Integrand size = 27, antiderivative size = 374 \[ \int \frac {A+B x+C x^2}{(d+e x)^2 \left (a+c x^2\right )^2} \, dx=-\frac {e \left (C d^2-B d e+A e^2\right )}{\left (c d^2+a e^2\right )^2 (d+e x)}-\frac {a \left (B c d^2-2 A c d e+2 a C d e-a B e^2\right )-\left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right ) x}{2 a \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}+\frac {\left (A c \left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right )+a \left (a^2 C e^4+c^2 d^3 (C d-2 B e)-6 a c d e^2 (C d-B e)\right )\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {c} \left (c d^2+a e^2\right )^3}-\frac {e \left (a e^2 (2 C d-B e)-c d \left (2 C d^2-e (3 B d-4 A e)\right )\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^3}+\frac {e \left (a e^2 (2 C d-B e)-c d \left (2 C d^2-e (3 B d-4 A e)\right )\right ) \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^3} \]
-e*(A*e^2-B*d*e+C*d^2)/(a*e^2+c*d^2)^2/(e*x+d)+1/2*(-a*(-2*A*c*d*e-B*a*e^2 +B*c*d^2+2*C*a*d*e)+(A*c*(-a*e^2+c*d^2)+a*(a*C*e^2-c*d*(-2*B*e+C*d)))*x)/a /(a*e^2+c*d^2)^2/(c*x^2+a)-e*(a*e^2*(-B*e+2*C*d)-c*d*(2*C*d^2-e*(-4*A*e+3* B*d)))*ln(e*x+d)/(a*e^2+c*d^2)^3+1/2*e*(a*e^2*(-B*e+2*C*d)-c*d*(2*C*d^2-e* (-4*A*e+3*B*d)))*ln(c*x^2+a)/(a*e^2+c*d^2)^3+1/2*(A*c*(-3*a^2*e^4+6*a*c*d^ 2*e^2+c^2*d^4)+a*(a^2*C*e^4+c^2*d^3*(-2*B*e+C*d)-6*a*c*d*e^2*(-B*e+C*d)))* arctan(x*c^(1/2)/a^(1/2))/a^(3/2)/(a*e^2+c*d^2)^3/c^(1/2)
Time = 0.25 (sec) , antiderivative size = 320, normalized size of antiderivative = 0.86 \[ \int \frac {A+B x+C x^2}{(d+e x)^2 \left (a+c x^2\right )^2} \, dx=\frac {-\frac {2 e \left (c d^2+a e^2\right ) \left (C d^2+e (-B d+A e)\right )}{d+e x}+\frac {\left (c d^2+a e^2\right ) \left (A c^2 d^2 x+a^2 e (-2 C d+B e+C e x)-a c \left (C d^2 x+B d (d-2 e x)+A e (-2 d+e x)\right )\right )}{a \left (a+c x^2\right )}+\frac {\left (A c \left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right )+a \left (a^2 C e^4+c^2 d^3 (C d-2 B e)+6 a c d e^2 (-C d+B e)\right )\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {c}}+2 e \left (2 c C d^3+c d e (-3 B d+4 A e)+a e^2 (-2 C d+B e)\right ) \log (d+e x)-e \left (2 c C d^3+c d e (-3 B d+4 A e)+a e^2 (-2 C d+B e)\right ) \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^3} \]
((-2*e*(c*d^2 + a*e^2)*(C*d^2 + e*(-(B*d) + A*e)))/(d + e*x) + ((c*d^2 + a *e^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)) + ((A*c*(c^2*d^4 + 6*a*c*d^ 2*e^2 - 3*a^2*e^4) + a*(a^2*C*e^4 + c^2*d^3*(C*d - 2*B*e) + 6*a*c*d*e^2*(- (C*d) + B*e)))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(a^(3/2)*Sqrt[c]) + 2*e*(2*c*C *d^3 + c*d*e*(-3*B*d + 4*A*e) + a*e^2*(-2*C*d + B*e))*Log[d + e*x] - e*(2* c*C*d^3 + c*d*e*(-3*B*d + 4*A*e) + a*e^2*(-2*C*d + B*e))*Log[a + c*x^2])/( 2*(c*d^2 + a*e^2)^3)
Time = 1.17 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2178, 25, 2160, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {A+B x+C x^2}{\left (a+c x^2\right )^2 (d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 2178 |
| \(\displaystyle -\frac {\int -\frac {\frac {c e^2 \left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right ) x^2}{\left (c d^2+a e^2\right )^2}+\frac {2 c e (A c d-a C d+a B e) x}{c d^2+a e^2}+\frac {c \left (A \left (c^2 d^4+5 a c e^2 d^2+2 a^2 e^4\right )-a d^2 \left (a C e^2-c d (C d-2 B e)\right )\right )}{\left (c d^2+a e^2\right )^2}}{(d+e x)^2 \left (c x^2+a\right )}dx}{2 a c}-\frac {a \left (-a B e^2+2 a C d e-2 A c d e+B c d^2\right )-x \left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right )}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\frac {c e^2 \left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right ) x^2}{\left (c d^2+a e^2\right )^2}+\frac {2 c e (A c d-a C d+a B e) x}{c d^2+a e^2}+\frac {c \left (A \left (c^2 d^4+5 a c e^2 d^2+2 a^2 e^4\right )-a d^2 \left (a C e^2-c d (C d-2 B e)\right )\right )}{\left (c d^2+a e^2\right )^2}}{(d+e x)^2 \left (c x^2+a\right )}dx}{2 a c}-\frac {a \left (-a B e^2+2 a C d e-2 A c d e+B c d^2\right )-x \left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right )}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )^2}\) |
| \(\Big \downarrow \) 2160 |
| \(\displaystyle \frac {\int \left (\frac {2 a c \left (2 c C d^3-c e (3 B d-4 A e) d-a e^2 (2 C d-B e)\right ) e^2}{\left (c d^2+a e^2\right )^3 (d+e x)}+\frac {2 a c \left (C d^2-B e d+A e^2\right ) e^2}{\left (c d^2+a e^2\right )^2 (d+e x)^2}+\frac {c \left (A c \left (c^2 d^4+6 a c e^2 d^2-3 a^2 e^4\right )+a \left (a^2 C e^4-6 a c d (C d-B e) e^2+c^2 d^3 (C d-2 B e)\right )-2 a c e \left (2 c C d^3-c e (3 B d-4 A e) d-a e^2 (2 C d-B e)\right ) x\right )}{\left (c d^2+a e^2\right )^3 \left (c x^2+a\right )}\right )dx}{2 a c}-\frac {a \left (-a B e^2+2 a C d e-2 A c d e+B c d^2\right )-x \left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right )}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\sqrt {c} \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (A c \left (-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right )+a \left (a^2 C e^4-6 a c d e^2 (C d-B e)+c^2 d^3 (C d-2 B e)\right )\right )}{\sqrt {a} \left (a e^2+c d^2\right )^3}-\frac {2 a c e \left (A e^2-B d e+C d^2\right )}{(d+e x) \left (a e^2+c d^2\right )^2}-\frac {a c e \log \left (a+c x^2\right ) \left (-a e^2 (2 C d-B e)-c d e (3 B d-4 A e)+2 c C d^3\right )}{\left (a e^2+c d^2\right )^3}+\frac {2 a c e \log (d+e x) \left (-a e^2 (2 C d-B e)-c d e (3 B d-4 A e)+2 c C d^3\right )}{\left (a e^2+c d^2\right )^3}}{2 a c}-\frac {a \left (-a B e^2+2 a C d e-2 A c d e+B c d^2\right )-x \left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right )}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )^2}\) |
-1/2*(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)/(a*(c*d^2 + a*e^2)^2*(a + c*x^2)) + ((-2*a*c*e*(C*d^2 - B*d*e + A*e^2))/((c*d^2 + a*e^2)^2*(d + e*x)) + (Sqrt [c]*(A*c*(c^2*d^4 + 6*a*c*d^2*e^2 - 3*a^2*e^4) + a*(a^2*C*e^4 + c^2*d^3*(C *d - 2*B*e) - 6*a*c*d*e^2*(C*d - B*e)))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(Sqrt [a]*(c*d^2 + a*e^2)^3) + (2*a*c*e*(2*c*C*d^3 - c*d*e*(3*B*d - 4*A*e) - a*e ^2*(2*C*d - B*e))*Log[d + e*x])/(c*d^2 + a*e^2)^3 - (a*c*e*(2*c*C*d^3 - c* d*e*(3*B*d - 4*A*e) - a*e^2*(2*C*d - B*e))*Log[a + c*x^2])/(c*d^2 + a*e^2) ^3)/(2*a*c)
3.1.55.3.1 Defintions of rubi rules used
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{Qx = PolynomialQuotient[(d + e*x)^m*Pq, a + b*x^2, x], R = Coeff[Po lynomialRemainder[(d + e*x)^m*Pq, a + b*x^2, x], x, 0], S = Coeff[Polynomia lRemainder[(d + e*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*S - b*R*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*b*(p + 1)) Int[(d + e*x )^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*b*(p + 1)*Qx)/(d + e*x)^m + (b*R*( 2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, d, e}, x] && PolyQ[Pq, x ] && NeQ[b*d^2 + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]
Time = 0.64 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.13
| method | result | size |
| default | \(-\frac {\frac {\frac {\left (A \,a^{2} c \,e^{4}-A \,d^{4} c^{3}-2 B \,a^{2} c d \,e^{3}-2 B a \,c^{2} d^{3} e -C \,a^{3} e^{4}+C a \,c^{2} d^{4}\right ) x}{2 a}-A a c d \,e^{3}-A \,c^{2} d^{3} e -\frac {B \,e^{4} a^{2}}{2}+\frac {B \,c^{2} d^{4}}{2}+C \,a^{2} d \,e^{3}+C a c \,d^{3} e}{c \,x^{2}+a}+\frac {\frac {\left (8 A a \,c^{2} d \,e^{3}+2 B \,e^{4} c \,a^{2}-6 B a \,c^{2} d^{2} e^{2}-4 C \,a^{2} c d \,e^{3}+4 C a \,c^{2} d^{3} e \right ) \ln \left (c \,x^{2}+a \right )}{2 c}+\frac {\left (3 A \,a^{2} c \,e^{4}-6 A a \,c^{2} d^{2} e^{2}-A \,d^{4} c^{3}-6 B \,a^{2} c d \,e^{3}+2 B a \,c^{2} d^{3} e -C \,a^{3} e^{4}+6 C \,a^{2} c \,d^{2} e^{2}-C a \,c^{2} d^{4}\right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}}{2 a}}{\left (e^{2} a +c \,d^{2}\right )^{3}}+\frac {e \left (4 A c d \,e^{2}+B \,e^{3} a -3 B c \,d^{2} e -2 C a d \,e^{2}+2 C c \,d^{3}\right ) \ln \left (e x +d \right )}{\left (e^{2} a +c \,d^{2}\right )^{3}}-\frac {e \left (A \,e^{2}-B d e +C \,d^{2}\right )}{\left (e^{2} a +c \,d^{2}\right )^{2} \left (e x +d \right )}\) | \(424\) |
| risch | \(\text {Expression too large to display}\) | \(1606\) |
-1/(a*e^2+c*d^2)^3*((1/2*(A*a^2*c*e^4-A*c^3*d^4-2*B*a^2*c*d*e^3-2*B*a*c^2* d^3*e-C*a^3*e^4+C*a*c^2*d^4)/a*x-A*a*c*d*e^3-A*c^2*d^3*e-1/2*B*e^4*a^2+1/2 *B*c^2*d^4+C*a^2*d*e^3+C*a*c*d^3*e)/(c*x^2+a)+1/2/a*(1/2*(8*A*a*c^2*d*e^3+ 2*B*a^2*c*e^4-6*B*a*c^2*d^2*e^2-4*C*a^2*c*d*e^3+4*C*a*c^2*d^3*e)/c*ln(c*x^ 2+a)+(3*A*a^2*c*e^4-6*A*a*c^2*d^2*e^2-A*c^3*d^4-6*B*a^2*c*d*e^3+2*B*a*c^2* d^3*e-C*a^3*e^4+6*C*a^2*c*d^2*e^2-C*a*c^2*d^4)/(a*c)^(1/2)*arctan(c*x/(a*c )^(1/2))))+e*(4*A*c*d*e^2+B*a*e^3-3*B*c*d^2*e-2*C*a*d*e^2+2*C*c*d^3)/(a*e^ 2+c*d^2)^3*ln(e*x+d)-e*(A*e^2-B*d*e+C*d^2)/(a*e^2+c*d^2)^2/(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 1447 vs. \(2 (360) = 720\).
Time = 130.34 (sec) , antiderivative size = 2916, normalized size of antiderivative = 7.80 \[ \int \frac {A+B x+C x^2}{(d+e x)^2 \left (a+c x^2\right )^2} \, dx=\text {Too large to display} \]
[-1/4*(2*B*a^2*c^3*d^5 - 4*B*a^3*c^2*d^3*e^2 + 8*C*a^4*c*d^2*e^3 - 6*B*a^4 *c*d*e^4 + 4*A*a^4*c*e^5 + 4*(2*C*a^3*c^2 - A*a^2*c^3)*d^4*e - 2*(4*B*a^2* c^3*d^3*e^2 + 4*B*a^3*c^2*d*e^4 - (3*C*a^2*c^3 - A*a*c^4)*d^4*e - 2*(C*a^3 *c^2 + A*a^2*c^3)*d^2*e^3 + (C*a^4*c - 3*A*a^3*c^2)*e^5)*x^2 + (2*B*a^2*c^ 2*d^4*e - 6*B*a^3*c*d^2*e^3 - (C*a^2*c^2 + A*a*c^3)*d^5 + 6*(C*a^3*c - A*a ^2*c^2)*d^3*e^2 - (C*a^4 - 3*A*a^3*c)*d*e^4 + (2*B*a*c^3*d^3*e^2 - 6*B*a^2 *c^2*d*e^4 - (C*a*c^3 + A*c^4)*d^4*e + 6*(C*a^2*c^2 - A*a*c^3)*d^2*e^3 - ( C*a^3*c - 3*A*a^2*c^2)*e^5)*x^3 + (2*B*a*c^3*d^4*e - 6*B*a^2*c^2*d^2*e^3 - (C*a*c^3 + A*c^4)*d^5 + 6*(C*a^2*c^2 - A*a*c^3)*d^3*e^2 - (C*a^3*c - 3*A* a^2*c^2)*d*e^4)*x^2 + (2*B*a^2*c^2*d^3*e^2 - 6*B*a^3*c*d*e^4 - (C*a^2*c^2 + A*a*c^3)*d^4*e + 6*(C*a^3*c - A*a^2*c^2)*d^2*e^3 - (C*a^4 - 3*A*a^3*c)*e ^5)*x)*sqrt(-a*c)*log((c*x^2 + 2*sqrt(-a*c)*x - a)/(c*x^2 + a)) - 2*(B*a^2 *c^3*d^4*e + 2*B*a^3*c^2*d^2*e^3 + B*a^4*c*e^5 - (C*a^2*c^3 - A*a*c^4)*d^5 - 2*(C*a^3*c^2 - A*a^2*c^3)*d^3*e^2 - (C*a^4*c - A*a^3*c^2)*d*e^4)*x + 2* (2*C*a^3*c^2*d^4*e - 3*B*a^3*c^2*d^3*e^2 + B*a^4*c*d*e^4 - 2*(C*a^4*c - 2* A*a^3*c^2)*d^2*e^3 + (2*C*a^2*c^3*d^3*e^2 - 3*B*a^2*c^3*d^2*e^3 + B*a^3*c^ 2*e^5 - 2*(C*a^3*c^2 - 2*A*a^2*c^3)*d*e^4)*x^3 + (2*C*a^2*c^3*d^4*e - 3*B* a^2*c^3*d^3*e^2 + B*a^3*c^2*d*e^4 - 2*(C*a^3*c^2 - 2*A*a^2*c^3)*d^2*e^3)*x ^2 + (2*C*a^3*c^2*d^3*e^2 - 3*B*a^3*c^2*d^2*e^3 + B*a^4*c*e^5 - 2*(C*a^4*c - 2*A*a^3*c^2)*d*e^4)*x)*log(c*x^2 + a) - 4*(2*C*a^3*c^2*d^4*e - 3*B*a...
Timed out. \[ \int \frac {A+B x+C x^2}{(d+e x)^2 \left (a+c x^2\right )^2} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 604, normalized size of antiderivative = 1.61 \[ \int \frac {A+B x+C x^2}{(d+e x)^2 \left (a+c x^2\right )^2} \, dx=-\frac {{\left (2 \, C c d^{3} e - 3 \, B c d^{2} e^{2} + B a e^{4} - 2 \, {\left (C a - 2 \, A c\right )} d e^{3}\right )} \log \left (c x^{2} + a\right )}{2 \, {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )}} + \frac {{\left (2 \, C c d^{3} e - 3 \, B c d^{2} e^{2} + B a e^{4} - 2 \, {\left (C a - 2 \, A c\right )} d e^{3}\right )} \log \left (e x + d\right )}{c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}} - \frac {{\left (2 \, B a c^{2} d^{3} e - 6 \, B a^{2} c d e^{3} - {\left (C a c^{2} + A c^{3}\right )} d^{4} + 6 \, {\left (C a^{2} c - A a c^{2}\right )} d^{2} e^{2} - {\left (C a^{3} - 3 \, A a^{2} c\right )} e^{4}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, {\left (a c^{3} d^{6} + 3 \, a^{2} c^{2} d^{4} e^{2} + 3 \, a^{3} c d^{2} e^{4} + a^{4} e^{6}\right )} \sqrt {a c}} - \frac {B a c d^{3} - 3 \, B a^{2} d e^{2} + 2 \, A a^{2} e^{3} + 2 \, {\left (2 \, C a^{2} - A a c\right )} d^{2} e - {\left (4 \, B a c d e^{2} - {\left (3 \, C a c - A c^{2}\right )} d^{2} e + {\left (C a^{2} - 3 \, A a c\right )} e^{3}\right )} x^{2} - {\left (B a c d^{2} e + B a^{2} e^{3} - {\left (C a c - A c^{2}\right )} d^{3} - {\left (C a^{2} - A a c\right )} d e^{2}\right )} x}{2 \, {\left (a^{2} c^{2} d^{5} + 2 \, a^{3} c d^{3} e^{2} + a^{4} d e^{4} + {\left (a c^{3} d^{4} e + 2 \, a^{2} c^{2} d^{2} e^{3} + a^{3} c e^{5}\right )} x^{3} + {\left (a c^{3} d^{5} + 2 \, a^{2} c^{2} d^{3} e^{2} + a^{3} c d e^{4}\right )} x^{2} + {\left (a^{2} c^{2} d^{4} e + 2 \, a^{3} c d^{2} e^{3} + a^{4} e^{5}\right )} x\right )}} \]
-1/2*(2*C*c*d^3*e - 3*B*c*d^2*e^2 + B*a*e^4 - 2*(C*a - 2*A*c)*d*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) + (2*C*c *d^3*e - 3*B*c*d^2*e^2 + B*a*e^4 - 2*(C*a - 2*A*c)*d*e^3)*log(e*x + d)/(c^ 3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6) - 1/2*(2*B*a*c^2*d^3* e - 6*B*a^2*c*d*e^3 - (C*a*c^2 + A*c^3)*d^4 + 6*(C*a^2*c - A*a*c^2)*d^2*e^ 2 - (C*a^3 - 3*A*a^2*c)*e^4)*arctan(c*x/sqrt(a*c))/((a*c^3*d^6 + 3*a^2*c^2 *d^4*e^2 + 3*a^3*c*d^2*e^4 + a^4*e^6)*sqrt(a*c)) - 1/2*(B*a*c*d^3 - 3*B*a^ 2*d*e^2 + 2*A*a^2*e^3 + 2*(2*C*a^2 - A*a*c)*d^2*e - (4*B*a*c*d*e^2 - (3*C* a*c - A*c^2)*d^2*e + (C*a^2 - 3*A*a*c)*e^3)*x^2 - (B*a*c*d^2*e + B*a^2*e^3 - (C*a*c - A*c^2)*d^3 - (C*a^2 - A*a*c)*d*e^2)*x)/(a^2*c^2*d^5 + 2*a^3*c* d^3*e^2 + a^4*d*e^4 + (a*c^3*d^4*e + 2*a^2*c^2*d^2*e^3 + a^3*c*e^5)*x^3 + (a*c^3*d^5 + 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4)*x^2 + (a^2*c^2*d^4*e + 2*a^3 *c*d^2*e^3 + a^4*e^5)*x)
Time = 0.28 (sec) , antiderivative size = 638, normalized size of antiderivative = 1.71 \[ \int \frac {A+B x+C x^2}{(d+e x)^2 \left (a+c x^2\right )^2} \, dx=-\frac {{\left (2 \, C c d^{3} e - 3 \, B c d^{2} e^{2} - 2 \, C a d e^{3} + 4 \, A c d e^{3} + B a e^{4}\right )} \log \left (c - \frac {2 \, c d}{e x + d} + \frac {c d^{2}}{{\left (e x + d\right )}^{2}} + \frac {a e^{2}}{{\left (e x + d\right )}^{2}}\right )}{2 \, {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )}} - \frac {\frac {C d^{2} e^{5}}{e x + d} - \frac {B d e^{6}}{e x + d} + \frac {A e^{7}}{e x + d}}{c^{2} d^{4} e^{4} + 2 \, a c d^{2} e^{6} + a^{2} e^{8}} + \frac {{\left (C a c^{2} d^{4} e^{2} + A c^{3} d^{4} e^{2} - 2 \, B a c^{2} d^{3} e^{3} - 6 \, C a^{2} c d^{2} e^{4} + 6 \, A a c^{2} d^{2} e^{4} + 6 \, B a^{2} c d e^{5} + C a^{3} e^{6} - 3 \, A a^{2} c e^{6}\right )} \arctan \left (\frac {c d - \frac {c d^{2}}{e x + d} - \frac {a e^{2}}{e x + d}}{\sqrt {a c} e}\right )}{2 \, {\left (a c^{3} d^{6} + 3 \, a^{2} c^{2} d^{4} e^{2} + 3 \, a^{3} c d^{2} e^{4} + a^{4} e^{6}\right )} \sqrt {a c} e^{2}} - \frac {\frac {C a c^{2} d^{3} e - A c^{3} d^{3} e - 3 \, B a c^{2} d^{2} e^{2} - 3 \, C a^{2} c d e^{3} + 3 \, A a c^{2} d e^{3} + B a^{2} c e^{4}}{c d^{2} + a e^{2}} - \frac {C a c^{2} d^{4} e^{2} - A c^{3} d^{4} e^{2} - 4 \, B a c^{2} d^{3} e^{3} - 6 \, C a^{2} c d^{2} e^{4} + 6 \, A a c^{2} d^{2} e^{4} + 4 \, B a^{2} c d e^{5} + C a^{3} e^{6} - A a^{2} c e^{6}}{{\left (c d^{2} + a e^{2}\right )} {\left (e x + d\right )} e}}{2 \, {\left (c d^{2} + a e^{2}\right )}^{2} a {\left (c - \frac {2 \, c d}{e x + d} + \frac {c d^{2}}{{\left (e x + d\right )}^{2}} + \frac {a e^{2}}{{\left (e x + d\right )}^{2}}\right )}} \]
-1/2*(2*C*c*d^3*e - 3*B*c*d^2*e^2 - 2*C*a*d*e^3 + 4*A*c*d*e^3 + B*a*e^4)*l og(c - 2*c*d/(e*x + d) + c*d^2/(e*x + d)^2 + a*e^2/(e*x + d)^2)/(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^5/(e*x + d) - B*d *e^6/(e*x + d) + A*e^7/(e*x + d))/(c^2*d^4*e^4 + 2*a*c*d^2*e^6 + a^2*e^8) + 1/2*(C*a*c^2*d^4*e^2 + A*c^3*d^4*e^2 - 2*B*a*c^2*d^3*e^3 - 6*C*a^2*c*d^2 *e^4 + 6*A*a*c^2*d^2*e^4 + 6*B*a^2*c*d*e^5 + C*a^3*e^6 - 3*A*a^2*c*e^6)*ar ctan((c*d - c*d^2/(e*x + d) - a*e^2/(e*x + d))/(sqrt(a*c)*e))/((a*c^3*d^6 + 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 + a^4*e^6)*sqrt(a*c)*e^2) - 1/2*((C* a*c^2*d^3*e - A*c^3*d^3*e - 3*B*a*c^2*d^2*e^2 - 3*C*a^2*c*d*e^3 + 3*A*a*c^ 2*d*e^3 + B*a^2*c*e^4)/(c*d^2 + a*e^2) - (C*a*c^2*d^4*e^2 - A*c^3*d^4*e^2 - 4*B*a*c^2*d^3*e^3 - 6*C*a^2*c*d^2*e^4 + 6*A*a*c^2*d^2*e^4 + 4*B*a^2*c*d* e^5 + C*a^3*e^6 - A*a^2*c*e^6)/((c*d^2 + a*e^2)*(e*x + d)*e))/((c*d^2 + a* e^2)^2*a*(c - 2*c*d/(e*x + d) + c*d^2/(e*x + d)^2 + a*e^2/(e*x + d)^2))
Time = 20.36 (sec) , antiderivative size = 2094, normalized size of antiderivative = 5.60 \[ \int \frac {A+B x+C x^2}{(d+e x)^2 \left (a+c x^2\right )^2} \, dx=\text {Too large to display} \]
((x^2*(C*a^2*e^3 - 3*A*a*c*e^3 + A*c^2*d^2*e + 4*B*a*c*d*e^2 - 3*C*a*c*d^2 *e))/(2*a*(a^2*e^4 + c^2*d^4 + 2*a*c*d^2*e^2)) - (2*A*a*e^3 + B*c*d^3 - 3* B*a*d*e^2 - 2*A*c*d^2*e + 4*C*a*d^2*e)/(2*(a*e^2 + c*d^2)^2) + (x*(A*c*d + B*a*e - C*a*d))/(2*a*(a*e^2 + c*d^2)))/(a*d + a*e*x + c*d*x^2 + c*e*x^3) - (log(3*A*e^6*(-a^3*c)^(3/2) - A*c^4*d^6*(-a^3*c)^(1/2) + C*a^4*e^6*(-a^3 *c)^(1/2) + 31*C*d^2*e^4*(-a^3*c)^(3/2) + 6*B*a^5*c*e^6 - 18*B*d*e^5*(-a^3 *c)^(3/2) - 6*B*e^6*x*(-a^3*c)^(3/2) - C*a^5*c*e^6*x + 14*C*d*e^5*x*(-a^3* c)^(3/2) - 2*A*a^2*c^4*d^5*e + 30*A*a^4*c^2*d*e^5 - 14*C*a^3*c^3*d^5*e + 3 *A*a^4*c^2*e^6*x + C*a^2*c^4*d^6*x - C*a*c^3*d^6*(-a^3*c)^(1/2) - 36*A*a^3 *c^3*d^3*e^3 + 22*B*a^3*c^3*d^4*e^2 - 36*B*a^4*c^2*d^2*e^4 + 36*C*a^4*c^2* d^3*e^3 - 14*C*a^5*c*d*e^5 + A*a*c^5*d^6*x + 5*A*a^2*c^4*d^4*e^2*x - 57*A* a^3*c^3*d^2*e^4*x + 44*B*a^3*c^3*d^3*e^3*x - 31*C*a^3*c^3*d^4*e^2*x + 31*C *a^4*c^2*d^2*e^4*x - 5*A*a*c^3*d^4*e^2*(-a^3*c)^(1/2) + 57*A*a^2*c^2*d^2*e ^4*(-a^3*c)^(1/2) - 44*B*a^2*c^2*d^3*e^3*(-a^3*c)^(1/2) + 31*C*a^2*c^2*d^4 *e^2*(-a^3*c)^(1/2) - 2*B*a^2*c^4*d^5*e*x - 18*B*a^4*c^2*d*e^5*x + 2*B*a*c ^3*d^5*e*(-a^3*c)^(1/2) - 2*A*c^4*d^5*e*x*(-a^3*c)^(1/2) - 36*B*a^2*c^2*d^ 2*e^4*x*(-a^3*c)^(1/2) + 36*C*a^2*c^2*d^3*e^3*x*(-a^3*c)^(1/2) - 14*C*a*c^ 3*d^5*e*x*(-a^3*c)^(1/2) - 36*A*a*c^3*d^3*e^3*x*(-a^3*c)^(1/2) + 30*A*a^2* c^2*d*e^5*x*(-a^3*c)^(1/2) + 22*B*a*c^3*d^4*e^2*x*(-a^3*c)^(1/2))*(c^2*(a* ((C*d^4*(-a^3*c)^(1/2))/4 + (3*A*d^2*e^2*(-a^3*c)^(1/2))/2 - (B*d^3*e*(...