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

Optimal. Leaf size=571 \[ -\frac {4 a^2 e \left (a e^2 (2 C d-B e)-c d \left (2 C d^2-e (3 B d-4 A e)\right )\right )-x \left (A c \left (-7 a^2 e^4+12 a c d^2 e^2+3 c^2 d^4\right )+a \left (3 a^2 C e^4-2 a c d e^2 (6 C d-7 B e)+c^2 d^3 (C d-2 B e)\right )\right )}{8 a^2 \left (a+c x^2\right ) \left (a e^2+c d^2\right )^3}+\frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (3 A c \left (-5 a^3 e^6+15 a^2 c d^2 e^4+5 a c^2 d^4 e^2+c^3 d^6\right )+a \left (3 a^3 C e^6-3 a^2 c d e^4 (11 C d-10 B e)+a c^2 d^3 e^2 (13 C d-20 B e)+c^3 d^5 (C d-2 B e)\right )\right )}{8 a^{5/2} \sqrt {c} \left (a e^2+c d^2\right )^4}-\frac {a \left (-a B e^2+2 a C d e-2 A c d e+B c d^2\right )-x \left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right )}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )^2}+\frac {e^3 \log \left (a+c x^2\right ) \left (a e^2 (2 C d-B e)-c d \left (4 C d^2-e (5 B d-6 A e)\right )\right )}{2 \left (a e^2+c d^2\right )^4}-\frac {e^3 \left (A e^2-B d e+C d^2\right )}{(d+e x) \left (a e^2+c d^2\right )^3}-\frac {e^3 \log (d+e x) \left (a e^2 (2 C d-B e)-c d \left (4 C d^2-e (5 B d-6 A e)\right )\right )}{\left (a e^2+c d^2\right )^4} \]

[Out]

-e^3*(A*e^2-B*d*e+C*d^2)/(a*e^2+c*d^2)^3/(e*x+d)+1/4*(-a*(-2*A*c*d*e-B*a*e^2+B*c*d^2+2*C*a*d*e)+(A*c*(-a*e^2+c
*d^2)+a*(a*C*e^2-c*d*(-2*B*e+C*d)))*x)/a/(a*e^2+c*d^2)^2/(c*x^2+a)^2+1/8*(-4*a^2*e*(a*e^2*(-B*e+2*C*d)-c*d*(2*
C*d^2-e*(-4*A*e+3*B*d)))+(A*c*(-7*a^2*e^4+12*a*c*d^2*e^2+3*c^2*d^4)+a*(3*a^2*C*e^4-2*a*c*d*e^2*(-7*B*e+6*C*d)+
c^2*d^3*(-2*B*e+C*d)))*x)/a^2/(a*e^2+c*d^2)^3/(c*x^2+a)-e^3*(a*e^2*(-B*e+2*C*d)-c*d*(4*C*d^2-e*(-6*A*e+5*B*d))
)*ln(e*x+d)/(a*e^2+c*d^2)^4+1/2*e^3*(a*e^2*(-B*e+2*C*d)-c*d*(4*C*d^2-e*(-6*A*e+5*B*d)))*ln(c*x^2+a)/(a*e^2+c*d
^2)^4+1/8*(3*A*c*(-5*a^3*e^6+15*a^2*c*d^2*e^4+5*a*c^2*d^4*e^2+c^3*d^6)+a*(3*a^3*C*e^6+a*c^2*d^3*e^2*(-20*B*e+1
3*C*d)-3*a^2*c*d*e^4*(-10*B*e+11*C*d)+c^3*d^5*(-2*B*e+C*d)))*arctan(x*c^(1/2)/a^(1/2))/a^(5/2)/(a*e^2+c*d^2)^4
/c^(1/2)

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Rubi [A]  time = 1.92, antiderivative size = 566, normalized size of antiderivative = 0.99, 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} \[ \frac {x \left (A c \left (-7 a^2 e^4+12 a c d^2 e^2+3 c^2 d^4\right )+a \left (3 a^2 C e^4-2 a c d e^2 (6 C d-7 B e)+c^2 d^3 (C d-2 B e)\right )\right )+4 a^2 e \left (-a e^2 (2 C d-B e)-c d e (3 B d-4 A e)+2 c C d^3\right )}{8 a^2 \left (a+c x^2\right ) \left (a e^2+c d^2\right )^3}+\frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (3 A c \left (15 a^2 c d^2 e^4-5 a^3 e^6+5 a c^2 d^4 e^2+c^3 d^6\right )+a \left (-3 a^2 c d e^4 (11 C d-10 B e)+3 a^3 C e^6+a c^2 d^3 e^2 (13 C d-20 B e)+c^3 d^5 (C d-2 B e)\right )\right )}{8 a^{5/2} \sqrt {c} \left (a e^2+c d^2\right )^4}-\frac {a \left (-a B e^2+2 a C d e-2 A c d e+B c d^2\right )-x \left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right )}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )^2}-\frac {e^3 \log \left (a+c x^2\right ) \left (-a e^2 (2 C d-B e)-c d e (5 B d-6 A e)+4 c C d^3\right )}{2 \left (a e^2+c d^2\right )^4}-\frac {e^3 \left (A e^2-B d e+C d^2\right )}{(d+e x) \left (a e^2+c d^2\right )^3}+\frac {e^3 \log (d+e x) \left (-a e^2 (2 C d-B e)-c d e (5 B d-6 A e)+4 c C d^3\right )}{\left (a e^2+c d^2\right )^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((e^3*(C*d^2 - B*d*e + A*e^2))/((c*d^2 + a*e^2)^3*(d + e*x))) - (a*(B*c*d^2 - 2*A*c*d*e + 2*a*C*d*e - a*B*e^2
) - (A*c*(c*d^2 - a*e^2) + a*(a*C*e^2 - c*d*(C*d - 2*B*e)))*x)/(4*a*(c*d^2 + a*e^2)^2*(a + c*x^2)^2) + (4*a^2*
e*(2*c*C*d^3 - c*d*e*(3*B*d - 4*A*e) - a*e^2*(2*C*d - B*e)) + (A*c*(3*c^2*d^4 + 12*a*c*d^2*e^2 - 7*a^2*e^4) +
a*(3*a^2*C*e^4 - 2*a*c*d*e^2*(6*C*d - 7*B*e) + c^2*d^3*(C*d - 2*B*e)))*x)/(8*a^2*(c*d^2 + a*e^2)^3*(a + c*x^2)
) + ((3*A*c*(c^3*d^6 + 5*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4 - 5*a^3*e^6) + a*(3*a^3*C*e^6 + a*c^2*d^3*e^2*(13*C*
d - 20*B*e) - 3*a^2*c*d*e^4*(11*C*d - 10*B*e) + c^3*d^5*(C*d - 2*B*e)))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2
)*Sqrt[c]*(c*d^2 + a*e^2)^4) + (e^3*(4*c*C*d^3 - c*d*e*(5*B*d - 6*A*e) - a*e^2*(2*C*d - B*e))*Log[d + e*x])/(c
*d^2 + a*e^2)^4 - (e^3*(4*c*C*d^3 - c*d*e*(5*B*d - 6*A*e) - a*e^2*(2*C*d - B*e))*Log[a + c*x^2])/(2*(c*d^2 + a
*e^2)^4)

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)^2 \left (a+c x^2\right )^3} \, dx &=-\frac {a \left (B c d^2-2 A c d e+2 a C d e-a B e^2\right )-\left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right ) x}{4 a \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}-\frac {\int \frac {-\frac {c \left (A \left (3 c^2 d^4+9 a c d^2 e^2+4 a^2 e^4\right )-a d^2 \left (a C e^2-c d (C d-2 B e)\right )\right )}{\left (c d^2+a e^2\right )^2}-\frac {2 c e \left (A c d \left (3 c d^2+a e^2\right )-a \left (c d^2 (3 C d-4 B e)+a e^2 (C d-2 B e)\right )\right ) x}{\left (c d^2+a e^2\right )^2}-\frac {3 c e^2 \left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right ) x^2}{\left (c d^2+a e^2\right )^2}}{(d+e x)^2 \left (a+c x^2\right )^2} \, dx}{4 a c}\\ &=-\frac {a \left (B c d^2-2 A c d e+2 a C d e-a B e^2\right )-\left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right ) x}{4 a \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac {4 a^2 e \left (2 c C d^3-c d e (3 B d-4 A e)-a e^2 (2 C d-B e)\right )+\left (A c \left (3 c^2 d^4+12 a c d^2 e^2-7 a^2 e^4\right )+a \left (3 a^2 C e^4-2 a c d e^2 (6 C d-7 B e)+c^2 d^3 (C d-2 B e)\right )\right ) x}{8 a^2 \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )}+\frac {\int \frac {\frac {c^2 \left (A \left (3 c^3 d^6+12 a c^2 d^4 e^2+33 a^2 c d^2 e^4+8 a^3 e^6\right )-a d^2 \left (5 a^2 C e^4-6 a c d e^2 (2 C d-3 B e)-c^2 d^3 (C d-2 B e)\right )\right )}{\left (c d^2+a e^2\right )^3}+\frac {2 c^2 e \left (3 A c d \left (c d^2+3 a e^2\right )-a \left (a e^2 (5 C d-4 B e)-c d^2 (C d-2 B e)\right )\right ) x}{\left (c d^2+a e^2\right )^2}+\frac {c^2 e^2 \left (A c \left (3 c^2 d^4+12 a c d^2 e^2-7 a^2 e^4\right )+a \left (3 a^2 C e^4-2 a c d e^2 (6 C d-7 B e)+c^2 d^3 (C d-2 B e)\right )\right ) x^2}{\left (c d^2+a e^2\right )^3}}{(d+e x)^2 \left (a+c x^2\right )} \, dx}{8 a^2 c^2}\\ &=-\frac {a \left (B c d^2-2 A c d e+2 a C d e-a B e^2\right )-\left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right ) x}{4 a \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac {4 a^2 e \left (2 c C d^3-c d e (3 B d-4 A e)-a e^2 (2 C d-B e)\right )+\left (A c \left (3 c^2 d^4+12 a c d^2 e^2-7 a^2 e^4\right )+a \left (3 a^2 C e^4-2 a c d e^2 (6 C d-7 B e)+c^2 d^3 (C d-2 B e)\right )\right ) x}{8 a^2 \left (c d^2+a e^2\right )^3 \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)^2}+\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)}+\frac {c^2 \left (3 A c \left (c^3 d^6+5 a c^2 d^4 e^2+15 a^2 c d^2 e^4-5 a^3 e^6\right )+a \left (3 a^3 C e^6+a c^2 d^3 e^2 (13 C d-20 B e)-3 a^2 c d e^4 (11 C d-10 B e)+c^3 d^5 (C d-2 B e)\right )-8 a^2 c 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 ) x\right )}{\left (c d^2+a e^2\right )^4 \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 )}{\left (c d^2+a e^2\right )^3 (d+e x)}-\frac {a \left (B c d^2-2 A c d e+2 a C d e-a B e^2\right )-\left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right ) x}{4 a \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac {4 a^2 e \left (2 c C d^3-c d e (3 B d-4 A e)-a e^2 (2 C d-B e)\right )+\left (A c \left (3 c^2 d^4+12 a c d^2 e^2-7 a^2 e^4\right )+a \left (3 a^2 C e^4-2 a c d e^2 (6 C d-7 B e)+c^2 d^3 (C d-2 B e)\right )\right ) x}{8 a^2 \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )}+\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 ) \log (d+e x)}{\left (c d^2+a e^2\right )^4}+\frac {\int \frac {3 A c \left (c^3 d^6+5 a c^2 d^4 e^2+15 a^2 c d^2 e^4-5 a^3 e^6\right )+a \left (3 a^3 C e^6+a c^2 d^3 e^2 (13 C d-20 B e)-3 a^2 c d e^4 (11 C d-10 B e)+c^3 d^5 (C d-2 B e)\right )-8 a^2 c 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 ) x}{a+c x^2} \, dx}{8 a^2 \left (c d^2+a e^2\right )^4}\\ &=-\frac {e^3 \left (C d^2-B d e+A e^2\right )}{\left (c d^2+a e^2\right )^3 (d+e x)}-\frac {a \left (B c d^2-2 A c d e+2 a C d e-a B e^2\right )-\left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right ) x}{4 a \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac {4 a^2 e \left (2 c C d^3-c d e (3 B d-4 A e)-a e^2 (2 C d-B e)\right )+\left (A c \left (3 c^2 d^4+12 a c d^2 e^2-7 a^2 e^4\right )+a \left (3 a^2 C e^4-2 a c d e^2 (6 C d-7 B e)+c^2 d^3 (C d-2 B e)\right )\right ) x}{8 a^2 \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )}+\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 ) \log (d+e x)}{\left (c d^2+a e^2\right )^4}-\frac {\left (c 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 )\right ) \int \frac {x}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^4}+\frac {\left (3 A c \left (c^3 d^6+5 a c^2 d^4 e^2+15 a^2 c d^2 e^4-5 a^3 e^6\right )+a \left (3 a^3 C e^6+a c^2 d^3 e^2 (13 C d-20 B e)-3 a^2 c d e^4 (11 C d-10 B e)+c^3 d^5 (C d-2 B e)\right )\right ) \int \frac {1}{a+c x^2} \, dx}{8 a^2 \left (c d^2+a e^2\right )^4}\\ &=-\frac {e^3 \left (C d^2-B d e+A e^2\right )}{\left (c d^2+a e^2\right )^3 (d+e x)}-\frac {a \left (B c d^2-2 A c d e+2 a C d e-a B e^2\right )-\left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right ) x}{4 a \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac {4 a^2 e \left (2 c C d^3-c d e (3 B d-4 A e)-a e^2 (2 C d-B e)\right )+\left (A c \left (3 c^2 d^4+12 a c d^2 e^2-7 a^2 e^4\right )+a \left (3 a^2 C e^4-2 a c d e^2 (6 C d-7 B e)+c^2 d^3 (C d-2 B e)\right )\right ) x}{8 a^2 \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )}+\frac {\left (3 A c \left (c^3 d^6+5 a c^2 d^4 e^2+15 a^2 c d^2 e^4-5 a^3 e^6\right )+a \left (3 a^3 C e^6+a c^2 d^3 e^2 (13 C d-20 B e)-3 a^2 c d e^4 (11 C d-10 B e)+c^3 d^5 (C d-2 B e)\right )\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {c} \left (c d^2+a e^2\right )^4}+\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 ) \log (d+e x)}{\left (c d^2+a e^2\right )^4}-\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 ) \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^4}\\ \end {align*}

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Mathematica [A]  time = 0.76, size = 498, normalized size = 0.87 \[ \frac {\frac {2 \left (a e^2+c d^2\right )^2 \left (a^2 e (B e-2 C d+C e x)-a c \left (A e (e x-2 d)+B d (d-2 e x)+C d^2 x\right )+A c^2 d^2 x\right )}{a \left (a+c x^2\right )^2}+\frac {\left (a e^2+c d^2\right ) \left (a^3 e^3 (4 B e-8 C d+3 C e x)+a^2 c e \left (e (A e (16 d-7 e x)-2 B d (6 d-7 e x))+4 C d^2 (2 d-3 e x)\right )+a c^2 d^2 x \left (2 e (6 A e-B d)+C d^2\right )+3 A c^3 d^4 x\right )}{a^2 \left (a+c x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (3 A c \left (-5 a^3 e^6+15 a^2 c d^2 e^4+5 a c^2 d^4 e^2+c^3 d^6\right )+a \left (3 a^3 C e^6+3 a^2 c d e^4 (10 B e-11 C d)+a c^2 d^3 e^2 (13 C d-20 B e)+c^3 d^5 (C d-2 B e)\right )\right )}{a^{5/2} \sqrt {c}}-4 e^3 \log \left (a+c x^2\right ) \left (a e^2 (B e-2 C d)+c d e (6 A e-5 B d)+4 c C d^3\right )+8 e^3 \log (d+e x) \left (a e^2 (B e-2 C d)+c d e (6 A e-5 B d)+4 c C d^3\right )-\frac {8 e^3 \left (a e^2+c d^2\right ) \left (e (A e-B d)+C d^2\right )}{d+e x}}{8 \left (a e^2+c d^2\right )^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-8*e^3*(c*d^2 + a*e^2)*(C*d^2 + e*(-(B*d) + A*e)))/(d + e*x) + (2*(c*d^2 + a*e^2)^2*(A*c^2*d^2*x + a^2*e*(-2
*C*d + B*e + C*e*x) - a*c*(C*d^2*x + B*d*(d - 2*e*x) + A*e*(-2*d + e*x))))/(a*(a + c*x^2)^2) + ((c*d^2 + a*e^2
)*(3*A*c^3*d^4*x + a*c^2*d^2*(C*d^2 + 2*e*(-(B*d) + 6*A*e))*x + a^3*e^3*(-8*C*d + 4*B*e + 3*C*e*x) + a^2*c*e*(
4*C*d^2*(2*d - 3*e*x) + e*(-2*B*d*(6*d - 7*e*x) + A*e*(16*d - 7*e*x)))))/(a^2*(a + c*x^2)) + ((3*A*c*(c^3*d^6
+ 5*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4 - 5*a^3*e^6) + a*(3*a^3*C*e^6 + a*c^2*d^3*e^2*(13*C*d - 20*B*e) + c^3*d^5
*(C*d - 2*B*e) + 3*a^2*c*d*e^4*(-11*C*d + 10*B*e)))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(a^(5/2)*Sqrt[c]) + 8*e^3*(4*
c*C*d^3 + c*d*e*(-5*B*d + 6*A*e) + a*e^2*(-2*C*d + B*e))*Log[d + e*x] - 4*e^3*(4*c*C*d^3 + c*d*e*(-5*B*d + 6*A
*e) + a*e^2*(-2*C*d + B*e))*Log[a + c*x^2])/(8*(c*d^2 + a*e^2)^4)

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

[Out]

Timed out

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giac [A]  time = 0.24, size = 1107, normalized size = 1.94 \[ \frac {{\left (C a c^{3} d^{6} e^{2} + 3 \, A c^{4} d^{6} e^{2} - 2 \, B a c^{3} d^{5} e^{3} + 13 \, C a^{2} c^{2} d^{4} e^{4} + 15 \, A a c^{3} d^{4} e^{4} - 20 \, B a^{2} c^{2} d^{3} e^{5} - 33 \, C a^{3} c d^{2} e^{6} + 45 \, A a^{2} c^{2} d^{2} e^{6} + 30 \, B a^{3} c d e^{7} + 3 \, C a^{4} e^{8} - 15 \, A a^{3} c e^{8}\right )} \arctan \left (\frac {{\left (c d - \frac {c d^{2}}{x e + d} - \frac {a e^{2}}{x e + d}\right )} e^{\left (-1\right )}}{\sqrt {a c}}\right ) e^{\left (-2\right )}}{8 \, {\left (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}\right )} \sqrt {a c}} - \frac {{\left (4 \, C c d^{3} e^{3} - 5 \, B c d^{2} e^{4} - 2 \, C a d e^{5} + 6 \, A c d e^{5} + B a e^{6}\right )} \log \left (c - \frac {2 \, c d}{x e + d} + \frac {c d^{2}}{{\left (x e + d\right )}^{2}} + \frac {a e^{2}}{{\left (x e + d\right )}^{2}}\right )}{2 \, {\left (c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )}} - \frac {\frac {C d^{2} e^{9}}{x e + d} - \frac {B d e^{10}}{x e + d} + \frac {A e^{11}}{x e + d}}{c^{3} d^{6} e^{6} + 3 \, a c^{2} d^{4} e^{8} + 3 \, a^{2} c d^{2} e^{10} + a^{3} e^{12}} + \frac {C a c^{4} d^{5} e + 3 \, A c^{5} d^{5} e - 2 \, B a c^{4} d^{4} e^{2} - 22 \, C a^{2} c^{3} d^{3} e^{3} + 14 \, A a c^{4} d^{3} e^{3} + 32 \, B a^{2} c^{3} d^{2} e^{4} + 17 \, C a^{3} c^{2} d e^{5} - 29 \, A a^{2} c^{3} d e^{5} - 6 \, B a^{3} c^{2} e^{6} - \frac {{\left (3 \, C a c^{4} d^{6} e^{2} + 9 \, A c^{5} d^{6} e^{2} - 6 \, B a c^{4} d^{5} e^{3} - 77 \, C a^{2} c^{3} d^{4} e^{4} + 41 \, A a c^{4} d^{4} e^{4} + 116 \, B a^{2} c^{3} d^{3} e^{5} + 77 \, C a^{3} c^{2} d^{2} e^{6} - 121 \, A a^{2} c^{3} d^{2} e^{6} - 38 \, B a^{3} c^{2} d e^{7} - 3 \, C a^{4} c e^{8} + 7 \, A a^{3} c^{2} e^{8}\right )} e^{\left (-1\right )}}{x e + d} + \frac {{\left (3 \, C a c^{4} d^{7} e^{3} + 9 \, A c^{5} d^{7} e^{3} - 6 \, B a c^{4} d^{6} e^{4} - 89 \, C a^{2} c^{3} d^{5} e^{5} + 45 \, A a c^{4} d^{5} e^{5} + 140 \, B a^{2} c^{3} d^{4} e^{6} + 85 \, C a^{3} c^{2} d^{3} e^{7} - 145 \, A a^{2} c^{3} d^{3} e^{7} - 22 \, B a^{3} c^{2} d^{2} e^{8} + 17 \, C a^{4} c d e^{9} - 21 \, A a^{3} c^{2} d e^{9} - 8 \, B a^{4} c e^{10}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac {{\left (C a c^{4} d^{8} e^{4} + 3 \, A c^{5} d^{8} e^{4} - 2 \, B a c^{4} d^{7} e^{5} - 34 \, C a^{2} c^{3} d^{6} e^{6} + 18 \, A a c^{4} d^{6} e^{6} + 58 \, B a^{2} c^{3} d^{5} e^{7} + 20 \, C a^{3} c^{2} d^{4} e^{8} - 60 \, A a^{2} c^{3} d^{4} e^{8} + 26 \, B a^{3} c^{2} d^{3} e^{9} + 50 \, C a^{4} c d^{2} e^{10} - 66 \, A a^{3} c^{2} d^{2} e^{10} - 34 \, B a^{4} c d e^{11} - 5 \, C a^{5} e^{12} + 9 \, A a^{4} c e^{12}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}}}{8 \, {\left (c d^{2} + a e^{2}\right )}^{4} a^{2} {\left (c - \frac {2 \, c d}{x e + d} + \frac {c d^{2}}{{\left (x e + d\right )}^{2}} + \frac {a e^{2}}{{\left (x e + d\right )}^{2}}\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(C*a*c^3*d^6*e^2 + 3*A*c^4*d^6*e^2 - 2*B*a*c^3*d^5*e^3 + 13*C*a^2*c^2*d^4*e^4 + 15*A*a*c^3*d^4*e^4 - 20*B*
a^2*c^2*d^3*e^5 - 33*C*a^3*c*d^2*e^6 + 45*A*a^2*c^2*d^2*e^6 + 30*B*a^3*c*d*e^7 + 3*C*a^4*e^8 - 15*A*a^3*c*e^8)
*arctan((c*d - c*d^2/(x*e + d) - a*e^2/(x*e + d))*e^(-1)/sqrt(a*c))*e^(-2)/((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)*sqrt(a*c)) - 1/2*(4*C*c*d^3*e^3 - 5*B*c*d^2*e^4 - 2*C*a*d*e^5
+ 6*A*c*d*e^5 + B*a*e^6)*log(c - 2*c*d/(x*e + d) + c*d^2/(x*e + d)^2 + a*e^2/(x*e + d)^2)/(c^4*d^8 + 4*a*c^3*d
^6*e^2 + 6*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 + a^4*e^8) - (C*d^2*e^9/(x*e + d) - B*d*e^10/(x*e + d) + A*e^11/(
x*e + d))/(c^3*d^6*e^6 + 3*a*c^2*d^4*e^8 + 3*a^2*c*d^2*e^10 + a^3*e^12) + 1/8*(C*a*c^4*d^5*e + 3*A*c^5*d^5*e -
2*B*a*c^4*d^4*e^2 - 22*C*a^2*c^3*d^3*e^3 + 14*A*a*c^4*d^3*e^3 + 32*B*a^2*c^3*d^2*e^4 + 17*C*a^3*c^2*d*e^5 - 2
9*A*a^2*c^3*d*e^5 - 6*B*a^3*c^2*e^6 - (3*C*a*c^4*d^6*e^2 + 9*A*c^5*d^6*e^2 - 6*B*a*c^4*d^5*e^3 - 77*C*a^2*c^3*
d^4*e^4 + 41*A*a*c^4*d^4*e^4 + 116*B*a^2*c^3*d^3*e^5 + 77*C*a^3*c^2*d^2*e^6 - 121*A*a^2*c^3*d^2*e^6 - 38*B*a^3
*c^2*d*e^7 - 3*C*a^4*c*e^8 + 7*A*a^3*c^2*e^8)*e^(-1)/(x*e + d) + (3*C*a*c^4*d^7*e^3 + 9*A*c^5*d^7*e^3 - 6*B*a*
c^4*d^6*e^4 - 89*C*a^2*c^3*d^5*e^5 + 45*A*a*c^4*d^5*e^5 + 140*B*a^2*c^3*d^4*e^6 + 85*C*a^3*c^2*d^3*e^7 - 145*A
*a^2*c^3*d^3*e^7 - 22*B*a^3*c^2*d^2*e^8 + 17*C*a^4*c*d*e^9 - 21*A*a^3*c^2*d*e^9 - 8*B*a^4*c*e^10)*e^(-2)/(x*e
+ d)^2 - (C*a*c^4*d^8*e^4 + 3*A*c^5*d^8*e^4 - 2*B*a*c^4*d^7*e^5 - 34*C*a^2*c^3*d^6*e^6 + 18*A*a*c^4*d^6*e^6 +
58*B*a^2*c^3*d^5*e^7 + 20*C*a^3*c^2*d^4*e^8 - 60*A*a^2*c^3*d^4*e^8 + 26*B*a^3*c^2*d^3*e^9 + 50*C*a^4*c*d^2*e^1
0 - 66*A*a^3*c^2*d^2*e^10 - 34*B*a^4*c*d*e^11 - 5*C*a^5*e^12 + 9*A*a^4*c*e^12)*e^(-3)/(x*e + d)^3)/((c*d^2 + a
*e^2)^4*a^2*(c - 2*c*d/(x*e + d) + c*d^2/(x*e + d)^2 + a*e^2/(x*e + d)^2)^2)

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maple [B]  time = 0.03, size = 2159, normalized size = 3.78 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/(a*e^2+c*d^2)^4*c*ln(c*x^2+a)*d*A*e^5+1/(a*e^2+c*d^2)^4*a*ln(c*x^2+a)*C*d*e^5+3/8/(a*e^2+c*d^2)^4*a^2/(a*c)
^(1/2)*arctan(1/(a*c)^(1/2)*c*x)*C*e^6-3/2/(a*e^2+c*d^2)^4/(c*x^2+a)^2*C*a^3*d*e^5+1/2/(a*e^2+c*d^2)^4/(c*x^2+
a)^2*A*c^3*d^5*e+5/8/(a*e^2+c*d^2)^4/(c*x^2+a)^2*a^3*C*e^6*x-1/8/(a*e^2+c*d^2)^4/(c*x^2+a)^2*C*c^3*d^6*x+5/2/(
a*e^2+c*d^2)^4*c*ln(c*x^2+a)*d^2*e^4*B-2/(a*e^2+c*d^2)^4*c*ln(c*x^2+a)*C*d^3*e^3+6*e^5/(a*e^2+c*d^2)^4*ln(e*x+
d)*A*c*d-5*e^4/(a*e^2+c*d^2)^4*ln(e*x+d)*B*c*d^2-2*e^5/(a*e^2+c*d^2)^4*ln(e*x+d)*C*a*d+4*e^3/(a*e^2+c*d^2)^4*l
n(e*x+d)*C*c*d^3-e^5/(a*e^2+c*d^2)^3/(e*x+d)*A-7/8/(a*e^2+c*d^2)^4/(c*x^2+a)^2*C*a^2*c*d^2*e^4*x-13/8/(a*e^2+c
*d^2)^4/(c*x^2+a)^2*C*a*c^2*d^4*e^2*x+7/4/(a*e^2+c*d^2)^4/(c*x^2+a)^2*B*x^3*a*c^2*d*e^5-9/8/(a*e^2+c*d^2)^4/(c
*x^2+a)^2*C*x^3*a*c^2*d^2*e^4+2/(a*e^2+c*d^2)^4/(c*x^2+a)^2*A*x^2*a*c^2*d*e^5-1/(a*e^2+c*d^2)^4/(c*x^2+a)^2*B*
x^2*a*c^2*d^2*e^4-1/(a*e^2+c*d^2)^4/(c*x^2+a)^2*C*x^2*a^2*c*d*e^5+15/8/(a*e^2+c*d^2)^4/a/(a*c)^(1/2)*arctan(1/
(a*c)^(1/2)*c*x)*A*c^3*d^4*e^2+15/4/(a*e^2+c*d^2)^4*a/(a*c)^(1/2)*arctan(1/(a*c)^(1/2)*c*x)*B*c*d*e^5-1/4/(a*e
^2+c*d^2)^4/a/(a*c)^(1/2)*arctan(1/(a*c)^(1/2)*c*x)*B*c^3*d^5*e-33/8/(a*e^2+c*d^2)^4*a/(a*c)^(1/2)*arctan(1/(a
*c)^(1/2)*c*x)*C*c*d^2*e^4+15/8/(a*e^2+c*d^2)^4/(c*x^2+a)^2*c^4/a*x^3*A*d^4*e^2-1/4/(a*e^2+c*d^2)^4/(c*x^2+a)^
2*c^4/a*x^3*B*d^5*e+3/8/(a*e^2+c*d^2)^4/(c*x^2+a)^2*A*a*c^2*d^2*e^4*x+9/4/(a*e^2+c*d^2)^4/(c*x^2+a)^2*d*a^2*c*
B*e^5*x+5/2/(a*e^2+c*d^2)^4/(c*x^2+a)^2*B*a*c^2*d^3*e^3*x+e^6/(a*e^2+c*d^2)^4*ln(e*x+d)*B*a+e^4/(a*e^2+c*d^2)^
3/(e*x+d)*B*d-e^3/(a*e^2+c*d^2)^3/(e*x+d)*C*d^2+3/4/(a*e^2+c*d^2)^4/(c*x^2+a)^2*B*a^3*e^6-1/4/(a*e^2+c*d^2)^4/
(c*x^2+a)^2*B*c^3*d^6-1/2/(a*e^2+c*d^2)^4*a*ln(c*x^2+a)*e^6*B+45/8/(a*e^2+c*d^2)^4/(a*c)^(1/2)*arctan(1/(a*c)^
(1/2)*c*x)*A*c^2*d^2*e^4-5/2/(a*e^2+c*d^2)^4/(a*c)^(1/2)*arctan(1/(a*c)^(1/2)*c*x)*B*c^2*d^3*e^3+13/8/(a*e^2+c
*d^2)^4/(a*c)^(1/2)*arctan(1/(a*c)^(1/2)*c*x)*C*c^2*d^4*e^2+3/2/(a*e^2+c*d^2)^4/(c*x^2+a)^2*B*x^3*c^3*d^3*e^3-
11/8/(a*e^2+c*d^2)^4/(c*x^2+a)^2*C*x^3*c^3*d^4*e^2+2/(a*e^2+c*d^2)^4/(c*x^2+a)^2*A*x^2*c^3*d^3*e^3+1/2/(a*e^2+
c*d^2)^4/(c*x^2+a)^2*B*x^2*a^2*c*e^6-3/2/(a*e^2+c*d^2)^4/(c*x^2+a)^2*B*x^2*c^3*d^4*e^2+1/(a*e^2+c*d^2)^4/(c*x^
2+a)^2*C*x^2*c^3*d^5*e-9/8/(a*e^2+c*d^2)^4/(c*x^2+a)^2*A*a^2*c*e^6*x+17/8/(a*e^2+c*d^2)^4/(c*x^2+a)^2*A*c^3*d^
4*e^2*x+1/4/(a*e^2+c*d^2)^4/(c*x^2+a)^2*B*c^3*d^5*e*x+3/8/(a*e^2+c*d^2)^4/(c*x^2+a)^2*C*x^3*a^2*c*e^6-7/8/(a*e
^2+c*d^2)^4/(c*x^2+a)^2*A*x^3*a*c^2*e^6+5/8/(a*e^2+c*d^2)^4/(c*x^2+a)^2*A*x^3*c^3*d^2*e^4+5/2/(a*e^2+c*d^2)^4/
(c*x^2+a)^2*A*a^2*c*d*e^5+3/(a*e^2+c*d^2)^4/(c*x^2+a)^2*A*a*c^2*d^3*e^3-3/4/(a*e^2+c*d^2)^4/(c*x^2+a)^2*B*a^2*
c*d^2*e^4-7/4/(a*e^2+c*d^2)^4/(c*x^2+a)^2*B*a*c^2*d^4*e^2-1/(a*e^2+c*d^2)^4/(c*x^2+a)^2*C*a^2*c*d^3*e^3+1/2/(a
*e^2+c*d^2)^4/(c*x^2+a)^2*C*a*c^2*d^5*e+3/8/(a*e^2+c*d^2)^4/(c*x^2+a)^2*c^5/a^2*x^3*A*d^6+1/8/(a*e^2+c*d^2)^4/
(c*x^2+a)^2*c^4/a*x^3*C*d^6+5/8/(a*e^2+c*d^2)^4/(c*x^2+a)^2/a*x*A*c^4*d^6-15/8/(a*e^2+c*d^2)^4*a/(a*c)^(1/2)*a
rctan(1/(a*c)^(1/2)*c*x)*A*c*e^6+3/8/(a*e^2+c*d^2)^4/a^2/(a*c)^(1/2)*arctan(1/(a*c)^(1/2)*c*x)*A*c^4*d^6+1/8/(
a*e^2+c*d^2)^4/a/(a*c)^(1/2)*arctan(1/(a*c)^(1/2)*c*x)*C*c^3*d^6

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maxima [B]  time = 1.24, size = 1196, normalized size = 2.09 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(4*C*c*d^3*e^3 - 5*B*c*d^2*e^4 + B*a*e^6 - 2*(C*a - 3*A*c)*d*e^5)*log(c*x^2 + a)/(c^4*d^8 + 4*a*c^3*d^6*e
^2 + 6*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 + a^4*e^8) + (4*C*c*d^3*e^3 - 5*B*c*d^2*e^4 + B*a*e^6 - 2*(C*a - 3*A*
c)*d*e^5)*log(e*x + d)/(c^4*d^8 + 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 + a^4*e^8) - 1/8*(2*B*
a*c^3*d^5*e + 20*B*a^2*c^2*d^3*e^3 - 30*B*a^3*c*d*e^5 - (C*a*c^3 + 3*A*c^4)*d^6 - (13*C*a^2*c^2 + 15*A*a*c^3)*
d^4*e^2 + 3*(11*C*a^3*c - 15*A*a^2*c^2)*d^2*e^4 - 3*(C*a^4 - 5*A*a^3*c)*e^6)*arctan(c*x/sqrt(a*c))/((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)*sqrt(a*c)) - 1/8*(2*B*a^2*c^2*d^5 + 12
*B*a^3*c*d^3*e^2 - 14*B*a^4*d*e^4 + 8*A*a^4*e^5 - 4*(C*a^3*c + A*a^2*c^2)*d^4*e + 20*(C*a^4 - A*a^3*c)*d^2*e^3
+ (2*B*a*c^3*d^3*e^2 - 22*B*a^2*c^2*d*e^4 - (C*a*c^3 + 3*A*c^4)*d^4*e + 4*(5*C*a^2*c^2 - 3*A*a*c^3)*d^2*e^3 -
3*(C*a^3*c - 5*A*a^2*c^2)*e^5)*x^4 + (2*B*a*c^3*d^4*e - 2*B*a^2*c^2*d^2*e^3 - 4*B*a^3*c*e^5 - (C*a*c^3 + 3*A*
c^4)*d^5 + 4*(C*a^2*c^2 - 3*A*a*c^3)*d^3*e^2 + (5*C*a^3*c - 9*A*a^2*c^2)*d*e^4)*x^3 + (10*B*a^2*c^2*d^3*e^2 -
38*B*a^3*c*d*e^4 - (7*C*a^2*c^2 + 5*A*a*c^3)*d^4*e + 4*(9*C*a^3*c - 7*A*a^2*c^2)*d^2*e^3 - 5*(C*a^4 - 5*A*a^3*
c)*e^5)*x^2 - (6*B*a^3*c*d^2*e^3 + 6*B*a^4*e^5 - (C*a^2*c^2 - 5*A*a*c^3)*d^5 - 8*(C*a^3*c - 2*A*a^2*c^2)*d^3*e
^2 - (7*C*a^4 - 11*A*a^3*c)*d*e^4)*x)/(a^4*c^3*d^7 + 3*a^5*c^2*d^5*e^2 + 3*a^6*c*d^3*e^4 + a^7*d*e^6 + (a^2*c^
5*d^6*e + 3*a^3*c^4*d^4*e^3 + 3*a^4*c^3*d^2*e^5 + a^5*c^2*e^7)*x^5 + (a^2*c^5*d^7 + 3*a^3*c^4*d^5*e^2 + 3*a^4*
c^3*d^3*e^4 + a^5*c^2*d*e^6)*x^4 + 2*(a^3*c^4*d^6*e + 3*a^4*c^3*d^4*e^3 + 3*a^5*c^2*d^2*e^5 + a^6*c*e^7)*x^3 +
2*(a^3*c^4*d^7 + 3*a^4*c^3*d^5*e^2 + 3*a^5*c^2*d^3*e^4 + a^6*c*d*e^6)*x^2 + (a^4*c^3*d^6*e + 3*a^5*c^2*d^4*e^
3 + 3*a^6*c*d^2*e^5 + a^7*e^7)*x)

________________________________________________________________________________________

mupad [B]  time = 6.66, size = 6848, normalized size = 11.99 \[ \text {result too large to display} \]

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)^2),x)

[Out]

symsum(log(root(17920*a^9*c^5*d^8*e^8*z^3 + 14336*a^10*c^4*d^6*e^10*z^3 + 14336*a^8*c^6*d^10*e^6*z^3 + 7168*a^
11*c^3*d^4*e^12*z^3 + 7168*a^7*c^7*d^12*e^4*z^3 + 2048*a^12*c^2*d^2*e^14*z^3 + 2048*a^6*c^8*d^14*e^2*z^3 + 256
*a^5*c^9*d^16*z^3 + 256*a^13*c*e^16*z^3 + 948*B*C*a^7*c*d*e^11*z - 12*A*B*a*c^7*d^11*e*z + 9768*B*C*a^5*c^3*d^
5*e^7*z - 7476*B*C*a^6*c^2*d^3*e^9*z - 328*B*C*a^4*c^4*d^7*e^5*z - 92*B*C*a^3*c^5*d^9*e^3*z - 12486*A*C*a^5*c^
3*d^4*e^8*z + 5868*A*C*a^6*c^2*d^2*e^10*z + 282*A*C*a^3*c^5*d^8*e^4*z + 168*A*C*a^4*c^4*d^6*e^6*z + 108*A*C*a^
2*c^6*d^10*e^2*z + 14820*A*B*a^5*c^3*d^3*e^9*z - 840*A*B*a^4*c^4*d^5*e^7*z - 600*A*B*a^3*c^5*d^7*e^5*z - 180*A
*B*a^2*c^6*d^9*e^3*z - 4*B*C*a^2*c^6*d^11*e*z - 3204*A*B*a^6*c^2*d*e^11*z + 4239*C^2*a^6*c^2*d^4*e^8*z - 3924*
C^2*a^5*c^3*d^6*e^6*z + 103*C^2*a^4*c^4*d^8*e^4*z + 26*C^2*a^3*c^5*d^10*e^2*z - 6000*B^2*a^5*c^3*d^4*e^8*z + 2
820*B^2*a^6*c^2*d^2*e^10*z + 280*B^2*a^4*c^4*d^6*e^6*z + 80*B^2*a^3*c^5*d^8*e^4*z + 4*B^2*a^2*c^6*d^10*e^2*z -
8262*A^2*a^5*c^3*d^2*e^10*z + 1575*A^2*a^4*c^4*d^4*e^8*z + 1260*A^2*a^3*c^5*d^6*e^6*z + 495*A^2*a^2*c^6*d^8*e
^4*z - 90*A*C*a^7*c*e^12*z + 6*A*C*a*c^7*d^12*z - 966*C^2*a^7*c*d^2*e^10*z + 90*A^2*a*c^7*d^10*e^2*z + C^2*a^2
*c^6*d^12*z + 225*A^2*a^6*c^2*e^12*z - 192*B^2*a^7*c*e^12*z + 9*A^2*c^8*d^12*z + 9*C^2*a^8*e^12*z + 78*A*B*C*a
*c^4*d^6*e^4 + 942*A*B*C*a^2*c^3*d^4*e^6 - 342*A*B*C*a^3*c^2*d^2*e^8 - 129*B*C^2*a^4*c*d^2*e^8 + 990*A^2*C*a^3
*c^2*d*e^9 - 234*A^2*C*a*c^4*d^5*e^5 - 24*A*C^2*a*c^4*d^7*e^3 + 333*A^2*B*a*c^4*d^4*e^6 - 252*A*B^2*a^3*c^2*d*
e^9 - 60*A*B^2*a*c^4*d^5*e^5 + 204*B^2*C*a^4*c*d*e^9 - 234*A*C^2*a^4*c*d*e^9 - 624*B^2*C*a^3*c^2*d^3*e^7 + 405
*B*C^2*a^3*c^2*d^4*e^6 - 36*B^2*C*a^2*c^3*d^5*e^5 + 21*B*C^2*a^2*c^3*d^6*e^4 - 1296*A^2*C*a^2*c^3*d^3*e^7 + 39
6*A*C^2*a^3*c^2*d^3*e^7 - 330*A*C^2*a^2*c^3*d^5*e^5 + 1863*A^2*B*a^2*c^3*d^2*e^8 - 672*A*B^2*a^2*c^3*d^3*e^7 +
90*A*B*C*a^4*c*e^10 + 8*C^3*a^4*c*d^3*e^7 - 1350*A^3*a^2*c^3*d*e^9 - 324*A^3*a*c^4*d^3*e^7 - 36*A^2*C*c^5*d^7
*e^3 + 45*A^2*B*c^5*d^6*e^4 - 225*A^2*B*a^3*c^2*e^10 - 86*C^3*a^3*c^2*d^5*e^5 - 4*C^3*a^2*c^3*d^7*e^3 + 316*B^
3*a^3*c^2*d^2*e^8 + 20*B^3*a^2*c^3*d^4*e^6 + 18*C^3*a^5*d*e^9 - 64*B^3*a^4*c*e^10 - 9*B*C^2*a^5*e^10 - 54*A^3*
c^5*d^5*e^5, z, k)*((120*A*a^8*c^2*e^13 - 24*C*a^9*c*e^13 + 24*A*a^2*c^8*d^12*e - 112*B*a^8*c^2*d*e^12 + 8*C*a
^3*c^7*d^12*e + 144*A*a^3*c^7*d^10*e^3 + 456*A*a^4*c^6*d^8*e^5 + 864*A*a^5*c^5*d^6*e^7 + 936*A*a^6*c^4*d^4*e^9
+ 528*A*a^7*c^3*d^2*e^11 - 16*B*a^3*c^7*d^11*e^2 - 176*B*a^4*c^6*d^9*e^4 - 544*B*a^5*c^5*d^7*e^6 - 736*B*a^6*
c^4*d^5*e^8 - 464*B*a^7*c^3*d^3*e^10 + 112*C*a^4*c^6*d^10*e^3 + 344*C*a^5*c^5*d^8*e^5 + 416*C*a^6*c^4*d^6*e^7
+ 184*C*a^7*c^3*d^4*e^9 - 16*C*a^8*c^2*d^2*e^11)/(64*(a^10*e^12 + a^4*c^6*d^12 + 6*a^9*c*d^2*e^10 + 6*a^5*c^5*
d^10*e^2 + 15*a^6*c^4*d^8*e^4 + 20*a^7*c^3*d^6*e^6 + 15*a^8*c^2*d^4*e^8)) + root(17920*a^9*c^5*d^8*e^8*z^3 + 1
4336*a^10*c^4*d^6*e^10*z^3 + 14336*a^8*c^6*d^10*e^6*z^3 + 7168*a^11*c^3*d^4*e^12*z^3 + 7168*a^7*c^7*d^12*e^4*z
^3 + 2048*a^12*c^2*d^2*e^14*z^3 + 2048*a^6*c^8*d^14*e^2*z^3 + 256*a^5*c^9*d^16*z^3 + 256*a^13*c*e^16*z^3 + 948
*B*C*a^7*c*d*e^11*z - 12*A*B*a*c^7*d^11*e*z + 9768*B*C*a^5*c^3*d^5*e^7*z - 7476*B*C*a^6*c^2*d^3*e^9*z - 328*B*
C*a^4*c^4*d^7*e^5*z - 92*B*C*a^3*c^5*d^9*e^3*z - 12486*A*C*a^5*c^3*d^4*e^8*z + 5868*A*C*a^6*c^2*d^2*e^10*z + 2
82*A*C*a^3*c^5*d^8*e^4*z + 168*A*C*a^4*c^4*d^6*e^6*z + 108*A*C*a^2*c^6*d^10*e^2*z + 14820*A*B*a^5*c^3*d^3*e^9*
z - 840*A*B*a^4*c^4*d^5*e^7*z - 600*A*B*a^3*c^5*d^7*e^5*z - 180*A*B*a^2*c^6*d^9*e^3*z - 4*B*C*a^2*c^6*d^11*e*z
- 3204*A*B*a^6*c^2*d*e^11*z + 4239*C^2*a^6*c^2*d^4*e^8*z - 3924*C^2*a^5*c^3*d^6*e^6*z + 103*C^2*a^4*c^4*d^8*e
^4*z + 26*C^2*a^3*c^5*d^10*e^2*z - 6000*B^2*a^5*c^3*d^4*e^8*z + 2820*B^2*a^6*c^2*d^2*e^10*z + 280*B^2*a^4*c^4*
d^6*e^6*z + 80*B^2*a^3*c^5*d^8*e^4*z + 4*B^2*a^2*c^6*d^10*e^2*z - 8262*A^2*a^5*c^3*d^2*e^10*z + 1575*A^2*a^4*c
^4*d^4*e^8*z + 1260*A^2*a^3*c^5*d^6*e^6*z + 495*A^2*a^2*c^6*d^8*e^4*z - 90*A*C*a^7*c*e^12*z + 6*A*C*a*c^7*d^12
*z - 966*C^2*a^7*c*d^2*e^10*z + 90*A^2*a*c^7*d^10*e^2*z + C^2*a^2*c^6*d^12*z + 225*A^2*a^6*c^2*e^12*z - 192*B^
2*a^7*c*e^12*z + 9*A^2*c^8*d^12*z + 9*C^2*a^8*e^12*z + 78*A*B*C*a*c^4*d^6*e^4 + 942*A*B*C*a^2*c^3*d^4*e^6 - 34
2*A*B*C*a^3*c^2*d^2*e^8 - 129*B*C^2*a^4*c*d^2*e^8 + 990*A^2*C*a^3*c^2*d*e^9 - 234*A^2*C*a*c^4*d^5*e^5 - 24*A*C
^2*a*c^4*d^7*e^3 + 333*A^2*B*a*c^4*d^4*e^6 - 252*A*B^2*a^3*c^2*d*e^9 - 60*A*B^2*a*c^4*d^5*e^5 + 204*B^2*C*a^4*
c*d*e^9 - 234*A*C^2*a^4*c*d*e^9 - 624*B^2*C*a^3*c^2*d^3*e^7 + 405*B*C^2*a^3*c^2*d^4*e^6 - 36*B^2*C*a^2*c^3*d^5
*e^5 + 21*B*C^2*a^2*c^3*d^6*e^4 - 1296*A^2*C*a^2*c^3*d^3*e^7 + 396*A*C^2*a^3*c^2*d^3*e^7 - 330*A*C^2*a^2*c^3*d
^5*e^5 + 1863*A^2*B*a^2*c^3*d^2*e^8 - 672*A*B^2*a^2*c^3*d^3*e^7 + 90*A*B*C*a^4*c*e^10 + 8*C^3*a^4*c*d^3*e^7 -
1350*A^3*a^2*c^3*d*e^9 - 324*A^3*a*c^4*d^3*e^7 - 36*A^2*C*c^5*d^7*e^3 + 45*A^2*B*c^5*d^6*e^4 - 225*A^2*B*a^3*c
^2*e^10 - 86*C^3*a^3*c^2*d^5*e^5 - 4*C^3*a^2*c^3*d^7*e^3 + 316*B^3*a^3*c^2*d^2*e^8 + 20*B^3*a^2*c^3*d^4*e^6 +
18*C^3*a^5*d*e^9 - 64*B^3*a^4*c*e^10 - 9*B*C^2*a^5*e^10 - 54*A^3*c^5*d^5*e^5, z, k)*((512*a^11*c^2*d*e^14 + 51
2*a^5*c^8*d^13*e^2 + 3072*a^6*c^7*d^11*e^4 + 7680*a^7*c^6*d^9*e^6 + 10240*a^8*c^5*d^7*e^8 + 7680*a^9*c^4*d^5*e
^10 + 3072*a^10*c^3*d^3*e^12)/(64*(a^10*e^12 + a^4*c^6*d^12 + 6*a^9*c*d^2*e^10 + 6*a^5*c^5*d^10*e^2 + 15*a^6*c
^4*d^8*e^4 + 20*a^7*c^3*d^6*e^6 + 15*a^8*c^2*d^4*e^8)) + (x*(384*a^11*c^2*e^15 - 128*a^4*c^9*d^14*e - 384*a^5*
c^8*d^12*e^3 + 384*a^6*c^7*d^10*e^5 + 3200*a^7*c^6*d^8*e^7 + 5760*a^8*c^5*d^6*e^9 + 4992*a^9*c^4*d^4*e^11 + 21
76*a^10*c^3*d^2*e^13))/(64*(a^10*e^12 + a^4*c^6*d^12 + 6*a^9*c*d^2*e^10 + 6*a^5*c^5*d^10*e^2 + 15*a^6*c^4*d^8*
e^4 + 20*a^7*c^3*d^6*e^6 + 15*a^8*c^2*d^4*e^8))) + (x*(192*B*a^8*c^2*e^13 + 912*A*a^7*c^3*d*e^12 - 336*C*a^8*c
^2*d*e^12 + 48*A*a^2*c^8*d^11*e^2 + 336*A*a^3*c^7*d^9*e^4 + 1632*A*a^4*c^6*d^7*e^6 + 3360*A*a^5*c^5*d^5*e^8 +
2928*A*a^6*c^4*d^3*e^10 - 32*B*a^3*c^7*d^10*e^3 - 704*B*a^4*c^6*d^8*e^5 - 1728*B*a^5*c^5*d^6*e^7 - 1280*B*a^6*
c^4*d^4*e^9 - 32*B*a^7*c^3*d^2*e^11 + 16*C*a^3*c^7*d^11*e^2 + 496*C*a^4*c^6*d^9*e^4 + 1056*C*a^5*c^5*d^7*e^6 +
352*C*a^6*c^4*d^5*e^8 - 560*C*a^7*c^3*d^3*e^10))/(64*(a^10*e^12 + a^4*c^6*d^12 + 6*a^9*c*d^2*e^10 + 6*a^5*c^5
*d^10*e^2 + 15*a^6*c^4*d^8*e^4 + 20*a^7*c^3*d^6*e^6 + 15*a^8*c^2*d^4*e^8))) + (9*A^2*c^7*d^9*e^2 + 198*A^2*a^2
*c^5*d^5*e^6 + 216*A^2*a^3*c^4*d^3*e^8 + 4*B^2*a^2*c^5*d^7*e^4 - 8*B^2*a^3*c^4*d^5*e^6 - 412*B^2*a^4*c^3*d^3*e
^8 + C^2*a^2*c^5*d^9*e^2 - 8*C^2*a^3*c^4*d^7*e^4 - 250*C^2*a^4*c^3*d^5*e^6 + 296*C^2*a^5*c^2*d^3*e^8 - 120*A*B
*a^5*c^2*e^11 - 39*C^2*a^6*c*d*e^10 + 72*A^2*a*c^6*d^7*e^4 - 495*A^2*a^4*c^3*d*e^10 + 176*B^2*a^5*c^2*d*e^10 +
24*B*C*a^6*c*e^11 - 12*A*B*a*c^6*d^8*e^3 + 6*A*C*a*c^6*d^9*e^2 + 294*A*C*a^5*c^2*d*e^10 - 36*A*B*a^2*c^5*d^6*
e^5 + 36*A*B*a^3*c^4*d^4*e^7 + 1092*A*B*a^4*c^3*d^2*e^9 - 108*A*C*a^3*c^4*d^5*e^6 - 960*A*C*a^4*c^3*d^3*e^8 -
4*B*C*a^2*c^5*d^8*e^3 + 20*B*C*a^3*c^4*d^6*e^5 + 652*B*C*a^4*c^3*d^4*e^7 - 500*B*C*a^5*c^2*d^2*e^9)/(64*(a^10*
e^12 + a^4*c^6*d^12 + 6*a^9*c*d^2*e^10 + 6*a^5*c^5*d^10*e^2 + 15*a^6*c^4*d^8*e^4 + 20*a^7*c^3*d^6*e^6 + 15*a^8
*c^2*d^4*e^8)) + (x*(225*A^2*a^4*c^3*e^11 + 9*A^2*c^7*d^8*e^3 + 9*C^2*a^6*c*e^11 + 54*A^2*a^2*c^5*d^4*e^7 - 36
0*A^2*a^3*c^4*d^2*e^9 + 4*B^2*a^2*c^5*d^6*e^5 - 88*B^2*a^3*c^4*d^4*e^7 + 484*B^2*a^4*c^3*d^2*e^9 + C^2*a^2*c^5
*d^8*e^3 - 40*C^2*a^3*c^4*d^6*e^5 + 406*C^2*a^4*c^3*d^4*e^7 - 120*C^2*a^5*c^2*d^2*e^9 - 90*A*C*a^5*c^2*e^11 +
72*A^2*a*c^6*d^6*e^5 - 12*A*B*a*c^6*d^7*e^4 - 660*A*B*a^4*c^3*d*e^10 + 6*A*C*a*c^6*d^8*e^3 + 132*B*C*a^5*c^2*d
*e^10 + 84*A*B*a^2*c^5*d^5*e^6 + 588*A*B*a^3*c^4*d^3*e^8 - 96*A*C*a^2*c^5*d^6*e^5 - 492*A*C*a^3*c^4*d^4*e^7 +
672*A*C*a^4*c^3*d^2*e^9 - 4*B*C*a^2*c^5*d^7*e^4 + 124*B*C*a^3*c^4*d^5*e^6 - 892*B*C*a^4*c^3*d^3*e^8))/(64*(a^1
0*e^12 + a^4*c^6*d^12 + 6*a^9*c*d^2*e^10 + 6*a^5*c^5*d^10*e^2 + 15*a^6*c^4*d^8*e^4 + 20*a^7*c^3*d^6*e^6 + 15*a
^8*c^2*d^4*e^8)))*root(17920*a^9*c^5*d^8*e^8*z^3 + 14336*a^10*c^4*d^6*e^10*z^3 + 14336*a^8*c^6*d^10*e^6*z^3 +
7168*a^11*c^3*d^4*e^12*z^3 + 7168*a^7*c^7*d^12*e^4*z^3 + 2048*a^12*c^2*d^2*e^14*z^3 + 2048*a^6*c^8*d^14*e^2*z^
3 + 256*a^5*c^9*d^16*z^3 + 256*a^13*c*e^16*z^3 + 948*B*C*a^7*c*d*e^11*z - 12*A*B*a*c^7*d^11*e*z + 9768*B*C*a^5
*c^3*d^5*e^7*z - 7476*B*C*a^6*c^2*d^3*e^9*z - 328*B*C*a^4*c^4*d^7*e^5*z - 92*B*C*a^3*c^5*d^9*e^3*z - 12486*A*C
*a^5*c^3*d^4*e^8*z + 5868*A*C*a^6*c^2*d^2*e^10*z + 282*A*C*a^3*c^5*d^8*e^4*z + 168*A*C*a^4*c^4*d^6*e^6*z + 108
*A*C*a^2*c^6*d^10*e^2*z + 14820*A*B*a^5*c^3*d^3*e^9*z - 840*A*B*a^4*c^4*d^5*e^7*z - 600*A*B*a^3*c^5*d^7*e^5*z
- 180*A*B*a^2*c^6*d^9*e^3*z - 4*B*C*a^2*c^6*d^11*e*z - 3204*A*B*a^6*c^2*d*e^11*z + 4239*C^2*a^6*c^2*d^4*e^8*z
- 3924*C^2*a^5*c^3*d^6*e^6*z + 103*C^2*a^4*c^4*d^8*e^4*z + 26*C^2*a^3*c^5*d^10*e^2*z - 6000*B^2*a^5*c^3*d^4*e^
8*z + 2820*B^2*a^6*c^2*d^2*e^10*z + 280*B^2*a^4*c^4*d^6*e^6*z + 80*B^2*a^3*c^5*d^8*e^4*z + 4*B^2*a^2*c^6*d^10*
e^2*z - 8262*A^2*a^5*c^3*d^2*e^10*z + 1575*A^2*a^4*c^4*d^4*e^8*z + 1260*A^2*a^3*c^5*d^6*e^6*z + 495*A^2*a^2*c^
6*d^8*e^4*z - 90*A*C*a^7*c*e^12*z + 6*A*C*a*c^7*d^12*z - 966*C^2*a^7*c*d^2*e^10*z + 90*A^2*a*c^7*d^10*e^2*z +
C^2*a^2*c^6*d^12*z + 225*A^2*a^6*c^2*e^12*z - 192*B^2*a^7*c*e^12*z + 9*A^2*c^8*d^12*z + 9*C^2*a^8*e^12*z + 78*
A*B*C*a*c^4*d^6*e^4 + 942*A*B*C*a^2*c^3*d^4*e^6 - 342*A*B*C*a^3*c^2*d^2*e^8 - 129*B*C^2*a^4*c*d^2*e^8 + 990*A^
2*C*a^3*c^2*d*e^9 - 234*A^2*C*a*c^4*d^5*e^5 - 24*A*C^2*a*c^4*d^7*e^3 + 333*A^2*B*a*c^4*d^4*e^6 - 252*A*B^2*a^3
*c^2*d*e^9 - 60*A*B^2*a*c^4*d^5*e^5 + 204*B^2*C*a^4*c*d*e^9 - 234*A*C^2*a^4*c*d*e^9 - 624*B^2*C*a^3*c^2*d^3*e^
7 + 405*B*C^2*a^3*c^2*d^4*e^6 - 36*B^2*C*a^2*c^3*d^5*e^5 + 21*B*C^2*a^2*c^3*d^6*e^4 - 1296*A^2*C*a^2*c^3*d^3*e
^7 + 396*A*C^2*a^3*c^2*d^3*e^7 - 330*A*C^2*a^2*c^3*d^5*e^5 + 1863*A^2*B*a^2*c^3*d^2*e^8 - 672*A*B^2*a^2*c^3*d^
3*e^7 + 90*A*B*C*a^4*c*e^10 + 8*C^3*a^4*c*d^3*e^7 - 1350*A^3*a^2*c^3*d*e^9 - 324*A^3*a*c^4*d^3*e^7 - 36*A^2*C*
c^5*d^7*e^3 + 45*A^2*B*c^5*d^6*e^4 - 225*A^2*B*a^3*c^2*e^10 - 86*C^3*a^3*c^2*d^5*e^5 - 4*C^3*a^2*c^3*d^7*e^3 +
316*B^3*a^3*c^2*d^2*e^8 + 20*B^3*a^2*c^3*d^4*e^6 + 18*C^3*a^5*d*e^9 - 64*B^3*a^4*c*e^10 - 9*B*C^2*a^5*e^10 -
54*A^3*c^5*d^5*e^5, z, k), k, 1, 3) + ((x^4*(3*C*a^3*c*e^5 + 3*A*c^4*d^4*e - 15*A*a^2*c^2*e^5 + 12*A*a*c^3*d^2
*e^3 - 2*B*a*c^3*d^3*e^2 + 22*B*a^2*c^2*d*e^4 - 20*C*a^2*c^2*d^2*e^3 + C*a*c^3*d^4*e))/(8*a^2*(a^3*e^6 + c^3*d
^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4)) - (4*A*a^2*e^5 + B*c^2*d^5 - 7*B*a^2*d*e^4 - 2*A*c^2*d^4*e + 10*C*a^2
*d^2*e^3 - 2*C*a*c*d^4*e - 10*A*a*c*d^2*e^3 + 6*B*a*c*d^3*e^2)/(4*(a*e^2 + c*d^2)*(a^2*e^4 + c^2*d^4 + 2*a*c*d
^2*e^2)) + (x^3*(3*A*c^3*d^3 + 4*B*a^2*c*e^3 + C*a*c^2*d^3 + 9*A*a*c^2*d*e^2 - 2*B*a*c^2*d^2*e - 5*C*a^2*c*d*e
^2))/(8*a^2*(a^2*e^4 + c^2*d^4 + 2*a*c*d^2*e^2)) + (x*(5*A*c^2*d^3 + 6*B*a^2*e^3 - C*a*c*d^3 - 7*C*a^2*d*e^2 +
11*A*a*c*d*e^2))/(8*a*(a^2*e^4 + c^2*d^4 + 2*a*c*d^2*e^2)) + (x^2*(5*C*a^3*e^5 - 25*A*a^2*c*e^5 + 5*A*c^3*d^4
*e + 28*A*a*c^2*d^2*e^3 - 10*B*a*c^2*d^3*e^2 - 36*C*a^2*c*d^2*e^3 + 38*B*a^2*c*d*e^4 + 7*C*a*c^2*d^4*e))/(8*a*
(a*e^2 + c*d^2)*(a^2*e^4 + c^2*d^4 + 2*a*c*d^2*e^2)))/(a^2*d + c^2*d*x^4 + c^2*e*x^5 + a^2*e*x + 2*a*c*d*x^2 +
2*a*c*e*x^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)**2/(c*x**2+a)**3,x)

[Out]

Timed out

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