3.63 \(\int \frac {A+B x+C x^2}{(d+e x)^3 (a+c x^2)^3} \, dx\)

Optimal. Leaf size=753 \[ -\frac {e^3 \log \left (a+c x^2\right ) \left (a^2 C e^4-a c e^2 \left (3 A e^2-9 B d e+13 C d^2\right )+c^2 d^2 \left (10 C d^2-3 e (5 B d-7 A e)\right )\right )}{2 \left (a e^2+c d^2\right )^5}+\frac {e^3 \log (d+e x) \left (a^2 C e^4-a c e^2 \left (3 A e^2-9 B d e+13 C d^2\right )+c^2 d^2 \left (10 C d^2-3 e (5 B d-7 A e)\right )\right )}{\left (a e^2+c d^2\right )^5}+\frac {4 a^2 e \left (a^2 C e^4-2 a c e^2 \left (4 C d^2-e (3 B d-A e)\right )+c^2 d^2 \left (3 C d^2-2 e (3 B d-5 A e)\right )\right )+c x \left (3 A c d \left (-11 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right )-a \left (-7 a^2 e^4 (3 C d-B e)+2 a c d^2 e^2 (13 C d-19 B e)-c^2 d^4 (C d-3 B e)\right )\right )}{8 a^2 \left (a+c x^2\right ) \left (a e^2+c d^2\right )^4}+\frac {\sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (3 A c d \left (-35 a^3 e^6+35 a^2 c d^2 e^4+7 a c^2 d^4 e^2+c^3 d^6\right )+a \left (15 a^3 e^6 (3 C d-B e)-5 a^2 c d^2 e^4 (25 C d-27 B e)+a c^2 d^4 e^2 (23 C d-45 B e)+c^3 d^6 (C d-3 B e)\right )\right )}{8 a^{5/2} \left (a e^2+c d^2\right )^5}-\frac {a \left (B c d \left (c d^2-3 a e^2\right )-e (A c-a C) \left (3 c d^2-a e^2\right )\right )-c x \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right )}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )^3}+\frac {e^3 \left (a e^2 (2 C d-B e)-c d \left (4 C d^2-e (5 B d-6 A e)\right )\right )}{(d+e x) \left (a e^2+c d^2\right )^4}-\frac {e^3 \left (A e^2-B d e+C d^2\right )}{2 (d+e x)^2 \left (a e^2+c d^2\right )^3} \]

[Out]

-1/2*e^3*(A*e^2-B*d*e+C*d^2)/(a*e^2+c*d^2)^3/(e*x+d)^2+e^3*(a*e^2*(-B*e+2*C*d)-c*d*(4*C*d^2-e*(-6*A*e+5*B*d)))
/(a*e^2+c*d^2)^4/(e*x+d)+1/4*(-a*(B*c*d*(-3*a*e^2+c*d^2)-(A*c-C*a)*e*(-a*e^2+3*c*d^2))+c*(A*c*d*(-3*a*e^2+c*d^
2)-a*(c*d^2*(-3*B*e+C*d)-a*e^2*(-B*e+3*C*d)))*x)/a/(a*e^2+c*d^2)^3/(c*x^2+a)^2+1/8*(4*a^2*e*(a^2*C*e^4+c^2*d^2
*(3*C*d^2-2*e*(-5*A*e+3*B*d))-2*a*c*e^2*(4*C*d^2-e*(-A*e+3*B*d)))+c*(3*A*c*d*(-11*a^2*e^4+6*a*c*d^2*e^2+c^2*d^
4)-a*(2*a*c*d^2*e^2*(-19*B*e+13*C*d)-c^2*d^4*(-3*B*e+C*d)-7*a^2*e^4*(-B*e+3*C*d)))*x)/a^2/(a*e^2+c*d^2)^4/(c*x
^2+a)+e^3*(a^2*C*e^4-a*c*e^2*(3*A*e^2-9*B*d*e+13*C*d^2)+c^2*d^2*(10*C*d^2-3*e*(-7*A*e+5*B*d)))*ln(e*x+d)/(a*e^
2+c*d^2)^5-1/2*e^3*(a^2*C*e^4-a*c*e^2*(3*A*e^2-9*B*d*e+13*C*d^2)+c^2*d^2*(10*C*d^2-3*e*(-7*A*e+5*B*d)))*ln(c*x
^2+a)/(a*e^2+c*d^2)^5+1/8*(3*A*c*d*(-35*a^3*e^6+35*a^2*c*d^2*e^4+7*a*c^2*d^4*e^2+c^3*d^6)+a*(a*c^2*d^4*e^2*(-4
5*B*e+23*C*d)-5*a^2*c*d^2*e^4*(-27*B*e+25*C*d)+c^3*d^6*(-3*B*e+C*d)+15*a^3*e^6*(-B*e+3*C*d)))*arctan(x*c^(1/2)
/a^(1/2))*c^(1/2)/a^(5/2)/(a*e^2+c*d^2)^5

________________________________________________________________________________________

Rubi [A]  time = 3.14, antiderivative size = 753, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1647, 1629, 635, 205, 260} \[ \frac {c x \left (3 A c d \left (-11 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right )-a \left (-7 a^2 e^4 (3 C d-B e)+2 a c d^2 e^2 (13 C d-19 B e)-c^2 d^4 (C d-3 B e)\right )\right )+4 a^2 e \left (a^2 C e^4-2 a c e^2 \left (4 C d^2-e (3 B d-A e)\right )+c^2 \left (3 C d^4-2 d^2 e (3 B d-5 A e)\right )\right )}{8 a^2 \left (a+c x^2\right ) \left (a e^2+c d^2\right )^4}-\frac {e^3 \log \left (a+c x^2\right ) \left (a^2 C e^4-a c e^2 \left (3 A e^2-9 B d e+13 C d^2\right )+c^2 \left (10 C d^4-3 d^2 e (5 B d-7 A e)\right )\right )}{2 \left (a e^2+c d^2\right )^5}+\frac {e^3 \log (d+e x) \left (a^2 C e^4-a c e^2 \left (3 A e^2-9 B d e+13 C d^2\right )+c^2 \left (10 C d^4-3 d^2 e (5 B d-7 A e)\right )\right )}{\left (a e^2+c d^2\right )^5}+\frac {\sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (3 A c d \left (35 a^2 c d^2 e^4-35 a^3 e^6+7 a c^2 d^4 e^2+c^3 d^6\right )+a \left (-5 a^2 c d^2 e^4 (25 C d-27 B e)+15 a^3 e^6 (3 C d-B e)+a c^2 d^4 e^2 (23 C d-45 B e)+c^3 d^6 (C d-3 B e)\right )\right )}{8 a^{5/2} \left (a e^2+c d^2\right )^5}-\frac {a \left (B c d \left (c d^2-3 a e^2\right )-e (A c-a C) \left (3 c d^2-a e^2\right )\right )-c x \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right )}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )^3}-\frac {e^3 \left (-a e^2 (2 C d-B e)-c d e (5 B d-6 A e)+4 c C d^3\right )}{(d+e x) \left (a e^2+c d^2\right )^4}-\frac {e^3 \left (A e^2-B d e+C d^2\right )}{2 (d+e x)^2 \left (a e^2+c d^2\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(e^3*(C*d^2 - B*d*e + A*e^2))/(2*(c*d^2 + a*e^2)^3*(d + e*x)^2) - (e^3*(4*c*C*d^3 - c*d*e*(5*B*d - 6*A*e) - a
*e^2*(2*C*d - B*e)))/((c*d^2 + a*e^2)^4*(d + e*x)) - (a*(B*c*d*(c*d^2 - 3*a*e^2) - (A*c - a*C)*e*(3*c*d^2 - a*
e^2)) - c*(A*c*d*(c*d^2 - 3*a*e^2) - a*(c*d^2*(C*d - 3*B*e) - a*e^2*(3*C*d - B*e)))*x)/(4*a*(c*d^2 + a*e^2)^3*
(a + c*x^2)^2) + (4*a^2*e*(a^2*C*e^4 + c^2*(3*C*d^4 - 2*d^2*e*(3*B*d - 5*A*e)) - 2*a*c*e^2*(4*C*d^2 - e*(3*B*d
- A*e))) + c*(3*A*c*d*(c^2*d^4 + 6*a*c*d^2*e^2 - 11*a^2*e^4) - a*(2*a*c*d^2*e^2*(13*C*d - 19*B*e) - c^2*d^4*(
C*d - 3*B*e) - 7*a^2*e^4*(3*C*d - B*e)))*x)/(8*a^2*(c*d^2 + a*e^2)^4*(a + c*x^2)) + (Sqrt[c]*(3*A*c*d*(c^3*d^6
+ 7*a*c^2*d^4*e^2 + 35*a^2*c*d^2*e^4 - 35*a^3*e^6) + a*(a*c^2*d^4*e^2*(23*C*d - 45*B*e) - 5*a^2*c*d^2*e^4*(25
*C*d - 27*B*e) + c^3*d^6*(C*d - 3*B*e) + 15*a^3*e^6*(3*C*d - B*e)))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*(c
*d^2 + a*e^2)^5) + (e^3*(a^2*C*e^4 - a*c*e^2*(13*C*d^2 - 9*B*d*e + 3*A*e^2) + c^2*(10*C*d^4 - 3*d^2*e*(5*B*d -
7*A*e)))*Log[d + e*x])/(c*d^2 + a*e^2)^5 - (e^3*(a^2*C*e^4 - a*c*e^2*(13*C*d^2 - 9*B*d*e + 3*A*e^2) + c^2*(10
*C*d^4 - 3*d^2*e*(5*B*d - 7*A*e)))*Log[a + c*x^2])/(2*(c*d^2 + a*e^2)^5)

Rule 205

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

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

Rule 1629

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*
Pq*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 1647

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(d +
e*x)^m*Pq, a + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 0], g = Coeff[Polyn
omialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 1]}, Simp[((a*g - c*f*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1))
, x] + Dist[1/(2*a*c*(p + 1)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*ExpandToSum[(2*a*c*(p + 1)*Q)/(d + e*x)^m +
(c*f*(2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] &
& LtQ[p, -1] && ILtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {A+B x+C x^2}{(d+e x)^3 \left (a+c x^2\right )^3} \, dx &=-\frac {a \left (B c d \left (c d^2-3 a e^2\right )-(A c-a C) e \left (3 c d^2-a e^2\right )\right )-c \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right ) x}{4 a \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )^2}-\frac {\int \frac {-\frac {c \left (A \left (3 c^3 d^6+15 a c^2 d^4 e^2+12 a^2 c d^2 e^4+4 a^3 e^6\right )+a c d^3 \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right )}{\left (c d^2+a e^2\right )^3}-\frac {c e \left (A c^2 d^3 \left (9 c d^2+5 a e^2\right )+a \left (4 a^2 B e^5-3 c^2 d^4 (3 C d-5 B e)-5 a c d^2 e^2 (C d-3 B e)\right )\right ) x}{\left (c d^2+a e^2\right )^3}-\frac {c e^2 \left (A c \left (9 c^2 d^4-15 a c d^2 e^2-4 a^2 e^4\right )+a \left (4 a^2 C e^4-c^2 d^3 (9 C d-23 B e)+3 a c d e^2 (5 C d+B e)\right )\right ) x^2}{\left (c d^2+a e^2\right )^3}-\frac {3 c^2 e^3 \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right ) x^3}{\left (c d^2+a e^2\right )^3}}{(d+e x)^3 \left (a+c x^2\right )^2} \, dx}{4 a c}\\ &=-\frac {a \left (B c d \left (c d^2-3 a e^2\right )-(A c-a C) e \left (3 c d^2-a e^2\right )\right )-c \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right ) x}{4 a \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )^2}+\frac {4 a^2 e \left (a^2 C e^4+c^2 \left (3 C d^4-2 d^2 e (3 B d-5 A e)\right )-2 a c e^2 \left (4 C d^2-e (3 B d-A e)\right )\right )+c \left (3 A c d \left (c^2 d^4+6 a c d^2 e^2-11 a^2 e^4\right )-a \left (2 a c d^2 e^2 (13 C d-19 B e)-c^2 d^4 (C d-3 B e)-7 a^2 e^4 (3 C d-B e)\right )\right ) x}{8 a^2 \left (c d^2+a e^2\right )^4 \left (a+c x^2\right )}+\frac {\int \frac {\frac {c^2 \left (A \left (3 c^4 d^8+18 a c^3 d^6 e^2+87 a^2 c^2 d^4 e^4+32 a^3 c d^2 e^6+8 a^4 e^8\right )+a c d^3 \left (2 a c d^2 e^2 (11 C d-21 B e)+c^2 d^4 (C d-3 B e)-9 a^2 e^4 (3 C d-B e)\right )\right )}{\left (c d^2+a e^2\right )^4}+\frac {c^2 e \left (3 A c^2 d^3 \left (3 c^2 d^4+18 a c d^2 e^2+31 a^2 e^4\right )+a \left (8 a^3 B e^7-a^2 c d^2 e^4 (65 C d-59 B e)+3 c^3 d^6 (C d-3 B e)-2 a c^2 d^4 e^2 (7 C d+3 B e)\right )\right ) x}{\left (c d^2+a e^2\right )^4}+\frac {c^2 e^2 \left (A c \left (9 c^3 d^6+54 a c^2 d^4 e^2-19 a^2 c d^2 e^4-16 a^3 e^6\right )+a \left (8 a^3 C e^6-a^2 c d e^4 (C d-27 B e)-6 a c^2 d^3 e^2 (9 C d-11 B e)+3 c^3 d^5 (C d-3 B e)\right )\right ) x^2}{\left (c d^2+a e^2\right )^4}+\frac {c^3 e^3 \left (3 A c d \left (c^2 d^4+6 a c d^2 e^2-11 a^2 e^4\right )-a \left (2 a c d^2 e^2 (13 C d-19 B e)-c^2 d^4 (C d-3 B e)-7 a^2 e^4 (3 C d-B e)\right )\right ) x^3}{\left (c d^2+a e^2\right )^4}}{(d+e x)^3 \left (a+c x^2\right )} \, dx}{8 a^2 c^2}\\ &=-\frac {a \left (B c d \left (c d^2-3 a e^2\right )-(A c-a C) e \left (3 c d^2-a e^2\right )\right )-c \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right ) x}{4 a \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )^2}+\frac {4 a^2 e \left (a^2 C e^4+c^2 \left (3 C d^4-2 d^2 e (3 B d-5 A e)\right )-2 a c e^2 \left (4 C d^2-e (3 B d-A e)\right )\right )+c \left (3 A c d \left (c^2 d^4+6 a c d^2 e^2-11 a^2 e^4\right )-a \left (2 a c d^2 e^2 (13 C d-19 B e)-c^2 d^4 (C d-3 B e)-7 a^2 e^4 (3 C d-B e)\right )\right ) x}{8 a^2 \left (c d^2+a e^2\right )^4 \left (a+c x^2\right )}+\frac {\int \left (\frac {8 a^2 c^2 e^4 \left (C d^2-B d e+A e^2\right )}{\left (c d^2+a e^2\right )^3 (d+e x)^3}+\frac {8 a^2 c^2 e^4 \left (4 c C d^3-c d e (5 B d-6 A e)-a e^2 (2 C d-B e)\right )}{\left (c d^2+a e^2\right )^4 (d+e x)^2}+\frac {8 a^2 c^2 e^4 \left (a^2 C e^4-a c e^2 \left (13 C d^2-9 B d e+3 A e^2\right )+c^2 \left (10 C d^4-3 d^2 e (5 B d-7 A e)\right )\right )}{\left (c d^2+a e^2\right )^5 (d+e x)}+\frac {c^3 \left (3 A c d \left (c^3 d^6+7 a c^2 d^4 e^2+35 a^2 c d^2 e^4-35 a^3 e^6\right )+a \left (a c^2 d^4 e^2 (23 C d-45 B e)-5 a^2 c d^2 e^4 (25 C d-27 B e)+c^3 d^6 (C d-3 B e)+15 a^3 e^6 (3 C d-B e)\right )-8 a^2 e^3 \left (a^2 C e^4-a c e^2 \left (13 C d^2-9 B d e+3 A e^2\right )+c^2 \left (10 C d^4-3 d^2 e (5 B d-7 A e)\right )\right ) x\right )}{\left (c d^2+a e^2\right )^5 \left (a+c x^2\right )}\right ) \, dx}{8 a^2 c^2}\\ &=-\frac {e^3 \left (C d^2-B d e+A e^2\right )}{2 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac {e^3 \left (4 c C d^3-c d e (5 B d-6 A e)-a e^2 (2 C d-B e)\right )}{\left (c d^2+a e^2\right )^4 (d+e x)}-\frac {a \left (B c d \left (c d^2-3 a e^2\right )-(A c-a C) e \left (3 c d^2-a e^2\right )\right )-c \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right ) x}{4 a \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )^2}+\frac {4 a^2 e \left (a^2 C e^4+c^2 \left (3 C d^4-2 d^2 e (3 B d-5 A e)\right )-2 a c e^2 \left (4 C d^2-e (3 B d-A e)\right )\right )+c \left (3 A c d \left (c^2 d^4+6 a c d^2 e^2-11 a^2 e^4\right )-a \left (2 a c d^2 e^2 (13 C d-19 B e)-c^2 d^4 (C d-3 B e)-7 a^2 e^4 (3 C d-B e)\right )\right ) x}{8 a^2 \left (c d^2+a e^2\right )^4 \left (a+c x^2\right )}+\frac {e^3 \left (a^2 C e^4-a c e^2 \left (13 C d^2-9 B d e+3 A e^2\right )+c^2 \left (10 C d^4-3 d^2 e (5 B d-7 A e)\right )\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^5}+\frac {c \int \frac {3 A c d \left (c^3 d^6+7 a c^2 d^4 e^2+35 a^2 c d^2 e^4-35 a^3 e^6\right )+a \left (a c^2 d^4 e^2 (23 C d-45 B e)-5 a^2 c d^2 e^4 (25 C d-27 B e)+c^3 d^6 (C d-3 B e)+15 a^3 e^6 (3 C d-B e)\right )-8 a^2 e^3 \left (a^2 C e^4-a c e^2 \left (13 C d^2-9 B d e+3 A e^2\right )+c^2 \left (10 C d^4-3 d^2 e (5 B d-7 A e)\right )\right ) x}{a+c x^2} \, dx}{8 a^2 \left (c d^2+a e^2\right )^5}\\ &=-\frac {e^3 \left (C d^2-B d e+A e^2\right )}{2 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac {e^3 \left (4 c C d^3-c d e (5 B d-6 A e)-a e^2 (2 C d-B e)\right )}{\left (c d^2+a e^2\right )^4 (d+e x)}-\frac {a \left (B c d \left (c d^2-3 a e^2\right )-(A c-a C) e \left (3 c d^2-a e^2\right )\right )-c \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right ) x}{4 a \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )^2}+\frac {4 a^2 e \left (a^2 C e^4+c^2 \left (3 C d^4-2 d^2 e (3 B d-5 A e)\right )-2 a c e^2 \left (4 C d^2-e (3 B d-A e)\right )\right )+c \left (3 A c d \left (c^2 d^4+6 a c d^2 e^2-11 a^2 e^4\right )-a \left (2 a c d^2 e^2 (13 C d-19 B e)-c^2 d^4 (C d-3 B e)-7 a^2 e^4 (3 C d-B e)\right )\right ) x}{8 a^2 \left (c d^2+a e^2\right )^4 \left (a+c x^2\right )}+\frac {e^3 \left (a^2 C e^4-a c e^2 \left (13 C d^2-9 B d e+3 A e^2\right )+c^2 \left (10 C d^4-3 d^2 e (5 B d-7 A e)\right )\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^5}-\frac {\left (c e^3 \left (a^2 C e^4-a c e^2 \left (13 C d^2-9 B d e+3 A e^2\right )+c^2 \left (10 C d^4-3 d^2 e (5 B d-7 A e)\right )\right )\right ) \int \frac {x}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^5}+\frac {\left (c \left (3 A c d \left (c^3 d^6+7 a c^2 d^4 e^2+35 a^2 c d^2 e^4-35 a^3 e^6\right )+a \left (a c^2 d^4 e^2 (23 C d-45 B e)-5 a^2 c d^2 e^4 (25 C d-27 B e)+c^3 d^6 (C d-3 B e)+15 a^3 e^6 (3 C d-B e)\right )\right )\right ) \int \frac {1}{a+c x^2} \, dx}{8 a^2 \left (c d^2+a e^2\right )^5}\\ &=-\frac {e^3 \left (C d^2-B d e+A e^2\right )}{2 \left (c d^2+a e^2\right )^3 (d+e x)^2}-\frac {e^3 \left (4 c C d^3-c d e (5 B d-6 A e)-a e^2 (2 C d-B e)\right )}{\left (c d^2+a e^2\right )^4 (d+e x)}-\frac {a \left (B c d \left (c d^2-3 a e^2\right )-(A c-a C) e \left (3 c d^2-a e^2\right )\right )-c \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right ) x}{4 a \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )^2}+\frac {4 a^2 e \left (a^2 C e^4+c^2 \left (3 C d^4-2 d^2 e (3 B d-5 A e)\right )-2 a c e^2 \left (4 C d^2-e (3 B d-A e)\right )\right )+c \left (3 A c d \left (c^2 d^4+6 a c d^2 e^2-11 a^2 e^4\right )-a \left (2 a c d^2 e^2 (13 C d-19 B e)-c^2 d^4 (C d-3 B e)-7 a^2 e^4 (3 C d-B e)\right )\right ) x}{8 a^2 \left (c d^2+a e^2\right )^4 \left (a+c x^2\right )}+\frac {\sqrt {c} \left (3 A c d \left (c^3 d^6+7 a c^2 d^4 e^2+35 a^2 c d^2 e^4-35 a^3 e^6\right )+a \left (a c^2 d^4 e^2 (23 C d-45 B e)-5 a^2 c d^2 e^4 (25 C d-27 B e)+c^3 d^6 (C d-3 B e)+15 a^3 e^6 (3 C d-B e)\right )\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} \left (c d^2+a e^2\right )^5}+\frac {e^3 \left (a^2 C e^4-a c e^2 \left (13 C d^2-9 B d e+3 A e^2\right )+c^2 \left (10 C d^4-3 d^2 e (5 B d-7 A e)\right )\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^5}-\frac {e^3 \left (a^2 C e^4-a c e^2 \left (13 C d^2-9 B d e+3 A e^2\right )+c^2 \left (10 C d^4-3 d^2 e (5 B d-7 A e)\right )\right ) \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^5}\\ \end {align*}

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Mathematica [A]  time = 1.08, size = 672, normalized size = 0.89 \[ \frac {-4 \log \left (a+c x^2\right ) \left (a^2 C e^7+a c e^5 \left (-3 A e^2+9 B d e-13 C d^2\right )+c^2 d^2 e^3 \left (3 e (7 A e-5 B d)+10 C d^2\right )\right )+8 \log (d+e x) \left (a^2 C e^7+a c e^5 \left (-3 A e^2+9 B d e-13 C d^2\right )+c^2 d^2 e^3 \left (3 e (7 A e-5 B d)+10 C d^2\right )\right )+\frac {2 \left (a e^2+c d^2\right )^2 \left (a^3 C e^3-a^2 c e (e (A e-3 B d+B e x)+3 C d (d-e x))-a c^2 d \left (3 A e (e x-d)+B d (d-3 e x)+C d^2 x\right )+A c^3 d^3 x\right )}{a \left (a+c x^2\right )^2}+\frac {\sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (3 A c d \left (-35 a^3 e^6+35 a^2 c d^2 e^4+7 a c^2 d^4 e^2+c^3 d^6\right )+a \left (-15 a^3 e^6 (B e-3 C d)-5 a^2 c d^2 e^4 (25 C d-27 B e)+a c^2 d^4 e^2 (23 C d-45 B e)+c^3 d^6 (C d-3 B e)\right )\right )}{a^{5/2}}+\frac {\left (a e^2+c d^2\right ) \left (4 a^4 C e^5+a^3 c e^3 (e (-8 A e+24 B d-7 B e x)+C d (21 e x-32 d))+a^2 c^2 d e \left (e \left (40 A d e-33 A e^2 x-24 B d^2+38 B d e x\right )+2 C d^2 (6 d-13 e x)\right )+a c^3 d^3 x \left (3 e (6 A e-B d)+C d^2\right )+3 A c^4 d^5 x\right )}{a^2 \left (a+c x^2\right )}-\frac {4 e^3 \left (a e^2+c d^2\right )^2 \left (e (A e-B d)+C d^2\right )}{(d+e x)^2}-\frac {8 e^3 \left (a e^2+c d^2\right ) \left (a e^2 (B e-2 C d)+c d e (6 A e-5 B d)+4 c C d^3\right )}{d+e x}}{8 \left (a e^2+c d^2\right )^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-4*e^3*(c*d^2 + a*e^2)^2*(C*d^2 + e*(-(B*d) + A*e)))/(d + e*x)^2 - (8*e^3*(c*d^2 + a*e^2)*(4*c*C*d^3 + c*d*e
*(-5*B*d + 6*A*e) + a*e^2*(-2*C*d + B*e)))/(d + e*x) + (2*(c*d^2 + a*e^2)^2*(a^3*C*e^3 + A*c^3*d^3*x - a*c^2*d
*(C*d^2*x + B*d*(d - 3*e*x) + 3*A*e*(-d + e*x)) - a^2*c*e*(3*C*d*(d - e*x) + e*(-3*B*d + A*e + B*e*x))))/(a*(a
+ c*x^2)^2) + ((c*d^2 + a*e^2)*(4*a^4*C*e^5 + 3*A*c^4*d^5*x + a*c^3*d^3*(C*d^2 + 3*e*(-(B*d) + 6*A*e))*x + a^
3*c*e^3*(C*d*(-32*d + 21*e*x) + e*(24*B*d - 8*A*e - 7*B*e*x)) + a^2*c^2*d*e*(2*C*d^2*(6*d - 13*e*x) + e*(-24*B
*d^2 + 40*A*d*e + 38*B*d*e*x - 33*A*e^2*x))))/(a^2*(a + c*x^2)) + (Sqrt[c]*(3*A*c*d*(c^3*d^6 + 7*a*c^2*d^4*e^2
+ 35*a^2*c*d^2*e^4 - 35*a^3*e^6) + a*(a*c^2*d^4*e^2*(23*C*d - 45*B*e) - 5*a^2*c*d^2*e^4*(25*C*d - 27*B*e) + c
^3*d^6*(C*d - 3*B*e) - 15*a^3*e^6*(-3*C*d + B*e)))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/a^(5/2) + 8*(a^2*C*e^7 + a*c*e
^5*(-13*C*d^2 + 9*B*d*e - 3*A*e^2) + c^2*d^2*e^3*(10*C*d^2 + 3*e*(-5*B*d + 7*A*e)))*Log[d + e*x] - 4*(a^2*C*e^
7 + a*c*e^5*(-13*C*d^2 + 9*B*d*e - 3*A*e^2) + c^2*d^2*e^3*(10*C*d^2 + 3*e*(-5*B*d + 7*A*e)))*Log[a + c*x^2])/(
8*(c*d^2 + a*e^2)^5)

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fricas [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((C*x^2+B*x+A)/(e*x+d)^3/(c*x^2+a)^3,x, algorithm="fricas")

[Out]

Timed out

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giac [B]  time = 0.20, size = 1532, normalized size = 2.03 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(10*C*c^2*d^4*e^3 - 15*B*c^2*d^3*e^4 - 13*C*a*c*d^2*e^5 + 21*A*c^2*d^2*e^5 + 9*B*a*c*d*e^6 + C*a^2*e^7 -
3*A*a*c*e^7)*log(c*x^2 + a)/(c^5*d^10 + 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 + 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^
2*e^8 + a^5*e^10) + (10*C*c^2*d^4*e^4 - 15*B*c^2*d^3*e^5 - 13*C*a*c*d^2*e^6 + 21*A*c^2*d^2*e^6 + 9*B*a*c*d*e^7
+ C*a^2*e^8 - 3*A*a*c*e^8)*log(abs(x*e + d))/(c^5*d^10*e + 5*a*c^4*d^8*e^3 + 10*a^2*c^3*d^6*e^5 + 10*a^3*c^2*
d^4*e^7 + 5*a^4*c*d^2*e^9 + a^5*e^11) + 1/8*(C*a*c^4*d^7 + 3*A*c^5*d^7 - 3*B*a*c^4*d^6*e + 23*C*a^2*c^3*d^5*e^
2 + 21*A*a*c^4*d^5*e^2 - 45*B*a^2*c^3*d^4*e^3 - 125*C*a^3*c^2*d^3*e^4 + 105*A*a^2*c^3*d^3*e^4 + 135*B*a^3*c^2*
d^2*e^5 + 45*C*a^4*c*d*e^6 - 105*A*a^3*c^2*d*e^6 - 15*B*a^4*c*e^7)*arctan(c*x/sqrt(a*c))/((a^2*c^5*d^10 + 5*a^
3*c^4*d^8*e^2 + 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6 + 5*a^6*c*d^2*e^8 + a^7*e^10)*sqrt(a*c)) + 1/8*(C*a*c^
4*d^5*x^5*e^2 + 3*A*c^5*d^5*x^5*e^2 + 2*C*a*c^4*d^6*x^4*e + 6*A*c^5*d^6*x^4*e + C*a*c^4*d^7*x^3 + 3*A*c^5*d^7*
x^3 - 3*B*a*c^4*d^4*x^5*e^3 - 6*B*a*c^4*d^5*x^4*e^2 - 3*B*a*c^4*d^6*x^3*e - 58*C*a^2*c^3*d^3*x^5*e^4 + 18*A*a*
c^4*d^3*x^5*e^4 - 76*C*a^2*c^3*d^4*x^4*e^3 + 36*A*a*c^4*d^4*x^4*e^3 - 3*C*a^2*c^3*d^5*x^3*e^2 + 23*A*a*c^4*d^5
*x^3*e^2 + 10*C*a^2*c^3*d^6*x^2*e + 10*A*a*c^4*d^6*x^2*e - C*a^2*c^3*d^7*x + 5*A*a*c^4*d^7*x + 78*B*a^2*c^3*d^
2*x^5*e^5 + 96*B*a^2*c^3*d^3*x^4*e^4 - 7*B*a^2*c^3*d^4*x^3*e^3 - 20*B*a^2*c^3*d^5*x^2*e^2 - B*a^2*c^3*d^6*x*e
- 2*B*a^2*c^3*d^7 + 37*C*a^3*c^2*d*x^5*e^6 - 81*A*a^2*c^3*d*x^5*e^6 + 22*C*a^3*c^2*d^2*x^4*e^5 - 78*A*a^2*c^3*
d^2*x^4*e^5 - 129*C*a^3*c^2*d^3*x^3*e^4 + 61*A*a^2*c^3*d^3*x^3*e^4 - 142*C*a^3*c^2*d^4*x^2*e^3 + 74*A*a^2*c^3*
d^4*x^2*e^3 - 10*C*a^3*c^2*d^5*x*e^2 + 26*A*a^2*c^3*d^5*x*e^2 + 6*C*a^3*c^2*d^6*e + 6*A*a^2*c^3*d^6*e - 15*B*a
^3*c^2*x^5*e^7 + 6*B*a^3*c^2*d*x^4*e^6 + 163*B*a^3*c^2*d^2*x^3*e^5 + 176*B*a^3*c^2*d^3*x^2*e^4 + 2*B*a^3*c^2*d
^4*x*e^3 - 20*B*a^3*c^2*d^5*e^2 + 4*C*a^4*c*x^4*e^7 - 12*A*a^3*c^2*x^4*e^7 + 67*C*a^4*c*d*x^3*e^6 - 151*A*a^3*
c^2*d*x^3*e^6 + 46*C*a^4*c*d^2*x^2*e^5 - 146*A*a^3*c^2*d^2*x^2*e^5 - 77*C*a^4*c*d^3*x*e^4 + 49*A*a^3*c^2*d^3*x
*e^4 - 72*C*a^4*c*d^4*e^3 + 44*A*a^3*c^2*d^4*e^3 - 25*B*a^4*c*x^3*e^7 + 4*B*a^4*c*d*x^2*e^6 + 91*B*a^4*c*d^2*x
*e^5 + 74*B*a^4*c*d^3*e^4 + 6*C*a^5*x^2*e^7 - 18*A*a^4*c*x^2*e^7 + 28*C*a^5*d*x*e^6 - 68*A*a^4*c*d*x*e^6 + 18*
C*a^5*d^2*e^5 - 62*A*a^4*c*d^2*e^5 - 8*B*a^5*x*e^7 - 4*B*a^5*d*e^6 - 4*A*a^5*e^7)/((a^2*c^4*d^8 + 4*a^3*c^3*d^
6*e^2 + 6*a^4*c^2*d^4*e^4 + 4*a^5*c*d^2*e^6 + a^6*e^8)*(c*x^3*e + c*d*x^2 + a*x*e + a*d)^2)

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maple [B]  time = 0.04, size = 2737, normalized size = 3.63 \[ \text {Expression too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((C*x^2+B*x+A)/(e*x+d)^3/(c*x^2+a)^3,x)

[Out]

15/2/(a*e^2+c*d^2)^5*c^2*ln(c*x^2+a)*d^3*e^4*B-5/(a*e^2+c*d^2)^5*c^2*ln(c*x^2+a)*C*d^4*e^3+3/2/(a*e^2+c*d^2)^5
*c*a*ln(c*x^2+a)*A*e^7+2*e^5/(a*e^2+c*d^2)^4/(e*x+d)*C*a*d-4*e^3/(a*e^2+c*d^2)^4/(e*x+d)*C*c*d^3-3*e^7/(a*e^2+
c*d^2)^5*ln(e*x+d)*A*a*c+21*e^5/(a*e^2+c*d^2)^5*ln(e*x+d)*A*c^2*d^2-15*e^4/(a*e^2+c*d^2)^5*ln(e*x+d)*B*c^2*d^3
+10*e^3/(a*e^2+c*d^2)^5*ln(e*x+d)*C*c^2*d^4-6*e^5/(a*e^2+c*d^2)^4/(e*x+d)*A*c*d+5*e^4/(a*e^2+c*d^2)^4/(e*x+d)*
B*c*d^2-5/4/(a*e^2+c*d^2)^5*c/(c*x^2+a)^2*A*e^7*a^3+3/4/(a*e^2+c*d^2)^5*c^4/(c*x^2+a)^2*A*d^6*e-1/8/(a*e^2+c*d
^2)^5*c^4/(c*x^2+a)^2*C*x*d^7-21/2/(a*e^2+c*d^2)^5*c^2*ln(c*x^2+a)*A*d^2*e^5-1/2*e^5/(a*e^2+c*d^2)^3/(e*x+d)^2
*A-7/2/(a*e^2+c*d^2)^5*c^2/(c*x^2+a)^2*C*x^2*a^2*d^2*e^5-5/2/(a*e^2+c*d^2)^5*c^3/(c*x^2+a)^2*C*x^2*a*d^4*e^3+2
1/8/(a*e^2+c*d^2)^5*c^5/(c*x^2+a)^2/a*x^3*A*d^5*e^2-3/8/(a*e^2+c*d^2)^5*c^5/(c*x^2+a)^2/a*x^3*B*d^6*e+31/8/(a*
e^2+c*d^2)^5*c^3/(c*x^2+a)^2*B*x^3*a*d^2*e^5-39/8/(a*e^2+c*d^2)^5*c^2/(c*x^2+a)^2*A*x*a^2*d*e^6-25/8/(a*e^2+c*
d^2)^5*c^3/(c*x^2+a)^2*A*x*a*d^3*e^4+33/8/(a*e^2+c*d^2)^5*c^2/(c*x^2+a)^2*B*x*a^2*d^2*e^5+45/8/(a*e^2+c*d^2)^5
*c^3/(c*x^2+a)^2*B*x*a*d^4*e^3+5/8/(a*e^2+c*d^2)^5*c^2/(c*x^2+a)^2*C*x*a^2*d^3*e^4-23/8/(a*e^2+c*d^2)^5*c^3/(c
*x^2+a)^2*C*x*a*d^5*e^2-105/8/(a*e^2+c*d^2)^5*c^2*a/(a*c)^(1/2)*arctan(1/(a*c)^(1/2)*c*x)*A*d*e^6+21/8/(a*e^2+
c*d^2)^5*c^4/a/(a*c)^(1/2)*arctan(1/(a*c)^(1/2)*c*x)*A*d^5*e^2+135/8/(a*e^2+c*d^2)^5*c^2*a/(a*c)^(1/2)*arctan(
1/(a*c)^(1/2)*c*x)*B*d^2*e^5-3/8/(a*e^2+c*d^2)^5*c^4/a/(a*c)^(1/2)*arctan(1/(a*c)^(1/2)*c*x)*B*d^6*e-125/8/(a*
e^2+c*d^2)^5*c^2*a/(a*c)^(1/2)*arctan(1/(a*c)^(1/2)*c*x)*C*d^3*e^4+45/8/(a*e^2+c*d^2)^5*c*a^2/(a*c)^(1/2)*arct
an(1/(a*c)^(1/2)*c*x)*C*d*e^6+27/8/(a*e^2+c*d^2)^5*c/(c*x^2+a)^2*C*x*a^3*d*e^6-33/8/(a*e^2+c*d^2)^5*c^3/(c*x^2
+a)^2*A*x^3*a*d*e^6+21/8/(a*e^2+c*d^2)^5*c^2/(c*x^2+a)^2*C*x^3*a^2*d*e^6-5/8/(a*e^2+c*d^2)^5*c^3/(c*x^2+a)^2*C
*x^3*a*d^3*e^4+4/(a*e^2+c*d^2)^5*c^3/(c*x^2+a)^2*A*x^2*a*d^2*e^5+3/(a*e^2+c*d^2)^5*c^2/(c*x^2+a)^2*B*x^2*a^2*d
*e^6-1/2/(a*e^2+c*d^2)^5*a^2*ln(c*x^2+a)*C*e^7-e^6/(a*e^2+c*d^2)^4/(e*x+d)*B*a+1/2*e^4/(a*e^2+c*d^2)^3/(e*x+d)
^2*B*d-1/2*e^3/(a*e^2+c*d^2)^3/(e*x+d)^2*C*d^2+e^7/(a*e^2+c*d^2)^5*ln(e*x+d)*a^2*C+3/4/(a*e^2+c*d^2)^5/(c*x^2+
a)^2*C*a^4*e^7-1/4/(a*e^2+c*d^2)^5*c^4/(c*x^2+a)^2*d^7*B-15/8/(a*e^2+c*d^2)^5*c^4/(c*x^2+a)^2*A*x^3*d^3*e^4+35
/8/(a*e^2+c*d^2)^5*c^4/(c*x^2+a)^2*B*x^3*d^4*e^3-25/8/(a*e^2+c*d^2)^5*c^4/(c*x^2+a)^2*C*x^3*d^5*e^2-1/(a*e^2+c
*d^2)^5*c^2/(c*x^2+a)^2*A*x^2*a^2*e^7+5/(a*e^2+c*d^2)^5*c^4/(c*x^2+a)^2*A*x^2*d^4*e^3-3/(a*e^2+c*d^2)^5*c^4/(c
*x^2+a)^2*B*x^2*d^5*e^2+3/2/(a*e^2+c*d^2)^5*c^4/(c*x^2+a)^2*C*x^2*d^6*e-7/8/(a*e^2+c*d^2)^5*c^2/(c*x^2+a)^2*B*
x^3*a^2*e^7+3/8/(a*e^2+c*d^2)^5*c^6/(c*x^2+a)^2/a^2*x^3*A*d^7+1/8/(a*e^2+c*d^2)^5*c^5/(c*x^2+a)^2/a*x^3*C*d^7+
5/8/(a*e^2+c*d^2)^5*c^5/(c*x^2+a)^2/a*x*A*d^7+17/4/(a*e^2+c*d^2)^5*c^2/(c*x^2+a)^2*A*d^2*e^5*a^2+25/4/(a*e^2+c
*d^2)^5*c^3/(c*x^2+a)^2*A*d^4*e^3*a+5/4/(a*e^2+c*d^2)^5*c^2/(c*x^2+a)^2*d^3*e^4*B*a^2-11/4/(a*e^2+c*d^2)^5*c^3
/(c*x^2+a)^2*d^5*e^2*B*a-15/4/(a*e^2+c*d^2)^5*c^2/(c*x^2+a)^2*C*a^2*d^4*e^3+3/4/(a*e^2+c*d^2)^5*c^3/(c*x^2+a)^
2*C*a*d^6*e+9*e^6/(a*e^2+c*d^2)^5*ln(e*x+d)*B*a*c*d-13*e^5/(a*e^2+c*d^2)^5*ln(e*x+d)*C*a*c*d^2+1/2/(a*e^2+c*d^
2)^5*c/(c*x^2+a)^2*C*x^2*a^3*e^7-9/8/(a*e^2+c*d^2)^5*c/(c*x^2+a)^2*B*x*a^3*e^7+15/4/(a*e^2+c*d^2)^5*c/(c*x^2+a
)^2*d*e^6*B*a^3-15/4/(a*e^2+c*d^2)^5*c/(c*x^2+a)^2*C*a^3*d^2*e^5+1/8/(a*e^2+c*d^2)^5*c^4/a/(a*c)^(1/2)*arctan(
1/(a*c)^(1/2)*c*x)*C*d^7-9/2/(a*e^2+c*d^2)^5*c*a*ln(c*x^2+a)*d*e^6*B+13/2/(a*e^2+c*d^2)^5*c*a*ln(c*x^2+a)*C*d^
2*e^5-15/8/(a*e^2+c*d^2)^5*c*a^2/(a*c)^(1/2)*arctan(1/(a*c)^(1/2)*c*x)*B*e^7+105/8/(a*e^2+c*d^2)^5*c^3/(a*c)^(
1/2)*arctan(1/(a*c)^(1/2)*c*x)*A*d^3*e^4-45/8/(a*e^2+c*d^2)^5*c^3/(a*c)^(1/2)*arctan(1/(a*c)^(1/2)*c*x)*B*d^4*
e^3+23/8/(a*e^2+c*d^2)^5*c^3/(a*c)^(1/2)*arctan(1/(a*c)^(1/2)*c*x)*C*d^5*e^2+3/8/(a*e^2+c*d^2)^5*c^5/a^2/(a*c)
^(1/2)*arctan(1/(a*c)^(1/2)*c*x)*A*d^7+19/8/(a*e^2+c*d^2)^5*c^4/(c*x^2+a)^2*A*x*d^5*e^2+3/8/(a*e^2+c*d^2)^5*c^
4/(c*x^2+a)^2*B*x*d^6*e

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maxima [B]  time = 1.27, size = 1835, normalized size = 2.44 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(10*C*c^2*d^4*e^3 - 15*B*c^2*d^3*e^4 + 9*B*a*c*d*e^6 - (13*C*a*c - 21*A*c^2)*d^2*e^5 + (C*a^2 - 3*A*a*c)*
e^7)*log(c*x^2 + a)/(c^5*d^10 + 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 + 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 +
a^5*e^10) + (10*C*c^2*d^4*e^3 - 15*B*c^2*d^3*e^4 + 9*B*a*c*d*e^6 - (13*C*a*c - 21*A*c^2)*d^2*e^5 + (C*a^2 - 3*
A*a*c)*e^7)*log(e*x + d)/(c^5*d^10 + 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 + 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e
^8 + a^5*e^10) - 1/8*(3*B*a*c^4*d^6*e + 45*B*a^2*c^3*d^4*e^3 - 135*B*a^3*c^2*d^2*e^5 + 15*B*a^4*c*e^7 - (C*a*c
^4 + 3*A*c^5)*d^7 - (23*C*a^2*c^3 + 21*A*a*c^4)*d^5*e^2 + 5*(25*C*a^3*c^2 - 21*A*a^2*c^3)*d^3*e^4 - 15*(3*C*a^
4*c - 7*A*a^3*c^2)*d*e^6)*arctan(c*x/sqrt(a*c))/((a^2*c^5*d^10 + 5*a^3*c^4*d^8*e^2 + 10*a^4*c^3*d^6*e^4 + 10*a
^5*c^2*d^4*e^6 + 5*a^6*c*d^2*e^8 + a^7*e^10)*sqrt(a*c)) - 1/8*(2*B*a^2*c^3*d^7 + 20*B*a^3*c^2*d^5*e^2 - 74*B*a
^4*c*d^3*e^4 + 4*B*a^5*d*e^6 + 4*A*a^5*e^7 - 6*(C*a^3*c^2 + A*a^2*c^3)*d^6*e + 4*(18*C*a^4*c - 11*A*a^3*c^2)*d
^4*e^3 - 2*(9*C*a^5 - 31*A*a^4*c)*d^2*e^5 + (3*B*a*c^4*d^4*e^3 - 78*B*a^2*c^3*d^2*e^5 + 15*B*a^3*c^2*e^7 - (C*
a*c^4 + 3*A*c^5)*d^5*e^2 + 2*(29*C*a^2*c^3 - 9*A*a*c^4)*d^3*e^4 - (37*C*a^3*c^2 - 81*A*a^2*c^3)*d*e^6)*x^5 + 2
*(3*B*a*c^4*d^5*e^2 - 48*B*a^2*c^3*d^3*e^4 - 3*B*a^3*c^2*d*e^6 - (C*a*c^4 + 3*A*c^5)*d^6*e + 2*(19*C*a^2*c^3 -
9*A*a*c^4)*d^4*e^3 - (11*C*a^3*c^2 - 39*A*a^2*c^3)*d^2*e^5 - 2*(C*a^4*c - 3*A*a^3*c^2)*e^7)*x^4 + (3*B*a*c^4*
d^6*e + 7*B*a^2*c^3*d^4*e^3 - 163*B*a^3*c^2*d^2*e^5 + 25*B*a^4*c*e^7 - (C*a*c^4 + 3*A*c^5)*d^7 + (3*C*a^2*c^3
- 23*A*a*c^4)*d^5*e^2 + (129*C*a^3*c^2 - 61*A*a^2*c^3)*d^3*e^4 - (67*C*a^4*c - 151*A*a^3*c^2)*d*e^6)*x^3 + 2*(
10*B*a^2*c^3*d^5*e^2 - 88*B*a^3*c^2*d^3*e^4 - 2*B*a^4*c*d*e^6 - 5*(C*a^2*c^3 + A*a*c^4)*d^6*e + (71*C*a^3*c^2
- 37*A*a^2*c^3)*d^4*e^3 - (23*C*a^4*c - 73*A*a^3*c^2)*d^2*e^5 - 3*(C*a^5 - 3*A*a^4*c)*e^7)*x^2 + (B*a^2*c^3*d^
6*e - 2*B*a^3*c^2*d^4*e^3 - 91*B*a^4*c*d^2*e^5 + 8*B*a^5*e^7 + (C*a^2*c^3 - 5*A*a*c^4)*d^7 + 2*(5*C*a^3*c^2 -
13*A*a^2*c^3)*d^5*e^2 + 7*(11*C*a^4*c - 7*A*a^3*c^2)*d^3*e^4 - 4*(7*C*a^5 - 17*A*a^4*c)*d*e^6)*x)/(a^4*c^4*d^1
0 + 4*a^5*c^3*d^8*e^2 + 6*a^6*c^2*d^6*e^4 + 4*a^7*c*d^4*e^6 + a^8*d^2*e^8 + (a^2*c^6*d^8*e^2 + 4*a^3*c^5*d^6*e
^4 + 6*a^4*c^4*d^4*e^6 + 4*a^5*c^3*d^2*e^8 + a^6*c^2*e^10)*x^6 + 2*(a^2*c^6*d^9*e + 4*a^3*c^5*d^7*e^3 + 6*a^4*
c^4*d^5*e^5 + 4*a^5*c^3*d^3*e^7 + a^6*c^2*d*e^9)*x^5 + (a^2*c^6*d^10 + 6*a^3*c^5*d^8*e^2 + 14*a^4*c^4*d^6*e^4
+ 16*a^5*c^3*d^4*e^6 + 9*a^6*c^2*d^2*e^8 + 2*a^7*c*e^10)*x^4 + 4*(a^3*c^5*d^9*e + 4*a^4*c^4*d^7*e^3 + 6*a^5*c^
3*d^5*e^5 + 4*a^6*c^2*d^3*e^7 + a^7*c*d*e^9)*x^3 + (2*a^3*c^5*d^10 + 9*a^4*c^4*d^8*e^2 + 16*a^5*c^3*d^6*e^4 +
14*a^6*c^2*d^4*e^6 + 6*a^7*c*d^2*e^8 + a^8*e^10)*x^2 + 2*(a^4*c^4*d^9*e + 4*a^5*c^3*d^7*e^3 + 6*a^6*c^2*d^5*e^
5 + 4*a^7*c*d^3*e^7 + a^8*d*e^9)*x)

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mupad [B]  time = 7.24, size = 8774, normalized size = 11.65 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x + C*x^2)/((a + c*x^2)^3*(d + e*x)^3),x)

[Out]

((x^5*(3*A*c^5*d^5*e^2 - 15*B*a^3*c^2*e^7 + 18*A*a*c^4*d^3*e^4 - 81*A*a^2*c^3*d*e^6 - 3*B*a*c^4*d^4*e^3 + C*a*
c^4*d^5*e^2 + 37*C*a^3*c^2*d*e^6 + 78*B*a^2*c^3*d^2*e^5 - 58*C*a^2*c^3*d^3*e^4))/(8*a^2*(a^4*e^8 + c^4*d^8 + 4
*a*c^3*d^6*e^2 + 4*a^3*c*d^2*e^6 + 6*a^2*c^2*d^4*e^4)) - (2*A*a^3*e^7 + B*c^3*d^7 + 2*B*a^3*d*e^6 - 3*A*c^3*d^
6*e - 9*C*a^3*d^2*e^5 - 22*A*a*c^2*d^4*e^3 + 31*A*a^2*c*d^2*e^5 + 10*B*a*c^2*d^5*e^2 - 37*B*a^2*c*d^3*e^4 + 36
*C*a^2*c*d^4*e^3 - 3*C*a*c^2*d^6*e)/(4*(a^4*e^8 + c^4*d^8 + 4*a*c^3*d^6*e^2 + 4*a^3*c*d^2*e^6 + 6*a^2*c^2*d^4*
e^4)) + (x*(5*A*c^4*d^7 - 8*B*a^4*e^7 - C*a*c^3*d^7 + 28*C*a^4*d*e^6 + 26*A*a*c^3*d^5*e^2 + 91*B*a^3*c*d^2*e^5
- 77*C*a^3*c*d^3*e^4 + 49*A*a^2*c^2*d^3*e^4 + 2*B*a^2*c^2*d^4*e^3 - 10*C*a^2*c^2*d^5*e^2 - 68*A*a^3*c*d*e^6 -
B*a*c^3*d^6*e))/(8*a*(a^4*e^8 + c^4*d^8 + 4*a*c^3*d^6*e^2 + 4*a^3*c*d^2*e^6 + 6*a^2*c^2*d^4*e^4)) + (x^2*(3*C
*a^4*e^7 - 9*A*a^3*c*e^7 + 5*A*c^4*d^6*e + 37*A*a*c^3*d^4*e^3 - 10*B*a*c^3*d^5*e^2 + 23*C*a^3*c*d^2*e^5 - 73*A
*a^2*c^2*d^2*e^5 + 88*B*a^2*c^2*d^3*e^4 - 71*C*a^2*c^2*d^4*e^3 + 2*B*a^3*c*d*e^6 + 5*C*a*c^3*d^6*e))/(4*a*(a^4
*e^8 + c^4*d^8 + 4*a*c^3*d^6*e^2 + 4*a^3*c*d^2*e^6 + 6*a^2*c^2*d^4*e^4)) + (x^3*(3*A*c^5*d^7 - 25*B*a^4*c*e^7
+ C*a*c^4*d^7 + 23*A*a*c^4*d^5*e^2 - 151*A*a^3*c^2*d*e^6 + 61*A*a^2*c^3*d^3*e^4 - 7*B*a^2*c^3*d^4*e^3 + 163*B*
a^3*c^2*d^2*e^5 - 3*C*a^2*c^3*d^5*e^2 - 129*C*a^3*c^2*d^3*e^4 - 3*B*a*c^4*d^6*e + 67*C*a^4*c*d*e^6))/(8*a^2*(a
^4*e^8 + c^4*d^8 + 4*a*c^3*d^6*e^2 + 4*a^3*c*d^2*e^6 + 6*a^2*c^2*d^4*e^4)) + (x^4*(2*C*a^4*c*e^7 + 3*A*c^5*d^6
*e - 6*A*a^3*c^2*e^7 + 18*A*a*c^4*d^4*e^3 - 3*B*a*c^4*d^5*e^2 + 3*B*a^3*c^2*d*e^6 - 39*A*a^2*c^3*d^2*e^5 + 48*
B*a^2*c^3*d^3*e^4 - 38*C*a^2*c^3*d^4*e^3 + 11*C*a^3*c^2*d^2*e^5 + C*a*c^4*d^6*e))/(4*a^2*(a^4*e^8 + c^4*d^8 +
4*a*c^3*d^6*e^2 + 4*a^3*c*d^2*e^6 + 6*a^2*c^2*d^4*e^4)))/(x^2*(a^2*e^2 + 2*a*c*d^2) + x^4*(c^2*d^2 + 2*a*c*e^2
) + a^2*d^2 + c^2*e^2*x^6 + 2*a^2*d*e*x + 2*c^2*d*e*x^5 + 4*a*c*d*e*x^3) + symsum(log(root(2560*a^14*c*d^2*e^1
8*z^3 + 64512*a^10*c^5*d^10*e^10*z^3 + 53760*a^11*c^4*d^8*e^12*z^3 + 53760*a^9*c^6*d^12*e^8*z^3 + 30720*a^12*c
^3*d^6*e^14*z^3 + 30720*a^8*c^7*d^14*e^6*z^3 + 11520*a^13*c^2*d^4*e^16*z^3 + 11520*a^7*c^8*d^16*e^4*z^3 + 2560
*a^6*c^9*d^18*e^2*z^3 + 256*a^5*c^10*d^20*z^3 + 256*a^15*e^20*z^3 - 4806*B*C*a^8*c*d*e^13*z - 18*A*B*a*c^8*d^1
3*e*z - 147930*B*C*a^6*c^3*d^5*e^9*z + 74760*B*C*a^5*c^4*d^7*e^7*z + 66588*B*C*a^7*c^2*d^3*e^11*z - 1050*B*C*a
^4*c^5*d^9*e^5*z - 228*B*C*a^3*c^6*d^11*e^3*z + 152052*A*C*a^6*c^3*d^4*e^10*z - 109830*A*C*a^5*c^4*d^6*e^8*z -
32490*A*C*a^7*c^2*d^2*e^12*z + 426*A*C*a^3*c^6*d^10*e^4*z - 360*A*C*a^4*c^5*d^8*e^6*z + 180*A*C*a^2*c^7*d^12*
e^2*z + 158130*A*B*a^5*c^4*d^5*e^9*z - 121356*A*B*a^6*c^3*d^3*e^11*z - 3240*A*B*a^4*c^5*d^7*e^7*z - 1710*A*B*a
^3*c^6*d^9*e^5*z - 396*A*B*a^2*c^7*d^11*e^3*z - 6*B*C*a^2*c^7*d^13*e*z + 13518*A*B*a^7*c^2*d*e^13*z + 67615*C^
2*a^6*c^3*d^6*e^8*z - 47538*C^2*a^7*c^2*d^4*e^10*z - 24860*C^2*a^5*c^4*d^8*e^6*z + 279*C^2*a^4*c^5*d^10*e^4*z
+ 46*C^2*a^3*c^6*d^12*e^2*z + 71415*B^2*a^6*c^3*d^4*e^10*z - 55260*B^2*a^5*c^4*d^6*e^8*z - 19602*B^2*a^7*c^2*d
^2*e^12*z + 1215*B^2*a^4*c^5*d^8*e^6*z + 270*B^2*a^3*c^6*d^10*e^4*z + 9*B^2*a^2*c^7*d^12*e^2*z - 106722*A^2*a^
5*c^4*d^4*e^10*z + 35217*A^2*a^6*c^3*d^2*e^12*z + 6615*A^2*a^4*c^5*d^6*e^8*z + 3780*A^2*a^3*c^6*d^8*e^6*z + 10
71*A^2*a^2*c^7*d^10*e^4*z + 1152*A*C*a^8*c*e^14*z + 6*A*C*a*c^8*d^14*z + 7017*C^2*a^8*c*d^2*e^12*z + 126*A^2*a
*c^8*d^12*e^2*z + C^2*a^2*c^7*d^14*z - 1728*A^2*a^7*c^2*e^14*z + 225*B^2*a^8*c*e^14*z + 9*A^2*c^9*d^14*z - 192
*C^2*a^9*e^14*z + 3168*A*B*C*a^4*c^2*d*e^10 + 270*A*B*C*a*c^5*d^7*e^4 - 6930*A*B*C*a^3*c^3*d^3*e^8 + 5148*A*B*
C*a^2*c^4*d^5*e^6 - 819*A^2*C*a*c^5*d^6*e^5 - 60*A*C^2*a*c^5*d^8*e^3 - 6102*A^2*B*a^3*c^3*d*e^10 + 1512*A^2*B*
a*c^5*d^5*e^6 - 270*A*B^2*a*c^5*d^6*e^5 - 378*B*C^2*a^5*c*d*e^10 - 5049*B^2*C*a^3*c^3*d^4*e^7 + 4698*B^2*C*a^4
*c^2*d^2*e^9 + 2508*B*C^2*a^3*c^3*d^5*e^6 - 1977*B*C^2*a^4*c^2*d^3*e^8 - 180*B^2*C*a^2*c^4*d^6*e^5 + 75*B*C^2*
a^2*c^4*d^7*e^4 + 15921*A^2*C*a^3*c^3*d^2*e^9 - 7848*A^2*C*a^2*c^4*d^4*e^7 - 6363*A*C^2*a^4*c^2*d^2*e^9 + 4926
*A*C^2*a^3*c^3*d^4*e^7 - 1443*A*C^2*a^2*c^4*d^6*e^5 + 14283*A^2*B*a^2*c^4*d^3*e^8 - 4617*A*B^2*a^2*c^4*d^4*e^7
- 1944*A*B^2*a^3*c^3*d^2*e^9 + 791*C^3*a^5*c*d^2*e^9 - 2025*B^3*a^4*c^2*d*e^10 - 1674*A^3*a*c^5*d^4*e^7 - 90*
A^2*C*c^6*d^8*e^3 + 135*A^2*B*c^6*d^7*e^4 - 1728*A^2*C*a^4*c^2*e^11 + 675*A*B^2*a^4*c^2*e^11 - 225*B^2*C*a^5*c
*e^11 + 576*A*C^2*a^5*c*e^11 - 397*C^3*a^3*c^3*d^6*e^5 - 108*C^3*a^4*c^2*d^4*e^7 - 10*C^3*a^2*c^4*d^8*e^3 + 32
94*B^3*a^3*c^3*d^3*e^8 + 135*B^3*a^2*c^4*d^5*e^6 - 11853*A^3*a^2*c^4*d^2*e^9 - 189*A^3*c^6*d^6*e^5 + 1728*A^3*
a^3*c^3*e^11 - 64*C^3*a^6*e^11, z, k)*(root(2560*a^14*c*d^2*e^18*z^3 + 64512*a^10*c^5*d^10*e^10*z^3 + 53760*a^
11*c^4*d^8*e^12*z^3 + 53760*a^9*c^6*d^12*e^8*z^3 + 30720*a^12*c^3*d^6*e^14*z^3 + 30720*a^8*c^7*d^14*e^6*z^3 +
11520*a^13*c^2*d^4*e^16*z^3 + 11520*a^7*c^8*d^16*e^4*z^3 + 2560*a^6*c^9*d^18*e^2*z^3 + 256*a^5*c^10*d^20*z^3 +
256*a^15*e^20*z^3 - 4806*B*C*a^8*c*d*e^13*z - 18*A*B*a*c^8*d^13*e*z - 147930*B*C*a^6*c^3*d^5*e^9*z + 74760*B*
C*a^5*c^4*d^7*e^7*z + 66588*B*C*a^7*c^2*d^3*e^11*z - 1050*B*C*a^4*c^5*d^9*e^5*z - 228*B*C*a^3*c^6*d^11*e^3*z +
152052*A*C*a^6*c^3*d^4*e^10*z - 109830*A*C*a^5*c^4*d^6*e^8*z - 32490*A*C*a^7*c^2*d^2*e^12*z + 426*A*C*a^3*c^6
*d^10*e^4*z - 360*A*C*a^4*c^5*d^8*e^6*z + 180*A*C*a^2*c^7*d^12*e^2*z + 158130*A*B*a^5*c^4*d^5*e^9*z - 121356*A
*B*a^6*c^3*d^3*e^11*z - 3240*A*B*a^4*c^5*d^7*e^7*z - 1710*A*B*a^3*c^6*d^9*e^5*z - 396*A*B*a^2*c^7*d^11*e^3*z -
6*B*C*a^2*c^7*d^13*e*z + 13518*A*B*a^7*c^2*d*e^13*z + 67615*C^2*a^6*c^3*d^6*e^8*z - 47538*C^2*a^7*c^2*d^4*e^1
0*z - 24860*C^2*a^5*c^4*d^8*e^6*z + 279*C^2*a^4*c^5*d^10*e^4*z + 46*C^2*a^3*c^6*d^12*e^2*z + 71415*B^2*a^6*c^3
*d^4*e^10*z - 55260*B^2*a^5*c^4*d^6*e^8*z - 19602*B^2*a^7*c^2*d^2*e^12*z + 1215*B^2*a^4*c^5*d^8*e^6*z + 270*B^
2*a^3*c^6*d^10*e^4*z + 9*B^2*a^2*c^7*d^12*e^2*z - 106722*A^2*a^5*c^4*d^4*e^10*z + 35217*A^2*a^6*c^3*d^2*e^12*z
+ 6615*A^2*a^4*c^5*d^6*e^8*z + 3780*A^2*a^3*c^6*d^8*e^6*z + 1071*A^2*a^2*c^7*d^10*e^4*z + 1152*A*C*a^8*c*e^14
*z + 6*A*C*a*c^8*d^14*z + 7017*C^2*a^8*c*d^2*e^12*z + 126*A^2*a*c^8*d^12*e^2*z + C^2*a^2*c^7*d^14*z - 1728*A^2
*a^7*c^2*e^14*z + 225*B^2*a^8*c*e^14*z + 9*A^2*c^9*d^14*z - 192*C^2*a^9*e^14*z + 3168*A*B*C*a^4*c^2*d*e^10 + 2
70*A*B*C*a*c^5*d^7*e^4 - 6930*A*B*C*a^3*c^3*d^3*e^8 + 5148*A*B*C*a^2*c^4*d^5*e^6 - 819*A^2*C*a*c^5*d^6*e^5 - 6
0*A*C^2*a*c^5*d^8*e^3 - 6102*A^2*B*a^3*c^3*d*e^10 + 1512*A^2*B*a*c^5*d^5*e^6 - 270*A*B^2*a*c^5*d^6*e^5 - 378*B
*C^2*a^5*c*d*e^10 - 5049*B^2*C*a^3*c^3*d^4*e^7 + 4698*B^2*C*a^4*c^2*d^2*e^9 + 2508*B*C^2*a^3*c^3*d^5*e^6 - 197
7*B*C^2*a^4*c^2*d^3*e^8 - 180*B^2*C*a^2*c^4*d^6*e^5 + 75*B*C^2*a^2*c^4*d^7*e^4 + 15921*A^2*C*a^3*c^3*d^2*e^9 -
7848*A^2*C*a^2*c^4*d^4*e^7 - 6363*A*C^2*a^4*c^2*d^2*e^9 + 4926*A*C^2*a^3*c^3*d^4*e^7 - 1443*A*C^2*a^2*c^4*d^6
*e^5 + 14283*A^2*B*a^2*c^4*d^3*e^8 - 4617*A*B^2*a^2*c^4*d^4*e^7 - 1944*A*B^2*a^3*c^3*d^2*e^9 + 791*C^3*a^5*c*d
^2*e^9 - 2025*B^3*a^4*c^2*d*e^10 - 1674*A^3*a*c^5*d^4*e^7 - 90*A^2*C*c^6*d^8*e^3 + 135*A^2*B*c^6*d^7*e^4 - 172
8*A^2*C*a^4*c^2*e^11 + 675*A*B^2*a^4*c^2*e^11 - 225*B^2*C*a^5*c*e^11 + 576*A*C^2*a^5*c*e^11 - 397*C^3*a^3*c^3*
d^6*e^5 - 108*C^3*a^4*c^2*d^4*e^7 - 10*C^3*a^2*c^4*d^8*e^3 + 3294*B^3*a^3*c^3*d^3*e^8 + 135*B^3*a^2*c^4*d^5*e^
6 - 11853*A^3*a^2*c^4*d^2*e^9 - 189*A^3*c^6*d^6*e^5 + 1728*A^3*a^3*c^3*e^11 - 64*C^3*a^6*e^11, z, k)*((512*a^1
3*c^2*d*e^18 + 512*a^5*c^10*d^17*e^2 + 4096*a^6*c^9*d^15*e^4 + 14336*a^7*c^8*d^13*e^6 + 28672*a^8*c^7*d^11*e^8
+ 35840*a^9*c^6*d^9*e^10 + 28672*a^10*c^5*d^7*e^12 + 14336*a^11*c^4*d^5*e^14 + 4096*a^12*c^3*d^3*e^16)/(64*(a
^12*e^16 + a^4*c^8*d^16 + 8*a^11*c*d^2*e^14 + 8*a^5*c^7*d^14*e^2 + 28*a^6*c^6*d^12*e^4 + 56*a^7*c^5*d^10*e^6 +
70*a^8*c^4*d^8*e^8 + 56*a^9*c^3*d^6*e^10 + 28*a^10*c^2*d^4*e^12)) + (x*(384*a^13*c^2*e^19 - 128*a^4*c^11*d^18
*e - 640*a^5*c^10*d^16*e^3 - 512*a^6*c^9*d^14*e^5 + 3584*a^7*c^8*d^12*e^7 + 12544*a^8*c^7*d^10*e^9 + 19712*a^9
*c^6*d^8*e^11 + 17920*a^10*c^5*d^6*e^13 + 9728*a^11*c^4*d^4*e^15 + 2944*a^12*c^3*d^2*e^17))/(64*(a^12*e^16 + a
^4*c^8*d^16 + 8*a^11*c*d^2*e^14 + 8*a^5*c^7*d^14*e^2 + 28*a^6*c^6*d^12*e^4 + 56*a^7*c^5*d^10*e^6 + 70*a^8*c^4*
d^8*e^8 + 56*a^9*c^3*d^6*e^10 + 28*a^10*c^2*d^4*e^12))) + (120*B*a^10*c^2*e^16 + 24*A*a^2*c^10*d^15*e + 456*A*
a^9*c^3*d*e^15 + 8*C*a^3*c^9*d^15*e - 232*C*a^10*c^2*d*e^15 + 216*A*a^3*c^9*d^13*e^3 + 1176*A*a^4*c^8*d^11*e^5
+ 3480*A*a^5*c^7*d^9*e^7 + 5640*A*a^6*c^6*d^7*e^9 + 5064*A*a^7*c^5*d^5*e^11 + 2376*A*a^8*c^4*d^3*e^13 - 24*B*
a^3*c^9*d^14*e^2 - 408*B*a^4*c^8*d^12*e^4 - 1560*B*a^5*c^7*d^10*e^6 - 2520*B*a^6*c^6*d^8*e^8 - 1800*B*a^7*c^5*
d^6*e^10 - 264*B*a^8*c^4*d^4*e^12 + 312*B*a^9*c^3*d^2*e^14 + 200*C*a^4*c^8*d^13*e^3 + 648*C*a^5*c^7*d^11*e^5 +
520*C*a^6*c^6*d^9*e^7 - 680*C*a^7*c^5*d^7*e^9 - 1512*C*a^8*c^4*d^5*e^11 - 1000*C*a^9*c^3*d^3*e^13)/(64*(a^12*
e^16 + a^4*c^8*d^16 + 8*a^11*c*d^2*e^14 + 8*a^5*c^7*d^14*e^2 + 28*a^6*c^6*d^12*e^4 + 56*a^7*c^5*d^10*e^6 + 70*
a^8*c^4*d^8*e^8 + 56*a^9*c^3*d^6*e^10 + 28*a^10*c^2*d^4*e^12)) + (x*(192*C*a^10*c^2*e^16 - 576*A*a^9*c^3*e^16
+ 1488*B*a^9*c^3*d*e^15 + 48*A*a^2*c^10*d^14*e^2 + 480*A*a^3*c^9*d^12*e^4 + 4176*A*a^4*c^8*d^10*e^6 + 12288*A*
a^5*c^7*d^8*e^8 + 15312*A*a^6*c^6*d^6*e^10 + 7776*A*a^7*c^5*d^4*e^12 + 432*A*a^8*c^4*d^2*e^14 - 48*B*a^3*c^9*d
^13*e^3 - 1824*B*a^4*c^8*d^11*e^5 - 5328*B*a^5*c^7*d^9*e^7 - 4032*B*a^6*c^6*d^7*e^9 + 2352*B*a^7*c^5*d^5*e^11
+ 4320*B*a^8*c^4*d^3*e^13 + 16*C*a^3*c^9*d^14*e^2 + 1056*C*a^4*c^8*d^12*e^4 + 2160*C*a^5*c^7*d^10*e^6 - 1408*C
*a^6*c^6*d^8*e^8 - 6672*C*a^7*c^5*d^6*e^10 - 5472*C*a^8*c^4*d^4*e^12 - 1136*C*a^9*c^3*d^2*e^14))/(64*(a^12*e^1
6 + a^4*c^8*d^16 + 8*a^11*c*d^2*e^14 + 8*a^5*c^7*d^14*e^2 + 28*a^6*c^6*d^12*e^4 + 56*a^7*c^5*d^10*e^6 + 70*a^8
*c^4*d^8*e^8 + 56*a^9*c^3*d^6*e^10 + 28*a^10*c^2*d^4*e^12))) + (9*A^2*c^9*d^11*e^2 + 342*A^2*a^2*c^7*d^7*e^6 +
36*A^2*a^3*c^6*d^5*e^8 - 7479*A^2*a^4*c^5*d^3*e^10 + 9*B^2*a^2*c^7*d^9*e^4 - 108*B^2*a^3*c^6*d^7*e^6 - 3402*B
^2*a^4*c^5*d^5*e^8 + 5076*B^2*a^5*c^4*d^3*e^10 + C^2*a^2*c^7*d^11*e^2 - 36*C^2*a^3*c^6*d^9*e^4 - 1306*C^2*a^4*
c^5*d^7*e^6 + 4708*C^2*a^5*c^4*d^5*e^8 - 2943*C^2*a^6*c^3*d^3*e^10 + 360*A*B*a^6*c^3*e^13 - 120*B*C*a^7*c^2*e^
13 + 108*A^2*a*c^8*d^9*e^4 + 1944*A^2*a^5*c^4*d*e^12 - 855*B^2*a^6*c^3*d*e^12 + 296*C^2*a^7*c^2*d*e^12 - 18*A*
B*a*c^8*d^10*e^3 + 6*A*C*a*c^8*d^11*e^2 - 1536*A*C*a^6*c^3*d*e^12 + 756*A*B*a^3*c^6*d^6*e^7 + 11016*A*B*a^4*c^
5*d^4*e^9 - 7794*A*B*a^5*c^4*d^2*e^11 - 72*A*C*a^2*c^7*d^9*e^4 - 732*A*C*a^3*c^6*d^7*e^6 - 7368*A*C*a^4*c^5*d^
5*e^8 + 10182*A*C*a^5*c^4*d^3*e^10 - 6*B*C*a^2*c^7*d^10*e^3 + 144*B*C*a^3*c^6*d^8*e^5 + 4284*B*C*a^4*c^5*d^6*e
^7 - 10440*B*C*a^5*c^4*d^4*e^9 + 3738*B*C*a^6*c^3*d^2*e^11)/(64*(a^12*e^16 + a^4*c^8*d^16 + 8*a^11*c*d^2*e^14
+ 8*a^5*c^7*d^14*e^2 + 28*a^6*c^6*d^12*e^4 + 56*a^7*c^5*d^10*e^6 + 70*a^8*c^4*d^8*e^8 + 56*a^9*c^3*d^6*e^10 +
28*a^10*c^2*d^4*e^12)) + (x*(225*B^2*a^6*c^3*e^13 + 9*A^2*c^9*d^10*e^3 - 162*A^2*a^2*c^7*d^6*e^7 - 2916*A^2*a^
3*c^6*d^4*e^9 + 6561*A^2*a^4*c^5*d^2*e^11 + 9*B^2*a^2*c^7*d^8*e^5 - 468*B^2*a^3*c^6*d^6*e^7 + 6174*B^2*a^4*c^5
*d^4*e^9 - 2340*B^2*a^5*c^4*d^2*e^11 + C^2*a^2*c^7*d^10*e^3 - 116*C^2*a^3*c^6*d^8*e^5 + 3438*C^2*a^4*c^5*d^6*e
^7 - 4292*C^2*a^5*c^4*d^4*e^9 + 1369*C^2*a^6*c^3*d^2*e^11 + 108*A^2*a*c^8*d^8*e^5 - 18*A*B*a*c^8*d^9*e^4 + 243
0*A*B*a^5*c^4*d*e^12 + 6*A*C*a*c^8*d^10*e^3 - 1110*B*C*a^6*c^3*d*e^12 + 360*A*B*a^2*c^7*d^7*e^6 + 3204*A*B*a^3
*c^6*d^5*e^8 - 13176*A*B*a^4*c^5*d^3*e^10 - 312*A*C*a^2*c^7*d^8*e^5 - 2028*A*C*a^3*c^6*d^6*e^7 + 10728*A*C*a^4
*c^5*d^4*e^9 - 5994*A*C*a^5*c^4*d^2*e^11 - 6*B*C*a^2*c^7*d^9*e^4 + 504*B*C*a^3*c^6*d^7*e^6 - 9300*B*C*a^4*c^5*
d^5*e^8 + 7512*B*C*a^5*c^4*d^3*e^10))/(64*(a^12*e^16 + a^4*c^8*d^16 + 8*a^11*c*d^2*e^14 + 8*a^5*c^7*d^14*e^2 +
28*a^6*c^6*d^12*e^4 + 56*a^7*c^5*d^10*e^6 + 70*a^8*c^4*d^8*e^8 + 56*a^9*c^3*d^6*e^10 + 28*a^10*c^2*d^4*e^12))
)*root(2560*a^14*c*d^2*e^18*z^3 + 64512*a^10*c^5*d^10*e^10*z^3 + 53760*a^11*c^4*d^8*e^12*z^3 + 53760*a^9*c^6*d
^12*e^8*z^3 + 30720*a^12*c^3*d^6*e^14*z^3 + 30720*a^8*c^7*d^14*e^6*z^3 + 11520*a^13*c^2*d^4*e^16*z^3 + 11520*a
^7*c^8*d^16*e^4*z^3 + 2560*a^6*c^9*d^18*e^2*z^3 + 256*a^5*c^10*d^20*z^3 + 256*a^15*e^20*z^3 - 4806*B*C*a^8*c*d
*e^13*z - 18*A*B*a*c^8*d^13*e*z - 147930*B*C*a^6*c^3*d^5*e^9*z + 74760*B*C*a^5*c^4*d^7*e^7*z + 66588*B*C*a^7*c
^2*d^3*e^11*z - 1050*B*C*a^4*c^5*d^9*e^5*z - 228*B*C*a^3*c^6*d^11*e^3*z + 152052*A*C*a^6*c^3*d^4*e^10*z - 1098
30*A*C*a^5*c^4*d^6*e^8*z - 32490*A*C*a^7*c^2*d^2*e^12*z + 426*A*C*a^3*c^6*d^10*e^4*z - 360*A*C*a^4*c^5*d^8*e^6
*z + 180*A*C*a^2*c^7*d^12*e^2*z + 158130*A*B*a^5*c^4*d^5*e^9*z - 121356*A*B*a^6*c^3*d^3*e^11*z - 3240*A*B*a^4*
c^5*d^7*e^7*z - 1710*A*B*a^3*c^6*d^9*e^5*z - 396*A*B*a^2*c^7*d^11*e^3*z - 6*B*C*a^2*c^7*d^13*e*z + 13518*A*B*a
^7*c^2*d*e^13*z + 67615*C^2*a^6*c^3*d^6*e^8*z - 47538*C^2*a^7*c^2*d^4*e^10*z - 24860*C^2*a^5*c^4*d^8*e^6*z + 2
79*C^2*a^4*c^5*d^10*e^4*z + 46*C^2*a^3*c^6*d^12*e^2*z + 71415*B^2*a^6*c^3*d^4*e^10*z - 55260*B^2*a^5*c^4*d^6*e
^8*z - 19602*B^2*a^7*c^2*d^2*e^12*z + 1215*B^2*a^4*c^5*d^8*e^6*z + 270*B^2*a^3*c^6*d^10*e^4*z + 9*B^2*a^2*c^7*
d^12*e^2*z - 106722*A^2*a^5*c^4*d^4*e^10*z + 35217*A^2*a^6*c^3*d^2*e^12*z + 6615*A^2*a^4*c^5*d^6*e^8*z + 3780*
A^2*a^3*c^6*d^8*e^6*z + 1071*A^2*a^2*c^7*d^10*e^4*z + 1152*A*C*a^8*c*e^14*z + 6*A*C*a*c^8*d^14*z + 7017*C^2*a^
8*c*d^2*e^12*z + 126*A^2*a*c^8*d^12*e^2*z + C^2*a^2*c^7*d^14*z - 1728*A^2*a^7*c^2*e^14*z + 225*B^2*a^8*c*e^14*
z + 9*A^2*c^9*d^14*z - 192*C^2*a^9*e^14*z + 3168*A*B*C*a^4*c^2*d*e^10 + 270*A*B*C*a*c^5*d^7*e^4 - 6930*A*B*C*a
^3*c^3*d^3*e^8 + 5148*A*B*C*a^2*c^4*d^5*e^6 - 819*A^2*C*a*c^5*d^6*e^5 - 60*A*C^2*a*c^5*d^8*e^3 - 6102*A^2*B*a^
3*c^3*d*e^10 + 1512*A^2*B*a*c^5*d^5*e^6 - 270*A*B^2*a*c^5*d^6*e^5 - 378*B*C^2*a^5*c*d*e^10 - 5049*B^2*C*a^3*c^
3*d^4*e^7 + 4698*B^2*C*a^4*c^2*d^2*e^9 + 2508*B*C^2*a^3*c^3*d^5*e^6 - 1977*B*C^2*a^4*c^2*d^3*e^8 - 180*B^2*C*a
^2*c^4*d^6*e^5 + 75*B*C^2*a^2*c^4*d^7*e^4 + 15921*A^2*C*a^3*c^3*d^2*e^9 - 7848*A^2*C*a^2*c^4*d^4*e^7 - 6363*A*
C^2*a^4*c^2*d^2*e^9 + 4926*A*C^2*a^3*c^3*d^4*e^7 - 1443*A*C^2*a^2*c^4*d^6*e^5 + 14283*A^2*B*a^2*c^4*d^3*e^8 -
4617*A*B^2*a^2*c^4*d^4*e^7 - 1944*A*B^2*a^3*c^3*d^2*e^9 + 791*C^3*a^5*c*d^2*e^9 - 2025*B^3*a^4*c^2*d*e^10 - 16
74*A^3*a*c^5*d^4*e^7 - 90*A^2*C*c^6*d^8*e^3 + 135*A^2*B*c^6*d^7*e^4 - 1728*A^2*C*a^4*c^2*e^11 + 675*A*B^2*a^4*
c^2*e^11 - 225*B^2*C*a^5*c*e^11 + 576*A*C^2*a^5*c*e^11 - 397*C^3*a^3*c^3*d^6*e^5 - 108*C^3*a^4*c^2*d^4*e^7 - 1
0*C^3*a^2*c^4*d^8*e^3 + 3294*B^3*a^3*c^3*d^3*e^8 + 135*B^3*a^2*c^4*d^5*e^6 - 11853*A^3*a^2*c^4*d^2*e^9 - 189*A
^3*c^6*d^6*e^5 + 1728*A^3*a^3*c^3*e^11 - 64*C^3*a^6*e^11, z, k), k, 1, 3)

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sympy [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((C*x**2+B*x+A)/(e*x+d)**3/(c*x**2+a)**3,x)

[Out]

Timed out

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