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

Optimal. Leaf size=353 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c d \left (15 a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right )+a \left (-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right ) (C d-B e)\right )}{8 a^{5/2} \sqrt{c} \left (a e^2+c d^2\right )^3}+\frac{4 a^2 e \left (A e^2-B d e+C d^2\right )+x \left (A c d \left (7 a e^2+3 c d^2\right )+a \left (c d^2-3 a e^2\right ) (C d-B e)\right )}{8 a^2 \left (a+c x^2\right ) \left (a e^2+c d^2\right )^2}-\frac{a (a C e-A c e+B c d)-c x (a B e-a C d+A c d)}{4 a c \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )}-\frac{e^3 \log \left (a+c x^2\right ) \left (A e^2-B d e+C d^2\right )}{2 \left (a e^2+c d^2\right )^3}+\frac{e^3 \log (d+e x) \left (A e^2-B d e+C d^2\right )}{\left (a e^2+c d^2\right )^3} \]

[Out]

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

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Rubi [A]  time = 0.734348, antiderivative size = 353, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {1647, 823, 801, 635, 205, 260} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c d \left (15 a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right )+a \left (-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right ) (C d-B e)\right )}{8 a^{5/2} \sqrt{c} \left (a e^2+c d^2\right )^3}+\frac{4 a^2 e \left (A e^2-B d e+C d^2\right )+x \left (A c d \left (7 a e^2+3 c d^2\right )+a \left (c d^2-3 a e^2\right ) (C d-B e)\right )}{8 a^2 \left (a+c x^2\right ) \left (a e^2+c d^2\right )^2}-\frac{a (a C e-A c e+B c d)-c x (a B e-a C d+A c d)}{4 a c \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )}-\frac{e^3 \log \left (a+c x^2\right ) \left (A e^2-B d e+C d^2\right )}{2 \left (a e^2+c d^2\right )^3}+\frac{e^3 \log (d+e x) \left (A e^2-B d e+C d^2\right )}{\left (a e^2+c d^2\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 823

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(
m + 1)*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), x] + Di
st[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^2*
(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 801

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

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

Mathematica [A]  time = 0.486328, size = 321, normalized size = 0.91 \[ \frac{\frac{2 \left (a e^2+c d^2\right )^2 \left (a^2 (-C) e+a c (A e-B d+B e x-C d x)+A c^2 d x\right )}{a c \left (a+c x^2\right )^2}+\frac{\left (a e^2+c d^2\right ) \left (a^2 e (e (4 A e-4 B d+3 B e x)+C d (4 d-3 e x))+a c d x \left (e (7 A e-B d)+C d^2\right )+3 A c^2 d^3 x\right )}{a^2 \left (a+c x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c d \left (15 a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right )+a \left (-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right ) (C d-B e)\right )}{a^{5/2} \sqrt{c}}-4 e^3 \log \left (a+c x^2\right ) \left (e (A e-B d)+C d^2\right )+8 e^3 \log (d+e x) \left (e (A e-B d)+C d^2\right )}{8 \left (a e^2+c d^2\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.069, size = 1598, normalized size = 4.5 \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)/(c*x^2+a)^3,x)

[Out]

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

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

[Out]

Timed out

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

[Out]

Timed out

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Giac [B]  time = 1.18124, size = 965, normalized size = 2.73 \begin{align*} -\frac{{\left (C d^{2} e^{3} - B d e^{4} + A e^{5}\right )} \log \left (c x^{2} + a\right )}{2 \,{\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )}} + \frac{{\left (C d^{2} e^{4} - B d e^{5} + A e^{6}\right )} \log \left ({\left | x e + d \right |}\right )}{c^{3} d^{6} e + 3 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} + a^{3} e^{7}} + \frac{{\left (C a c^{2} d^{5} + 3 \, A c^{3} d^{5} - B a c^{2} d^{4} e + 6 \, C a^{2} c d^{3} e^{2} + 10 \, A a c^{2} d^{3} e^{2} - 6 \, B a^{2} c d^{2} e^{3} - 3 \, C a^{3} d e^{4} + 15 \, A a^{2} c d e^{4} + 3 \, B a^{3} e^{5}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{8 \,{\left (a^{2} c^{3} d^{6} + 3 \, a^{3} c^{2} d^{4} e^{2} + 3 \, a^{4} c d^{2} e^{4} + a^{5} e^{6}\right )} \sqrt{a c}} - \frac{2 \, B a^{2} c^{3} d^{5} - 2 \, C a^{3} c^{2} d^{4} e - 2 \, A a^{2} c^{3} d^{4} e + 8 \, B a^{3} c^{2} d^{3} e^{2} - 8 \, A a^{3} c^{2} d^{2} e^{3} + 6 \, B a^{4} c d e^{4} + 2 \, C a^{5} e^{5} - 6 \, A a^{4} c e^{5} -{\left (C a c^{4} d^{5} + 3 \, A c^{5} d^{5} - B a c^{4} d^{4} e - 2 \, C a^{2} c^{3} d^{3} e^{2} + 10 \, A a c^{4} d^{3} e^{2} + 2 \, B a^{2} c^{3} d^{2} e^{3} - 3 \, C a^{3} c^{2} d e^{4} + 7 \, A a^{2} c^{3} d e^{4} + 3 \, B a^{3} c^{2} e^{5}\right )} x^{3} - 4 \,{\left (C a^{2} c^{3} d^{4} e - B a^{2} c^{3} d^{3} e^{2} + C a^{3} c^{2} d^{2} e^{3} + A a^{2} c^{3} d^{2} e^{3} - B a^{3} c^{2} d e^{4} + A a^{3} c^{2} e^{5}\right )} x^{2} +{\left (C a^{2} c^{3} d^{5} - 5 \, A a c^{4} d^{5} - B a^{2} c^{3} d^{4} e + 6 \, C a^{3} c^{2} d^{3} e^{2} - 14 \, A a^{2} c^{3} d^{3} e^{2} - 6 \, B a^{3} c^{2} d^{2} e^{3} + 5 \, C a^{4} c d e^{4} - 9 \, A a^{3} c^{2} d e^{4} - 5 \, B a^{4} c e^{5}\right )} x}{8 \,{\left (c d^{2} + a e^{2}\right )}^{3}{\left (c x^{2} + a\right )}^{2} a^{2} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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