3.23.22 \(\int \frac {(d+e x)^4}{(a+b x+c x^2)^5} \, dx\) [2222]

Optimal. Leaf size=545 \[ -\frac {(b+2 c x) (d+e x)^4}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac {(d+e x)^3 \left (7 b c d-2 b^2 e-6 a c e+7 c (2 c d-b e) x\right )}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}+\frac {(d+e x)^2 \left (28 b^2 c d e+28 a c^2 d e-3 b^3 e^2-b c \left (35 c d^2+23 a e^2\right )-c \left (70 c^2 d^2+13 b^2 e^2-2 c e (35 b d-9 a e)\right ) x\right )}{6 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}-\frac {6 b^4 d e^3+16 a c^2 d e \left (35 c d^2+16 a e^2\right )+4 b^2 c d e \left (70 c d^2+83 a e^2\right )-5 b^3 \left (19 c d^2 e^2+5 a e^4\right )-10 b c \left (21 c^2 d^4+88 a c d^2 e^2+11 a^2 e^4\right )-\left (420 c^4 d^4+19 b^4 e^4-40 c^3 d^2 e (21 b d-2 a e)-38 b^2 c e^3 (5 b d-a e)+2 c^2 e^2 \left (305 b^2 d^2-40 a b d e-18 a^2 e^2\right )\right ) x}{6 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac {2 \left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2}} \]

[Out]

-1/4*(2*c*x+b)*(e*x+d)^4/(-4*a*c+b^2)/(c*x^2+b*x+a)^4+1/6*(e*x+d)^3*(7*b*c*d-2*b^2*e-6*a*c*e+7*c*(-b*e+2*c*d)*
x)/(-4*a*c+b^2)^2/(c*x^2+b*x+a)^3+1/6*(e*x+d)^2*(28*b^2*c*d*e+28*a*c^2*d*e-3*b^3*e^2-b*c*(23*a*e^2+35*c*d^2)-c
*(70*c^2*d^2+13*b^2*e^2-2*c*e*(-9*a*e+35*b*d))*x)/(-4*a*c+b^2)^3/(c*x^2+b*x+a)^2+1/6*(-6*b^4*d*e^3-16*a*c^2*d*
e*(16*a*e^2+35*c*d^2)-4*b^2*c*d*e*(83*a*e^2+70*c*d^2)+5*b^3*(5*a*e^4+19*c*d^2*e^2)+10*b*c*(11*a^2*e^4+88*a*c*d
^2*e^2+21*c^2*d^4)+(420*c^4*d^4+19*b^4*e^4-40*c^3*d^2*e*(-2*a*e+21*b*d)-38*b^2*c*e^3*(-a*e+5*b*d)+2*c^2*e^2*(-
18*a^2*e^2-40*a*b*d*e+305*b^2*d^2))*x)/(-4*a*c+b^2)^4/(c*x^2+b*x+a)-2*(70*c^4*d^4+b^4*e^4-4*b^2*c*e^3*(-3*a*e+
5*b*d)-20*c^3*d^2*e*(-3*a*e+7*b*d)+6*c^2*e^2*(a^2*e^2-10*a*b*d*e+15*b^2*d^2))*arctanh((2*c*x+b)/(-4*a*c+b^2)^(
1/2))/(-4*a*c+b^2)^(9/2)

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Rubi [A]
time = 0.56, antiderivative size = 545, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {750, 834, 791, 632, 212} \begin {gather*} -\frac {2 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2}}-\frac {-x \left (2 c^2 e^2 \left (-18 a^2 e^2-40 a b d e+305 b^2 d^2\right )-38 b^2 c e^3 (5 b d-a e)-40 c^3 d^2 e (21 b d-2 a e)+19 b^4 e^4+420 c^4 d^4\right )-10 b c \left (11 a^2 e^4+88 a c d^2 e^2+21 c^2 d^4\right )-5 b^3 \left (5 a e^4+19 c d^2 e^2\right )+4 b^2 c d e \left (83 a e^2+70 c d^2\right )+16 a c^2 d e \left (16 a e^2+35 c d^2\right )+6 b^4 d e^3}{6 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac {(b+2 c x) (d+e x)^4}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac {(d+e x)^3 \left (-6 a c e-2 b^2 e+7 c x (2 c d-b e)+7 b c d\right )}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}+\frac {(d+e x)^2 \left (-c x \left (-2 c e (35 b d-9 a e)+13 b^2 e^2+70 c^2 d^2\right )-b c \left (23 a e^2+35 c d^2\right )+28 a c^2 d e-3 b^3 e^2+28 b^2 c d e\right )}{6 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^4/(a + b*x + c*x^2)^5,x]

[Out]

-1/4*((b + 2*c*x)*(d + e*x)^4)/((b^2 - 4*a*c)*(a + b*x + c*x^2)^4) + ((d + e*x)^3*(7*b*c*d - 2*b^2*e - 6*a*c*e
+ 7*c*(2*c*d - b*e)*x))/(6*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^3) + ((d + e*x)^2*(28*b^2*c*d*e + 28*a*c^2*d*e -
3*b^3*e^2 - b*c*(35*c*d^2 + 23*a*e^2) - c*(70*c^2*d^2 + 13*b^2*e^2 - 2*c*e*(35*b*d - 9*a*e))*x))/(6*(b^2 - 4*
a*c)^3*(a + b*x + c*x^2)^2) - (6*b^4*d*e^3 + 16*a*c^2*d*e*(35*c*d^2 + 16*a*e^2) + 4*b^2*c*d*e*(70*c*d^2 + 83*a
*e^2) - 5*b^3*(19*c*d^2*e^2 + 5*a*e^4) - 10*b*c*(21*c^2*d^4 + 88*a*c*d^2*e^2 + 11*a^2*e^4) - (420*c^4*d^4 + 19
*b^4*e^4 - 40*c^3*d^2*e*(21*b*d - 2*a*e) - 38*b^2*c*e^3*(5*b*d - a*e) + 2*c^2*e^2*(305*b^2*d^2 - 40*a*b*d*e -
18*a^2*e^2))*x)/(6*(b^2 - 4*a*c)^4*(a + b*x + c*x^2)) - (2*(70*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(5*b*d - 3*a*e)
- 20*c^3*d^2*e*(7*b*d - 3*a*e) + 6*c^2*e^2*(15*b^2*d^2 - 10*a*b*d*e + a^2*e^2))*ArcTanh[(b + 2*c*x)/Sqrt[b^2
- 4*a*c]])/(b^2 - 4*a*c)^(9/2)

Rule 212

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

Rule 632

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

Rule 750

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

Rule 791

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

Rule 834

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

Rubi steps

\begin {align*} \int \frac {(d+e x)^4}{\left (a+b x+c x^2\right )^5} \, dx &=-\frac {(b+2 c x) (d+e x)^4}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac {\int \frac {(d+e x)^3 (-14 c d+4 b e-6 c e x)}{\left (a+b x+c x^2\right )^4} \, dx}{4 \left (b^2-4 a c\right )}\\ &=-\frac {(b+2 c x) (d+e x)^4}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac {(d+e x)^3 \left (7 b c d-2 b^2 e-6 a c e+7 c (2 c d-b e) x\right )}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac {\int \frac {(d+e x)^2 \left (-4 \left (35 c^2 d^2+3 b^2 e^2-c e (28 b d-9 a e)\right )-28 c e (2 c d-b e) x\right )}{\left (a+b x+c x^2\right )^3} \, dx}{12 \left (b^2-4 a c\right )^2}\\ &=-\frac {(b+2 c x) (d+e x)^4}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac {(d+e x)^3 \left (7 b c d-2 b^2 e-6 a c e+7 c (2 c d-b e) x\right )}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}+\frac {(d+e x)^2 \left (28 b^2 c d e+28 a c^2 d e-3 b^3 e^2-b c \left (35 c d^2+23 a e^2\right )-c \left (70 c^2 d^2+13 b^2 e^2-2 c e (35 b d-9 a e)\right ) x\right )}{6 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}+\frac {\int \frac {(d+e x) \left (-4 \left (210 c^3 d^3-6 b^3 e^3+b c e^2 (95 b d-46 a e)-10 c^2 d e (28 b d-11 a e)\right )-4 c e \left (70 c^2 d^2+13 b^2 e^2-2 c e (35 b d-9 a e)\right ) x\right )}{\left (a+b x+c x^2\right )^2} \, dx}{24 \left (b^2-4 a c\right )^3}\\ &=-\frac {(b+2 c x) (d+e x)^4}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac {(d+e x)^3 \left (7 b c d-2 b^2 e-6 a c e+7 c (2 c d-b e) x\right )}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}+\frac {(d+e x)^2 \left (28 b^2 c d e+28 a c^2 d e-3 b^3 e^2-b c \left (35 c d^2+23 a e^2\right )-c \left (70 c^2 d^2+13 b^2 e^2-2 c e (35 b d-9 a e)\right ) x\right )}{6 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}-\frac {6 b^4 d e^3+16 a c^2 d e \left (35 c d^2+16 a e^2\right )+4 b^2 c d e \left (70 c d^2+83 a e^2\right )-5 b^3 \left (19 c d^2 e^2+5 a e^4\right )-10 b c \left (21 c^2 d^4+88 a c d^2 e^2+11 a^2 e^4\right )-\left (420 c^4 d^4+19 b^4 e^4-40 c^3 d^2 e (21 b d-2 a e)-38 b^2 c e^3 (5 b d-a e)+2 c^2 e^2 \left (305 b^2 d^2-40 a b d e-18 a^2 e^2\right )\right ) x}{6 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}+\frac {\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^4}\\ &=-\frac {(b+2 c x) (d+e x)^4}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac {(d+e x)^3 \left (7 b c d-2 b^2 e-6 a c e+7 c (2 c d-b e) x\right )}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}+\frac {(d+e x)^2 \left (28 b^2 c d e+28 a c^2 d e-3 b^3 e^2-b c \left (35 c d^2+23 a e^2\right )-c \left (70 c^2 d^2+13 b^2 e^2-2 c e (35 b d-9 a e)\right ) x\right )}{6 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}-\frac {6 b^4 d e^3+16 a c^2 d e \left (35 c d^2+16 a e^2\right )+4 b^2 c d e \left (70 c d^2+83 a e^2\right )-5 b^3 \left (19 c d^2 e^2+5 a e^4\right )-10 b c \left (21 c^2 d^4+88 a c d^2 e^2+11 a^2 e^4\right )-\left (420 c^4 d^4+19 b^4 e^4-40 c^3 d^2 e (21 b d-2 a e)-38 b^2 c e^3 (5 b d-a e)+2 c^2 e^2 \left (305 b^2 d^2-40 a b d e-18 a^2 e^2\right )\right ) x}{6 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac {\left (2 \left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right )\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{\left (b^2-4 a c\right )^4}\\ &=-\frac {(b+2 c x) (d+e x)^4}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac {(d+e x)^3 \left (7 b c d-2 b^2 e-6 a c e+7 c (2 c d-b e) x\right )}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}+\frac {(d+e x)^2 \left (28 b^2 c d e+28 a c^2 d e-3 b^3 e^2-b c \left (35 c d^2+23 a e^2\right )-c \left (70 c^2 d^2+13 b^2 e^2-2 c e (35 b d-9 a e)\right ) x\right )}{6 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}-\frac {6 b^4 d e^3+16 a c^2 d e \left (35 c d^2+16 a e^2\right )+4 b^2 c d e \left (70 c d^2+83 a e^2\right )-5 b^3 \left (19 c d^2 e^2+5 a e^4\right )-10 b c \left (21 c^2 d^4+88 a c d^2 e^2+11 a^2 e^4\right )-\left (420 c^4 d^4+19 b^4 e^4-40 c^3 d^2 e (21 b d-2 a e)-38 b^2 c e^3 (5 b d-a e)+2 c^2 e^2 \left (305 b^2 d^2-40 a b d e-18 a^2 e^2\right )\right ) x}{6 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac {2 \left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 1.15, size = 713, normalized size = 1.31 \begin {gather*} \frac {1}{12} \left (\frac {\left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (b+2 c x)}{c^2 \left (-b^2+4 a c\right )^3 (a+x (b+c x))^2}+\frac {6 \left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) (b+2 c x)}{c \left (b^2-4 a c\right )^4 (a+x (b+c x))}+\frac {3 \left (b^4 e^4 x+b^3 e^3 (a e-4 c d x)+2 b^2 c e^2 \left (3 c d^2 x-2 a e (d+e x)\right )+b c \left (-3 a^2 e^4+c^2 d^3 (d-4 e x)+6 a c d e^2 (d+2 e x)\right )+2 c^2 \left (c^2 d^4 x+a^2 e^3 (4 d+e x)-2 a c d^2 e (2 d+3 e x)\right )\right )}{c^3 \left (-b^2+4 a c\right ) (a+x (b+c x))^4}+\frac {3 b^5 e^4-2 b^4 c e^3 (6 d+e x)+2 b^3 c e^2 \left (-10 a e^2+c d (9 d-4 e x)\right )+2 b c^2 \left (23 a^2 e^4+7 c^2 d^3 (d-4 e x)+6 a c d e^2 (d-2 e x)\right )+4 b^2 c^2 e \left (a e^2 (13 d+6 e x)+c d^2 (-7 d+9 e x)\right )-4 c^3 \left (-7 c^2 d^4 x-6 a c d^2 e^2 x+a^2 e^3 (32 d+9 e x)\right )}{c^3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^3}+\frac {24 \left (70 c^4 d^4+b^4 e^4-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+6 c^2 e^2 \left (15 b^2 d^2-10 a b d e+a^2 e^2\right )\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{9/2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^4/(a + b*x + c*x^2)^5,x]

[Out]

(((70*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(5*b*d - 3*a*e) - 20*c^3*d^2*e*(7*b*d - 3*a*e) + 6*c^2*e^2*(15*b^2*d^2 -
10*a*b*d*e + a^2*e^2))*(b + 2*c*x))/(c^2*(-b^2 + 4*a*c)^3*(a + x*(b + c*x))^2) + (6*(70*c^4*d^4 + b^4*e^4 - 4
*b^2*c*e^3*(5*b*d - 3*a*e) - 20*c^3*d^2*e*(7*b*d - 3*a*e) + 6*c^2*e^2*(15*b^2*d^2 - 10*a*b*d*e + a^2*e^2))*(b
+ 2*c*x))/(c*(b^2 - 4*a*c)^4*(a + x*(b + c*x))) + (3*(b^4*e^4*x + b^3*e^3*(a*e - 4*c*d*x) + 2*b^2*c*e^2*(3*c*d
^2*x - 2*a*e*(d + e*x)) + b*c*(-3*a^2*e^4 + c^2*d^3*(d - 4*e*x) + 6*a*c*d*e^2*(d + 2*e*x)) + 2*c^2*(c^2*d^4*x
+ a^2*e^3*(4*d + e*x) - 2*a*c*d^2*e*(2*d + 3*e*x))))/(c^3*(-b^2 + 4*a*c)*(a + x*(b + c*x))^4) + (3*b^5*e^4 - 2
*b^4*c*e^3*(6*d + e*x) + 2*b^3*c*e^2*(-10*a*e^2 + c*d*(9*d - 4*e*x)) + 2*b*c^2*(23*a^2*e^4 + 7*c^2*d^3*(d - 4*
e*x) + 6*a*c*d*e^2*(d - 2*e*x)) + 4*b^2*c^2*e*(a*e^2*(13*d + 6*e*x) + c*d^2*(-7*d + 9*e*x)) - 4*c^3*(-7*c^2*d^
4*x - 6*a*c*d^2*e^2*x + a^2*e^3*(32*d + 9*e*x)))/(c^3*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^3) + (24*(70*c^4*d^4 +
b^4*e^4 - 4*b^2*c*e^3*(5*b*d - 3*a*e) - 20*c^3*d^2*e*(7*b*d - 3*a*e) + 6*c^2*e^2*(15*b^2*d^2 - 10*a*b*d*e + a
^2*e^2))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(9/2))/12

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1831\) vs. \(2(532)=1064\).
time = 0.86, size = 1832, normalized size = 3.36

method	result	size
default	\(\text {Expression too large to display}\)	\(1832\)
risch	\(\text {Expression too large to display}\)	\(3864\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^4/(c*x^2+b*x+a)^5,x,method=_RETURNVERBOSE)

[Out]

((6*a^2*c^2*e^4+12*a*b^2*c*e^4-60*a*b*c^2*d*e^3+60*a*c^3*d^2*e^2+b^4*e^4-20*b^3*c*d*e^3+90*b^2*c^2*d^2*e^2-140
*b*c^3*d^3*e+70*c^4*d^4)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*c^3*x^7+7/2*(6*a^2*c^2*e^
4+12*a*b^2*c*e^4-60*a*b*c^2*d*e^3+60*a*c^3*d^2*e^2+b^4*e^4-20*b^3*c*d*e^3+90*b^2*c^2*d^2*e^2-140*b*c^3*d^3*e+7
0*c^4*d^4)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*b*c^2*x^6+1/3*c*(11*a*c+13*b^2)*(6*a^2*
c^2*e^4+12*a*b^2*c*e^4-60*a*b*c^2*d*e^3+60*a*c^3*d^2*e^2+b^4*e^4-20*b^3*c*d*e^3+90*b^2*c^2*d^2*e^2-140*b*c^3*d
^3*e+70*c^4*d^4)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*x^5+5/12*b*(22*a*c+5*b^2)*(6*a^2*
c^2*e^4+12*a*b^2*c*e^4-60*a*b*c^2*d*e^3+60*a*c^3*d^2*e^2+b^4*e^4-20*b^3*c*d*e^3+90*b^2*c^2*d^2*e^2-140*b*c^3*d
^3*e+70*c^4*d^4)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*x^4-1/3*(66*a^4*c^3*e^4-450*a^3*b
^2*c^2*e^4+876*a^3*b*c^3*d*e^3-876*a^3*c^4*d^2*e^2-203*a^2*b^4*c*e^4+1504*a^2*b^3*c^2*d*e^3-2526*a^2*b^2*c^3*d
^2*e^2+2044*a^2*b*c^4*d^3*e-1022*a^2*c^5*d^4-37*a*b^6*e^4+440*a*b^5*c*d*e^3-1854*a*b^4*c^2*d^2*e^2+2828*a*b^3*
c^3*d^3*e-1414*a*b^2*c^4*d^4+12*b^7*d*e^3-54*b^6*c*d^2*e^2+84*b^5*c^2*d^3*e-42*b^4*c^3*d^4)/(256*a^4*c^4-256*a
^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*x^3+1/6*(314*a^4*b*c^2*e^4-1024*a^4*c^3*d*e^3+508*a^3*b^3*c*e^4-1604
*a^3*b^2*c^2*d*e^3+2628*a^3*b*c^3*d^2*e^2+129*a^2*b^5*e^4-1596*a^2*b^4*c*d*e^3+4278*a^2*b^3*c^2*d^2*e^2-6132*a
^2*b^2*c^3*d^3*e+3066*a^2*b*c^4*d^4-36*a*b^6*d*e^3+492*a*b^5*c*d^2*e^2-784*a*b^4*c^2*d^3*e+392*a*b^3*c^3*d^4-1
8*b^7*d^2*e^2+28*b^6*c*d^3*e-14*b^5*c^2*d^4)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*x^2-1
/3*(18*a^5*c^2*e^4-184*a^4*b^2*c*e^4+332*a^4*b*c^2*d*e^3+180*a^4*c^3*d^2*e^2-47*a^3*b^4*e^4+604*a^3*b^3*c*d*e^
3-1674*a^3*b^2*c^2*d^2*e^2+1116*a^3*b*c^3*d^3*e-558*a^3*c^4*d^4+12*a^2*b^5*d*e^3-168*a^2*b^4*c*d^2*e^2+696*a^2
*b^3*c^2*d^3*e-348*a^2*b^2*c^3*d^4+6*a*b^6*d^2*e^2-76*a*b^5*c*d^3*e+38*a*b^4*c^2*d^4+4*b^7*d^3*e-2*b^6*c*d^4)/
(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*x+1/12*(220*a^5*b*c*e^4-512*a^5*c^2*d*e^3+50*a^4*b
^3*e^4-664*a^4*b^2*c*d*e^3+1944*a^4*b*c^2*d^2*e^2-1536*a^4*c^3*d^3*e-12*a^3*b^4*d*e^3+168*a^3*b^3*c*d^2*e^2-69
6*a^3*b^2*c^2*d^3*e+1116*a^3*b*c^3*d^4-6*a^2*b^5*d^2*e^2+76*a^2*b^4*c*d^3*e-326*a^2*b^3*c^2*d^4-4*a*b^6*d^3*e+
50*a*b^5*c*d^4-3*b^7*d^4)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8))/(c*x^2+b*x+a)^4+2*(6*a^
2*c^2*e^4+12*a*b^2*c*e^4-60*a*b*c^2*d*e^3+60*a*c^3*d^2*e^2+b^4*e^4-20*b^3*c*d*e^3+90*b^2*c^2*d^2*e^2-140*b*c^3
*d^3*e+70*c^4*d^4)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/(4*a*c-b^2)^(1/2)*arctan((2*c*x
+b)/(4*a*c-b^2)^(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((e*x+d)^4/(c*x^2+b*x+a)^5,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(4*a*c-b^2>0)', see `assume?` f
or more deta

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3604 vs. \(2 (535) = 1070\).
time = 3.00, size = 7229, normalized size = 13.26 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4/(c*x^2+b*x+a)^5,x, algorithm="fricas")

[Out]

[1/12*(840*(b^2*c^7 - 4*a*c^8)*d^4*x^7 + 2940*(b^3*c^6 - 4*a*b*c^7)*d^4*x^6 + 280*(13*b^4*c^5 - 41*a*b^2*c^6 -
44*a^2*c^7)*d^4*x^5 + 350*(5*b^5*c^4 + 2*a*b^3*c^5 - 88*a^2*b*c^6)*d^4*x^4 + 56*(3*b^6*c^3 + 89*a*b^4*c^4 - 3
31*a^2*b^2*c^5 - 292*a^3*c^6)*d^4*x^3 - 28*(b^7*c^2 - 32*a*b^5*c^3 - 107*a^2*b^3*c^4 + 876*a^3*b*c^5)*d^4*x^2
+ 8*(b^8*c - 23*a*b^6*c^2 + 250*a^2*b^4*c^3 - 417*a^3*b^2*c^4 - 1116*a^4*c^5)*d^4*x - (3*b^9 - 62*a*b^7*c + 52
6*a^2*b^5*c^2 - 2420*a^3*b^3*c^3 + 4464*a^4*b*c^4)*d^4 + 12*(70*c^8*d^4*x^8 + 280*b*c^7*d^4*x^7 + 280*a^3*b*c^
4*d^4*x + 70*a^4*c^4*d^4 + 140*(3*b^2*c^6 + 2*a*c^7)*d^4*x^6 + 280*(b^3*c^5 + 3*a*b*c^6)*d^4*x^5 + 70*(b^4*c^4
+ 12*a*b^2*c^5 + 6*a^2*c^6)*d^4*x^4 + 280*(a*b^3*c^4 + 3*a^2*b*c^5)*d^4*x^3 + 140*(3*a^2*b^2*c^4 + 2*a^3*c^5)
*d^4*x^2 + ((b^4*c^4 + 12*a*b^2*c^5 + 6*a^2*c^6)*x^8 + a^4*b^4 + 12*a^5*b^2*c + 6*a^6*c^2 + 4*(b^5*c^3 + 12*a*
b^3*c^4 + 6*a^2*b*c^5)*x^7 + 2*(3*b^6*c^2 + 38*a*b^4*c^3 + 42*a^2*b^2*c^4 + 12*a^3*c^5)*x^6 + 4*(b^7*c + 15*a*
b^5*c^2 + 42*a^2*b^3*c^3 + 18*a^3*b*c^4)*x^5 + (b^8 + 24*a*b^6*c + 156*a^2*b^4*c^2 + 144*a^3*b^2*c^3 + 36*a^4*
c^4)*x^4 + 4*(a*b^7 + 15*a^2*b^5*c + 42*a^3*b^3*c^2 + 18*a^4*b*c^3)*x^3 + 2*(3*a^2*b^6 + 38*a^3*b^4*c + 42*a^4
*b^2*c^2 + 12*a^5*c^3)*x^2 + 4*(a^3*b^5 + 12*a^4*b^3*c + 6*a^5*b*c^2)*x)*e^4 - 20*((b^3*c^5 + 3*a*b*c^6)*d*x^8
+ 4*(b^4*c^4 + 3*a*b^2*c^5)*d*x^7 + 2*(3*b^5*c^3 + 11*a*b^3*c^4 + 6*a^2*b*c^5)*d*x^6 + 4*(b^6*c^2 + 6*a*b^4*c
^3 + 9*a^2*b^2*c^4)*d*x^5 + (b^7*c + 15*a*b^5*c^2 + 42*a^2*b^3*c^3 + 18*a^3*b*c^4)*d*x^4 + 4*(a*b^6*c + 6*a^2*
b^4*c^2 + 9*a^3*b^2*c^3)*d*x^3 + 2*(3*a^2*b^5*c + 11*a^3*b^3*c^2 + 6*a^4*b*c^3)*d*x^2 + 4*(a^3*b^4*c + 3*a^4*b
^2*c^2)*d*x + (a^4*b^3*c + 3*a^5*b*c^2)*d)*e^3 + 30*((3*b^2*c^6 + 2*a*c^7)*d^2*x^8 + 4*(3*b^3*c^5 + 2*a*b*c^6)
*d^2*x^7 + 2*(9*b^4*c^4 + 12*a*b^2*c^5 + 4*a^2*c^6)*d^2*x^6 + 4*(3*b^5*c^3 + 11*a*b^3*c^4 + 6*a^2*b*c^5)*d^2*x
^5 + (3*b^6*c^2 + 38*a*b^4*c^3 + 42*a^2*b^2*c^4 + 12*a^3*c^5)*d^2*x^4 + 4*(3*a*b^5*c^2 + 11*a^2*b^3*c^3 + 6*a^
3*b*c^4)*d^2*x^3 + 2*(9*a^2*b^4*c^2 + 12*a^3*b^2*c^3 + 4*a^4*c^4)*d^2*x^2 + 4*(3*a^3*b^3*c^2 + 2*a^4*b*c^3)*d^
2*x + (3*a^4*b^2*c^2 + 2*a^5*c^3)*d^2)*e^2 - 140*(b*c^7*d^3*x^8 + 4*b^2*c^6*d^3*x^7 + 4*a^3*b^2*c^3*d^3*x + a^
4*b*c^3*d^3 + 2*(3*b^3*c^5 + 2*a*b*c^6)*d^3*x^6 + 4*(b^4*c^4 + 3*a*b^2*c^5)*d^3*x^5 + (b^5*c^3 + 12*a*b^3*c^4
+ 6*a^2*b*c^5)*d^3*x^4 + 4*(a*b^4*c^3 + 3*a^2*b^2*c^4)*d^3*x^3 + 2*(3*a^2*b^3*c^3 + 2*a^3*b*c^4)*d^3*x^2)*e)*s
qrt(b^2 - 4*a*c)*log((2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c - sqrt(b^2 - 4*a*c)*(2*c*x + b))/(c*x^2 + b*x + a)) +
(50*a^4*b^5 + 20*a^5*b^3*c - 880*a^6*b*c^2 + 12*(b^6*c^3 + 8*a*b^4*c^4 - 42*a^2*b^2*c^5 - 24*a^3*c^6)*x^7 + 42
*(b^7*c^2 + 8*a*b^5*c^3 - 42*a^2*b^3*c^4 - 24*a^3*b*c^5)*x^6 + 4*(13*b^8*c + 115*a*b^6*c^2 - 458*a^2*b^4*c^3 -
774*a^3*b^2*c^4 - 264*a^4*c^5)*x^5 + 5*(5*b^9 + 62*a*b^7*c - 34*a^2*b^5*c^2 - 1044*a^3*b^3*c^3 - 528*a^4*b*c^
4)*x^4 + 4*(37*a*b^8 + 55*a^2*b^6*c - 362*a^3*b^4*c^2 - 1866*a^4*b^2*c^3 + 264*a^5*c^4)*x^3 + 2*(129*a^2*b^7 -
8*a^3*b^5*c - 1718*a^4*b^3*c^2 - 1256*a^5*b*c^3)*x^2 + 4*(47*a^3*b^6 - 4*a^4*b^4*c - 754*a^5*b^2*c^2 + 72*a^6
*c^3)*x)*e^4 - 4*(60*(b^5*c^4 - a*b^3*c^5 - 12*a^2*b*c^6)*d*x^7 + 210*(b^6*c^3 - a*b^4*c^4 - 12*a^2*b^2*c^5)*d
*x^6 + 20*(13*b^7*c^2 - 2*a*b^5*c^3 - 167*a^2*b^3*c^4 - 132*a^3*b*c^5)*d*x^5 + 25*(5*b^8*c + 17*a*b^6*c^2 - 82
*a^2*b^4*c^3 - 264*a^3*b^2*c^4)*d*x^4 + 4*(3*b^9 + 98*a*b^7*c - 64*a^2*b^5*c^2 - 1285*a^3*b^3*c^3 - 876*a^4*b*
c^4)*d*x^3 + 2*(9*a*b^8 + 363*a^2*b^6*c - 1195*a^3*b^4*c^2 - 1348*a^4*b^2*c^3 - 1024*a^5*c^4)*d*x^2 + 4*(3*a^2
*b^7 + 139*a^3*b^5*c - 521*a^4*b^3*c^2 - 332*a^5*b*c^3)*d*x + (3*a^3*b^6 + 154*a^4*b^4*c - 536*a^5*b^2*c^2 - 5
12*a^6*c^3)*d)*e^3 + 6*(60*(3*b^4*c^5 - 10*a*b^2*c^6 - 8*a^2*c^7)*d^2*x^7 + 210*(3*b^5*c^4 - 10*a*b^3*c^5 - 8*
a^2*b*c^6)*d^2*x^6 + 20*(39*b^6*c^3 - 97*a*b^4*c^4 - 214*a^2*b^2*c^5 - 88*a^3*c^6)*d^2*x^5 + 25*(15*b^7*c^2 +
16*a*b^5*c^3 - 260*a^2*b^3*c^4 - 176*a^3*b*c^5)*d^2*x^4 + 4*(9*b^8*c + 273*a*b^6*c^2 - 815*a^2*b^4*c^3 - 1538*
a^3*b^2*c^4 - 584*a^4*c^5)*d^2*x^3 - 2*(3*b^9 - 94*a*b^7*c - 385*a^2*b^5*c^2 + 2414*a^3*b^3*c^3 + 1752*a^4*b*c
^4)*d^2*x^2 - 4*(a*b^8 - 32*a^2*b^6*c - 167*a^3*b^4*c^2 + 1146*a^4*b^2*c^3 - 120*a^5*c^4)*d^2*x - (a^2*b^7 - 3
2*a^3*b^5*c - 212*a^4*b^3*c^2 + 1296*a^5*b*c^3)*d^2)*e^2 - 4*(420*(b^3*c^6 - 4*a*b*c^7)*d^3*x^7 + 1470*(b^4*c^
5 - 4*a*b^2*c^6)*d^3*x^6 + 140*(13*b^5*c^4 - 41*a*b^3*c^5 - 44*a^2*b*c^6)*d^3*x^5 + 175*(5*b^6*c^3 + 2*a*b^4*c
^4 - 88*a^2*b^2*c^5)*d^3*x^4 + 28*(3*b^7*c^2 + 89*a*b^5*c^3 - 331*a^2*b^3*c^4 - 292*a^3*b*c^5)*d^3*x^3 - 14*(b
^8*c - 32*a*b^6*c^2 - 107*a^2*b^4*c^3 + 876*a^3*b^2*c^4)*d^3*x^2 + 4*(b^9 - 23*a*b^7*c + 250*a^2*b^5*c^2 - 417
*a^3*b^3*c^3 - 1116*a^4*b*c^4)*d^3*x + (a*b^8 - 23*a^2*b^6*c + 250*a^3*b^4*c^2 - 312*a^4*b^2*c^3 - 1536*a^5*c^
4)*d^3)*e)/(a^4*b^10 - 20*a^5*b^8*c + 160*a^6*b^6*c^2 - 640*a^7*b^4*c^3 + 1280*a^8*b^2*c^4 - 1024*a^9*c^5 + (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)*x^8 + 4*(b^11*c^
3 - 20*a*b^9*c^4 + 160*a^2*b^7*c^5 - 640*a^3*b^...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**4/(c*x**2+b*x+a)**5,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1839 vs. \(2 (535) = 1070\).
time = 0.96, size = 1839, normalized size = 3.37 \begin {gather*} \frac {2 \, {\left (70 \, c^{4} d^{4} - 140 \, b c^{3} d^{3} e + 90 \, b^{2} c^{2} d^{2} e^{2} + 60 \, a c^{3} d^{2} e^{2} - 20 \, b^{3} c d e^{3} - 60 \, a b c^{2} d e^{3} + b^{4} e^{4} + 12 \, a b^{2} c e^{4} + 6 \, a^{2} c^{2} e^{4}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {840 \, c^{7} d^{4} x^{7} - 1680 \, b c^{6} d^{3} x^{7} e + 2940 \, b c^{6} d^{4} x^{6} + 1080 \, b^{2} c^{5} d^{2} x^{7} e^{2} + 720 \, a c^{6} d^{2} x^{7} e^{2} - 5880 \, b^{2} c^{5} d^{3} x^{6} e + 3640 \, b^{2} c^{5} d^{4} x^{5} + 3080 \, a c^{6} d^{4} x^{5} - 240 \, b^{3} c^{4} d x^{7} e^{3} - 720 \, a b c^{5} d x^{7} e^{3} + 3780 \, b^{3} c^{4} d^{2} x^{6} e^{2} + 2520 \, a b c^{5} d^{2} x^{6} e^{2} - 7280 \, b^{3} c^{4} d^{3} x^{5} e - 6160 \, a b c^{5} d^{3} x^{5} e + 1750 \, b^{3} c^{4} d^{4} x^{4} + 7700 \, a b c^{5} d^{4} x^{4} + 12 \, b^{4} c^{3} x^{7} e^{4} + 144 \, a b^{2} c^{4} x^{7} e^{4} + 72 \, a^{2} c^{5} x^{7} e^{4} - 840 \, b^{4} c^{3} d x^{6} e^{3} - 2520 \, a b^{2} c^{4} d x^{6} e^{3} + 4680 \, b^{4} c^{3} d^{2} x^{5} e^{2} + 7080 \, a b^{2} c^{4} d^{2} x^{5} e^{2} + 2640 \, a^{2} c^{5} d^{2} x^{5} e^{2} - 3500 \, b^{4} c^{3} d^{3} x^{4} e - 15400 \, a b^{2} c^{4} d^{3} x^{4} e + 168 \, b^{4} c^{3} d^{4} x^{3} + 5656 \, a b^{2} c^{4} d^{4} x^{3} + 4088 \, a^{2} c^{5} d^{4} x^{3} + 42 \, b^{5} c^{2} x^{6} e^{4} + 504 \, a b^{3} c^{3} x^{6} e^{4} + 252 \, a^{2} b c^{4} x^{6} e^{4} - 1040 \, b^{5} c^{2} d x^{5} e^{3} - 4000 \, a b^{3} c^{3} d x^{5} e^{3} - 2640 \, a^{2} b c^{4} d x^{5} e^{3} + 2250 \, b^{5} c^{2} d^{2} x^{4} e^{2} + 11400 \, a b^{3} c^{3} d^{2} x^{4} e^{2} + 6600 \, a^{2} b c^{4} d^{2} x^{4} e^{2} - 336 \, b^{5} c^{2} d^{3} x^{3} e - 11312 \, a b^{3} c^{3} d^{3} x^{3} e - 8176 \, a^{2} b c^{4} d^{3} x^{3} e - 28 \, b^{5} c^{2} d^{4} x^{2} + 784 \, a b^{3} c^{3} d^{4} x^{2} + 6132 \, a^{2} b c^{4} d^{4} x^{2} + 52 \, b^{6} c x^{5} e^{4} + 668 \, a b^{4} c^{2} x^{5} e^{4} + 840 \, a^{2} b^{2} c^{3} x^{5} e^{4} + 264 \, a^{3} c^{4} x^{5} e^{4} - 500 \, b^{6} c d x^{4} e^{3} - 3700 \, a b^{4} c^{2} d x^{4} e^{3} - 6600 \, a^{2} b^{2} c^{3} d x^{4} e^{3} + 216 \, b^{6} c d^{2} x^{3} e^{2} + 7416 \, a b^{4} c^{2} d^{2} x^{3} e^{2} + 10104 \, a^{2} b^{2} c^{3} d^{2} x^{3} e^{2} + 3504 \, a^{3} c^{4} d^{2} x^{3} e^{2} + 56 \, b^{6} c d^{3} x^{2} e - 1568 \, a b^{4} c^{2} d^{3} x^{2} e - 12264 \, a^{2} b^{2} c^{3} d^{3} x^{2} e + 8 \, b^{6} c d^{4} x - 152 \, a b^{4} c^{2} d^{4} x + 1392 \, a^{2} b^{2} c^{3} d^{4} x + 2232 \, a^{3} c^{4} d^{4} x + 25 \, b^{7} x^{4} e^{4} + 410 \, a b^{5} c x^{4} e^{4} + 1470 \, a^{2} b^{3} c^{2} x^{4} e^{4} + 660 \, a^{3} b c^{3} x^{4} e^{4} - 48 \, b^{7} d x^{3} e^{3} - 1760 \, a b^{5} c d x^{3} e^{3} - 6016 \, a^{2} b^{3} c^{2} d x^{3} e^{3} - 3504 \, a^{3} b c^{3} d x^{3} e^{3} - 36 \, b^{7} d^{2} x^{2} e^{2} + 984 \, a b^{5} c d^{2} x^{2} e^{2} + 8556 \, a^{2} b^{3} c^{2} d^{2} x^{2} e^{2} + 5256 \, a^{3} b c^{3} d^{2} x^{2} e^{2} - 16 \, b^{7} d^{3} x e + 304 \, a b^{5} c d^{3} x e - 2784 \, a^{2} b^{3} c^{2} d^{3} x e - 4464 \, a^{3} b c^{3} d^{3} x e - 3 \, b^{7} d^{4} + 50 \, a b^{5} c d^{4} - 326 \, a^{2} b^{3} c^{2} d^{4} + 1116 \, a^{3} b c^{3} d^{4} + 148 \, a b^{6} x^{3} e^{4} + 812 \, a^{2} b^{4} c x^{3} e^{4} + 1800 \, a^{3} b^{2} c^{2} x^{3} e^{4} - 264 \, a^{4} c^{3} x^{3} e^{4} - 72 \, a b^{6} d x^{2} e^{3} - 3192 \, a^{2} b^{4} c d x^{2} e^{3} - 3208 \, a^{3} b^{2} c^{2} d x^{2} e^{3} - 2048 \, a^{4} c^{3} d x^{2} e^{3} - 24 \, a b^{6} d^{2} x e^{2} + 672 \, a^{2} b^{4} c d^{2} x e^{2} + 6696 \, a^{3} b^{2} c^{2} d^{2} x e^{2} - 720 \, a^{4} c^{3} d^{2} x e^{2} - 4 \, a b^{6} d^{3} e + 76 \, a^{2} b^{4} c d^{3} e - 696 \, a^{3} b^{2} c^{2} d^{3} e - 1536 \, a^{4} c^{3} d^{3} e + 258 \, a^{2} b^{5} x^{2} e^{4} + 1016 \, a^{3} b^{3} c x^{2} e^{4} + 628 \, a^{4} b c^{2} x^{2} e^{4} - 48 \, a^{2} b^{5} d x e^{3} - 2416 \, a^{3} b^{3} c d x e^{3} - 1328 \, a^{4} b c^{2} d x e^{3} - 6 \, a^{2} b^{5} d^{2} e^{2} + 168 \, a^{3} b^{3} c d^{2} e^{2} + 1944 \, a^{4} b c^{2} d^{2} e^{2} + 188 \, a^{3} b^{4} x e^{4} + 736 \, a^{4} b^{2} c x e^{4} - 72 \, a^{5} c^{2} x e^{4} - 12 \, a^{3} b^{4} d e^{3} - 664 \, a^{4} b^{2} c d e^{3} - 512 \, a^{5} c^{2} d e^{3} + 50 \, a^{4} b^{3} e^{4} + 220 \, a^{5} b c e^{4}}{12 \, {\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} {\left (c x^{2} + b x + a\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4/(c*x^2+b*x+a)^5,x, algorithm="giac")

[Out]

2*(70*c^4*d^4 - 140*b*c^3*d^3*e + 90*b^2*c^2*d^2*e^2 + 60*a*c^3*d^2*e^2 - 20*b^3*c*d*e^3 - 60*a*b*c^2*d*e^3 +
b^4*e^4 + 12*a*b^2*c*e^4 + 6*a^2*c^2*e^4)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^8 - 16*a*b^6*c + 96*a^2*b
^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^4)*sqrt(-b^2 + 4*a*c)) + 1/12*(840*c^7*d^4*x^7 - 1680*b*c^6*d^3*x^7*e + 2
940*b*c^6*d^4*x^6 + 1080*b^2*c^5*d^2*x^7*e^2 + 720*a*c^6*d^2*x^7*e^2 - 5880*b^2*c^5*d^3*x^6*e + 3640*b^2*c^5*d
^4*x^5 + 3080*a*c^6*d^4*x^5 - 240*b^3*c^4*d*x^7*e^3 - 720*a*b*c^5*d*x^7*e^3 + 3780*b^3*c^4*d^2*x^6*e^2 + 2520*
a*b*c^5*d^2*x^6*e^2 - 7280*b^3*c^4*d^3*x^5*e - 6160*a*b*c^5*d^3*x^5*e + 1750*b^3*c^4*d^4*x^4 + 7700*a*b*c^5*d^
4*x^4 + 12*b^4*c^3*x^7*e^4 + 144*a*b^2*c^4*x^7*e^4 + 72*a^2*c^5*x^7*e^4 - 840*b^4*c^3*d*x^6*e^3 - 2520*a*b^2*c
^4*d*x^6*e^3 + 4680*b^4*c^3*d^2*x^5*e^2 + 7080*a*b^2*c^4*d^2*x^5*e^2 + 2640*a^2*c^5*d^2*x^5*e^2 - 3500*b^4*c^3
*d^3*x^4*e - 15400*a*b^2*c^4*d^3*x^4*e + 168*b^4*c^3*d^4*x^3 + 5656*a*b^2*c^4*d^4*x^3 + 4088*a^2*c^5*d^4*x^3 +
42*b^5*c^2*x^6*e^4 + 504*a*b^3*c^3*x^6*e^4 + 252*a^2*b*c^4*x^6*e^4 - 1040*b^5*c^2*d*x^5*e^3 - 4000*a*b^3*c^3*
d*x^5*e^3 - 2640*a^2*b*c^4*d*x^5*e^3 + 2250*b^5*c^2*d^2*x^4*e^2 + 11400*a*b^3*c^3*d^2*x^4*e^2 + 6600*a^2*b*c^4
*d^2*x^4*e^2 - 336*b^5*c^2*d^3*x^3*e - 11312*a*b^3*c^3*d^3*x^3*e - 8176*a^2*b*c^4*d^3*x^3*e - 28*b^5*c^2*d^4*x
^2 + 784*a*b^3*c^3*d^4*x^2 + 6132*a^2*b*c^4*d^4*x^2 + 52*b^6*c*x^5*e^4 + 668*a*b^4*c^2*x^5*e^4 + 840*a^2*b^2*c
^3*x^5*e^4 + 264*a^3*c^4*x^5*e^4 - 500*b^6*c*d*x^4*e^3 - 3700*a*b^4*c^2*d*x^4*e^3 - 6600*a^2*b^2*c^3*d*x^4*e^3
+ 216*b^6*c*d^2*x^3*e^2 + 7416*a*b^4*c^2*d^2*x^3*e^2 + 10104*a^2*b^2*c^3*d^2*x^3*e^2 + 3504*a^3*c^4*d^2*x^3*e
^2 + 56*b^6*c*d^3*x^2*e - 1568*a*b^4*c^2*d^3*x^2*e - 12264*a^2*b^2*c^3*d^3*x^2*e + 8*b^6*c*d^4*x - 152*a*b^4*c
^2*d^4*x + 1392*a^2*b^2*c^3*d^4*x + 2232*a^3*c^4*d^4*x + 25*b^7*x^4*e^4 + 410*a*b^5*c*x^4*e^4 + 1470*a^2*b^3*c
^2*x^4*e^4 + 660*a^3*b*c^3*x^4*e^4 - 48*b^7*d*x^3*e^3 - 1760*a*b^5*c*d*x^3*e^3 - 6016*a^2*b^3*c^2*d*x^3*e^3 -
3504*a^3*b*c^3*d*x^3*e^3 - 36*b^7*d^2*x^2*e^2 + 984*a*b^5*c*d^2*x^2*e^2 + 8556*a^2*b^3*c^2*d^2*x^2*e^2 + 5256*
a^3*b*c^3*d^2*x^2*e^2 - 16*b^7*d^3*x*e + 304*a*b^5*c*d^3*x*e - 2784*a^2*b^3*c^2*d^3*x*e - 4464*a^3*b*c^3*d^3*x
*e - 3*b^7*d^4 + 50*a*b^5*c*d^4 - 326*a^2*b^3*c^2*d^4 + 1116*a^3*b*c^3*d^4 + 148*a*b^6*x^3*e^4 + 812*a^2*b^4*c
*x^3*e^4 + 1800*a^3*b^2*c^2*x^3*e^4 - 264*a^4*c^3*x^3*e^4 - 72*a*b^6*d*x^2*e^3 - 3192*a^2*b^4*c*d*x^2*e^3 - 32
08*a^3*b^2*c^2*d*x^2*e^3 - 2048*a^4*c^3*d*x^2*e^3 - 24*a*b^6*d^2*x*e^2 + 672*a^2*b^4*c*d^2*x*e^2 + 6696*a^3*b^
2*c^2*d^2*x*e^2 - 720*a^4*c^3*d^2*x*e^2 - 4*a*b^6*d^3*e + 76*a^2*b^4*c*d^3*e - 696*a^3*b^2*c^2*d^3*e - 1536*a^
4*c^3*d^3*e + 258*a^2*b^5*x^2*e^4 + 1016*a^3*b^3*c*x^2*e^4 + 628*a^4*b*c^2*x^2*e^4 - 48*a^2*b^5*d*x*e^3 - 2416
*a^3*b^3*c*d*x*e^3 - 1328*a^4*b*c^2*d*x*e^3 - 6*a^2*b^5*d^2*e^2 + 168*a^3*b^3*c*d^2*e^2 + 1944*a^4*b*c^2*d^2*e
^2 + 188*a^3*b^4*x*e^4 + 736*a^4*b^2*c*x*e^4 - 72*a^5*c^2*x*e^4 - 12*a^3*b^4*d*e^3 - 664*a^4*b^2*c*d*e^3 - 512
*a^5*c^2*d*e^3 + 50*a^4*b^3*e^4 + 220*a^5*b*c*e^4)/((b^8 - 16*a*b^6*c + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256
*a^4*c^4)*(c*x^2 + b*x + a)^4)

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Mupad [B]
time = 2.54, size = 2337, normalized size = 4.29 \begin {gather*} \frac {\frac {x^2\,\left (314\,a^4\,b\,c^2\,e^4-1024\,a^4\,c^3\,d\,e^3+508\,a^3\,b^3\,c\,e^4-1604\,a^3\,b^2\,c^2\,d\,e^3+2628\,a^3\,b\,c^3\,d^2\,e^2+129\,a^2\,b^5\,e^4-1596\,a^2\,b^4\,c\,d\,e^3+4278\,a^2\,b^3\,c^2\,d^2\,e^2-6132\,a^2\,b^2\,c^3\,d^3\,e+3066\,a^2\,b\,c^4\,d^4-36\,a\,b^6\,d\,e^3+492\,a\,b^5\,c\,d^2\,e^2-784\,a\,b^4\,c^2\,d^3\,e+392\,a\,b^3\,c^3\,d^4-18\,b^7\,d^2\,e^2+28\,b^6\,c\,d^3\,e-14\,b^5\,c^2\,d^4\right )}{6\,\left (256\,a^4\,c^4-256\,a^3\,b^2\,c^3+96\,a^2\,b^4\,c^2-16\,a\,b^6\,c+b^8\right )}-\frac {x\,\left (18\,a^5\,c^2\,e^4-184\,a^4\,b^2\,c\,e^4+332\,a^4\,b\,c^2\,d\,e^3+180\,a^4\,c^3\,d^2\,e^2-47\,a^3\,b^4\,e^4+604\,a^3\,b^3\,c\,d\,e^3-1674\,a^3\,b^2\,c^2\,d^2\,e^2+1116\,a^3\,b\,c^3\,d^3\,e-558\,a^3\,c^4\,d^4+12\,a^2\,b^5\,d\,e^3-168\,a^2\,b^4\,c\,d^2\,e^2+696\,a^2\,b^3\,c^2\,d^3\,e-348\,a^2\,b^2\,c^3\,d^4+6\,a\,b^6\,d^2\,e^2-76\,a\,b^5\,c\,d^3\,e+38\,a\,b^4\,c^2\,d^4+4\,b^7\,d^3\,e-2\,b^6\,c\,d^4\right )}{3\,\left (256\,a^4\,c^4-256\,a^3\,b^2\,c^3+96\,a^2\,b^4\,c^2-16\,a\,b^6\,c+b^8\right )}-\frac {-220\,a^5\,b\,c\,e^4+512\,a^5\,c^2\,d\,e^3-50\,a^4\,b^3\,e^4+664\,a^4\,b^2\,c\,d\,e^3-1944\,a^4\,b\,c^2\,d^2\,e^2+1536\,a^4\,c^3\,d^3\,e+12\,a^3\,b^4\,d\,e^3-168\,a^3\,b^3\,c\,d^2\,e^2+696\,a^3\,b^2\,c^2\,d^3\,e-1116\,a^3\,b\,c^3\,d^4+6\,a^2\,b^5\,d^2\,e^2-76\,a^2\,b^4\,c\,d^3\,e+326\,a^2\,b^3\,c^2\,d^4+4\,a\,b^6\,d^3\,e-50\,a\,b^5\,c\,d^4+3\,b^7\,d^4}{12\,\left (256\,a^4\,c^4-256\,a^3\,b^2\,c^3+96\,a^2\,b^4\,c^2-16\,a\,b^6\,c+b^8\right )}+\frac {x^3\,\left (-66\,a^4\,c^3\,e^4+450\,a^3\,b^2\,c^2\,e^4-876\,a^3\,b\,c^3\,d\,e^3+876\,a^3\,c^4\,d^2\,e^2+203\,a^2\,b^4\,c\,e^4-1504\,a^2\,b^3\,c^2\,d\,e^3+2526\,a^2\,b^2\,c^3\,d^2\,e^2-2044\,a^2\,b\,c^4\,d^3\,e+1022\,a^2\,c^5\,d^4+37\,a\,b^6\,e^4-440\,a\,b^5\,c\,d\,e^3+1854\,a\,b^4\,c^2\,d^2\,e^2-2828\,a\,b^3\,c^3\,d^3\,e+1414\,a\,b^2\,c^4\,d^4-12\,b^7\,d\,e^3+54\,b^6\,c\,d^2\,e^2-84\,b^5\,c^2\,d^3\,e+42\,b^4\,c^3\,d^4\right )}{3\,\left (256\,a^4\,c^4-256\,a^3\,b^2\,c^3+96\,a^2\,b^4\,c^2-16\,a\,b^6\,c+b^8\right )}+\frac {5\,x^4\,\left (5\,b^3+22\,a\,c\,b\right )\,\left (6\,a^2\,c^2\,e^4+12\,a\,b^2\,c\,e^4-60\,a\,b\,c^2\,d\,e^3+60\,a\,c^3\,d^2\,e^2+b^4\,e^4-20\,b^3\,c\,d\,e^3+90\,b^2\,c^2\,d^2\,e^2-140\,b\,c^3\,d^3\,e+70\,c^4\,d^4\right )}{12\,\left (256\,a^4\,c^4-256\,a^3\,b^2\,c^3+96\,a^2\,b^4\,c^2-16\,a\,b^6\,c+b^8\right )}+\frac {x^5\,\left (13\,b^2\,c+11\,a\,c^2\right )\,\left (6\,a^2\,c^2\,e^4+12\,a\,b^2\,c\,e^4-60\,a\,b\,c^2\,d\,e^3+60\,a\,c^3\,d^2\,e^2+b^4\,e^4-20\,b^3\,c\,d\,e^3+90\,b^2\,c^2\,d^2\,e^2-140\,b\,c^3\,d^3\,e+70\,c^4\,d^4\right )}{3\,\left (256\,a^4\,c^4-256\,a^3\,b^2\,c^3+96\,a^2\,b^4\,c^2-16\,a\,b^6\,c+b^8\right )}+\frac {c^3\,x^7\,\left (6\,a^2\,c^2\,e^4+12\,a\,b^2\,c\,e^4-60\,a\,b\,c^2\,d\,e^3+60\,a\,c^3\,d^2\,e^2+b^4\,e^4-20\,b^3\,c\,d\,e^3+90\,b^2\,c^2\,d^2\,e^2-140\,b\,c^3\,d^3\,e+70\,c^4\,d^4\right )}{256\,a^4\,c^4-256\,a^3\,b^2\,c^3+96\,a^2\,b^4\,c^2-16\,a\,b^6\,c+b^8}+\frac {7\,b\,c^2\,x^6\,\left (6\,a^2\,c^2\,e^4+12\,a\,b^2\,c\,e^4-60\,a\,b\,c^2\,d\,e^3+60\,a\,c^3\,d^2\,e^2+b^4\,e^4-20\,b^3\,c\,d\,e^3+90\,b^2\,c^2\,d^2\,e^2-140\,b\,c^3\,d^3\,e+70\,c^4\,d^4\right )}{2\,\left (256\,a^4\,c^4-256\,a^3\,b^2\,c^3+96\,a^2\,b^4\,c^2-16\,a\,b^6\,c+b^8\right )}}{x^4\,\left (6\,a^2\,c^2+12\,a\,b^2\,c+b^4\right )+a^4+c^4\,x^8+x^2\,\left (4\,c\,a^3+6\,a^2\,b^2\right )+x^6\,\left (6\,b^2\,c^2+4\,a\,c^3\right )+x^3\,\left (12\,c\,a^2\,b+4\,a\,b^3\right )+x^5\,\left (4\,b^3\,c+12\,a\,b\,c^2\right )+4\,b\,c^3\,x^7+4\,a^3\,b\,x}+\frac {2\,\mathrm {atan}\left (\frac {\left (\frac {2\,c\,x\,\left (6\,a^2\,c^2\,e^4+12\,a\,b^2\,c\,e^4-60\,a\,b\,c^2\,d\,e^3+60\,a\,c^3\,d^2\,e^2+b^4\,e^4-20\,b^3\,c\,d\,e^3+90\,b^2\,c^2\,d^2\,e^2-140\,b\,c^3\,d^3\,e+70\,c^4\,d^4\right )}{{\left (4\,a\,c-b^2\right )}^{9/2}}+\frac {\left (256\,a^4\,b\,c^4-256\,a^3\,b^3\,c^3+96\,a^2\,b^5\,c^2-16\,a\,b^7\,c+b^9\right )\,\left (6\,a^2\,c^2\,e^4+12\,a\,b^2\,c\,e^4-60\,a\,b\,c^2\,d\,e^3+60\,a\,c^3\,d^2\,e^2+b^4\,e^4-20\,b^3\,c\,d\,e^3+90\,b^2\,c^2\,d^2\,e^2-140\,b\,c^3\,d^3\,e+70\,c^4\,d^4\right )}{{\left (4\,a\,c-b^2\right )}^{9/2}\,\left (256\,a^4\,c^4-256\,a^3\,b^2\,c^3+96\,a^2\,b^4\,c^2-16\,a\,b^6\,c+b^8\right )}\right )\,\left (256\,a^4\,c^4-256\,a^3\,b^2\,c^3+96\,a^2\,b^4\,c^2-16\,a\,b^6\,c+b^8\right )}{6\,a^2\,c^2\,e^4+12\,a\,b^2\,c\,e^4-60\,a\,b\,c^2\,d\,e^3+60\,a\,c^3\,d^2\,e^2+b^4\,e^4-20\,b^3\,c\,d\,e^3+90\,b^2\,c^2\,d^2\,e^2-140\,b\,c^3\,d^3\,e+70\,c^4\,d^4}\right )\,\left (6\,a^2\,c^2\,e^4+12\,a\,b^2\,c\,e^4-60\,a\,b\,c^2\,d\,e^3+60\,a\,c^3\,d^2\,e^2+b^4\,e^4-20\,b^3\,c\,d\,e^3+90\,b^2\,c^2\,d^2\,e^2-140\,b\,c^3\,d^3\,e+70\,c^4\,d^4\right )}{{\left (4\,a\,c-b^2\right )}^{9/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^4/(a + b*x + c*x^2)^5,x)

[Out]

((x^2*(129*a^2*b^5*e^4 - 14*b^5*c^2*d^4 - 18*b^7*d^2*e^2 + 392*a*b^3*c^3*d^4 + 3066*a^2*b*c^4*d^4 + 508*a^3*b^
3*c*e^4 + 314*a^4*b*c^2*e^4 - 1024*a^4*c^3*d*e^3 - 36*a*b^6*d*e^3 + 28*b^6*c*d^3*e + 4278*a^2*b^3*c^2*d^2*e^2
- 784*a*b^4*c^2*d^3*e + 492*a*b^5*c*d^2*e^2 - 1596*a^2*b^4*c*d*e^3 - 6132*a^2*b^2*c^3*d^3*e + 2628*a^3*b*c^3*d
^2*e^2 - 1604*a^3*b^2*c^2*d*e^3))/(6*(b^8 + 256*a^4*c^4 + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 - 16*a*b^6*c)) - (x
*(4*b^7*d^3*e - 2*b^6*c*d^4 - 47*a^3*b^4*e^4 - 558*a^3*c^4*d^4 + 18*a^5*c^2*e^4 + 38*a*b^4*c^2*d^4 - 184*a^4*b
^2*c*e^4 + 6*a*b^6*d^2*e^2 + 12*a^2*b^5*d*e^3 - 348*a^2*b^2*c^3*d^4 + 180*a^4*c^3*d^2*e^2 - 76*a*b^5*c*d^3*e -
1674*a^3*b^2*c^2*d^2*e^2 + 1116*a^3*b*c^3*d^3*e + 604*a^3*b^3*c*d*e^3 + 332*a^4*b*c^2*d*e^3 + 696*a^2*b^3*c^2
*d^3*e - 168*a^2*b^4*c*d^2*e^2))/(3*(b^8 + 256*a^4*c^4 + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 - 16*a*b^6*c)) - (3*
b^7*d^4 - 50*a^4*b^3*e^4 - 1116*a^3*b*c^3*d^4 + 12*a^3*b^4*d*e^3 + 1536*a^4*c^3*d^3*e + 512*a^5*c^2*d*e^3 + 32
6*a^2*b^3*c^2*d^4 + 6*a^2*b^5*d^2*e^2 - 50*a*b^5*c*d^4 - 220*a^5*b*c*e^4 + 4*a*b^6*d^3*e - 76*a^2*b^4*c*d^3*e
+ 664*a^4*b^2*c*d*e^3 + 696*a^3*b^2*c^2*d^3*e - 168*a^3*b^3*c*d^2*e^2 - 1944*a^4*b*c^2*d^2*e^2)/(12*(b^8 + 256
*a^4*c^4 + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 - 16*a*b^6*c)) + (x^3*(37*a*b^6*e^4 - 12*b^7*d*e^3 + 1022*a^2*c^5*
d^4 - 66*a^4*c^3*e^4 + 42*b^4*c^3*d^4 + 1414*a*b^2*c^4*d^4 + 203*a^2*b^4*c*e^4 - 84*b^5*c^2*d^3*e + 54*b^6*c*d
^2*e^2 + 450*a^3*b^2*c^2*e^4 + 876*a^3*c^4*d^2*e^2 - 440*a*b^5*c*d*e^3 + 2526*a^2*b^2*c^3*d^2*e^2 - 2828*a*b^3
*c^3*d^3*e - 2044*a^2*b*c^4*d^3*e - 876*a^3*b*c^3*d*e^3 + 1854*a*b^4*c^2*d^2*e^2 - 1504*a^2*b^3*c^2*d*e^3))/(3
*(b^8 + 256*a^4*c^4 + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 - 16*a*b^6*c)) + (5*x^4*(5*b^3 + 22*a*b*c)*(b^4*e^4 + 7
0*c^4*d^4 + 6*a^2*c^2*e^4 + 60*a*c^3*d^2*e^2 + 90*b^2*c^2*d^2*e^2 + 12*a*b^2*c*e^4 - 140*b*c^3*d^3*e - 20*b^3*
c*d*e^3 - 60*a*b*c^2*d*e^3))/(12*(b^8 + 256*a^4*c^4 + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 - 16*a*b^6*c)) + (x^5*(
11*a*c^2 + 13*b^2*c)*(b^4*e^4 + 70*c^4*d^4 + 6*a^2*c^2*e^4 + 60*a*c^3*d^2*e^2 + 90*b^2*c^2*d^2*e^2 + 12*a*b^2*
c*e^4 - 140*b*c^3*d^3*e - 20*b^3*c*d*e^3 - 60*a*b*c^2*d*e^3))/(3*(b^8 + 256*a^4*c^4 + 96*a^2*b^4*c^2 - 256*a^3
*b^2*c^3 - 16*a*b^6*c)) + (c^3*x^7*(b^4*e^4 + 70*c^4*d^4 + 6*a^2*c^2*e^4 + 60*a*c^3*d^2*e^2 + 90*b^2*c^2*d^2*e
^2 + 12*a*b^2*c*e^4 - 140*b*c^3*d^3*e - 20*b^3*c*d*e^3 - 60*a*b*c^2*d*e^3))/(b^8 + 256*a^4*c^4 + 96*a^2*b^4*c^
2 - 256*a^3*b^2*c^3 - 16*a*b^6*c) + (7*b*c^2*x^6*(b^4*e^4 + 70*c^4*d^4 + 6*a^2*c^2*e^4 + 60*a*c^3*d^2*e^2 + 90
*b^2*c^2*d^2*e^2 + 12*a*b^2*c*e^4 - 140*b*c^3*d^3*e - 20*b^3*c*d*e^3 - 60*a*b*c^2*d*e^3))/(2*(b^8 + 256*a^4*c^
4 + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 - 16*a*b^6*c)))/(x^4*(b^4 + 6*a^2*c^2 + 12*a*b^2*c) + a^4 + c^4*x^8 + x^2
*(4*a^3*c + 6*a^2*b^2) + x^6*(4*a*c^3 + 6*b^2*c^2) + x^3*(4*a*b^3 + 12*a^2*b*c) + x^5*(4*b^3*c + 12*a*b*c^2) +
4*b*c^3*x^7 + 4*a^3*b*x) + (2*atan((((2*c*x*(b^4*e^4 + 70*c^4*d^4 + 6*a^2*c^2*e^4 + 60*a*c^3*d^2*e^2 + 90*b^2
*c^2*d^2*e^2 + 12*a*b^2*c*e^4 - 140*b*c^3*d^3*e - 20*b^3*c*d*e^3 - 60*a*b*c^2*d*e^3))/(4*a*c - b^2)^(9/2) + ((
b^9 + 256*a^4*b*c^4 + 96*a^2*b^5*c^2 - 256*a^3*b^3*c^3 - 16*a*b^7*c)*(b^4*e^4 + 70*c^4*d^4 + 6*a^2*c^2*e^4 + 6
0*a*c^3*d^2*e^2 + 90*b^2*c^2*d^2*e^2 + 12*a*b^2*c*e^4 - 140*b*c^3*d^3*e - 20*b^3*c*d*e^3 - 60*a*b*c^2*d*e^3))/
((4*a*c - b^2)^(9/2)*(b^8 + 256*a^4*c^4 + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 - 16*a*b^6*c)))*(b^8 + 256*a^4*c^4
+ 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 - 16*a*b^6*c))/(b^4*e^4 + 70*c^4*d^4 + 6*a^2*c^2*e^4 + 60*a*c^3*d^2*e^2 + 9
0*b^2*c^2*d^2*e^2 + 12*a*b^2*c*e^4 - 140*b*c^3*d^3*e - 20*b^3*c*d*e^3 - 60*a*b*c^2*d*e^3))*(b^4*e^4 + 70*c^4*d
^4 + 6*a^2*c^2*e^4 + 60*a*c^3*d^2*e^2 + 90*b^2*c^2*d^2*e^2 + 12*a*b^2*c*e^4 - 140*b*c^3*d^3*e - 20*b^3*c*d*e^3
- 60*a*b*c^2*d*e^3))/(4*a*c - b^2)^(9/2)

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