c^5)*d^3*e^6 + (45*b^7*c - 582*a*b^5*c^2 + 2072*a^2*b^3*c^3 - 3264*a^3*b
*c^4)*d^2*e^7 - 3*(35*b^8 - 250*a*b^6*c + 516*a^2*b^4*c^2 - 848*a^3*b^2*c^3 + 1536*a^4*c^4)*d*e^8 + 3*(35*a*b^
7 - 230*a^2*b^5*c + 232*a^3*b^3*c^2 + 448*a^4*b*c^3)*e^9)*x^3 - (192*b*c^7*d^9 - 192*(3*b^2*c^6 - 2*a*c^7)*d^8
*e + 8*(35*b^3*c^5 + 36*a*b*c^6)*d^7*e^2 + 8*(65*b^4*c^4 - 330*a*b^2*c^5 + 504*a^2*c^6)*d^6*e^3 - 8*(55*b^5*c^
3 - 235*a*b^3*c^4 + 732*a^2*b*c^5)*d^5*e^4 - 16*(22*b^6*c^2 - 160*a*b^4*c^3 + 57*a^2*b^2*c^4 + 28*a^3*c^5)*d^4
*e^5 + (551*b^7*c - 3894*a*b^5*c^2 + 5992*a^2*b^3*c^3 - 1408*a^3*b*c^4)*d^3*e^6 - (175*b^8 - 1282*a*b^6*c + 27
24*a^2*b^4*c^2 - 3344*a^3*b^2*c^3 + 4736*a^4*c^4)*d^2*e^7 + (35*a*b^7 - 402*a^2*b^5*c + 1608*a^3*b^3*c^2 - 217
6*a^4*b*c^3)*d*e^8 + 4*(35*a^2*b^6 - 279*a^3*b^4*c + 588*a^4*b^2*c^2 - 160*a^5*c^3)*e^9)*x^2 - (48*(b^2*c^6 +
4*a*c^7)*d^9 - 8*(23*b^3*c^5 + 60*a*b*c^6)*d^8*e + 8*(25*b^4*c^4 + 54*a*b^2*c^5 + 168*a^2*c^6)*d^7*e^2 + 8*(10
*b^5*c^3 - 145*a*b^3*c^4 - 84*a^2*b*c^5)*d^6*e^3 - 80*(4*b^6*c^2 - 31*a*b^4*c^3 + 57*a^2*b^2*c^4 - 44*a^3*c^5)
*d^5*e^4 + 8*(29*b^7*c - 294*a*b^5*c^2 + 1015*a^2*b^3*c^3 - 1156*a^3*b*c^4)*d^4*e^5 - 2*(28*b^8 - 535*a*b^6*c
+ 2805*a^2*b^4*c^2 - 4064*a^3*b^2*c^3 - 208*a^4*c^4)*d^3*e^6 - (182*a*b^7 - 951*a^2*b^5*c - 624*a^3*b^3*c^2 +
5360*a^4*b*c^3)*d^2*e^7 + (217*a^2*b^6 - 1770*a^3*b^4*c + 4032*a^4*b^2*c^2 - 1952*a^5*c^3)*d*e^8 + 21*(a^3*b^5
 - 8*a^4*b^3*c + 16*a^5*b*c^2)*e^9)*x)*sqrt(c*x^2 + b*x + a))/((a^2*b^4*c^5 - 8*a^3*b^2*c^6 + 16*a^4*c^7)*d^12
 - 5*(a^2*b^5*c^4 - 8*a^3*b^3*c^5 + 16*a^4*b*c^6)*d^11*e + 5*(2*a^2*b^6*c^3 - 15*a^3*b^4*c^4 + 24*a^4*b^2*c^5
+ 16*a^5*c^6)*d^10*e^2 - 10*(a^2*b^7*c^2 - 6*a^3*b^5*c^3 + 32*a^5*b*c^5)*d^9*e^3 + 5*(a^2*b^8*c - 2*a^3*b^6*c^
2 - 30*a^4*b^4*c^3 + 80*a^5*b^2*c^4 + 32*a^6*c^5)*d^8*e^4 - (a^2*b^9 + 12*a^3*b^7*c - 114*a^4*b^5*c^2 + 80*a^5
*b^3*c^3 + 480*a^6*b*c^4)*d^7*e^5 + 5*(a^3*b^8 - 2*a^4*b^6*c - 30*a^5*b^4*c^2 + 80*a^6*b^2*c^3 + 32*a^7*c^4)*d
^6*e^6 - 10*(a^4*b^7 - 6*a^5*b^5*c + 32*a^7*b*c^3)*d^5*e^7 + 5*(2*a^5*b^6 - 15*a^6*b^4*c + 24*a^7*b^2*c^2 + 16
*a^8*c^3)*d^4*e^8 - 5*(a^6*b^5 - 8*a^7*b^3*c + 16*a^8*b*c^2)*d^3*e^9 + (a^7*b^4 - 8*a^8*b^2*c + 16*a^9*c^2)*d^
2*e^10 + ((b^4*c^7 - 8*a*b^2*c^8 + 16*a^2*c^9)*d^10*e^2 - 5*(b^5*c^6 - 8*a*b^3*c^7 + 16*a^2*b*c^8)*d^9*e^3 + 5
*(2*b^6*c^5 - 15*a*b^4*c^6 + 24*a^2*b^2*c^7 + 16*a^3*c^8)*d^8*e^4 - 10*(b^7*c^4 - 6*a*b^5*c^5 + 32*a^3*b*c^7)*
d^7*e^5 + 5*(b^8*c^3 - 2*a*b^6*c^4 - 30*a^2*b^4*c^5 + 80*a^3*b^2*c^6 + 32*a^4*c^7)*d^6*e^6 - (b^9*c^2 + 12*a*b
^7*c^3 - 114*a^2*b^5*c^4 + 80*a^3*b^3*c^5 + 480*a^4*b*c^6)*d^5*e^7 + 5*(a*b^8*c^2 - 2*a^2*b^6*c^3 - 30*a^3*b^4
*c^4 + 80*a^4*b^2*c^5 + 32*a^5*c^6)*d^4*e^8 - 10*(a^2*b^7*c^2 - 6*a^3*b^5*c^3 + 32*a^5*b*c^5)*d^3*e^9 + 5*(2*a
^3*b^6*c^2 - 15*a^4*b^4*c^3 + 24*a^5*b^2*c^4 + 16*a^6*c^5)*d^2*e^10 - 5*(a^4*b^5*c^2 - 8*a^5*b^3*c^3 + 16*a^6*
b*c^4)*d*e^11 + (a^5*b^4*c^2 - 8*a^6*b^2*c^3 + 16*a^7*c^4)*e^12)*x^6 + 2*((b^4*c^7 - 8*a*b^2*c^8 + 16*a^2*c^9)
*d^11*e - 4*(b^5*c^6 - 8*a*b^3*c^7 + 16*a^2*b*c^8)*d^10*e^2 + 5*(b^6*c^5 - 7*a*b^4*c^6 + 8*a^2*b^2*c^7 + 16*a^
3*c^8)*d^9*e^3 - 15*(a*b^5*c^5 - 8*a^2*b^3*c^6 + 16*a^3*b*c^7)*d^8*e^4 - 5*(b^8*c^3 - 10*a*b^6*c^4 + 30*a^2*b^
4*c^5 - 16*a^3*b^2*c^6 - 32*a^4*c^7)*d^7*e^5 + 2*(2*b^9*c^2 - 11*a*b^7*c^3 - 18*a^2*b^5*c^4 + 160*a^3*b^3*c^5
- 160*a^4*b*c^6)*d^6*e^6 - (b^10*c + 7*a*b^8*c^2 - 104*a^2*b^6*c^3 + 230*a^3*b^4*c^4 + 80*a^4*b^2*c^5 - 160*a^
5*c^6)*d^5*e^7 + 5*(a*b^9*c - 4*a^2*b^7*c^2 - 18*a^3*b^5*c^3 + 80*a^4*b^3*c^4 - 32*a^5*b*c^5)*d^4*e^8 - 5*(2*a
^2*b^8*c - 14*a^3*b^6*c^2 + 15*a^4*b^4*c^3 + 40*a^5*b^2*c^4 - 16*a^6*c^5)*d^3*e^9 + 10*(a^3*b^7*c - 8*a^4*b^5*
c^2 + 16*a^5*b^3*c^3)*d^2*e^10 - (5*a^4*b^6*c - 41*a^5*b^4*c^2 + 88*a^6*b^2*c^3 - 16*a^7*c^4)*d*e^11 + (a^5*b^
5*c - 8*a^6*b^3*c^2 + 16*a^7*b*c^3)*e^12)*x^5 + ((b^4*c^7 - 8*a*b^2*c^8 + 16*a^2*c^9)*d^12 - (b^5*c^6 - 8*a*b^
3*c^7 + 16*a^2*b*c^8)*d^11*e - (9*b^6*c^5 - 79*a*b^4*c^6 + 200*a^2*b^2*c^7 - 112*a^3*c^8)*d^10*e^2 + 5*(5*b^7*
c^4 - 42*a*b^5*c^5 + 96*a^2*b^3*c^6 - 32*a^3*b*c^7)*d^9*e^3 - 5*(5*b^8*c^3 - 35*a*b^6*c^4 + 36*a^2*b^4*c^5 + 1
12*a^3*b^2*c^6 - 64*a^4*c^7)*d^8*e^4 + 3*(3*b^9*c^2 - 4*a*b^7*c^3 - 122*a^2*b^5*c^4 + 400*a^3*b^3*c^5 - 160*a^
4*b*c^6)*d^7*e^5 + (b^10*c - 43*a*b^8*c^2 + 276*a^2*b^6*c^3 - 370*a^3*b^4*c^4 - 560*a^4*b^2*c^5 + 480*a^5*c^6)
*d^6*e^6 - (b^11 - 6*a*b^9*c - 40*a^2*b^7*c^2 + 392*a^3*b^5*c^3 - 960*a^4*b^3*c^4 + 640*a^5*b*c^5)*d^5*e^7 + 5
*(a*b^10 - 8*a^2*b^8*c + 16*a^3*b^6*c^2 + 5*a^4*b^4*c^3 - 40*a^5*b^2*c^4 + 80*a^6*c^5)*d^4*e^8 - 5*(2*a^2*b^9
- 16*a^3*b^7*c + 37*a^4*b^5*c^2 - 40*a^5*b^3*c^3 + 80*a^6*b*c^4)*d^3*e^9 + (10*a^3*b^8 - 75*a^4*b^6*c + 131*a^
5*b^4*c^2 - 8*a^6*b^2*c^3 + 176*a^7*c^4)*d^2*e^10 - (5*a^4*b^7 - 34*a^5*b^5*c + 32*a^6*b^3*c^2 + 96*a^7*b*c^3)
*d*e^11 + (a^5*b^6 - 6*a^6*b^4*c + 32*a^8*c^3)*e^12)*x^4 + 2*((b^5*c^6 - 8*a*b^3*c^7 + 16*a^2*b*c^8)*d^12 - 2*
(2*b^6*c^5 - 17*a*b^4*c^6 + 40*a^2*b^2*c^7 - 16*a^3*c^8)*d^11*e + (5*b^7*c^4 - 44*a*b^5*c^5 + 112*a^2*b^3*c^6
- 64*a^3*b*c^7)*d^10*e^2 + 10*(a^2*b^4*c^5 - 8*a^3*b^2*c^6 + 16*a^4*c^7)*d^9*e^3 - 5*(b^9*c^2 - 8*a*b^7*c^3 +
21*a^2*b^5*c^4 - 40*a^3*b^3*c^5 + 80*a^4*b*c^6)*d^8*e^4 + 2*(2*b^10*c - 11*a*b^8*c^2 + 2*a^2*b^6*c^3 + 10*a^3*
b^4*c^4 + 80*a^4*b^2*c^5 + 160*a^5*c^6)*d^7*e^5 - (b^11 + 4*a*b^9*c - 70*a^2*b^7*c^2 + 152*a^3*b^5*c^3 - 160*a
^4*b^3*c^4 + 640*a^5*b*c^5)*d^6*e^6 + 2*(2*a*b^10 - 11*a^2*b^8*c + 2*a^3*b^6*c^2 + 10*a^4*b^4*c^3 + 80*a^5*b^2
*c^4 + 160*a^6*c^5)*d^5*e^7 - 5*(a^2*b^9 - 8*a^3*b^7*c + 21*a^4*b^5*c^2 - 40*a^5*b^3*c^3 + 80*a^6*b*c^4)*d^4*e
^8 + 10*(a^5*b^4*c^2 - 8*a^6*b^2*c^3 + 16*a^7*c^4)*d^3*e^9 + (5*a^4*b^7 - 44*a^5*b^5*c + 112*a^6*b^3*c^2 - 64*
a^7*b*c^3)*d^2*e^10 - 2*(2*a^5*b^6 - 17*a^6*b^4*c + 40*a^7*b^2*c^2 - 16*a^8*c^3)*d*e^11 + (a^6*b^5 - 8*a^7*b^3
*c + 16*a^8*b*c^2)*e^12)*x^3 + ((b^6*c^5 - 6*a*b^4*c^6 + 32*a^3*c^8)*d^12 - (5*b^7*c^4 - 34*a*b^5*c^5 + 32*a^2
*b^3*c^6 + 96*a^3*b*c^7)*d^11*e + (10*b^8*c^3 - 75*a*b^6*c^4 + 131*a^2*b^4*c^5 - 8*a^3*b^2*c^6 + 176*a^4*c^7)*
d^10*e^2 - 5*(2*b^9*c^2 - 16*a*b^7*c^3 + 37*a^2*b^5*c^4 - 40*a^3*b^3*c^5 + 80*a^4*b*c^6)*d^9*e^3 + 5*(b^10*c -
 8*a*b^8*c^2 + 16*a^2*b^6*c^3 + 5*a^3*b^4*c^4 - 40*a^4*b^2*c^5 + 80*a^5*c^6)*d^8*e^4 - (b^11 - 6*a*b^9*c - 40*
a^2*b^7*c^2 + 392*a^3*b^5*c^3 - 960*a^4*b^3*c^4 + 640*a^5*b*c^5)*d^7*e^5 + (a*b^10 - 43*a^2*b^8*c + 276*a^3*b^
6*c^2 - 370*a^4*b^4*c^3 - 560*a^5*b^2*c^4 + 480*a^6*c^5)*d^6*e^6 + 3*(3*a^2*b^9 - 4*a^3*b^7*c - 122*a^4*b^5*c^
2 + 400*a^5*b^3*c^3 - 160*a^6*b*c^4)*d^5*e^7 - 5*(5*a^3*b^8 - 35*a^4*b^6*c + 36*a^5*b^4*c^2 + 112*a^6*b^2*c^3
- 64*a^7*c^4)*d^4*e^8 + 5*(5*a^4*b^7 - 42*a^5*b^5*c + 96*a^6*b^3*c^2 - 32*a^7*b*c^3)*d^3*e^9 - (9*a^5*b^6 - 79
*a^6*b^4*c + 200*a^7*b^2*c^2 - 112*a^8*c^3)*d^2*e^10 - (a^6*b^5 - 8*a^7*b^3*c + 16*a^8*b*c^2)*d*e^11 + (a^7*b^
4 - 8*a^8*b^2*c + 16*a^9*c^2)*e^12)*x^2 + 2*((a*b^5*c^5 - 8*a^2*b^3*c^6 + 16*a^3*b*c^7)*d^12 - (5*a*b^6*c^4 -
41*a^2*b^4*c^5 + 88*a^3*b^2*c^6 - 16*a^4*c^7)*d^11*e + 10*(a*b^7*c^3 - 8*a^2*b^5*c^4 + 16*a^3*b^3*c^5)*d^10*e^
2 - 5*(2*a*b^8*c^2 - 14*a^2*b^6*c^3 + 15*a^3*b^4*c^4 + 40*a^4*b^2*c^5 - 16*a^5*c^6)*d^9*e^3 + 5*(a*b^9*c - 4*a
^2*b^7*c^2 - 18*a^3*b^5*c^3 + 80*a^4*b^3*c^4 - 32*a^5*b*c^5)*d^8*e^4 - (a*b^10 + 7*a^2*b^8*c - 104*a^3*b^6*c^2
 + 230*a^4*b^4*c^3 + 80*a^5*b^2*c^4 - 160*a^6*c^5)*d^7*e^5 + 2*(2*a^2*b^9 - 11*a^3*b^7*c - 18*a^4*b^5*c^2 + 16
0*a^5*b^3*c^3 - 160*a^6*b*c^4)*d^6*e^6 - 5*(a^3*b^8 - 10*a^4*b^6*c + 30*a^5*b^4*c^2 - 16*a^6*b^2*c^3 - 32*a^7*
c^4)*d^5*e^7 - 15*(a^5*b^5*c - 8*a^6*b^3*c^2 + 16*a^7*b*c^3)*d^4*e^8 + 5*(a^5*b^6 - 7*a^6*b^4*c + 8*a^7*b^2*c^
2 + 16*a^8*c^3)*d^3*e^9 - 4*(a^6*b^5 - 8*a^7*b^3*c + 16*a^8*b*c^2)*d^2*e^10 + (a^7*b^4 - 8*a^8*b^2*c + 16*a^9*
c^2)*d*e^11)*x), 1/24*(15*(24*(a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*d^4*e^4 - 24*(a^2*b^5*c - 8*a^3*b^3*c
^2 + 16*a^4*b*c^3)*d^3*e^5 + (7*a^2*b^6 - 60*a^3*b^4*c + 144*a^4*b^2*c^2 - 64*a^5*c^3)*d^2*e^6 + (24*(b^4*c^4
- 8*a*b^2*c^5 + 16*a^2*c^6)*d^2*e^6 - 24*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*d*e^7 + (7*b^6*c^2 - 60*a*b^4*
c^3 + 144*a^2*b^2*c^4 - 64*a^3*c^5)*e^8)*x^6 + 2*(24*(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*d^3*e^5 - (17*b^6*c^
2 - 132*a*b^4*c^3 + 240*a^2*b^2*c^4 + 64*a^3*c^5)*d*e^7 + (7*b^7*c - 60*a*b^5*c^2 + 144*a^2*b^3*c^3 - 64*a^3*b
*c^4)*e^8)*x^5 + (24*(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*d^4*e^4 + 72*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*
d^3*e^5 - (65*b^6*c^2 - 564*a*b^4*c^3 + 1392*a^2*b^2*c^4 - 704*a^3*c^5)*d^2*e^6 + 4*(b^7*c - 24*a*b^5*c^2 + 14
4*a^2*b^3*c^3 - 256*a^3*b*c^4)*d*e^7 + (7*b^8 - 46*a*b^6*c + 24*a^2*b^4*c^2 + 224*a^3*b^2*c^3 - 128*a^4*c^4)*e
^8)*x^4 + 2*(24*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*d^4*e^4 + 48*(a*b^4*c^3 - 8*a^2*b^2*c^4 + 16*a^3*c^5)*d
^3*e^5 - (17*b^7*c - 108*a*b^5*c^2 + 48*a^2*b^3*c^3 + 448*a^3*b*c^4)*d^2*e^6 + (7*b^8 - 70*a*b^6*c + 216*a^2*b
^4*c^2 - 160*a^3*b^2*c^3 - 128*a^4*c^4)*d*e^7 + (7*a*b^7 - 60*a^2*b^5*c + 144*a^3*b^3*c^2 - 64*a^4*b*c^3)*e^8)
*x^3 + (24*(b^6*c^2 - 6*a*b^4*c^3 + 32*a^3*c^5)*d^4*e^4 - 24*(b^7*c - 10*a*b^5*c^2 + 32*a^2*b^3*c^3 - 32*a^3*b
*c^4)*d^3*e^5 + (7*b^8 - 142*a*b^6*c + 816*a^2*b^4*c^2 - 1504*a^3*b^2*c^3 + 256*a^4*c^4)*d^2*e^6 + 4*(7*a*b^7
- 66*a^2*b^5*c + 192*a^3*b^3*c^2 - 160*a^4*b*c^3)*d*e^7 + (7*a^2*b^6 - 60*a^3*b^4*c + 144*a^4*b^2*c^2 - 64*a^5
*c^3)*e^8)*x^2 + 2*(24*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c^4)*d^4*e^4 - 24*(a*b^6*c - 9*a^2*b^4*c^2 + 24*a
^3*b^2*c^3 - 16*a^4*c^4)*d^3*e^5 + 7*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d^2*e^6 + (7*a^2*b
^6 - 60*a^3*b^4*c + 144*a^4*b^2*c^2 - 64*a^5*c^3)*d*e^7)*x)*sqrt(-c*d^2 + b*d*e - a*e^2)*arctan(-1/2*sqrt(-c*d
^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x)/(a*c*d^2 - a*b*d*e + a^2*e^2 + (c^2*
d^2 - b*c*d*e + a*c*e^2)*x^2 + (b*c*d^2 - b^2*d*e + a*b*e^2)*x)) - 2*(8*(b^3*c^5 - 12*a*b*c^6)*d^9 - 8*(5*b^4*
c^4 - 54*a*b^2*c^5 + 24*a^2*c^6)*d^8*e + 8*(10*b^5*c^3 - 85*a*b^3*c^4 + 12*a^2*b*c^5)*d^7*e^2 - 80*(b^6*c^2 -
4*a*b^4*c^3 - 21*a^2*b^2*c^4 + 28*a^3*c^5)*d^6*e^3 + 40*(b^7*c + 6*a*b^5*c^2 - 85*a^2*b^3*c^3 + 124*a^3*b*c^4)
*d^5*e^4 - 8*(b^8 + 38*a*b^6*c - 300*a^2*b^4*c^2 + 430*a^3*b^2*c^3 + 56*a^4*c^4)*d^4*e^5 + (88*a*b^7 - 543*a^2
*b^5*c + 480*a^3*b^3*c^2 + 688*a^4*b*c^3)*d^3*e^6 - (41*a^2*b^6 - 390*a^3*b^4*c + 1296*a^4*b^2*c^2 - 1696*a^5*
c^3)*d^2*e^7 - 45*(a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*d*e^8 + 6*(a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2)*e^9 -
(128*c^8*d^7*e^2 - 448*b*c^7*d^6*e^3 + 32*(13*b^2*c^6 + 32*a*c^7)*d^5*e^4 + 80*(b^3*c^5 - 32*a*b*c^6)*d^4*e^5
- 2*(233*b^4*c^4 - 1704*a*b^2*c^5 + 848*a^2*c^6)*d^3*e^6 + (395*b^5*c^3 - 2552*a*b^3*c^4 + 2544*a^2*b*c^5)*d^2
*e^7 - (105*b^6*c^2 - 470*a*b^4*c^3 - 672*a^2*b^2*c^4 + 2592*a^3*c^5)*d*e^8 + (105*a*b^5*c^2 - 760*a^2*b^3*c^3
 + 1296*a^3*b*c^4)*e^9)*x^5 - (256*c^8*d^8*e - 704*b*c^7*d^7*e^2 + 32*(5*b^2*c^6 + 64*a*c^7)*d^6*e^3 + 112*(7*
b^3*c^5 - 32*a*b*c^6)*d^5*e^4 - 8*(79*b^4*c^4 - 192*a*b^2*c^5 + 64*a^2*c^6)*d^4*e^5 - (269*b^5*c^3 - 2888*a*b^
3*c^4 + 3216*a^2*b*c^5)*d^3*e^6 + (615*b^6*c^2 - 4598*a*b^4*c^3 + 7680*a^2*b^2*c^4 - 2784*a^3*c^5)*d^2*e^7 - (
210*b^7*c - 1185*a*b^5*c^2 + 152*a^2*b^3*c^3 + 3696*a^3*b*c^4)*d*e^8 + 6*(35*a*b^6*c - 265*a^2*b^4*c^2 + 504*a
^3*b^2*c^3 - 80*a^4*c^4)*e^9)*x^4 - (128*c^8*d^9 - 64*b*c^7*d^8*e - 16*(55*b^2*c^6 - 76*a*c^7)*d^7*e^2 + 40*(2
9*b^3*c^5 - 4*a*b*c^6)*d^6*e^3 + 8*(25*b^4*c^4 - 678*a*b^2*c^5 + 520*a^2*c^6)*d^5*e^4 - 56*(23*b^5*c^3 - 167*a
*b^3*c^4 + 204*a^2*b*c^5)*d^4*e^5 + 4*(201*b^6*c^2 - 1393*a*b^4*c^3 + 2340*a^2*b^2*c^4 - 384*a^3*c^5)*d^3*e^6
+ (45*b^7*c - 582*a*b^5*c^2 + 2072*a^2*b^3*c^3 - 3264*a^3*b*c^4)*d^2*e^7 - 3*(35*b^8 - 250*a*b^6*c + 516*a^2*b
^4*c^2 - 848*a^3*b^2*c^3 + 1536*a^4*c^4)*d*e^8 + 3*(35*a*b^7 - 230*a^2*b^5*c + 232*a^3*b^3*c^2 + 448*a^4*b*c^3
)*e^9)*x^3 - (192*b*c^7*d^9 - 192*(3*b^2*c^6 - 2*a*c^7)*d^8*e + 8*(35*b^3*c^5 + 36*a*b*c^6)*d^7*e^2 + 8*(65*b^
4*c^4 - 330*a*b^2*c^5 + 504*a^2*c^6)*d^6*e^3 - 8*(55*b^5*c^3 - 235*a*b^3*c^4 + 732*a^2*b*c^5)*d^5*e^4 - 16*(22
*b^6*c^2 - 160*a*b^4*c^3 + 57*a^2*b^2*c^4 + 28*a^3*c^5)*d^4*e^5 + (551*b^7*c - 3894*a*b^5*c^2 + 5992*a^2*b^3*c
^3 - 1408*a^3*b*c^4)*d^3*e^6 - (175*b^8 - 1282*a*b^6*c + 2724*a^2*b^4*c^2 - 3344*a^3*b^2*c^3 + 4736*a^4*c^4)*d
^2*e^7 + (35*a*b^7 - 402*a^2*b^5*c + 1608*a^3*b^3*c^2 - 2176*a^4*b*c^3)*d*e^8 + 4*(35*a^2*b^6 - 279*a^3*b^4*c
+ 588*a^4*b^2*c^2 - 160*a^5*c^3)*e^9)*x^2 - (48*(b^2*c^6 + 4*a*c^7)*d^9 - 8*(23*b^3*c^5 + 60*a*b*c^6)*d^8*e +
8*(25*b^4*c^4 + 54*a*b^2*c^5 + 168*a^2*c^6)*d^7*e^2 + 8*(10*b^5*c^3 - 145*a*b^3*c^4 - 84*a^2*b*c^5)*d^6*e^3 -
80*(4*b^6*c^2 - 31*a*b^4*c^3 + 57*a^2*b^2*c^4 - 44*a^3*c^5)*d^5*e^4 + 8*(29*b^7*c - 294*a*b^5*c^2 + 1015*a^2*b
^3*c^3 - 1156*a^3*b*c^4)*d^4*e^5 - 2*(28*b^8 - 535*a*b^6*c + 2805*a^2*b^4*c^2 - 4064*a^3*b^2*c^3 - 208*a^4*c^4
)*d^3*e^6 - (182*a*b^7 - 951*a^2*b^5*c - 624*a^3*b^3*c^2 + 5360*a^4*b*c^3)*d^2*e^7 + (217*a^2*b^6 - 1770*a^3*b
^4*c + 4032*a^4*b^2*c^2 - 1952*a^5*c^3)*d*e^8 + 21*(a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*e^9)*x)*sqrt(c*x^2 +
 b*x + a))/((a^2*b^4*c^5 - 8*a^3*b^2*c^6 + 16*a^4*c^7)*d^12 - 5*(a^2*b^5*c^4 - 8*a^3*b^3*c^5 + 16*a^4*b*c^6)*d
^11*e + 5*(2*a^2*b^6*c^3 - 15*a^3*b^4*c^4 + 24*a^4*b^2*c^5 + 16*a^5*c^6)*d^10*e^2 - 10*(a^2*b^7*c^2 - 6*a^3*b^
5*c^3 + 32*a^5*b*c^5)*d^9*e^3 + 5*(a^2*b^8*c - 2*a^3*b^6*c^2 - 30*a^4*b^4*c^3 + 80*a^5*b^2*c^4 + 32*a^6*c^5)*d
^8*e^4 - (a^2*b^9 + 12*a^3*b^7*c - 114*a^4*b^5*c^2 + 80*a^5*b^3*c^3 + 480*a^6*b*c^4)*d^7*e^5 + 5*(a^3*b^8 - 2*
a^4*b^6*c - 30*a^5*b^4*c^2 + 80*a^6*b^2*c^3 + 32*a^7*c^4)*d^6*e^6 - 10*(a^4*b^7 - 6*a^5*b^5*c + 32*a^7*b*c^3)*
d^5*e^7 + 5*(2*a^5*b^6 - 15*a^6*b^4*c + 24*a^7*b^2*c^2 + 16*a^8*c^3)*d^4*e^8 - 5*(a^6*b^5 - 8*a^7*b^3*c + 16*a
^8*b*c^2)*d^3*e^9 + (a^7*b^4 - 8*a^8*b^2*c + 16*a^9*c^2)*d^2*e^10 + ((b^4*c^7 - 8*a*b^2*c^8 + 16*a^2*c^9)*d^10
*e^2 - 5*(b^5*c^6 - 8*a*b^3*c^7 + 16*a^2*b*c^8)*d^9*e^3 + 5*(2*b^6*c^5 - 15*a*b^4*c^6 + 24*a^2*b^2*c^7 + 16*a^
3*c^8)*d^8*e^4 - 10*(b^7*c^4 - 6*a*b^5*c^5 + 32*a^3*b*c^7)*d^7*e^5 + 5*(b^8*c^3 - 2*a*b^6*c^4 - 30*a^2*b^4*c^5
 + 80*a^3*b^2*c^6 + 32*a^4*c^7)*d^6*e^6 - (b^9*c^2 + 12*a*b^7*c^3 - 114*a^2*b^5*c^4 + 80*a^3*b^3*c^5 + 480*a^4
*b*c^6)*d^5*e^7 + 5*(a*b^8*c^2 - 2*a^2*b^6*c^3 - 30*a^3*b^4*c^4 + 80*a^4*b^2*c^5 + 32*a^5*c^6)*d^4*e^8 - 10*(a
^2*b^7*c^2 - 6*a^3*b^5*c^3 + 32*a^5*b*c^5)*d^3*e^9 + 5*(2*a^3*b^6*c^2 - 15*a^4*b^4*c^3 + 24*a^5*b^2*c^4 + 16*a
^6*c^5)*d^2*e^10 - 5*(a^4*b^5*c^2 - 8*a^5*b^3*c^3 + 16*a^6*b*c^4)*d*e^11 + (a^5*b^4*c^2 - 8*a^6*b^2*c^3 + 16*a
^7*c^4)*e^12)*x^6 + 2*((b^4*c^7 - 8*a*b^2*c^8 + 16*a^2*c^9)*d^11*e - 4*(b^5*c^6 - 8*a*b^3*c^7 + 16*a^2*b*c^8)*
d^10*e^2 + 5*(b^6*c^5 - 7*a*b^4*c^6 + 8*a^2*b^2*c^7 + 16*a^3*c^8)*d^9*e^3 - 15*(a*b^5*c^5 - 8*a^2*b^3*c^6 + 16
*a^3*b*c^7)*d^8*e^4 - 5*(b^8*c^3 - 10*a*b^6*c^4 + 30*a^2*b^4*c^5 - 16*a^3*b^2*c^6 - 32*a^4*c^7)*d^7*e^5 + 2*(2
*b^9*c^2 - 11*a*b^7*c^3 - 18*a^2*b^5*c^4 + 160*a^3*b^3*c^5 - 160*a^4*b*c^6)*d^6*e^6 - (b^10*c + 7*a*b^8*c^2 -
104*a^2*b^6*c^3 + 230*a^3*b^4*c^4 + 80*a^4*b^2*c^5 - 160*a^5*c^6)*d^5*e^7 + 5*(a*b^9*c - 4*a^2*b^7*c^2 - 18*a^
3*b^5*c^3 + 80*a^4*b^3*c^4 - 32*a^5*b*c^5)*d^4*e^8 - 5*(2*a^2*b^8*c - 14*a^3*b^6*c^2 + 15*a^4*b^4*c^3 + 40*a^5
*b^2*c^4 - 16*a^6*c^5)*d^3*e^9 + 10*(a^3*b^7*c - 8*a^4*b^5*c^2 + 16*a^5*b^3*c^3)*d^2*e^10 - (5*a^4*b^6*c - 41*
a^5*b^4*c^2 + 88*a^6*b^2*c^3 - 16*a^7*c^4)*d*e^11 + (a^5*b^5*c - 8*a^6*b^3*c^2 + 16*a^7*b*c^3)*e^12)*x^5 + ((b
^4*c^7 - 8*a*b^2*c^8 + 16*a^2*c^9)*d^12 - (b^5*c^6 - 8*a*b^3*c^7 + 16*a^2*b*c^8)*d^11*e - (9*b^6*c^5 - 79*a*b^
4*c^6 + 200*a^2*b^2*c^7 - 112*a^3*c^8)*d^10*e^2 + 5*(5*b^7*c^4 - 42*a*b^5*c^5 + 96*a^2*b^3*c^6 - 32*a^3*b*c^7)
*d^9*e^3 - 5*(5*b^8*c^3 - 35*a*b^6*c^4 + 36*a^2*b^4*c^5 + 112*a^3*b^2*c^6 - 64*a^4*c^7)*d^8*e^4 + 3*(3*b^9*c^2
 - 4*a*b^7*c^3 - 122*a^2*b^5*c^4 + 400*a^3*b^3*c^5 - 160*a^4*b*c^6)*d^7*e^5 + (b^10*c - 43*a*b^8*c^2 + 276*a^2
*b^6*c^3 - 370*a^3*b^4*c^4 - 560*a^4*b^2*c^5 + 480*a^5*c^6)*d^6*e^6 - (b^11 - 6*a*b^9*c - 40*a^2*b^7*c^2 + 392
*a^3*b^5*c^3 - 960*a^4*b^3*c^4 + 640*a^5*b*c^5)*d^5*e^7 + 5*(a*b^10 - 8*a^2*b^8*c + 16*a^3*b^6*c^2 + 5*a^4*b^4
*c^3 - 40*a^5*b^2*c^4 + 80*a^6*c^5)*d^4*e^8 - 5*(2*a^2*b^9 - 16*a^3*b^7*c + 37*a^4*b^5*c^2 - 40*a^5*b^3*c^3 +
80*a^6*b*c^4)*d^3*e^9 + (10*a^3*b^8 - 75*a^4*b^6*c + 131*a^5*b^4*c^2 - 8*a^6*b^2*c^3 + 176*a^7*c^4)*d^2*e^10 -
 (5*a^4*b^7 - 34*a^5*b^5*c + 32*a^6*b^3*c^2 + 96*a^7*b*c^3)*d*e^11 + (a^5*b^6 - 6*a^6*b^4*c + 32*a^8*c^3)*e^12
)*x^4 + 2*((b^5*c^6 - 8*a*b^3*c^7 + 16*a^2*b*c^8)*d^12 - 2*(2*b^6*c^5 - 17*a*b^4*c^6 + 40*a^2*b^2*c^7 - 16*a^3
*c^8)*d^11*e + (5*b^7*c^4 - 44*a*b^5*c^5 + 112*a^2*b^3*c^6 - 64*a^3*b*c^7)*d^10*e^2 + 10*(a^2*b^4*c^5 - 8*a^3*
b^2*c^6 + 16*a^4*c^7)*d^9*e^3 - 5*(b^9*c^2 - 8*a*b^7*c^3 + 21*a^2*b^5*c^4 - 40*a^3*b^3*c^5 + 80*a^4*b*c^6)*d^8
*e^4 + 2*(2*b^10*c - 11*a*b^8*c^2 + 2*a^2*b^6*c^3 + 10*a^3*b^4*c^4 + 80*a^4*b^2*c^5 + 160*a^5*c^6)*d^7*e^5 - (
b^11 + 4*a*b^9*c - 70*a^2*b^7*c^2 + 152*a^3*b^5*c^3 - 160*a^4*b^3*c^4 + 640*a^5*b*c^5)*d^6*e^6 + 2*(2*a*b^10 -
 11*a^2*b^8*c + 2*a^3*b^6*c^2 + 10*a^4*b^4*c^3 + 80*a^5*b^2*c^4 + 160*a^6*c^5)*d^5*e^7 - 5*(a^2*b^9 - 8*a^3*b^
7*c + 21*a^4*b^5*c^2 - 40*a^5*b^3*c^3 + 80*a^6*b*c^4)*d^4*e^8 + 10*(a^5*b^4*c^2 - 8*a^6*b^2*c^3 + 16*a^7*c^4)*
d^3*e^9 + (5*a^4*b^7 - 44*a^5*b^5*c + 112*a^6*b^3*c^2 - 64*a^7*b*c^3)*d^2*e^10 - 2*(2*a^5*b^6 - 17*a^6*b^4*c +
 40*a^7*b^2*c^2 - 16*a^8*c^3)*d*e^11 + (a^6*b^5 - 8*a^7*b^3*c + 16*a^8*b*c^2)*e^12)*x^3 + ((b^6*c^5 - 6*a*b^4*
c^6 + 32*a^3*c^8)*d^12 - (5*b^7*c^4 - 34*a*b^5*c^5 + 32*a^2*b^3*c^6 + 96*a^3*b*c^7)*d^11*e + (10*b^8*c^3 - 75*
a*b^6*c^4 + 131*a^2*b^4*c^5 - 8*a^3*b^2*c^6 + 176*a^4*c^7)*d^10*e^2 - 5*(2*b^9*c^2 - 16*a*b^7*c^3 + 37*a^2*b^5
*c^4 - 40*a^3*b^3*c^5 + 80*a^4*b*c^6)*d^9*e^3 + 5*(b^10*c - 8*a*b^8*c^2 + 16*a^2*b^6*c^3 + 5*a^3*b^4*c^4 - 40*
a^4*b^2*c^5 + 80*a^5*c^6)*d^8*e^4 - (b^11 - 6*a*b^9*c - 40*a^2*b^7*c^2 + 392*a^3*b^5*c^3 - 960*a^4*b^3*c^4 + 6
40*a^5*b*c^5)*d^7*e^5 + (a*b^10 - 43*a^2*b^8*c + 276*a^3*b^6*c^2 - 370*a^4*b^4*c^3 - 560*a^5*b^2*c^4 + 480*a^6
*c^5)*d^6*e^6 + 3*(3*a^2*b^9 - 4*a^3*b^7*c - 122*a^4*b^5*c^2 + 400*a^5*b^3*c^3 - 160*a^6*b*c^4)*d^5*e^7 - 5*(5
*a^3*b^8 - 35*a^4*b^6*c + 36*a^5*b^4*c^2 + 112*a^6*b^2*c^3 - 64*a^7*c^4)*d^4*e^8 + 5*(5*a^4*b^7 - 42*a^5*b^5*c
 + 96*a^6*b^3*c^2 - 32*a^7*b*c^3)*d^3*e^9 - (9*a^5*b^6 - 79*a^6*b^4*c + 200*a^7*b^2*c^2 - 112*a^8*c^3)*d^2*e^1
0 - (a^6*b^5 - 8*a^7*b^3*c + 16*a^8*b*c^2)*d*e^11 + (a^7*b^4 - 8*a^8*b^2*c + 16*a^9*c^2)*e^12)*x^2 + 2*((a*b^5
*c^5 - 8*a^2*b^3*c^6 + 16*a^3*b*c^7)*d^12 - (5*a*b^6*c^4 - 41*a^2*b^4*c^5 + 88*a^3*b^2*c^6 - 16*a^4*c^7)*d^11*
e + 10*(a*b^7*c^3 - 8*a^2*b^5*c^4 + 16*a^3*b^3*c^5)*d^10*e^2 - 5*(2*a*b^8*c^2 - 14*a^2*b^6*c^3 + 15*a^3*b^4*c^
4 + 40*a^4*b^2*c^5 - 16*a^5*c^6)*d^9*e^3 + 5*(a*b^9*c - 4*a^2*b^7*c^2 - 18*a^3*b^5*c^3 + 80*a^4*b^3*c^4 - 32*a
^5*b*c^5)*d^8*e^4 - (a*b^10 + 7*a^2*b^8*c - 104*a^3*b^6*c^2 + 230*a^4*b^4*c^3 + 80*a^5*b^2*c^4 - 160*a^6*c^5)*
d^7*e^5 + 2*(2*a^2*b^9 - 11*a^3*b^7*c - 18*a^4*b^5*c^2 + 160*a^5*b^3*c^3 - 160*a^6*b*c^4)*d^6*e^6 - 5*(a^3*b^8
 - 10*a^4*b^6*c + 30*a^5*b^4*c^2 - 16*a^6*b^2*c^3 - 32*a^7*c^4)*d^5*e^7 - 15*(a^5*b^5*c - 8*a^6*b^3*c^2 + 16*a
^7*b*c^3)*d^4*e^8 + 5*(a^5*b^6 - 7*a^6*b^4*c + 8*a^7*b^2*c^2 + 16*a^8*c^3)*d^3*e^9 - 4*(a^6*b^5 - 8*a^7*b^3*c
+ 16*a^8*b*c^2)*d^2*e^10 + (a^7*b^4 - 8*a^8*b^2*c + 16*a^9*c^2)*d*e^11)*x)]

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.07, size = 4942, normalized size = 7.96 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^3/(c*x^2+b*x+a)^(5/2),x)

[Out]

5/2*e^3*c/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*
b^2-35/3*e^3/(a*e^2-b*d*e+c*d^2)^3*c/(4*a*c-b^2)^2/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)
^(1/2)*b^4-35/2*e^3/(a*e^2-b*d*e+c*d^2)^4/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln(((b*e-2*c*d)*(x+d/e)/e+2*(a*e^2-b
*d*e+c*d^2)/e^2+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^
(1/2))/(x+d/e))*c^2*d^2+11/2*e/(a*e^2-b*d*e+c*d^2)^2*c/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b
*d*e+c*d^2)/e^2)^(3/2)*b^2-35/6*e^2/(a*e^2-b*d*e+c*d^2)^3/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^
2)/e^2)^(3/2)*b*c*d+44*e/(a*e^2-b*d*e+c*d^2)^2*c^2/(4*a*c-b^2)^2/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d
*e+c*d^2)/e^2)^(1/2)*b^2-35/2*e^4/(a*e^2-b*d*e+c*d^2)^4/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)
/e^2)^(1/2)*c*d*b-22/(a*e^2-b*d*e+c*d^2)^2*c^3/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d
^2)/e^2)^(3/2)*x*d-11/(a*e^2-b*d*e+c*d^2)^2*c^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*
d^2)/e^2)^(3/2)*b*d+280/3/(a*e^2-b*d*e+c*d^2)^3*c^4/(4*a*c-b^2)^2/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*
d*e+c*d^2)/e^2)^(1/2)*b*d^3+70/3/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b
*d*e+c*d^2)/e^2)^(3/2)*c^4*x*d^3-88/(a*e^2-b*d*e+c*d^2)^2*c^3/(4*a*c-b^2)^2/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e
+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b*d+35/3/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+
(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b*c^3*d^3-176/(a*e^2-b*d*e+c*d^2)^2*c^4/(4*a*c-b^2)^2/((x+d/e)^2*c+(b*e-2*c*d)*
(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*d+560/3/(a*e^2-b*d*e+c*d^2)^3*c^5/(4*a*c-b^2)^2/((x+d/e)^2*c+(b*e-2
*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*d^3-280*e/(a*e^2-b*d*e+c*d^2)^3*c^4/(4*a*c-b^2)^2/((x+d/e)^2*
c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b*d^2+35/2*e^2/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/((x+
d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*c^2*x*b^2*d+105/2*e^4/(a*e^2-b*d*e+c*d^2)^4/(4*a
*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b^2*c^2*d-105*e^3/(a*e^2-b*d*e+c*d
^2)^4/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b*c^3*d^2-35*e/(a*e^2-b*
d*e+c*d^2)^3/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*c^3*x*b*d^2+140*e^2
/(a*e^2-b*d*e+c*d^2)^3*c^3/(4*a*c-b^2)^2/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b
^2*d+35/8*e^5/(a*e^2-b*d*e+c*d^2)^4/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2-5/6*
e*c/(a*e^2-b*d*e+c*d^2)^2/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)-5/2*e^3*c/(a*e^2-b
*d*e+c*d^2)^3/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)-1/2/e/(a*e^2-b*d*e+c*d^2)/(x+d
/e)^2/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)+7/4*e/(a*e^2-b*d*e+c*d^2)^2/(x+d/e)/((
x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b+35/6*e/(a*e^2-b*d*e+c*d^2)^3/((x+d/e)^2*c+(b
*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*c^2*d^2-35/24*e^3/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/((x+d/e
)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b^4+35/2*e^3/(a*e^2-b*d*e+c*d^2)^4/((x+d/e)^2*c+(b*
e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*c^2*d^2-35/8*e^5/(a*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)/((x+d/e)^
2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^4-35/8*e^5/(a*e^2-b*d*e+c*d^2)^4/((a*e^2-b*d*e+c*d^
2)/e^2)^(1/2)*ln(((b*e-2*c*d)*(x+d/e)/e+2*(a*e^2-b*d*e+c*d^2)/e^2+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2
*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))*b^2+5/2*e^3*c/(a*e^2-b*d*e+c*d^2)^3/((a*e^2-
b*d*e+c*d^2)/e^2)^(1/2)*ln(((b*e-2*c*d)*(x+d/e)/e+2*(a*e^2-b*d*e+c*d^2)/e^2+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*
((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))-7/2/(a*e^2-b*d*e+c*d^2)^2/(x+d/e)/
((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*c*d+35/24*e^3/(a*e^2-b*d*e+c*d^2)^3/((x+d/e)
^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b^2-35/2*e/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/((x+d/e
)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b^2*c^2*d^2-70/3*e^3/(a*e^2-b*d*e+c*d^2)^3*c^2/(4*a
*c-b^2)^2/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b^3+11*e/(a*e^2-b*d*e+c*d^2)^2*c
^2/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*x*b+88*e/(a*e^2-b*d*e+c*d^2)^
2*c^3/(4*a*c-b^2)^2/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b+70*e^2/(a*e^2-b*d*e+
c*d^2)^4/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*c^4*d^3-35/12*e^3/(a*
e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*c*x*b^3-105/2
*e^3/(a*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2*c
^2*d^2+35*e^2/(a*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1
/2)*b*c^3*d^3+35/4*e^2/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2
)/e^2)^(3/2)*b^3*c*d+5*e^3*c^2/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d
*e+c*d^2)/e^2)^(1/2)*x*b-10*e^2*c^3/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^
2-b*d*e+c*d^2)/e^2)^(1/2)*x*d-5*e^2*c^2/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(
a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b*d-140*e/(a*e^2-b*d*e+c*d^2)^3*c^3/(4*a*c-b^2)^2/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/
e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2*d^2+105/4*e^4/(a*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)/((x+d/e)^2*c+(b*e-2*c*
d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^3*c*d+35/2*e^4/(a*e^2-b*d*e+c*d^2)^4/((a*e^2-b*d*e+c*d^2)/e^2)^(
1/2)*ln(((b*e-2*c*d)*(x+d/e)/e+2*(a*e^2-b*d*e+c*d^2)/e^2+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2
*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))*b*c*d+70*e^2/(a*e^2-b*d*e+c*d^2)^3*c^2/(4*a*c-b^2)^2/
((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^3*d-35/4*e^5/(a*e^2-b*d*e+c*d^2)^4/(4*a*c-
b^2)/((x+d/e)^2*c+(b*e-2*c*d)*(x+d/e)/e+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b^3*c

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^3/(c*x^2+b*x+a)^(5/2),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(a*e^2-b*d*e>0)', see `assume?`
 for more details)Is a*e^2-b*d*e                            +c*d^2    positive, negative or zero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (d+e\,x\right )}^3\,{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d + e*x)^3*(a + b*x + c*x^2)^(5/2)),x)

[Out]

int(1/((d + e*x)^3*(a + b*x + c*x^2)^(5/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d + e x\right )^{3} \left (a + b x + c x^{2}\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**3/(c*x**2+b*x+a)**(5/2),x)

[Out]

Integral(1/((d + e*x)**3*(a + b*x + c*x**2)**(5/2)), x)

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