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]

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

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Rubi [A]  time = 1.55, antiderivative size = 524, normalized size of antiderivative = 1.00, 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 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 )^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*}

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Mathematica [A]  time = 0.63, size = 466, normalized size = 0.89 \[ \frac {-\log \left (a+c x^2\right ) \left (a^2 C e^5-2 a c e^3 \left (e (A e-3 B d)+4 C d^2\right )+c^2 d^2 e \left (2 e (5 A e-3 B d)+3 C d^2\right )\right )+2 \log (d+e x) \left (a^2 C e^5-2 a c e^3 \left (e (A e-3 B d)+4 C d^2\right )+c^2 d^2 e \left (2 e (5 A e-3 B d)+3 C d^2\right )\right )+\frac {\sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (A c d \left (-15 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right )+a \left (-3 a^2 e^4 (B e-3 C d)-2 a c d^2 e^2 (7 C d-9 B e)+c^2 d^4 (C d-3 B e)\right )\right )}{a^{3/2}}+\frac {\left (a e^2+c d^2\right ) \left (a^3 C e^3-a^2 c e (e (A e-3 B d+B e x)+3 C d (d-e x))-a c^2 d \left (3 A e (e x-d)+B d (d-3 e x)+C d^2 x\right )+A c^3 d^3 x\right )}{a \left (a+c x^2\right )}-\frac {e \left (a e^2+c d^2\right )^2 \left (e (A e-B d)+C d^2\right )}{(d+e x)^2}-\frac {2 e \left (a e^2+c d^2\right ) \left (a e^2 (B e-2 C d)+c d e (4 A e-3 B d)+2 c C d^3\right )}{d+e x}}{2 \left (a e^2+c d^2\right )^4} \]

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)

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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)^2,x, algorithm="fricas")

[Out]

Timed out

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giac [A]  time = 0.19, size = 957, normalized size = 1.83 \[ -\frac {{\left (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}\right )} \log \left (c x^{2} + a\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 {{\left (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}\right )} \log \left ({\left | x e + d \right |}\right )}{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}} + \frac {{\left (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}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, {\left (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}\right )} \sqrt {a c}} - \frac {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} + {\left (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}\right )} x^{3} + {\left (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}\right )} x^{2} + {\left (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}\right )} x}{2 \, {\left (c d^{2} + a e^{2}\right )}^{4} {\left (c x^{2} + a\right )} {\left (x e + d\right )}^{2} a} \]

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)

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maple [B]  time = 0.03, size = 1588, normalized size = 3.03 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.22, size = 1030, normalized size = 1.97 \[ -\frac {{\left (3 \, C c^{2} d^{4} e - 6 \, B c^{2} d^{3} e^{2} + 6 \, B a c d e^{4} - 2 \, {\left (4 \, C a c - 5 \, A c^{2}\right )} d^{2} e^{3} + {\left (C a^{2} - 2 \, A a c\right )} e^{5}\right )} \log \left (c x^{2} + a\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 {{\left (3 \, C c^{2} d^{4} e - 6 \, B c^{2} d^{3} e^{2} + 6 \, B a c d e^{4} - 2 \, {\left (4 \, C a c - 5 \, A c^{2}\right )} d^{2} e^{3} + {\left (C a^{2} - 2 \, A a c\right )} e^{5}\right )} \log \left (e x + d\right )}{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}} - \frac {{\left (3 \, B a c^{3} d^{4} e - 18 \, B a^{2} c^{2} d^{2} e^{3} + 3 \, B a^{3} c e^{5} - {\left (C a c^{3} + A c^{4}\right )} d^{5} + 2 \, {\left (7 \, C a^{2} c^{2} - 5 \, A a c^{3}\right )} d^{3} e^{2} - 3 \, {\left (3 \, C a^{3} c - 5 \, A a^{2} c^{2}\right )} d e^{4}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, {\left (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}\right )} \sqrt {a c}} - \frac {B a c^{2} d^{5} - 10 \, B a^{2} c d^{3} e^{2} + B a^{3} d e^{4} + A a^{3} e^{5} + {\left (8 \, C a^{2} c - 3 \, A a c^{2}\right )} d^{4} e - 2 \, {\left (2 \, C a^{3} - 5 \, A a^{2} c\right )} d^{2} e^{3} - {\left (9 \, B a c^{2} d^{2} e^{3} - 3 \, B a^{2} c e^{5} - {\left (5 \, C a c^{2} - A c^{3}\right )} d^{3} e^{2} + {\left (7 \, C a^{2} c - 11 \, A a c^{2}\right )} d e^{4}\right )} x^{3} - {\left (12 \, B a c^{2} d^{3} e^{2} - {\left (7 \, C a c^{2} - 2 \, A c^{3}\right )} d^{4} e + 6 \, {\left (C a^{2} c - 2 \, A a c^{2}\right )} d^{2} e^{3} + {\left (C a^{3} - 2 \, A a^{2} c\right )} e^{5}\right )} x^{2} - {\left (B a c^{2} d^{4} e + 11 \, B a^{2} c d^{2} e^{3} - 2 \, B a^{3} e^{5} - {\left (C a c^{2} - A c^{3}\right )} d^{5} - {\left (7 \, C a^{2} c - 3 \, A a c^{2}\right )} d^{3} e^{2} + 2 \, {\left (3 \, C a^{3} - 5 \, A a^{2} c\right )} d e^{4}\right )} x}{2 \, {\left (a^{2} c^{3} d^{8} + 3 \, a^{3} c^{2} d^{6} e^{2} + 3 \, a^{4} c d^{4} e^{4} + a^{5} d^{2} e^{6} + {\left (a c^{4} d^{6} e^{2} + 3 \, a^{2} c^{3} d^{4} e^{4} + 3 \, a^{3} c^{2} d^{2} e^{6} + a^{4} c e^{8}\right )} x^{4} + 2 \, {\left (a c^{4} d^{7} e + 3 \, a^{2} c^{3} d^{5} e^{3} + 3 \, a^{3} c^{2} d^{3} e^{5} + a^{4} c d e^{7}\right )} x^{3} + {\left (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}\right )} x^{2} + 2 \, {\left (a^{2} c^{3} d^{7} e + 3 \, a^{3} c^{2} d^{5} e^{3} + 3 \, a^{4} c d^{3} e^{5} + a^{5} d e^{7}\right )} x\right )}} \]

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]

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

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mupad [B]  time = 14.48, size = 2828, normalized size = 5.40 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(C*c^2*d^7*(-a^3*c)^(3/2) - 3*B*a^6*e^7*(-a^3*c)^(1/2) - 6*C*a^8*e^7 + 12*A*a^7*c*e^7 - 3*B*a^7*c*e^7*x +
2*A*a^4*c^4*d^6*e + 20*C*a^5*c^3*d^6*e + 72*C*a^7*c*d^2*e^5 - A*a^3*c^5*d^7*x - C*a^4*c^4*d^7*x + 39*A*a^2*d*e
^6*(-a^3*c)^(3/2) + 21*C*a^6*d*e^6*(-a^3*c)^(1/2) - 3*B*c^2*d^6*e*(-a^3*c)^(3/2) + 12*A*a^2*e^7*x*(-a^3*c)^(3/
2) + 6*C*a^6*e^7*x*(-a^3*c)^(1/2) + 80*A*a^5*c^3*d^4*e^3 - 102*A*a^6*c^2*d^2*e^5 - 42*B*a^5*c^3*d^5*e^2 + 108*
B*a^6*c^2*d^3*e^4 - 94*C*a^6*c^2*d^4*e^3 - A*a^2*c^4*d^7*(-a^3*c)^(1/2) - 93*B*a^2*d^2*e^5*(-a^3*c)^(3/2) + 9*
A*c^2*d^5*e^2*(-a^3*c)^(3/2) + 119*C*a^2*d^3*e^4*(-a^3*c)^(3/2) - 42*B*a^7*c*d*e^6 - 9*A*a^4*c^4*d^5*e^2*x + 1
45*A*a^5*c^3*d^3*e^4*x - 93*B*a^5*c^3*d^4*e^3*x + 93*B*a^6*c^2*d^2*e^5*x + 51*C*a^5*c^3*d^5*e^2*x - 119*C*a^6*
c^2*d^3*e^4*x + 80*A*c^2*d^4*e^3*x*(-a^3*c)^(3/2) + 72*C*a^2*d^2*e^5*x*(-a^3*c)^(3/2) - 42*B*c^2*d^5*e^2*x*(-a
^3*c)^(3/2) + 21*C*a^7*c*d*e^6*x - 39*A*a^6*c^2*d*e^6*x + 3*B*a^4*c^4*d^6*e*x - 145*A*a*c*d^3*e^4*(-a^3*c)^(3/
2) + 93*B*a*c*d^4*e^3*(-a^3*c)^(3/2) - 51*C*a*c*d^5*e^2*(-a^3*c)^(3/2) - 42*B*a^2*d*e^6*x*(-a^3*c)^(3/2) + 20*
C*c^2*d^6*e*x*(-a^3*c)^(3/2) - 102*A*a*c*d^2*e^5*x*(-a^3*c)^(3/2) + 108*B*a*c*d^3*e^4*x*(-a^3*c)^(3/2) - 94*C*
a*c*d^4*e^3*x*(-a^3*c)^(3/2) - 2*A*a^2*c^4*d^6*e*x*(-a^3*c)^(1/2))*(e^2*(3*B*a^3*c^2*d^3 + (5*A*a*c^2*d^3*(-a^
3*c)^(1/2))/2 - (7*C*a^2*c*d^3*(-a^3*c)^(1/2))/2) + e^3*(4*C*a^4*c*d^2 - 5*A*a^3*c^2*d^2 + (9*B*a^2*c*d^2*(-a^
3*c)^(1/2))/2) - e^4*(3*B*a^4*c*d - (9*C*a^3*d*(-a^3*c)^(1/2))/4 + (15*A*a^2*c*d*(-a^3*c)^(1/2))/4) - e*((3*C*
a^3*c^2*d^4)/2 + (3*B*a*c^2*d^4*(-a^3*c)^(1/2))/4) - e^5*((C*a^5)/2 + (3*B*a^3*(-a^3*c)^(1/2))/4 - A*a^4*c) +
(A*c^3*d^5*(-a^3*c)^(1/2))/4 + (C*a*c^2*d^5*(-a^3*c)^(1/2))/4))/(a^7*e^8 + a^3*c^4*d^8 + 4*a^6*c*d^2*e^6 + 4*a
^4*c^3*d^6*e^2 + 6*a^5*c^2*d^4*e^4) - (log(3*B*a^6*e^7*(-a^3*c)^(1/2) - 6*C*a^8*e^7 - C*c^2*d^7*(-a^3*c)^(3/2)
+ 12*A*a^7*c*e^7 - 3*B*a^7*c*e^7*x + 2*A*a^4*c^4*d^6*e + 20*C*a^5*c^3*d^6*e + 72*C*a^7*c*d^2*e^5 - A*a^3*c^5*
d^7*x - C*a^4*c^4*d^7*x - 39*A*a^2*d*e^6*(-a^3*c)^(3/2) - 21*C*a^6*d*e^6*(-a^3*c)^(1/2) + 3*B*c^2*d^6*e*(-a^3*
c)^(3/2) - 12*A*a^2*e^7*x*(-a^3*c)^(3/2) - 6*C*a^6*e^7*x*(-a^3*c)^(1/2) + 80*A*a^5*c^3*d^4*e^3 - 102*A*a^6*c^2
*d^2*e^5 - 42*B*a^5*c^3*d^5*e^2 + 108*B*a^6*c^2*d^3*e^4 - 94*C*a^6*c^2*d^4*e^3 + A*a^2*c^4*d^7*(-a^3*c)^(1/2)
+ 93*B*a^2*d^2*e^5*(-a^3*c)^(3/2) - 9*A*c^2*d^5*e^2*(-a^3*c)^(3/2) - 119*C*a^2*d^3*e^4*(-a^3*c)^(3/2) - 42*B*a
^7*c*d*e^6 - 9*A*a^4*c^4*d^5*e^2*x + 145*A*a^5*c^3*d^3*e^4*x - 93*B*a^5*c^3*d^4*e^3*x + 93*B*a^6*c^2*d^2*e^5*x
+ 51*C*a^5*c^3*d^5*e^2*x - 119*C*a^6*c^2*d^3*e^4*x - 80*A*c^2*d^4*e^3*x*(-a^3*c)^(3/2) - 72*C*a^2*d^2*e^5*x*(
-a^3*c)^(3/2) + 42*B*c^2*d^5*e^2*x*(-a^3*c)^(3/2) + 21*C*a^7*c*d*e^6*x - 39*A*a^6*c^2*d*e^6*x + 3*B*a^4*c^4*d^
6*e*x + 145*A*a*c*d^3*e^4*(-a^3*c)^(3/2) - 93*B*a*c*d^4*e^3*(-a^3*c)^(3/2) + 51*C*a*c*d^5*e^2*(-a^3*c)^(3/2) +
42*B*a^2*d*e^6*x*(-a^3*c)^(3/2) - 20*C*c^2*d^6*e*x*(-a^3*c)^(3/2) + 102*A*a*c*d^2*e^5*x*(-a^3*c)^(3/2) - 108*
B*a*c*d^3*e^4*x*(-a^3*c)^(3/2) + 94*C*a*c*d^4*e^3*x*(-a^3*c)^(3/2) + 2*A*a^2*c^4*d^6*e*x*(-a^3*c)^(1/2))*(e^3*
(5*A*a^3*c^2*d^2 - 4*C*a^4*c*d^2 + (9*B*a^2*c*d^2*(-a^3*c)^(1/2))/2) - e^2*(3*B*a^3*c^2*d^3 - (5*A*a*c^2*d^3*(
-a^3*c)^(1/2))/2 + (7*C*a^2*c*d^3*(-a^3*c)^(1/2))/2) + e^4*(3*B*a^4*c*d + (9*C*a^3*d*(-a^3*c)^(1/2))/4 - (15*A
*a^2*c*d*(-a^3*c)^(1/2))/4) + e*((3*C*a^3*c^2*d^4)/2 - (3*B*a*c^2*d^4*(-a^3*c)^(1/2))/4) - e^5*((3*B*a^3*(-a^3
*c)^(1/2))/4 - (C*a^5)/2 + A*a^4*c) + (A*c^3*d^5*(-a^3*c)^(1/2))/4 + (C*a*c^2*d^5*(-a^3*c)^(1/2))/4))/(a^7*e^8
+ a^3*c^4*d^8 + 4*a^6*c*d^2*e^6 + 4*a^4*c^3*d^6*e^2 + 6*a^5*c^2*d^4*e^4) - ((A*a^2*e^5 + B*c^2*d^5 + B*a^2*d*
e^4 - 3*A*c^2*d^4*e - 4*C*a^2*d^2*e^3 + 8*C*a*c*d^4*e + 10*A*a*c*d^2*e^3 - 10*B*a*c*d^3*e^2)/(2*(a*e^2 + c*d^2
)*(a^2*e^4 + c^2*d^4 + 2*a*c*d^2*e^2)) + (x^3*(3*B*a^2*c*e^5 - A*c^3*d^3*e^2 - 9*B*a*c^2*d^2*e^3 + 5*C*a*c^2*d
^3*e^2 + 11*A*a*c^2*d*e^4 - 7*C*a^2*c*d*e^4))/(2*a*(a^3*e^6 + c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4)) -
(x*(A*c^3*d^5 - 2*B*a^3*e^5 - C*a*c^2*d^5 + 6*C*a^3*d*e^4 + 3*A*a*c^2*d^3*e^2 + 11*B*a^2*c*d^2*e^3 - 7*C*a^2*c
*d^3*e^2 - 10*A*a^2*c*d*e^4 + B*a*c^2*d^4*e))/(2*a*(a*e^2 + c*d^2)*(a^2*e^4 + c^2*d^4 + 2*a*c*d^2*e^2)) - (x^2
*(C*a^3*e^5 - 2*A*a^2*c*e^5 + 2*A*c^3*d^4*e - 12*A*a*c^2*d^2*e^3 + 12*B*a*c^2*d^3*e^2 + 6*C*a^2*c*d^2*e^3 - 7*
C*a*c^2*d^4*e))/(2*a*(a*e^2 + c*d^2)*(a^2*e^4 + c^2*d^4 + 2*a*c*d^2*e^2)))/(a*d^2 + x^2*(a*e^2 + c*d^2) + c*e^
2*x^4 + 2*a*d*e*x + 2*c*d*e*x^3) + (log(d + e*x)*(c^2*(10*A*d^2*e^3 - 6*B*d^3*e^2 + 3*C*d^4*e) - c*(2*A*a*e^5
- 6*B*a*d*e^4 + 8*C*a*d^2*e^3) + C*a^2*e^5))/(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)

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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)**2,x)

[Out]

Timed out

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