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

Optimal. Leaf size=524 \[ -\frac{e \log \left (a+c x^2\right ) \left (a^2 C e^4-2 a c e^2 \left (4 C d^2-e (3 B d-A e)\right )+c^2 d^2 \left (3 C d^2-2 e (3 B d-5 A e)\right )\right )}{2 \left (a e^2+c d^2\right )^4}+\frac{e \log (d+e x) \left (a^2 C e^4-2 a c e^2 \left (4 C d^2-e (3 B d-A e)\right )+c^2 d^2 \left (3 C d^2-2 e (3 B d-5 A e)\right )\right )}{\left (a e^2+c d^2\right )^4}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c d \left (-15 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right )-a \left (-3 a^2 e^4 (3 C d-B e)+2 a c d^2 e^2 (7 C d-9 B e)-c^2 d^4 (C d-3 B e)\right )\right )}{2 a^{3/2} \left (a e^2+c d^2\right )^4}-\frac{a \left (B c d \left (c d^2-3 a e^2\right )-e (A c-a C) \left (3 c d^2-a e^2\right )\right )-c x \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right )}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )^3}-\frac{e \left (A e^2-B d e+C d^2\right )}{2 (d+e x)^2 \left (a e^2+c d^2\right )^2}+\frac{e \left (a e^2 (2 C d-B e)-c d \left (2 C d^2-e (3 B d-4 A e)\right )\right )}{(d+e x) \left (a e^2+c d^2\right )^3} \]

[Out]

-(e*(C*d^2 - B*d*e + A*e^2))/(2*(c*d^2 + a*e^2)^2*(d + e*x)^2) + (e*(a*e^2*(2*C*d - B*e) - c*d*(2*C*d^2 - e*(3
*B*d - 4*A*e))))/((c*d^2 + a*e^2)^3*(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)/(2*a*(c*d^2 + a*e^2)^3*(a +
c*x^2)) + (Sqrt[c]*(A*c*d*(c^2*d^4 + 10*a*c*d^2*e^2 - 15*a^2*e^4) - a*(2*a*c*d^2*e^2*(7*C*d - 9*B*e) - c^2*d^
4*(C*d - 3*B*e) - 3*a^2*e^4*(3*C*d - B*e)))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(2*a^(3/2)*(c*d^2 + a*e^2)^4) + (e*(a
^2*C*e^4 + c^2*d^2*(3*C*d^2 - 2*e*(3*B*d - 5*A*e)) - 2*a*c*e^2*(4*C*d^2 - e*(3*B*d - A*e)))*Log[d + e*x])/(c*d
^2 + a*e^2)^4 - (e*(a^2*C*e^4 + c^2*d^2*(3*C*d^2 - 2*e*(3*B*d - 5*A*e)) - 2*a*c*e^2*(4*C*d^2 - e*(3*B*d - A*e)
))*Log[a + c*x^2])/(2*(c*d^2 + a*e^2)^4)

________________________________________________________________________________________

Rubi [A]  time = 1.55223, antiderivative size = 524, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ -\frac{e \log \left (a+c x^2\right ) \left (a^2 C e^4-2 a c e^2 \left (4 C d^2-e (3 B d-A e)\right )+c^2 \left (3 C d^4-2 d^2 e (3 B d-5 A e)\right )\right )}{2 \left (a e^2+c d^2\right )^4}+\frac{e \log (d+e x) \left (a^2 C e^4-2 a c e^2 \left (4 C d^2-e (3 B d-A e)\right )+c^2 \left (3 C d^4-2 d^2 e (3 B d-5 A e)\right )\right )}{\left (a e^2+c d^2\right )^4}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c d \left (-15 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right )-a \left (-3 a^2 e^4 (3 C d-B e)+2 a c d^2 e^2 (7 C d-9 B e)-c^2 d^4 (C d-3 B e)\right )\right )}{2 a^{3/2} \left (a e^2+c d^2\right )^4}-\frac{a \left (B c d \left (c d^2-3 a e^2\right )-e (A c-a C) \left (3 c d^2-a e^2\right )\right )-c x \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right )}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )^3}-\frac{e \left (A e^2-B d e+C d^2\right )}{2 (d+e x)^2 \left (a e^2+c d^2\right )^2}-\frac{e \left (-a e^2 (2 C d-B e)-c d e (3 B d-4 A e)+2 c C d^3\right )}{(d+e x) \left (a e^2+c d^2\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(e*(C*d^2 - B*d*e + A*e^2))/(2*(c*d^2 + a*e^2)^2*(d + e*x)^2) - (e*(2*c*C*d^3 - c*d*e*(3*B*d - 4*A*e) - a*e^2
*(2*C*d - B*e)))/((c*d^2 + a*e^2)^3*(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)/(2*a*(c*d^2 + a*e^2)^3*(a +
c*x^2)) + (Sqrt[c]*(A*c*d*(c^2*d^4 + 10*a*c*d^2*e^2 - 15*a^2*e^4) - a*(2*a*c*d^2*e^2*(7*C*d - 9*B*e) - c^2*d^
4*(C*d - 3*B*e) - 3*a^2*e^4*(3*C*d - B*e)))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(2*a^(3/2)*(c*d^2 + a*e^2)^4) + (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)))*Log[d + e*x])/(c*d
^2 + a*e^2)^4 - (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)
))*Log[a + c*x^2])/(2*(c*d^2 + a*e^2)^4)

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]

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 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 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]

Rubi steps

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

Mathematica [A]  time = 0.706129, size = 466, normalized size = 0.89 \[ \frac{\frac{\left (a e^2+c d^2\right ) \left (-a^2 c e (e (A e-3 B d+B e x)+3 C d (d-e x))+a^3 C e^3-a c^2 d \left (3 A e (e x-d)+B d (d-3 e x)+C d^2 x\right )+A c^3 d^3 x\right )}{a \left (a+c x^2\right )}-\log \left (a+c x^2\right ) \left (a^2 C e^5-2 a c e^3 \left (e (A e-3 B d)+4 C d^2\right )+c^2 d^2 e \left (2 e (5 A e-3 B d)+3 C d^2\right )\right )+2 \log (d+e x) \left (a^2 C e^5-2 a c e^3 \left (e (A e-3 B d)+4 C d^2\right )+c^2 d^2 e \left (2 e (5 A e-3 B d)+3 C d^2\right )\right )+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c d \left (-15 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right )+a \left (-3 a^2 e^4 (B e-3 C d)-2 a c d^2 e^2 (7 C d-9 B e)+c^2 d^4 (C d-3 B e)\right )\right )}{a^{3/2}}-\frac{e \left (a e^2+c d^2\right )^2 \left (e (A e-B d)+C d^2\right )}{(d+e x)^2}-\frac{2 e \left (a e^2+c d^2\right ) \left (a e^2 (B e-2 C d)+c d e (4 A e-3 B d)+2 c C d^3\right )}{d+e x}}{2 \left (a e^2+c d^2\right )^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-((e*(c*d^2 + a*e^2)^2*(C*d^2 + e*(-(B*d) + A*e)))/(d + e*x)^2) - (2*e*(c*d^2 + a*e^2)*(2*c*C*d^3 + c*d*e*(-3
*B*d + 4*A*e) + a*e^2*(-2*C*d + B*e)))/(d + e*x) + ((c*d^2 + a*e^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
)) + (Sqrt[c]*(A*c*d*(c^2*d^4 + 10*a*c*d^2*e^2 - 15*a^2*e^4) + a*(-2*a*c*d^2*e^2*(7*C*d - 9*B*e) + c^2*d^4*(C*
d - 3*B*e) - 3*a^2*e^4*(-3*C*d + B*e)))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/a^(3/2) + 2*(a^2*C*e^5 - 2*a*c*e^3*(4*C*d
^2 + e*(-3*B*d + A*e)) + c^2*d^2*e*(3*C*d^2 + 2*e*(-3*B*d + 5*A*e)))*Log[d + e*x] - (a^2*C*e^5 - 2*a*c*e^3*(4*
C*d^2 + e*(-3*B*d + A*e)) + c^2*d^2*e*(3*C*d^2 + 2*e*(-3*B*d + 5*A*e)))*Log[a + c*x^2])/(2*(c*d^2 + a*e^2)^4)

________________________________________________________________________________________

Maple [B]  time = 0.074, size = 1588, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}

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)^2,x)

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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)^2,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

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

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)**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.18891, size = 1292, normalized size = 2.47 \begin{align*} \text{result too large to display} \end{align*}

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)^2,x, algorithm="giac")

[Out]

-1/2*(3*C*c^2*d^4*e - 6*B*c^2*d^3*e^2 - 8*C*a*c*d^2*e^3 + 10*A*c^2*d^2*e^3 + 6*B*a*c*d*e^4 + C*a^2*e^5 - 2*A*a
*c*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) + (3*C*c^2*
d^4*e^2 - 6*B*c^2*d^3*e^3 - 8*C*a*c*d^2*e^4 + 10*A*c^2*d^2*e^4 + 6*B*a*c*d*e^5 + C*a^2*e^6 - 2*A*a*c*e^6)*log(
abs(x*e + d))/(c^4*d^8*e + 4*a*c^3*d^6*e^3 + 6*a^2*c^2*d^4*e^5 + 4*a^3*c*d^2*e^7 + a^4*e^9) + 1/2*(C*a*c^3*d^5
+ A*c^4*d^5 - 3*B*a*c^3*d^4*e - 14*C*a^2*c^2*d^3*e^2 + 10*A*a*c^3*d^3*e^2 + 18*B*a^2*c^2*d^2*e^3 + 9*C*a^3*c*
d*e^4 - 15*A*a^2*c^2*d*e^4 - 3*B*a^3*c*e^5)*arctan(c*x/sqrt(a*c))/((a*c^4*d^8 + 4*a^2*c^3*d^6*e^2 + 6*a^3*c^2*
d^4*e^4 + 4*a^4*c*d^2*e^6 + a^5*e^8)*sqrt(a*c)) - 1/2*(B*a*c^3*d^7 + 8*C*a^2*c^2*d^6*e - 3*A*a*c^3*d^6*e - 9*B
*a^2*c^2*d^5*e^2 + 4*C*a^3*c*d^4*e^3 + 7*A*a^2*c^2*d^4*e^3 - 9*B*a^3*c*d^3*e^4 - 4*C*a^4*d^2*e^5 + 11*A*a^3*c*
d^2*e^5 + B*a^4*d*e^6 + A*a^4*e^7 + (5*C*a*c^3*d^5*e^2 - A*c^4*d^5*e^2 - 9*B*a*c^3*d^4*e^3 - 2*C*a^2*c^2*d^3*e
^4 + 10*A*a*c^3*d^3*e^4 - 6*B*a^2*c^2*d^2*e^5 - 7*C*a^3*c*d*e^6 + 11*A*a^2*c^2*d*e^6 + 3*B*a^3*c*e^7)*x^3 + (7
*C*a*c^3*d^6*e - 2*A*c^4*d^6*e - 12*B*a*c^3*d^5*e^2 + C*a^2*c^2*d^4*e^3 + 10*A*a*c^3*d^4*e^3 - 12*B*a^2*c^2*d^
3*e^4 - 7*C*a^3*c*d^2*e^5 + 14*A*a^2*c^2*d^2*e^5 - C*a^4*e^7 + 2*A*a^3*c*e^7)*x^2 + (C*a*c^3*d^7 - A*c^4*d^7 -
B*a*c^3*d^6*e + 8*C*a^2*c^2*d^5*e^2 - 4*A*a*c^3*d^5*e^2 - 12*B*a^2*c^2*d^4*e^3 + C*a^3*c*d^3*e^4 + 7*A*a^2*c^
2*d^3*e^4 - 9*B*a^3*c*d^2*e^5 - 6*C*a^4*d*e^6 + 10*A*a^3*c*d*e^6 + 2*B*a^4*e^7)*x)/((c*d^2 + a*e^2)^4*(c*x^2 +
a)*(x*e + d)^2*a)