\(\int \frac {A+B x+C x^2}{(d+e x)^2 (a+c x^2)} \, dx\) [48]

Optimal result

Integrand size = 27, antiderivative size = 214 \[ \int \frac {A+B x+C x^2}{(d+e x)^2 \left (a+c x^2\right )} \, dx=-\frac {C d^2-B d e+A e^2}{e \left (c d^2+a e^2\right ) (d+e x)}+\frac {\left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c} \left (c d^2+a e^2\right )^2}-\frac {\left (B c d^2-2 A c d e+2 a C d e-a B e^2\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^2}+\frac {\left (B c d^2-2 A c d e+2 a C d e-a B e^2\right ) \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1643, 649, 211, 266} \[ \int \frac {A+B x+C x^2}{(d+e x)^2 \left (a+c x^2\right )} \, dx=\frac {\arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right )}{\sqrt {a} \sqrt {c} \left (a e^2+c d^2\right )^2}+\frac {\log \left (a+c x^2\right ) \left (-a B e^2+2 a C d e-2 A c d e+B c d^2\right )}{2 \left (a e^2+c d^2\right )^2}-\frac {A e^2-B d e+C d^2}{e (d+e x) \left (a e^2+c d^2\right )}-\frac {\log (d+e x) \left (-a B e^2+2 a C d e-2 A c d e+B c d^2\right )}{\left (a e^2+c d^2\right )^2} \]

[In]

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

[Out]

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

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 266

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 649

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 1643

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]

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

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.88 \[ \int \frac {A+B x+C x^2}{(d+e x)^2 \left (a+c x^2\right )} \, dx=\frac {-\frac {2 \left (c d^2+a e^2\right ) \left (C d^2+e (-B d+A e)\right )}{e (d+e x)}+\frac {2 \left (A c \left (c d^2-a e^2\right )+a \left (a C e^2+c d (-C d+2 B e)\right )\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c}}+\left (-2 B c d^2+4 A c d e-4 a C d e+2 a B e^2\right ) \log (d+e x)+\left (B c d^2-2 A c d e+2 a C d e-a B e^2\right ) \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.62 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.95

method	result	size
default	\(\frac {\frac {\left (-2 A \,c^{2} d e -B \,e^{2} a c +B \,c^{2} d^{2}+2 a c d e C \right ) \ln \left (c \,x^{2}+a \right )}{2 c}+\frac {\left (-A a c \,e^{2}+A \,c^{2} d^{2}+2 B a c d e +a^{2} C \,e^{2}-C a c \,d^{2}\right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}}{\left (e^{2} a +c \,d^{2}\right )^{2}}-\frac {A \,e^{2}-B d e +C \,d^{2}}{\left (e^{2} a +c \,d^{2}\right ) e \left (e x +d \right )}+\frac {\left (2 A c d e +B a \,e^{2}-B c \,d^{2}-2 a d e C \right ) \ln \left (e x +d \right )}{\left (e^{2} a +c \,d^{2}\right )^{2}}\)	\(204\)
risch	\(\text {Expression too large to display}\)	\(15347\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 442 vs. \(2 (204) = 408\).

Time = 25.70 (sec) , antiderivative size = 904, normalized size of antiderivative = 4.22 \[ \int \frac {A+B x+C x^2}{(d+e x)^2 \left (a+c x^2\right )} \, dx=\left [-\frac {2 \, C a c^{2} d^{4} - 2 \, B a c^{2} d^{3} e - 2 \, B a^{2} c d e^{3} + 2 \, A a^{2} c e^{4} + 2 \, {\left (C a^{2} c + A a c^{2}\right )} d^{2} e^{2} - {\left (2 \, B a c d^{2} e^{2} - {\left (C a c - A c^{2}\right )} d^{3} e + {\left (C a^{2} - A a c\right )} d e^{3} + {\left (2 \, B a c d e^{3} - {\left (C a c - A c^{2}\right )} d^{2} e^{2} + {\left (C a^{2} - A a c\right )} e^{4}\right )} x\right )} \sqrt {-a c} \log \left (\frac {c x^{2} + 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) - {\left (B a c^{2} d^{3} e - B a^{2} c d e^{3} + 2 \, {\left (C a^{2} c - A a c^{2}\right )} d^{2} e^{2} + {\left (B a c^{2} d^{2} e^{2} - B a^{2} c e^{4} + 2 \, {\left (C a^{2} c - A a c^{2}\right )} d e^{3}\right )} x\right )} \log \left (c x^{2} + a\right ) + 2 \, {\left (B a c^{2} d^{3} e - B a^{2} c d e^{3} + 2 \, {\left (C a^{2} c - A a c^{2}\right )} d^{2} e^{2} + {\left (B a c^{2} d^{2} e^{2} - B a^{2} c e^{4} + 2 \, {\left (C a^{2} c - A a c^{2}\right )} d e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (a c^{3} d^{5} e + 2 \, a^{2} c^{2} d^{3} e^{3} + a^{3} c d e^{5} + {\left (a c^{3} d^{4} e^{2} + 2 \, a^{2} c^{2} d^{2} e^{4} + a^{3} c e^{6}\right )} x\right )}}, -\frac {2 \, C a c^{2} d^{4} - 2 \, B a c^{2} d^{3} e - 2 \, B a^{2} c d e^{3} + 2 \, A a^{2} c e^{4} + 2 \, {\left (C a^{2} c + A a c^{2}\right )} d^{2} e^{2} - 2 \, {\left (2 \, B a c d^{2} e^{2} - {\left (C a c - A c^{2}\right )} d^{3} e + {\left (C a^{2} - A a c\right )} d e^{3} + {\left (2 \, B a c d e^{3} - {\left (C a c - A c^{2}\right )} d^{2} e^{2} + {\left (C a^{2} - A a c\right )} e^{4}\right )} x\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) - {\left (B a c^{2} d^{3} e - B a^{2} c d e^{3} + 2 \, {\left (C a^{2} c - A a c^{2}\right )} d^{2} e^{2} + {\left (B a c^{2} d^{2} e^{2} - B a^{2} c e^{4} + 2 \, {\left (C a^{2} c - A a c^{2}\right )} d e^{3}\right )} x\right )} \log \left (c x^{2} + a\right ) + 2 \, {\left (B a c^{2} d^{3} e - B a^{2} c d e^{3} + 2 \, {\left (C a^{2} c - A a c^{2}\right )} d^{2} e^{2} + {\left (B a c^{2} d^{2} e^{2} - B a^{2} c e^{4} + 2 \, {\left (C a^{2} c - A a c^{2}\right )} d e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (a c^{3} d^{5} e + 2 \, a^{2} c^{2} d^{3} e^{3} + a^{3} c d e^{5} + {\left (a c^{3} d^{4} e^{2} + 2 \, a^{2} c^{2} d^{2} e^{4} + a^{3} c e^{6}\right )} x\right )}}\right ] \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2}{(d+e x)^2 \left (a+c x^2\right )} \, dx=\text {Timed out} \]

[In]

integrate((C*x**2+B*x+A)/(e*x+d)**2/(c*x**2+a),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.19 \[ \int \frac {A+B x+C x^2}{(d+e x)^2 \left (a+c x^2\right )} \, dx=\frac {{\left (B c d^{2} - B a e^{2} + 2 \, {\left (C a - A c\right )} d e\right )} \log \left (c x^{2} + a\right )}{2 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}} - \frac {{\left (B c d^{2} - B a e^{2} + 2 \, {\left (C a - A c\right )} d e\right )} \log \left (e x + d\right )}{c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}} + \frac {{\left (2 \, B a c d e - {\left (C a c - A c^{2}\right )} d^{2} + {\left (C a^{2} - A a c\right )} e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {a c}} - \frac {C d^{2} - B d e + A e^{2}}{c d^{3} e + a d e^{3} + {\left (c d^{2} e^{2} + a e^{4}\right )} x} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.29 \[ \int \frac {A+B x+C x^2}{(d+e x)^2 \left (a+c x^2\right )} \, dx=\frac {{\left (B c d^{2} + 2 \, C a d e - 2 \, A c d e - B a e^{2}\right )} \log \left (c - \frac {2 \, c d}{e x + d} + \frac {c d^{2}}{{\left (e x + d\right )}^{2}} + \frac {a e^{2}}{{\left (e x + d\right )}^{2}}\right )}{2 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}} - \frac {\frac {C d^{2} e}{e x + d} - \frac {B d e^{2}}{e x + d} + \frac {A e^{3}}{e x + d}}{c d^{2} e^{2} + a e^{4}} - \frac {{\left (C a c d^{2} e^{2} - A c^{2} d^{2} e^{2} - 2 \, B a c d e^{3} - C a^{2} e^{4} + A a c e^{4}\right )} \arctan \left (\frac {c d - \frac {c d^{2}}{e x + d} - \frac {a e^{2}}{e x + d}}{\sqrt {a c} e}\right )}{{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {a c} e^{2}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 16.31 (sec) , antiderivative size = 1199, normalized size of antiderivative = 5.60 \[ \int \frac {A+B x+C x^2}{(d+e x)^2 \left (a+c x^2\right )} \, dx=\frac {\ln \left (C\,c\,d^4\,{\left (-a\,c\right )}^{3/2}-A\,a\,e^4\,{\left (-a\,c\right )}^{3/2}+3\,B\,a\,c^3\,d^4+3\,B\,a^3\,c\,e^4+A\,c^4\,d^4\,x+A\,c^3\,d^4\,\sqrt {-a\,c}-C\,a^3\,e^4\,\sqrt {-a\,c}-C\,a\,c^3\,d^4\,x-C\,a^3\,c\,e^4\,x+14\,A\,c\,d^2\,e^2\,{\left (-a\,c\right )}^{3/2}-14\,C\,a\,d^2\,e^2\,{\left (-a\,c\right )}^{3/2}-3\,B\,c^3\,d^4\,x\,\sqrt {-a\,c}+8\,A\,a^2\,c^2\,d\,e^3+8\,C\,a^2\,c^2\,d^3\,e+A\,a^2\,c^2\,e^4\,x-10\,B\,a^2\,c^2\,d^2\,e^2+8\,B\,a\,d\,e^3\,{\left (-a\,c\right )}^{3/2}-8\,B\,c\,d^3\,e\,{\left (-a\,c\right )}^{3/2}+3\,B\,a\,e^4\,x\,{\left (-a\,c\right )}^{3/2}-8\,A\,a\,c^3\,d^3\,e-8\,C\,a^3\,c\,d\,e^3+14\,C\,a^2\,c^2\,d^2\,e^2\,x+8\,A\,c\,d\,e^3\,x\,{\left (-a\,c\right )}^{3/2}-8\,C\,a\,d\,e^3\,x\,{\left (-a\,c\right )}^{3/2}+8\,C\,c\,d^3\,e\,x\,{\left (-a\,c\right )}^{3/2}+8\,B\,a\,c^3\,d^3\,e\,x+8\,A\,c^3\,d^3\,e\,x\,\sqrt {-a\,c}-10\,B\,c\,d^2\,e^2\,x\,{\left (-a\,c\right )}^{3/2}-14\,A\,a\,c^3\,d^2\,e^2\,x-8\,B\,a^2\,c^2\,d\,e^3\,x\right )\,\left (c^2\,\left (a\,\left (\frac {B\,d^2}{2}-A\,d\,e\right )+\frac {A\,d^2\,\sqrt {-a\,c}}{2}\right )-c\,\left (a^2\,\left (\frac {B\,e^2}{2}-C\,d\,e\right )+a\,\left (\frac {A\,e^2\,\sqrt {-a\,c}}{2}+\frac {C\,d^2\,\sqrt {-a\,c}}{2}-B\,d\,e\,\sqrt {-a\,c}\right )\right )+\frac {C\,a^2\,e^2\,\sqrt {-a\,c}}{2}\right )}{a^3\,c\,e^4+2\,a^2\,c^2\,d^2\,e^2+a\,c^3\,d^4}-\frac {\ln \left (d+e\,x\right )\,\left (c\,\left (B\,d^2-2\,A\,d\,e\right )-a\,\left (B\,e^2-2\,C\,d\,e\right )\right )}{a^2\,e^4+2\,a\,c\,d^2\,e^2+c^2\,d^4}-\frac {\ln \left (A\,a\,e^4\,{\left (-a\,c\right )}^{3/2}-C\,c\,d^4\,{\left (-a\,c\right )}^{3/2}+3\,B\,a\,c^3\,d^4+3\,B\,a^3\,c\,e^4+A\,c^4\,d^4\,x-A\,c^3\,d^4\,\sqrt {-a\,c}+C\,a^3\,e^4\,\sqrt {-a\,c}-C\,a\,c^3\,d^4\,x-C\,a^3\,c\,e^4\,x-14\,A\,c\,d^2\,e^2\,{\left (-a\,c\right )}^{3/2}+14\,C\,a\,d^2\,e^2\,{\left (-a\,c\right )}^{3/2}+3\,B\,c^3\,d^4\,x\,\sqrt {-a\,c}+8\,A\,a^2\,c^2\,d\,e^3+8\,C\,a^2\,c^2\,d^3\,e+A\,a^2\,c^2\,e^4\,x-10\,B\,a^2\,c^2\,d^2\,e^2-8\,B\,a\,d\,e^3\,{\left (-a\,c\right )}^{3/2}+8\,B\,c\,d^3\,e\,{\left (-a\,c\right )}^{3/2}-3\,B\,a\,e^4\,x\,{\left (-a\,c\right )}^{3/2}-8\,A\,a\,c^3\,d^3\,e-8\,C\,a^3\,c\,d\,e^3+14\,C\,a^2\,c^2\,d^2\,e^2\,x-8\,A\,c\,d\,e^3\,x\,{\left (-a\,c\right )}^{3/2}+8\,C\,a\,d\,e^3\,x\,{\left (-a\,c\right )}^{3/2}-8\,C\,c\,d^3\,e\,x\,{\left (-a\,c\right )}^{3/2}+8\,B\,a\,c^3\,d^3\,e\,x-8\,A\,c^3\,d^3\,e\,x\,\sqrt {-a\,c}+10\,B\,c\,d^2\,e^2\,x\,{\left (-a\,c\right )}^{3/2}-14\,A\,a\,c^3\,d^2\,e^2\,x-8\,B\,a^2\,c^2\,d\,e^3\,x\right )\,\left (c\,\left (a^2\,\left (\frac {B\,e^2}{2}-C\,d\,e\right )-a\,\left (\frac {A\,e^2\,\sqrt {-a\,c}}{2}+\frac {C\,d^2\,\sqrt {-a\,c}}{2}-B\,d\,e\,\sqrt {-a\,c}\right )\right )-c^2\,\left (a\,\left (\frac {B\,d^2}{2}-A\,d\,e\right )-\frac {A\,d^2\,\sqrt {-a\,c}}{2}\right )+\frac {C\,a^2\,e^2\,\sqrt {-a\,c}}{2}\right )}{a^3\,c\,e^4+2\,a^2\,c^2\,d^2\,e^2+a\,c^3\,d^4}-\frac {C\,d^2-B\,d\,e+A\,e^2}{e\,\left (c\,d^2+a\,e^2\right )\,\left (d+e\,x\right )} \]

[In]

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

[Out]

(log(C*c*d^4*(-a*c)^(3/2) - A*a*e^4*(-a*c)^(3/2) + 3*B*a*c^3*d^4 + 3*B*a^3*c*e^4 + A*c^4*d^4*x + A*c^3*d^4*(-a
*c)^(1/2) - C*a^3*e^4*(-a*c)^(1/2) - C*a*c^3*d^4*x - C*a^3*c*e^4*x + 14*A*c*d^2*e^2*(-a*c)^(3/2) - 14*C*a*d^2*
e^2*(-a*c)^(3/2) - 3*B*c^3*d^4*x*(-a*c)^(1/2) + 8*A*a^2*c^2*d*e^3 + 8*C*a^2*c^2*d^3*e + A*a^2*c^2*e^4*x - 10*B
*a^2*c^2*d^2*e^2 + 8*B*a*d*e^3*(-a*c)^(3/2) - 8*B*c*d^3*e*(-a*c)^(3/2) + 3*B*a*e^4*x*(-a*c)^(3/2) - 8*A*a*c^3*
d^3*e - 8*C*a^3*c*d*e^3 + 14*C*a^2*c^2*d^2*e^2*x + 8*A*c*d*e^3*x*(-a*c)^(3/2) - 8*C*a*d*e^3*x*(-a*c)^(3/2) + 8
*C*c*d^3*e*x*(-a*c)^(3/2) + 8*B*a*c^3*d^3*e*x + 8*A*c^3*d^3*e*x*(-a*c)^(1/2) - 10*B*c*d^2*e^2*x*(-a*c)^(3/2) -
14*A*a*c^3*d^2*e^2*x - 8*B*a^2*c^2*d*e^3*x)*(c^2*(a*((B*d^2)/2 - A*d*e) + (A*d^2*(-a*c)^(1/2))/2) - c*(a^2*((
B*e^2)/2 - C*d*e) + a*((A*e^2*(-a*c)^(1/2))/2 + (C*d^2*(-a*c)^(1/2))/2 - B*d*e*(-a*c)^(1/2))) + (C*a^2*e^2*(-a
*c)^(1/2))/2))/(a*c^3*d^4 + a^3*c*e^4 + 2*a^2*c^2*d^2*e^2) - (log(d + e*x)*(c*(B*d^2 - 2*A*d*e) - a*(B*e^2 - 2
*C*d*e)))/(a^2*e^4 + c^2*d^4 + 2*a*c*d^2*e^2) - (log(A*a*e^4*(-a*c)^(3/2) - C*c*d^4*(-a*c)^(3/2) + 3*B*a*c^3*d
^4 + 3*B*a^3*c*e^4 + A*c^4*d^4*x - A*c^3*d^4*(-a*c)^(1/2) + C*a^3*e^4*(-a*c)^(1/2) - C*a*c^3*d^4*x - C*a^3*c*e
^4*x - 14*A*c*d^2*e^2*(-a*c)^(3/2) + 14*C*a*d^2*e^2*(-a*c)^(3/2) + 3*B*c^3*d^4*x*(-a*c)^(1/2) + 8*A*a^2*c^2*d*
e^3 + 8*C*a^2*c^2*d^3*e + A*a^2*c^2*e^4*x - 10*B*a^2*c^2*d^2*e^2 - 8*B*a*d*e^3*(-a*c)^(3/2) + 8*B*c*d^3*e*(-a*
c)^(3/2) - 3*B*a*e^4*x*(-a*c)^(3/2) - 8*A*a*c^3*d^3*e - 8*C*a^3*c*d*e^3 + 14*C*a^2*c^2*d^2*e^2*x - 8*A*c*d*e^3
*x*(-a*c)^(3/2) + 8*C*a*d*e^3*x*(-a*c)^(3/2) - 8*C*c*d^3*e*x*(-a*c)^(3/2) + 8*B*a*c^3*d^3*e*x - 8*A*c^3*d^3*e*
x*(-a*c)^(1/2) + 10*B*c*d^2*e^2*x*(-a*c)^(3/2) - 14*A*a*c^3*d^2*e^2*x - 8*B*a^2*c^2*d*e^3*x)*(c*(a^2*((B*e^2)/
2 - C*d*e) - a*((A*e^2*(-a*c)^(1/2))/2 + (C*d^2*(-a*c)^(1/2))/2 - B*d*e*(-a*c)^(1/2))) - c^2*(a*((B*d^2)/2 - A
*d*e) - (A*d^2*(-a*c)^(1/2))/2) + (C*a^2*e^2*(-a*c)^(1/2))/2))/(a*c^3*d^4 + a^3*c*e^4 + 2*a^2*c^2*d^2*e^2) - (
A*e^2 + C*d^2 - B*d*e)/(e*(a*e^2 + c*d^2)*(d + e*x))