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

Optimal. Leaf size=305 \[ \frac{\log \left (a+c x^2\right ) \left (B c d \left (c d^2-3 a e^2\right )-e (A c-a C) \left (3 c d^2-a e^2\right )\right )}{2 \left (a e^2+c d^2\right )^3}-\frac{A e^2-B d e+C d^2}{2 e (d+e x)^2 \left (a e^2+c d^2\right )}+\frac{-a B e^2+2 a C d e-2 A c d e+B c d^2}{(d+e x) \left (a e^2+c d^2\right )^2}-\frac{\log (d+e x) \left (B c d \left (c d^2-3 a e^2\right )-e (A c-a C) \left (3 c d^2-a e^2\right )\right )}{\left (a e^2+c d^2\right )^3}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right )}{\sqrt{a} \left (a e^2+c d^2\right )^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.650946, antiderivative size = 305, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {1629, 635, 205, 260} \[ \frac{\log \left (a+c x^2\right ) \left (B c d \left (c d^2-3 a e^2\right )-e (A c-a C) \left (3 c d^2-a e^2\right )\right )}{2 \left (a e^2+c d^2\right )^3}-\frac{A e^2-B d e+C d^2}{2 e (d+e x)^2 \left (a e^2+c d^2\right )}+\frac{-a B e^2+2 a C d e-2 A c d e+B c d^2}{(d+e x) \left (a e^2+c d^2\right )^2}-\frac{\log (d+e x) \left (B c d \left (c d^2-3 a e^2\right )-e (A c-a C) \left (3 c d^2-a e^2\right )\right )}{\left (a e^2+c d^2\right )^3}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right )}{\sqrt{a} \left (a e^2+c d^2\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 )} \, dx &=\int \left (\frac{C d^2-B d e+A e^2}{\left (c d^2+a e^2\right ) (d+e x)^3}+\frac{e \left (-B c d^2+2 A c d e-2 a C d e+a B e^2\right )}{\left (c d^2+a e^2\right )^2 (d+e x)^2}+\frac{e \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 )}{\left (c d^2+a e^2\right )^3 (d+e x)}+\frac{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 )+\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 ) x\right )}{\left (c d^2+a e^2\right )^3 \left (a+c x^2\right )}\right ) \, dx\\ &=-\frac{C d^2-B d e+A e^2}{2 e \left (c d^2+a e^2\right ) (d+e x)^2}+\frac{B c d^2-2 A c d e+2 a C d e-a B e^2}{\left (c d^2+a e^2\right )^2 (d+e x)}-\frac{\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 ) \log (d+e x)}{\left (c d^2+a e^2\right )^3}+\frac{c \int \frac{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 )+\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 ) x}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^3}\\ &=-\frac{C d^2-B d e+A e^2}{2 e \left (c d^2+a e^2\right ) (d+e x)^2}+\frac{B c d^2-2 A c d e+2 a C d e-a B e^2}{\left (c d^2+a e^2\right )^2 (d+e x)}-\frac{\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 ) \log (d+e x)}{\left (c d^2+a e^2\right )^3}+\frac{\left (c \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 )\right ) \int \frac{x}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^3}+\frac{\left (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 )\right ) \int \frac{1}{a+c x^2} \, dx}{\left (c d^2+a e^2\right )^3}\\ &=-\frac{C d^2-B d e+A e^2}{2 e \left (c d^2+a e^2\right ) (d+e x)^2}+\frac{B c d^2-2 A c d e+2 a C d e-a B e^2}{\left (c d^2+a e^2\right )^2 (d+e x)}+\frac{\sqrt{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 ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} \left (c d^2+a e^2\right )^3}-\frac{\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 ) \log (d+e x)}{\left (c d^2+a e^2\right )^3}+\frac{\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 ) \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^3}\\ \end{align*}

Mathematica [A]  time = 0.348172, size = 277, normalized size = 0.91 \[ \frac{\log \left (a+c x^2\right ) \left (B c d \left (c d^2-3 a e^2\right )-e (A c-a C) \left (3 c d^2-a e^2\right )\right )-\frac{\left (a e^2+c d^2\right )^2 \left (e (A e-B d)+C d^2\right )}{e (d+e x)^2}+\frac{2 \left (a e^2+c d^2\right ) \left (-a B e^2+2 a C d e-2 A c d e+B c d^2\right )}{d+e x}-2 \log (d+e x) \left (B c d \left (c d^2-3 a e^2\right )-e (A c-a C) \left (3 c d^2-a e^2\right )\right )+\frac{2 \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c d \left (c d^2-3 a e^2\right )+a \left (a e^2 (3 C d-B e)+c d^2 (3 B e-C d)\right )\right )}{\sqrt{a}}}{2 \left (a e^2+c d^2\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.061, size = 754, normalized size = 2.5 \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)^3/(c*x^2+a),x)

[Out]

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

________________________________________________________________________________________

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),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),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),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.16342, size = 660, normalized size = 2.16 \begin{align*} \frac{{\left (B c^{2} d^{3} + 3 \, C a c d^{2} e - 3 \, A c^{2} d^{2} e - 3 \, B a c d e^{2} - C a^{2} e^{3} + A a c e^{3}\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 (B c^{2} d^{3} e + 3 \, C a c d^{2} e^{2} - 3 \, A c^{2} d^{2} e^{2} - 3 \, B a c d e^{3} - C a^{2} e^{4} + A a c e^{4}\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^{3} - A c^{3} d^{3} - 3 \, B a c^{2} d^{2} e - 3 \, C a^{2} c d e^{2} + 3 \, A a c^{2} d e^{2} + B a^{2} c e^{3}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{{\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 )} \sqrt{a c}} - \frac{{\left (C c^{2} d^{6} - 3 \, B c^{2} d^{5} e - 2 \, C a c d^{4} e^{2} + 5 \, A c^{2} d^{4} e^{2} - 2 \, B a c d^{3} e^{3} - 3 \, C a^{2} d^{2} e^{4} + 6 \, A a c d^{2} e^{4} + B a^{2} d e^{5} + A a^{2} e^{6} - 2 \,{\left (B c^{2} d^{4} e^{2} + 2 \, C a c d^{3} e^{3} - 2 \, A c^{2} d^{3} e^{3} + 2 \, C a^{2} d e^{5} - 2 \, A a c d e^{5} - B a^{2} e^{6}\right )} x\right )} e^{\left (-1\right )}}{2 \,{\left (c d^{2} + a e^{2}\right )}^{3}{\left (x e + d\right )}^{2}} \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),x, algorithm="giac")

[Out]

1/2*(B*c^2*d^3 + 3*C*a*c*d^2*e - 3*A*c^2*d^2*e - 3*B*a*c*d*e^2 - C*a^2*e^3 + A*a*c*e^3)*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) - (B*c^2*d^3*e + 3*C*a*c*d^2*e^2 - 3*A*c^2*d^2*e^2 - 3*B*a*c*
d*e^3 - C*a^2*e^4 + A*a*c*e^4)*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) - (
C*a*c^2*d^3 - A*c^3*d^3 - 3*B*a*c^2*d^2*e - 3*C*a^2*c*d*e^2 + 3*A*a*c^2*d*e^2 + B*a^2*c*e^3)*arctan(c*x/sqrt(a
*c))/((c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6)*sqrt(a*c)) - 1/2*(C*c^2*d^6 - 3*B*c^2*d^5*e - 2*
C*a*c*d^4*e^2 + 5*A*c^2*d^4*e^2 - 2*B*a*c*d^3*e^3 - 3*C*a^2*d^2*e^4 + 6*A*a*c*d^2*e^4 + B*a^2*d*e^5 + A*a^2*e^
6 - 2*(B*c^2*d^4*e^2 + 2*C*a*c*d^3*e^3 - 2*A*c^2*d^3*e^3 + 2*C*a^2*d*e^5 - 2*A*a*c*d*e^5 - B*a^2*e^6)*x)*e^(-1
)/((c*d^2 + a*e^2)^3*(x*e + d)^2)