3.23.23 \(\int \frac {(d+e x)^3}{(a+b x+c x^2)^5} \, dx\) [2223]

Optimal. Leaf size=378 \[ -\frac {(b+2 c x) (d+e x)^3}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac {(d+e x)^2 \left (14 b c d-3 b^2 e-16 a c e+14 c (2 c d-b e) x\right )}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac {3 b^3 d e^2-32 a c e \left (7 c d^2+a e^2\right )+2 b c d \left (35 c d^2+99 a e^2\right )-b^2 \left (49 c d^2 e+27 a e^3\right )+2 (2 c d-b e) \left (35 c^2 d^2+12 b^2 e^2-c e (35 b d+13 a e)\right ) x}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}+\frac {5 (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (b+2 c x)}{2 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac {10 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2}} \]

[Out]

-1/4*(2*c*x+b)*(e*x+d)^3/(-4*a*c+b^2)/(c*x^2+b*x+a)^4+1/12*(e*x+d)^2*(14*b*c*d-3*b^2*e-16*a*c*e+14*c*(-b*e+2*c
*d)*x)/(-4*a*c+b^2)^2/(c*x^2+b*x+a)^3+1/12*(-3*b^3*d*e^2+32*a*c*e*(a*e^2+7*c*d^2)-2*b*c*d*(99*a*e^2+35*c*d^2)+
b^2*(27*a*e^3+49*c*d^2*e)-2*(-b*e+2*c*d)*(35*c^2*d^2+12*b^2*e^2-c*e*(13*a*e+35*b*d))*x)/(-4*a*c+b^2)^3/(c*x^2+
b*x+a)^2+5/2*(-b*e+2*c*d)*(7*c^2*d^2+b^2*e^2-c*e*(-3*a*e+7*b*d))*(2*c*x+b)/(-4*a*c+b^2)^4/(c*x^2+b*x+a)-10*c*(
-b*e+2*c*d)*(7*c^2*d^2+b^2*e^2-c*e*(-3*a*e+7*b*d))*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 = 0.32, antiderivative size = 378, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {750, 834, 791, 628, 632, 212} \begin {gather*} \frac {5 (b+2 c x) (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{2 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac {10 c (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\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 {(b+2 c x) (d+e x)^3}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac {(d+e x)^2 \left (-16 a c e-3 b^2 e+14 c x (2 c d-b e)+14 b c d\right )}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac {2 x (2 c d-b e) \left (-c e (13 a e+35 b d)+12 b^2 e^2+35 c^2 d^2\right )-b^2 \left (27 a e^3+49 c d^2 e\right )+2 b c d \left (99 a e^2+35 c d^2\right )-32 a c e \left (a e^2+7 c d^2\right )+3 b^3 d e^2}{12 \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)^3/(a + b*x + c*x^2)^5,x]

[Out]

-1/4*((b + 2*c*x)*(d + e*x)^3)/((b^2 - 4*a*c)*(a + b*x + c*x^2)^4) + ((d + e*x)^2*(14*b*c*d - 3*b^2*e - 16*a*c
*e + 14*c*(2*c*d - b*e)*x))/(12*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^3) - (3*b^3*d*e^2 - 32*a*c*e*(7*c*d^2 + a*e^
2) + 2*b*c*d*(35*c*d^2 + 99*a*e^2) - b^2*(49*c*d^2*e + 27*a*e^3) + 2*(2*c*d - b*e)*(35*c^2*d^2 + 12*b^2*e^2 -
c*e*(35*b*d + 13*a*e))*x)/(12*(b^2 - 4*a*c)^3*(a + b*x + c*x^2)^2) + (5*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c
*e*(7*b*d - 3*a*e))*(b + 2*c*x))/(2*(b^2 - 4*a*c)^4*(a + b*x + c*x^2)) - (10*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*
e^2 - c*e*(7*b*d - 3*a*e))*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 628

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

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)^3}{\left (a+b x+c x^2\right )^5} \, dx &=-\frac {(b+2 c x) (d+e x)^3}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac {\int \frac {(d+e x)^2 (-14 c d+3 b e-8 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)^3}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac {(d+e x)^2 \left (14 b c d-3 b^2 e-16 a c e+14 c (2 c d-b e) x\right )}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac {\int \frac {(d+e x) \left (-2 \left (70 c^2 d^2+3 b^2 e^2-c e (49 b d-16 a e)\right )-42 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)^3}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac {(d+e x)^2 \left (14 b c d-3 b^2 e-16 a c e+14 c (2 c d-b e) x\right )}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac {3 b^3 d e^2-32 a c e \left (7 c d^2+a e^2\right )+2 b c d \left (35 c d^2+99 a e^2\right )-b^2 \left (49 c d^2 e+27 a e^3\right )+2 (2 c d-b e) \left (35 c^2 d^2+12 b^2 e^2-c e (35 b d+13 a e)\right ) x}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}-\frac {\left (5 (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right )\right ) \int \frac {1}{\left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right )^3}\\ &=-\frac {(b+2 c x) (d+e x)^3}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac {(d+e x)^2 \left (14 b c d-3 b^2 e-16 a c e+14 c (2 c d-b e) x\right )}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac {3 b^3 d e^2-32 a c e \left (7 c d^2+a e^2\right )+2 b c d \left (35 c d^2+99 a e^2\right )-b^2 \left (49 c d^2 e+27 a e^3\right )+2 (2 c d-b e) \left (35 c^2 d^2+12 b^2 e^2-c e (35 b d+13 a e)\right ) x}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}+\frac {5 (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (b+2 c x)}{2 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}+\frac {\left (5 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\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)^3}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac {(d+e x)^2 \left (14 b c d-3 b^2 e-16 a c e+14 c (2 c d-b e) x\right )}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac {3 b^3 d e^2-32 a c e \left (7 c d^2+a e^2\right )+2 b c d \left (35 c d^2+99 a e^2\right )-b^2 \left (49 c d^2 e+27 a e^3\right )+2 (2 c d-b e) \left (35 c^2 d^2+12 b^2 e^2-c e (35 b d+13 a e)\right ) x}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}+\frac {5 (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (b+2 c x)}{2 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac {\left (10 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\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)^3}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac {(d+e x)^2 \left (14 b c d-3 b^2 e-16 a c e+14 c (2 c d-b e) x\right )}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac {3 b^3 d e^2-32 a c e \left (7 c d^2+a e^2\right )+2 b c d \left (35 c d^2+99 a e^2\right )-b^2 \left (49 c d^2 e+27 a e^3\right )+2 (2 c d-b e) \left (35 c^2 d^2+12 b^2 e^2-c e (35 b d+13 a e)\right ) x}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}+\frac {5 (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\right ) (b+2 c x)}{2 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac {10 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2-c e (7 b d-3 a e)\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 = 0.75, size = 467, normalized size = 1.24 \begin {gather*} \frac {1}{12} \left (\frac {5 (2 c d-b e) \left (7 c^2 d^2+b^2 e^2+c e (-7 b d+3 a e)\right ) (b+2 c x)}{c \left (-b^2+4 a c\right )^3 (a+x (b+c x))^2}+\frac {30 (2 c d-b e) \left (7 c^2 d^2+b^2 e^2+c e (-7 b d+3 a e)\right ) (b+2 c x)}{\left (b^2-4 a c\right )^4 (a+x (b+c x))}+\frac {-3 b^4 e^3+b^3 c e^2 (9 d-2 e x)+4 c^2 \left (-8 a^2 e^3+7 c^2 d^3 x+3 a c d e^2 x\right )+b^2 c e \left (13 a e^2-3 c d (7 d-6 e x)\right )+2 b c^2 \left (7 c d^2 (d-3 e x)+3 a e^2 (d-e x)\right )}{c^2 \left (b^2-4 a c\right )^2 (a+x (b+c x))^3}+\frac {3 \left (-b^3 e^3 x+b^2 e^2 (-a e+3 c d x)+2 c \left (a^2 e^3+c^2 d^3 x-3 a c d e (d+e x)\right )+b c \left (c d^2 (d-3 e x)+3 a e^2 (d+e x)\right )\right )}{c^2 \left (-b^2+4 a c\right ) (a+x (b+c x))^4}+\frac {120 c (2 c d-b e) \left (7 c^2 d^2+b^2 e^2+c e (-7 b d+3 a e)\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)^3/(a + b*x + c*x^2)^5,x]

[Out]

((5*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 + c*e*(-7*b*d + 3*a*e))*(b + 2*c*x))/(c*(-b^2 + 4*a*c)^3*(a + x*(b + c*
x))^2) + (30*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 + c*e*(-7*b*d + 3*a*e))*(b + 2*c*x))/((b^2 - 4*a*c)^4*(a + x*(
b + c*x))) + (-3*b^4*e^3 + b^3*c*e^2*(9*d - 2*e*x) + 4*c^2*(-8*a^2*e^3 + 7*c^2*d^3*x + 3*a*c*d*e^2*x) + b^2*c*
e*(13*a*e^2 - 3*c*d*(7*d - 6*e*x)) + 2*b*c^2*(7*c*d^2*(d - 3*e*x) + 3*a*e^2*(d - e*x)))/(c^2*(b^2 - 4*a*c)^2*(
a + x*(b + c*x))^3) + (3*(-(b^3*e^3*x) + b^2*e^2*(-(a*e) + 3*c*d*x) + 2*c*(a^2*e^3 + c^2*d^3*x - 3*a*c*d*e*(d
+ e*x)) + b*c*(c*d^2*(d - 3*e*x) + 3*a*e^2*(d + e*x))))/(c^2*(-b^2 + 4*a*c)*(a + x*(b + c*x))^4) + (120*c*(2*c
*d - b*e)*(7*c^2*d^2 + b^2*e^2 + c*e*(-7*b*d + 3*a*e))*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. \(1364\) vs. \(2(365)=730\).
time = 0.98, size = 1365, normalized size = 3.61

method	result	size
default	\(\frac {-\frac {5 c^{4} \left (3 a b c \,e^{3}-6 d \,e^{2} c^{2} a +b^{3} e^{3}-9 b^{2} d \,e^{2} c +21 b \,c^{2} d^{2} e -14 c^{3} d^{3}\right ) x^{7}}{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 {35 c^{3} \left (3 a b c \,e^{3}-6 d \,e^{2} c^{2} a +b^{3} e^{3}-9 b^{2} d \,e^{2} c +21 b \,c^{2} d^{2} e -14 c^{3} d^{3}\right ) b \,x^{6}}{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 )}-\frac {5 c^{2} \left (11 a c +13 b^{2}\right ) \left (3 a b c \,e^{3}-6 d \,e^{2} c^{2} a +b^{3} e^{3}-9 b^{2} d \,e^{2} c +21 b \,c^{2} d^{2} e -14 c^{3} d^{3}\right ) x^{5}}{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 {25 b \left (22 a c +5 b^{2}\right ) c \left (3 a b c \,e^{3}-6 d \,e^{2} c^{2} a +b^{3} e^{3}-9 b^{2} d \,e^{2} c +21 b \,c^{2} d^{2} e -14 c^{3} d^{3}\right ) x^{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 {\left (73 a^{2} c^{2}+101 a c \,b^{2}+3 b^{4}\right ) \left (3 a b c \,e^{3}-6 d \,e^{2} c^{2} a +b^{3} e^{3}-9 b^{2} d \,e^{2} c +21 b \,c^{2} d^{2} e -14 c^{3} d^{3}\right ) x^{3}}{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 {\left (256 a^{4} c^{3} e^{3}+401 a^{3} b^{2} c^{2} e^{3}-1314 a^{3} b \,c^{3} d \,e^{2}+399 a^{2} b^{4} c \,e^{3}-2139 a^{2} b^{3} c^{2} d \,e^{2}+4599 a^{2} b^{2} c^{3} d^{2} e -3066 a^{2} b \,c^{4} d^{3}+9 a \,b^{6} e^{3}-246 a \,b^{5} c d \,e^{2}+588 a \,b^{4} c^{2} d^{2} e -392 a \,b^{3} c^{3} d^{3}+9 b^{7} d \,e^{2}-21 b^{6} c \,d^{2} e +14 b^{5} c^{2} d^{3}\right ) x^{2}}{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 {\left (83 a^{4} b \,c^{2} e^{3}+90 a^{4} c^{3} d \,e^{2}+151 a^{3} b^{3} c \,e^{3}-837 a^{3} b^{2} c^{2} d \,e^{2}+837 a^{3} b \,c^{3} d^{2} e -558 a^{3} c^{4} d^{3}+3 a^{2} b^{5} e^{3}-84 a^{2} b^{4} c d \,e^{2}+522 a^{2} b^{3} c^{2} d^{2} e -348 b^{2} c^{3} d^{3} a^{2}+3 a \,b^{6} d \,e^{2}-57 a \,b^{5} c \,d^{2} e +38 a \,b^{4} c^{2} d^{3}+3 b^{7} d^{2} e -2 b^{6} c \,d^{3}\right ) x}{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 {128 a^{5} c^{2} e^{3}+166 a^{4} b^{2} c \,e^{3}-972 a^{4} b \,c^{2} d \,e^{2}+1152 a^{4} c^{3} d^{2} e +3 a^{3} b^{4} e^{3}-84 a^{3} b^{3} c d \,e^{2}+522 a^{3} b^{2} c^{2} d^{2} e -1116 a^{3} b \,c^{3} d^{3}+3 a^{2} b^{5} d \,e^{2}-57 a^{2} b^{4} c \,d^{2} e +326 a^{2} b^{3} c^{2} d^{3}+3 a \,b^{6} d^{2} e -50 a \,b^{5} c \,d^{3}+3 b^{7} d^{3}}{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 )}}{\left (c \,x^{2}+b x +a \right )^{4}}-\frac {10 c \left (3 a b c \,e^{3}-6 d \,e^{2} c^{2} a +b^{3} e^{3}-9 b^{2} d \,e^{2} c +21 b \,c^{2} d^{2} e -14 c^{3} d^{3}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\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 ) \sqrt {4 a c -b^{2}}}\)	\(1365\)
risch	\(\text {Expression too large to display}\)	\(2719\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-5*c^4*(3*a*b*c*e^3-6*a*c^2*d*e^2+b^3*e^3-9*b^2*c*d*e^2+21*b*c^2*d^2*e-14*c^3*d^3)/(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^7-35/2*c^3*(3*a*b*c*e^3-6*a*c^2*d*e^2+b^3*e^3-9*b^2*c*d*e^2+21*b*c^2*d^2*e
-14*c^3*d^3)/(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*x^6-5/3*c^2*(11*a*c+13*b^2)*(3*a*b*
c*e^3-6*a*c^2*d*e^2+b^3*e^3-9*b^2*c*d*e^2+21*b*c^2*d^2*e-14*c^3*d^3)/(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^5-25/12*b*(22*a*c+5*b^2)*c*(3*a*b*c*e^3-6*a*c^2*d*e^2+b^3*e^3-9*b^2*c*d*e^2+21*b*c^2*d^2*
e-14*c^3*d^3)/(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*(73*a^2*c^2+101*a*b^2*c+3*b^
4)*(3*a*b*c*e^3-6*a*c^2*d*e^2+b^3*e^3-9*b^2*c*d*e^2+21*b*c^2*d^2*e-14*c^3*d^3)/(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*(256*a^4*c^3*e^3+401*a^3*b^2*c^2*e^3-1314*a^3*b*c^3*d*e^2+399*a^2*b^4*c*e
^3-2139*a^2*b^3*c^2*d*e^2+4599*a^2*b^2*c^3*d^2*e-3066*a^2*b*c^4*d^3+9*a*b^6*e^3-246*a*b^5*c*d*e^2+588*a*b^4*c^
2*d^2*e-392*a*b^3*c^3*d^3+9*b^7*d*e^2-21*b^6*c*d^2*e+14*b^5*c^2*d^3)/(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*(83*a^4*b*c^2*e^3+90*a^4*c^3*d*e^2+151*a^3*b^3*c*e^3-837*a^3*b^2*c^2*d*e^2+837*a^3*
b*c^3*d^2*e-558*a^3*c^4*d^3+3*a^2*b^5*e^3-84*a^2*b^4*c*d*e^2+522*a^2*b^3*c^2*d^2*e-348*a^2*b^2*c^3*d^3+3*a*b^6
*d*e^2-57*a*b^5*c*d^2*e+38*a*b^4*c^2*d^3+3*b^7*d^2*e-2*b^6*c*d^3)/(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*(128*a^5*c^2*e^3+166*a^4*b^2*c*e^3-972*a^4*b*c^2*d*e^2+1152*a^4*c^3*d^2*e+3*a^3*b^4*e^3
-84*a^3*b^3*c*d*e^2+522*a^3*b^2*c^2*d^2*e-1116*a^3*b*c^3*d^3+3*a^2*b^5*d*e^2-57*a^2*b^4*c*d^2*e+326*a^2*b^3*c^
2*d^3+3*a*b^6*d^2*e-50*a*b^5*c*d^3+3*b^7*d^3)/(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)^4-10*c*(3*a*b*c*e^3-6*a*c^2*d*e^2+b^3*e^3-9*b^2*c*d*e^2+21*b*c^2*d^2*e-14*c^3*d^3)/(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))

________________________________________________________________________________________

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)^3/(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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2876 vs. \(2 (375) = 750\).
time = 3.11, size = 5772, normalized size = 15.27 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/12*(840*(b^2*c^7 - 4*a*c^8)*d^3*x^7 + 2940*(b^3*c^6 - 4*a*b*c^7)*d^3*x^6 + 280*(13*b^4*c^5 - 41*a*b^2*c^6 -
44*a^2*c^7)*d^3*x^5 + 350*(5*b^5*c^4 + 2*a*b^3*c^5 - 88*a^2*b*c^6)*d^3*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^3*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^3*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^3*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^3 - 60*(14*c^8*d^3*x^8 + 56*b*c^7*d^3*x^7 + 56*a^3*b*c^4*
d^3*x + 14*a^4*c^4*d^3 + 28*(3*b^2*c^6 + 2*a*c^7)*d^3*x^6 + 56*(b^3*c^5 + 3*a*b*c^6)*d^3*x^5 + 14*(b^4*c^4 + 1
2*a*b^2*c^5 + 6*a^2*c^6)*d^3*x^4 + 56*(a*b^3*c^4 + 3*a^2*b*c^5)*d^3*x^3 + 28*(3*a^2*b^2*c^4 + 2*a^3*c^5)*d^3*x
^2 - ((b^3*c^5 + 3*a*b*c^6)*x^8 + a^4*b^3*c + 3*a^5*b*c^2 + 4*(b^4*c^4 + 3*a*b^2*c^5)*x^7 + 2*(3*b^5*c^3 + 11*
a*b^3*c^4 + 6*a^2*b*c^5)*x^6 + 4*(b^6*c^2 + 6*a*b^4*c^3 + 9*a^2*b^2*c^4)*x^5 + (b^7*c + 15*a*b^5*c^2 + 42*a^2*
b^3*c^3 + 18*a^3*b*c^4)*x^4 + 4*(a*b^6*c + 6*a^2*b^4*c^2 + 9*a^3*b^2*c^3)*x^3 + 2*(3*a^2*b^5*c + 11*a^3*b^3*c^
2 + 6*a^4*b*c^3)*x^2 + 4*(a^3*b^4*c + 3*a^4*b^2*c^2)*x)*e^3 + 3*((3*b^2*c^6 + 2*a*c^7)*d*x^8 + 4*(3*b^3*c^5 +
2*a*b*c^6)*d*x^7 + 2*(9*b^4*c^4 + 12*a*b^2*c^5 + 4*a^2*c^6)*d*x^6 + 4*(3*b^5*c^3 + 11*a*b^3*c^4 + 6*a^2*b*c^5)
*d*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*x^4 + 4*(3*a*b^5*c^2 + 11*a^2*b^3*c^3 + 6*
a^3*b*c^4)*d*x^3 + 2*(9*a^2*b^4*c^2 + 12*a^3*b^2*c^3 + 4*a^4*c^4)*d*x^2 + 4*(3*a^3*b^3*c^2 + 2*a^4*b*c^3)*d*x
+ (3*a^4*b^2*c^2 + 2*a^5*c^3)*d)*e^2 - 21*(b*c^7*d^2*x^8 + 4*b^2*c^6*d^2*x^7 + 4*a^3*b^2*c^3*d^2*x + a^4*b*c^3
*d^2 + 2*(3*b^3*c^5 + 2*a*b*c^6)*d^2*x^6 + 4*(b^4*c^4 + 3*a*b^2*c^5)*d^2*x^5 + (b^5*c^3 + 12*a*b^3*c^4 + 6*a^2
*b*c^5)*d^2*x^4 + 4*(a*b^4*c^3 + 3*a^2*b^2*c^4)*d^2*x^3 + 2*(3*a^2*b^3*c^3 + 2*a^3*b*c^4)*d^2*x^2)*e)*sqrt(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)) - (3*a^3*
b^6 + 154*a^4*b^4*c - 536*a^5*b^2*c^2 - 512*a^6*c^3 + 60*(b^5*c^4 - a*b^3*c^5 - 12*a^2*b*c^6)*x^7 + 210*(b^6*c
^3 - a*b^4*c^4 - 12*a^2*b^2*c^5)*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)*x^5 + 2
5*(5*b^8*c + 17*a*b^6*c^2 - 82*a^2*b^4*c^3 - 264*a^3*b^2*c^4)*x^4 + 4*(3*b^9 + 98*a*b^7*c - 64*a^2*b^5*c^2 - 1
285*a^3*b^3*c^3 - 876*a^4*b*c^4)*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)*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)*x)*e^3 + 3*(60*(3*b^4*c^5 - 10
*a*b^2*c^6 - 8*a^2*c^7)*d*x^7 + 210*(3*b^5*c^4 - 10*a*b^3*c^5 - 8*a^2*b*c^6)*d*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*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*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*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*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*x - (a^2*b^7 - 32*a^3*b^5*c - 212*a^4*b^3*c^2 + 1296*a^5*b*c^3)*d)*e^2 -
3*(420*(b^3*c^6 - 4*a*b*c^7)*d^2*x^7 + 1470*(b^4*c^5 - 4*a*b^2*c^6)*d^2*x^6 + 140*(13*b^5*c^4 - 41*a*b^3*c^5
- 44*a^2*b*c^6)*d^2*x^5 + 175*(5*b^6*c^3 + 2*a*b^4*c^4 - 88*a^2*b^2*c^5)*d^2*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^2*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
^2*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^2*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^2)*e)/(a^4*b^10 - 20*a^5*b^8*c + 160*a^6*b^6*c^2 - 64
0*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^5*c^6 + 128
0*a^4*b^3*c^7 - 1024*a^5*b*c^8)*x^7 + 2*(3*b^12*c^2 - 58*a*b^10*c^3 + 440*a^2*b^8*c^4 - 1600*a^3*b^6*c^5 + 256
0*a^4*b^4*c^6 - 512*a^5*b^2*c^7 - 2048*a^6*c^8)*x^6 + 4*(b^13*c - 17*a*b^11*c^2 + 100*a^2*b^9*c^3 - 160*a^3*b^
7*c^4 - 640*a^4*b^5*c^5 + 2816*a^5*b^3*c^6 - 3072*a^6*b*c^7)*x^5 + (b^14 - 8*a*b^12*c - 74*a^2*b^10*c^2 + 1160
*a^3*b^8*c^3 - 5440*a^4*b^6*c^4 + 10496*a^5*b^4*c^5 - 4608*a^6*b^2*c^6 - 6144*a^7*c^7)*x^4 + 4*(a*b^13 - 17*a^
2*b^11*c + 100*a^3*b^9*c^2 - 160*a^4*b^7*c^3 - 640*a^5*b^5*c^4 + 2816*a^6*b^3*c^5 - 3072*a^7*b*c^6)*x^3 + 2*(3
*a^2*b^12 - 58*a^3*b^10*c + 440*a^4*b^8*c^2 - 1600*a^5*b^6*c^3 + 2560*a^6*b^4*c^4 - 512*a^7*b^2*c^5 - 2048*a^8
*c^6)*x^2 + 4*(a^3*b^11 - 20*a^4*b^9*c + 160*a^5*b^7*c^2 - 640*a^6*b^5*c^3 + 1280*a^7*b^3*c^4 - 1024*a^8*b*c^5
)*x), 1/12*(840*(b^2*c^7 - 4*a*c^8)*d^3*x^7 + 2940*(b^3*c^6 - 4*a*b*c^7)*d^3*x^6 + 280*(13*b^4*c^5 - 41*a*b^2*
c^6 - 44*a^2*c^7)*d^3*x^5 + 350*(5*b^5*c^4 + 2*a*b^3*c^5 - 88*a^2*b*c^6)*d^3*x^4 + 56*(3*b^6*c^3 + 89*a*b^4*c^
4 - 331*a^2*b^2*c^5 - 292*a^3*c^6)*d^3*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^3
*x^2 + 8*(b^8*c - 23*a*b^6*c^2 + 250*a^2*b^4*c^...

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 2994 vs. \(2 (379) = 758\).
time = 60.80, size = 2994, normalized size = 7.92 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*c*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d)*(3*a*c*e**2 + b**2*e**2 - 7*b*c*d*e + 7*c**2*d**2)*log(x + (-5120
*a**5*c**6*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d)*(3*a*c*e**2 + b**2*e**2 - 7*b*c*d*e + 7*c**2*d**2) + 6400*
a**4*b**2*c**5*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d)*(3*a*c*e**2 + b**2*e**2 - 7*b*c*d*e + 7*c**2*d**2) - 3
200*a**3*b**4*c**4*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d)*(3*a*c*e**2 + b**2*e**2 - 7*b*c*d*e + 7*c**2*d**2)
+ 800*a**2*b**6*c**3*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d)*(3*a*c*e**2 + b**2*e**2 - 7*b*c*d*e + 7*c**2*d*
*2) - 100*a*b**8*c**2*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d)*(3*a*c*e**2 + b**2*e**2 - 7*b*c*d*e + 7*c**2*d*
*2) + 15*a*b**2*c**2*e**3 - 30*a*b*c**3*d*e**2 + 5*b**10*c*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d)*(3*a*c*e**
2 + b**2*e**2 - 7*b*c*d*e + 7*c**2*d**2) + 5*b**4*c*e**3 - 45*b**3*c**2*d*e**2 + 105*b**2*c**3*d**2*e - 70*b*c
**4*d**3)/(30*a*b*c**3*e**3 - 60*a*c**4*d*e**2 + 10*b**3*c**2*e**3 - 90*b**2*c**3*d*e**2 + 210*b*c**4*d**2*e -
140*c**5*d**3)) - 5*c*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d)*(3*a*c*e**2 + b**2*e**2 - 7*b*c*d*e + 7*c**2*d
**2)*log(x + (5120*a**5*c**6*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d)*(3*a*c*e**2 + b**2*e**2 - 7*b*c*d*e + 7*
c**2*d**2) - 6400*a**4*b**2*c**5*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d)*(3*a*c*e**2 + b**2*e**2 - 7*b*c*d*e
+ 7*c**2*d**2) + 3200*a**3*b**4*c**4*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d)*(3*a*c*e**2 + b**2*e**2 - 7*b*c*
d*e + 7*c**2*d**2) - 800*a**2*b**6*c**3*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d)*(3*a*c*e**2 + b**2*e**2 - 7*b
*c*d*e + 7*c**2*d**2) + 100*a*b**8*c**2*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d)*(3*a*c*e**2 + b**2*e**2 - 7*b
*c*d*e + 7*c**2*d**2) + 15*a*b**2*c**2*e**3 - 30*a*b*c**3*d*e**2 - 5*b**10*c*sqrt(-1/(4*a*c - b**2)**9)*(b*e -
2*c*d)*(3*a*c*e**2 + b**2*e**2 - 7*b*c*d*e + 7*c**2*d**2) + 5*b**4*c*e**3 - 45*b**3*c**2*d*e**2 + 105*b**2*c*
*3*d**2*e - 70*b*c**4*d**3)/(30*a*b*c**3*e**3 - 60*a*c**4*d*e**2 + 10*b**3*c**2*e**3 - 90*b**2*c**3*d*e**2 + 2
10*b*c**4*d**2*e - 140*c**5*d**3)) + (-128*a**5*c**2*e**3 - 166*a**4*b**2*c*e**3 + 972*a**4*b*c**2*d*e**2 - 11
52*a**4*c**3*d**2*e - 3*a**3*b**4*e**3 + 84*a**3*b**3*c*d*e**2 - 522*a**3*b**2*c**2*d**2*e + 1116*a**3*b*c**3*
d**3 - 3*a**2*b**5*d*e**2 + 57*a**2*b**4*c*d**2*e - 326*a**2*b**3*c**2*d**3 - 3*a*b**6*d**2*e + 50*a*b**5*c*d*
*3 - 3*b**7*d**3 + x**7*(-180*a*b*c**5*e**3 + 360*a*c**6*d*e**2 - 60*b**3*c**4*e**3 + 540*b**2*c**5*d*e**2 - 1
260*b*c**6*d**2*e + 840*c**7*d**3) + x**6*(-630*a*b**2*c**4*e**3 + 1260*a*b*c**5*d*e**2 - 210*b**4*c**3*e**3 +
1890*b**3*c**4*d*e**2 - 4410*b**2*c**5*d**2*e + 2940*b*c**6*d**3) + x**5*(-660*a**2*b*c**4*e**3 + 1320*a**2*c
**5*d*e**2 - 1000*a*b**3*c**3*e**3 + 3540*a*b**2*c**4*d*e**2 - 4620*a*b*c**5*d**2*e + 3080*a*c**6*d**3 - 260*b
**5*c**2*e**3 + 2340*b**4*c**3*d*e**2 - 5460*b**3*c**4*d**2*e + 3640*b**2*c**5*d**3) + x**4*(-1650*a**2*b**2*c
**3*e**3 + 3300*a**2*b*c**4*d*e**2 - 925*a*b**4*c**2*e**3 + 5700*a*b**3*c**3*d*e**2 - 11550*a*b**2*c**4*d**2*e
+ 7700*a*b*c**5*d**3 - 125*b**6*c*e**3 + 1125*b**5*c**2*d*e**2 - 2625*b**4*c**3*d**2*e + 1750*b**3*c**4*d**3)
+ x**3*(-876*a**3*b*c**3*e**3 + 1752*a**3*c**4*d*e**2 - 1504*a**2*b**3*c**2*e**3 + 5052*a**2*b**2*c**3*d*e**2
- 6132*a**2*b*c**4*d**2*e + 4088*a**2*c**5*d**3 - 440*a*b**5*c*e**3 + 3708*a*b**4*c**2*d*e**2 - 8484*a*b**3*c
**3*d**2*e + 5656*a*b**2*c**4*d**3 - 12*b**7*e**3 + 108*b**6*c*d*e**2 - 252*b**5*c**2*d**2*e + 168*b**4*c**3*d
**3) + x**2*(-512*a**4*c**3*e**3 - 802*a**3*b**2*c**2*e**3 + 2628*a**3*b*c**3*d*e**2 - 798*a**2*b**4*c*e**3 +
4278*a**2*b**3*c**2*d*e**2 - 9198*a**2*b**2*c**3*d**2*e + 6132*a**2*b*c**4*d**3 - 18*a*b**6*e**3 + 492*a*b**5*
c*d*e**2 - 1176*a*b**4*c**2*d**2*e + 784*a*b**3*c**3*d**3 - 18*b**7*d*e**2 + 42*b**6*c*d**2*e - 28*b**5*c**2*d
**3) + x*(-332*a**4*b*c**2*e**3 - 360*a**4*c**3*d*e**2 - 604*a**3*b**3*c*e**3 + 3348*a**3*b**2*c**2*d*e**2 - 3
348*a**3*b*c**3*d**2*e + 2232*a**3*c**4*d**3 - 12*a**2*b**5*e**3 + 336*a**2*b**4*c*d*e**2 - 2088*a**2*b**3*c**
2*d**2*e + 1392*a**2*b**2*c**3*d**3 - 12*a*b**6*d*e**2 + 228*a*b**5*c*d**2*e - 152*a*b**4*c**2*d**3 - 12*b**7*
d**2*e + 8*b**6*c*d**3))/(3072*a**8*c**4 - 3072*a**7*b**2*c**3 + 1152*a**6*b**4*c**2 - 192*a**5*b**6*c + 12*a*
*4*b**8 + x**8*(3072*a**4*c**8 - 3072*a**3*b**2*c**7 + 1152*a**2*b**4*c**6 - 192*a*b**6*c**5 + 12*b**8*c**4) +
x**7*(12288*a**4*b*c**7 - 12288*a**3*b**3*c**6 + 4608*a**2*b**5*c**5 - 768*a*b**7*c**4 + 48*b**9*c**3) + x**6
*(12288*a**5*c**7 + 6144*a**4*b**2*c**6 - 13824*a**3*b**4*c**5 + 6144*a**2*b**6*c**4 - 1104*a*b**8*c**3 + 72*b
**10*c**2) + x**5*(36864*a**5*b*c**6 - 24576*a**4*b**3*c**5 + 1536*a**3*b**5*c**4 + 2304*a**2*b**7*c**3 - 624*
a*b**9*c**2 + 48*b**11*c) + x**4*(18432*a**6*c**6 + 18432*a**5*b**2*c**5 - 26880*a**4*b**4*c**4 + 9600*a**3*b*
*6*c**3 - 1080*a**2*b**8*c**2 - 48*a*b**10*c + 12*b**12) + x**3*(36864*a**6*b*c**5 - 24576*a**5*b**3*c**4 + 15
36*a**4*b**5*c**3 + 2304*a**3*b**7*c**2 - 624*a**2*b**9*c + 48*a*b**11) + x**2*(12288*a**7*c**5 + 6144*a**6*b*
*2*c**4 - 13824*a**5*b**4*c**3 + 6144*a**4*b**6...

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1399 vs. \(2 (375) = 750\).
time = 1.51, size = 1399, normalized size = 3.70 \begin {gather*} \frac {10 \, {\left (14 \, c^{4} d^{3} - 21 \, b c^{3} d^{2} e + 9 \, b^{2} c^{2} d e^{2} + 6 \, a c^{3} d e^{2} - b^{3} c e^{3} - 3 \, a b c^{2} e^{3}\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^{3} x^{7} - 1260 \, b c^{6} d^{2} x^{7} e + 2940 \, b c^{6} d^{3} x^{6} + 540 \, b^{2} c^{5} d x^{7} e^{2} + 360 \, a c^{6} d x^{7} e^{2} - 4410 \, b^{2} c^{5} d^{2} x^{6} e + 3640 \, b^{2} c^{5} d^{3} x^{5} + 3080 \, a c^{6} d^{3} x^{5} - 60 \, b^{3} c^{4} x^{7} e^{3} - 180 \, a b c^{5} x^{7} e^{3} + 1890 \, b^{3} c^{4} d x^{6} e^{2} + 1260 \, a b c^{5} d x^{6} e^{2} - 5460 \, b^{3} c^{4} d^{2} x^{5} e - 4620 \, a b c^{5} d^{2} x^{5} e + 1750 \, b^{3} c^{4} d^{3} x^{4} + 7700 \, a b c^{5} d^{3} x^{4} - 210 \, b^{4} c^{3} x^{6} e^{3} - 630 \, a b^{2} c^{4} x^{6} e^{3} + 2340 \, b^{4} c^{3} d x^{5} e^{2} + 3540 \, a b^{2} c^{4} d x^{5} e^{2} + 1320 \, a^{2} c^{5} d x^{5} e^{2} - 2625 \, b^{4} c^{3} d^{2} x^{4} e - 11550 \, a b^{2} c^{4} d^{2} x^{4} e + 168 \, b^{4} c^{3} d^{3} x^{3} + 5656 \, a b^{2} c^{4} d^{3} x^{3} + 4088 \, a^{2} c^{5} d^{3} x^{3} - 260 \, b^{5} c^{2} x^{5} e^{3} - 1000 \, a b^{3} c^{3} x^{5} e^{3} - 660 \, a^{2} b c^{4} x^{5} e^{3} + 1125 \, b^{5} c^{2} d x^{4} e^{2} + 5700 \, a b^{3} c^{3} d x^{4} e^{2} + 3300 \, a^{2} b c^{4} d x^{4} e^{2} - 252 \, b^{5} c^{2} d^{2} x^{3} e - 8484 \, a b^{3} c^{3} d^{2} x^{3} e - 6132 \, a^{2} b c^{4} d^{2} x^{3} e - 28 \, b^{5} c^{2} d^{3} x^{2} + 784 \, a b^{3} c^{3} d^{3} x^{2} + 6132 \, a^{2} b c^{4} d^{3} x^{2} - 125 \, b^{6} c x^{4} e^{3} - 925 \, a b^{4} c^{2} x^{4} e^{3} - 1650 \, a^{2} b^{2} c^{3} x^{4} e^{3} + 108 \, b^{6} c d x^{3} e^{2} + 3708 \, a b^{4} c^{2} d x^{3} e^{2} + 5052 \, a^{2} b^{2} c^{3} d x^{3} e^{2} + 1752 \, a^{3} c^{4} d x^{3} e^{2} + 42 \, b^{6} c d^{2} x^{2} e - 1176 \, a b^{4} c^{2} d^{2} x^{2} e - 9198 \, a^{2} b^{2} c^{3} d^{2} x^{2} e + 8 \, b^{6} c d^{3} x - 152 \, a b^{4} c^{2} d^{3} x + 1392 \, a^{2} b^{2} c^{3} d^{3} x + 2232 \, a^{3} c^{4} d^{3} x - 12 \, b^{7} x^{3} e^{3} - 440 \, a b^{5} c x^{3} e^{3} - 1504 \, a^{2} b^{3} c^{2} x^{3} e^{3} - 876 \, a^{3} b c^{3} x^{3} e^{3} - 18 \, b^{7} d x^{2} e^{2} + 492 \, a b^{5} c d x^{2} e^{2} + 4278 \, a^{2} b^{3} c^{2} d x^{2} e^{2} + 2628 \, a^{3} b c^{3} d x^{2} e^{2} - 12 \, b^{7} d^{2} x e + 228 \, a b^{5} c d^{2} x e - 2088 \, a^{2} b^{3} c^{2} d^{2} x e - 3348 \, a^{3} b c^{3} d^{2} x e - 3 \, b^{7} d^{3} + 50 \, a b^{5} c d^{3} - 326 \, a^{2} b^{3} c^{2} d^{3} + 1116 \, a^{3} b c^{3} d^{3} - 18 \, a b^{6} x^{2} e^{3} - 798 \, a^{2} b^{4} c x^{2} e^{3} - 802 \, a^{3} b^{2} c^{2} x^{2} e^{3} - 512 \, a^{4} c^{3} x^{2} e^{3} - 12 \, a b^{6} d x e^{2} + 336 \, a^{2} b^{4} c d x e^{2} + 3348 \, a^{3} b^{2} c^{2} d x e^{2} - 360 \, a^{4} c^{3} d x e^{2} - 3 \, a b^{6} d^{2} e + 57 \, a^{2} b^{4} c d^{2} e - 522 \, a^{3} b^{2} c^{2} d^{2} e - 1152 \, a^{4} c^{3} d^{2} e - 12 \, a^{2} b^{5} x e^{3} - 604 \, a^{3} b^{3} c x e^{3} - 332 \, a^{4} b c^{2} x e^{3} - 3 \, a^{2} b^{5} d e^{2} + 84 \, a^{3} b^{3} c d e^{2} + 972 \, a^{4} b c^{2} d e^{2} - 3 \, a^{3} b^{4} e^{3} - 166 \, a^{4} b^{2} c e^{3} - 128 \, a^{5} c^{2} e^{3}}{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)^3/(c*x^2+b*x+a)^5,x, algorithm="giac")

[Out]

10*(14*c^4*d^3 - 21*b*c^3*d^2*e + 9*b^2*c^2*d*e^2 + 6*a*c^3*d*e^2 - b^3*c*e^3 - 3*a*b*c^2*e^3)*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^3*x^7 - 1260*b*c^6*d^2*x^7*e + 2940*b*c^6*d^3*x^6 + 540*b^2*c^5*d*x^7*e^2 + 360*a*c^6*d*x
^7*e^2 - 4410*b^2*c^5*d^2*x^6*e + 3640*b^2*c^5*d^3*x^5 + 3080*a*c^6*d^3*x^5 - 60*b^3*c^4*x^7*e^3 - 180*a*b*c^5
*x^7*e^3 + 1890*b^3*c^4*d*x^6*e^2 + 1260*a*b*c^5*d*x^6*e^2 - 5460*b^3*c^4*d^2*x^5*e - 4620*a*b*c^5*d^2*x^5*e +
1750*b^3*c^4*d^3*x^4 + 7700*a*b*c^5*d^3*x^4 - 210*b^4*c^3*x^6*e^3 - 630*a*b^2*c^4*x^6*e^3 + 2340*b^4*c^3*d*x^
5*e^2 + 3540*a*b^2*c^4*d*x^5*e^2 + 1320*a^2*c^5*d*x^5*e^2 - 2625*b^4*c^3*d^2*x^4*e - 11550*a*b^2*c^4*d^2*x^4*e
+ 168*b^4*c^3*d^3*x^3 + 5656*a*b^2*c^4*d^3*x^3 + 4088*a^2*c^5*d^3*x^3 - 260*b^5*c^2*x^5*e^3 - 1000*a*b^3*c^3*
x^5*e^3 - 660*a^2*b*c^4*x^5*e^3 + 1125*b^5*c^2*d*x^4*e^2 + 5700*a*b^3*c^3*d*x^4*e^2 + 3300*a^2*b*c^4*d*x^4*e^2
- 252*b^5*c^2*d^2*x^3*e - 8484*a*b^3*c^3*d^2*x^3*e - 6132*a^2*b*c^4*d^2*x^3*e - 28*b^5*c^2*d^3*x^2 + 784*a*b^
3*c^3*d^3*x^2 + 6132*a^2*b*c^4*d^3*x^2 - 125*b^6*c*x^4*e^3 - 925*a*b^4*c^2*x^4*e^3 - 1650*a^2*b^2*c^3*x^4*e^3
+ 108*b^6*c*d*x^3*e^2 + 3708*a*b^4*c^2*d*x^3*e^2 + 5052*a^2*b^2*c^3*d*x^3*e^2 + 1752*a^3*c^4*d*x^3*e^2 + 42*b^
6*c*d^2*x^2*e - 1176*a*b^4*c^2*d^2*x^2*e - 9198*a^2*b^2*c^3*d^2*x^2*e + 8*b^6*c*d^3*x - 152*a*b^4*c^2*d^3*x +
1392*a^2*b^2*c^3*d^3*x + 2232*a^3*c^4*d^3*x - 12*b^7*x^3*e^3 - 440*a*b^5*c*x^3*e^3 - 1504*a^2*b^3*c^2*x^3*e^3
- 876*a^3*b*c^3*x^3*e^3 - 18*b^7*d*x^2*e^2 + 492*a*b^5*c*d*x^2*e^2 + 4278*a^2*b^3*c^2*d*x^2*e^2 + 2628*a^3*b*c
^3*d*x^2*e^2 - 12*b^7*d^2*x*e + 228*a*b^5*c*d^2*x*e - 2088*a^2*b^3*c^2*d^2*x*e - 3348*a^3*b*c^3*d^2*x*e - 3*b^
7*d^3 + 50*a*b^5*c*d^3 - 326*a^2*b^3*c^2*d^3 + 1116*a^3*b*c^3*d^3 - 18*a*b^6*x^2*e^3 - 798*a^2*b^4*c*x^2*e^3 -
802*a^3*b^2*c^2*x^2*e^3 - 512*a^4*c^3*x^2*e^3 - 12*a*b^6*d*x*e^2 + 336*a^2*b^4*c*d*x*e^2 + 3348*a^3*b^2*c^2*d
*x*e^2 - 360*a^4*c^3*d*x*e^2 - 3*a*b^6*d^2*e + 57*a^2*b^4*c*d^2*e - 522*a^3*b^2*c^2*d^2*e - 1152*a^4*c^3*d^2*e
- 12*a^2*b^5*x*e^3 - 604*a^3*b^3*c*x*e^3 - 332*a^4*b*c^2*x*e^3 - 3*a^2*b^5*d*e^2 + 84*a^3*b^3*c*d*e^2 + 972*a
^4*b*c^2*d*e^2 - 3*a^3*b^4*e^3 - 166*a^4*b^2*c*e^3 - 128*a^5*c^2*e^3)/((b^8 - 16*a*b^6*c + 96*a^2*b^4*c^2 - 25
6*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 = 2.20, size = 1711, normalized size = 4.53 \begin {gather*} \frac {10\,c\,\mathrm {atan}\left (\frac {\left (\frac {10\,c^2\,x\,\left (b\,e-2\,c\,d\right )\,\left (b^2\,e^2-7\,b\,c\,d\,e+7\,c^2\,d^2+3\,a\,c\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{9/2}}+\frac {5\,c\,\left (b\,e-2\,c\,d\right )\,\left (b^2\,e^2-7\,b\,c\,d\,e+7\,c^2\,d^2+3\,a\,c\,e^2\right )\,\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 (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 )}{-5\,b^3\,c\,e^3+45\,b^2\,c^2\,d\,e^2-105\,b\,c^3\,d^2\,e-15\,a\,b\,c^2\,e^3+70\,c^4\,d^3+30\,a\,c^3\,d\,e^2}\right )\,\left (b\,e-2\,c\,d\right )\,\left (b^2\,e^2-7\,b\,c\,d\,e+7\,c^2\,d^2+3\,a\,c\,e^2\right )}{{\left (4\,a\,c-b^2\right )}^{9/2}}-\frac {\frac {128\,a^5\,c^2\,e^3+166\,a^4\,b^2\,c\,e^3-972\,a^4\,b\,c^2\,d\,e^2+1152\,a^4\,c^3\,d^2\,e+3\,a^3\,b^4\,e^3-84\,a^3\,b^3\,c\,d\,e^2+522\,a^3\,b^2\,c^2\,d^2\,e-1116\,a^3\,b\,c^3\,d^3+3\,a^2\,b^5\,d\,e^2-57\,a^2\,b^4\,c\,d^2\,e+326\,a^2\,b^3\,c^2\,d^3+3\,a\,b^6\,d^2\,e-50\,a\,b^5\,c\,d^3+3\,b^7\,d^3}{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\,\left (83\,a^4\,b\,c^2\,e^3+90\,a^4\,c^3\,d\,e^2+151\,a^3\,b^3\,c\,e^3-837\,a^3\,b^2\,c^2\,d\,e^2+837\,a^3\,b\,c^3\,d^2\,e-558\,a^3\,c^4\,d^3+3\,a^2\,b^5\,e^3-84\,a^2\,b^4\,c\,d\,e^2+522\,a^2\,b^3\,c^2\,d^2\,e-348\,a^2\,b^2\,c^3\,d^3+3\,a\,b^6\,d\,e^2-57\,a\,b^5\,c\,d^2\,e+38\,a\,b^4\,c^2\,d^3+3\,b^7\,d^2\,e-2\,b^6\,c\,d^3\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 {x^2\,\left (256\,a^4\,c^3\,e^3+401\,a^3\,b^2\,c^2\,e^3-1314\,a^3\,b\,c^3\,d\,e^2+399\,a^2\,b^4\,c\,e^3-2139\,a^2\,b^3\,c^2\,d\,e^2+4599\,a^2\,b^2\,c^3\,d^2\,e-3066\,a^2\,b\,c^4\,d^3+9\,a\,b^6\,e^3-246\,a\,b^5\,c\,d\,e^2+588\,a\,b^4\,c^2\,d^2\,e-392\,a\,b^3\,c^3\,d^3+9\,b^7\,d\,e^2-21\,b^6\,c\,d^2\,e+14\,b^5\,c^2\,d^3\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 {5\,c^4\,x^7\,\left (b^3\,e^3-9\,b^2\,c\,d\,e^2+21\,b\,c^2\,d^2\,e+3\,a\,b\,c\,e^3-14\,c^3\,d^3-6\,a\,c^2\,d\,e^2\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 {5\,x^5\,\left (13\,b^2\,c^2+11\,a\,c^3\right )\,\left (b^3\,e^3-9\,b^2\,c\,d\,e^2+21\,b\,c^2\,d^2\,e+3\,a\,b\,c\,e^3-14\,c^3\,d^3-6\,a\,c^2\,d\,e^2\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 {25\,x^4\,\left (5\,b^3\,c+22\,a\,b\,c^2\right )\,\left (b^3\,e^3-9\,b^2\,c\,d\,e^2+21\,b\,c^2\,d^2\,e+3\,a\,b\,c\,e^3-14\,c^3\,d^3-6\,a\,c^2\,d\,e^2\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^3\,\left (73\,a^2\,c^2+101\,a\,b^2\,c+3\,b^4\right )\,\left (b^3\,e^3-9\,b^2\,c\,d\,e^2+21\,b\,c^2\,d^2\,e+3\,a\,b\,c\,e^3-14\,c^3\,d^3-6\,a\,c^2\,d\,e^2\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 {35\,b\,c^3\,x^6\,\left (b^3\,e^3-9\,b^2\,c\,d\,e^2+21\,b\,c^2\,d^2\,e+3\,a\,b\,c\,e^3-14\,c^3\,d^3-6\,a\,c^2\,d\,e^2\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(10*c*atan((((10*c^2*x*(b*e - 2*c*d)*(b^2*e^2 + 7*c^2*d^2 + 3*a*c*e^2 - 7*b*c*d*e))/(4*a*c - b^2)^(9/2) + (5*c
*(b*e - 2*c*d)*(b^2*e^2 + 7*c^2*d^2 + 3*a*c*e^2 - 7*b*c*d*e)*(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))/((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))/(70*c^4*d^3 - 5*b^3*c*e^3 + 45*b^2*c^2
*d*e^2 - 15*a*b*c^2*e^3 + 30*a*c^3*d*e^2 - 105*b*c^3*d^2*e))*(b*e - 2*c*d)*(b^2*e^2 + 7*c^2*d^2 + 3*a*c*e^2 -
7*b*c*d*e))/(4*a*c - b^2)^(9/2) - ((3*b^7*d^3 + 3*a^3*b^4*e^3 + 128*a^5*c^2*e^3 - 1116*a^3*b*c^3*d^3 + 166*a^4
*b^2*c*e^3 + 3*a^2*b^5*d*e^2 + 1152*a^4*c^3*d^2*e + 326*a^2*b^3*c^2*d^3 - 50*a*b^5*c*d^3 + 3*a*b^6*d^2*e - 57*
a^2*b^4*c*d^2*e - 84*a^3*b^3*c*d*e^2 - 972*a^4*b*c^2*d*e^2 + 522*a^3*b^2*c^2*d^2*e)/(12*(b^8 + 256*a^4*c^4 + 9
6*a^2*b^4*c^2 - 256*a^3*b^2*c^3 - 16*a*b^6*c)) + (x*(3*b^7*d^2*e - 2*b^6*c*d^3 + 3*a^2*b^5*e^3 - 558*a^3*c^4*d
^3 + 38*a*b^4*c^2*d^3 + 151*a^3*b^3*c*e^3 + 83*a^4*b*c^2*e^3 + 90*a^4*c^3*d*e^2 - 348*a^2*b^2*c^3*d^3 + 3*a*b^
6*d*e^2 - 57*a*b^5*c*d^2*e - 84*a^2*b^4*c*d*e^2 + 837*a^3*b*c^3*d^2*e + 522*a^2*b^3*c^2*d^2*e - 837*a^3*b^2*c^
2*d*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)) + (x^2*(9*a*b^6*e^3 + 9*b^7*
d*e^2 + 256*a^4*c^3*e^3 + 14*b^5*c^2*d^3 - 392*a*b^3*c^3*d^3 - 3066*a^2*b*c^4*d^3 + 399*a^2*b^4*c*e^3 + 401*a^
3*b^2*c^2*e^3 - 21*b^6*c*d^2*e - 246*a*b^5*c*d*e^2 + 588*a*b^4*c^2*d^2*e - 1314*a^3*b*c^3*d*e^2 + 4599*a^2*b^2
*c^3*d^2*e - 2139*a^2*b^3*c^2*d*e^2))/(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))
+ (5*c^4*x^7*(b^3*e^3 - 14*c^3*d^3 + 3*a*b*c*e^3 - 6*a*c^2*d*e^2 + 21*b*c^2*d^2*e - 9*b^2*c*d*e^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) + (5*x^5*(11*a*c^3 + 13*b^2*c^2)*(b^3*e^3 - 14*c^3*d
^3 + 3*a*b*c*e^3 - 6*a*c^2*d*e^2 + 21*b*c^2*d^2*e - 9*b^2*c*d*e^2))/(3*(b^8 + 256*a^4*c^4 + 96*a^2*b^4*c^2 - 2
56*a^3*b^2*c^3 - 16*a*b^6*c)) + (25*x^4*(5*b^3*c + 22*a*b*c^2)*(b^3*e^3 - 14*c^3*d^3 + 3*a*b*c*e^3 - 6*a*c^2*d
*e^2 + 21*b*c^2*d^2*e - 9*b^2*c*d*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*(3*b^4 + 73*a^2*c^2 + 101*a*b^2*c)*(b^3*e^3 - 14*c^3*d^3 + 3*a*b*c*e^3 - 6*a*c^2*d*e^2 + 21*b*c^2*d^
2*e - 9*b^2*c*d*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)) + (35*b*c^3*x^6*
(b^3*e^3 - 14*c^3*d^3 + 3*a*b*c*e^3 - 6*a*c^2*d*e^2 + 21*b*c^2*d^2*e - 9*b^2*c*d*e^2))/(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)

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