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

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

[Out]

-((e^3*(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*(a*e^2*(2*C*d - B*e) - c*d*(2*C*d^2 - e*(3*B*d - 4*A*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*(a*e^2*(2*C*d - B*e) - c*d*(4*C*d^2 - e*(5*B*d - 6*A*e)))*Log[d + e*x])/(c
*d^2 + a*e^2)^4 + (e^3*(a*e^2*(2*C*d - B*e) - c*d*(4*C*d^2 - e*(5*B*d - 6*A*e)))*Log[a + c*x^2])/(2*(c*d^2 + a
*e^2)^4)

________________________________________________________________________________________

Rubi [A]  time = 1.92477, 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 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)^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*}

Mathematica [A]  time = 0.817795, size = 498, normalized size = 0.87 \[ \frac{\frac{2 \left (a e^2+c d^2\right )^2 \left (a^2 e (B e-2 C d+C e x)-a c \left (A e (e x-2 d)+B d (d-2 e x)+C d^2 x\right )+A c^2 d^2 x\right )}{a \left (a+c x^2\right )^2}+\frac{\left (a e^2+c d^2\right ) \left (a^2 c e \left (e (A e (16 d-7 e x)-2 B d (6 d-7 e x))+4 C d^2 (2 d-3 e x)\right )+a^3 e^3 (4 B e-8 C d+3 C e x)+a c^2 d^2 x \left (2 e (6 A e-B d)+C d^2\right )+3 A c^3 d^4 x\right )}{a^2 \left (a+c x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (3 A c \left (15 a^2 c d^2 e^4-5 a^3 e^6+5 a c^2 d^4 e^2+c^3 d^6\right )+a \left (3 a^2 c d e^4 (10 B e-11 C d)+3 a^3 C e^6+a c^2 d^3 e^2 (13 C d-20 B e)+c^3 d^5 (C d-2 B e)\right )\right )}{a^{5/2} \sqrt{c}}-4 e^3 \log \left (a+c x^2\right ) \left (a e^2 (B e-2 C d)+c d e (6 A e-5 B d)+4 c C d^3\right )-\frac{8 e^3 \left (a e^2+c d^2\right ) \left (e (A e-B d)+C d^2\right )}{d+e x}+8 e^3 \log (d+e x) \left (a e^2 (B e-2 C d)+c d e (6 A e-5 B d)+4 c C d^3\right )}{8 \left (a e^2+c d^2\right )^4} \]

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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Maple [B]  time = 0.076, size = 2159, normalized size = 3.8 \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)^2/(c*x^2+a)^3,x)

[Out]

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

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

[Out]

Exception raised: ValueError

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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)^2/(c*x^2+a)^3,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)**2/(c*x**2+a)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.21416, size = 1494, normalized size = 2.62 \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)^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)