Integrand size = 27, antiderivative size = 524 \[ \int \frac {A+B x+C x^2}{(d+e x)^3 \left (a+c x^2\right )^2} \, dx=-\frac {e \left (C d^2-B d e+A e^2\right )}{2 \left (c d^2+a e^2\right )^2 (d+e x)^2}+\frac {e \left (a e^2 (2 C d-B e)-c d \left (2 C d^2-e (3 B d-4 A e)\right )\right )}{\left (c d^2+a e^2\right )^3 (d+e x)}-\frac {a \left (B c d \left (c d^2-3 a e^2\right )-(A c-a C) e \left (3 c d^2-a e^2\right )\right )-c \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right ) x}{2 a \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )}+\frac {\sqrt {c} \left (A c d \left (c^2 d^4+10 a c d^2 e^2-15 a^2 e^4\right )-a \left (2 a c d^2 e^2 (7 C d-9 B e)-c^2 d^4 (C d-3 B e)-3 a^2 e^4 (3 C d-B e)\right )\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} \left (c d^2+a e^2\right )^4}+\frac {e \left (a^2 C e^4+c^2 d^2 \left (3 C d^2-2 e (3 B d-5 A e)\right )-2 a c e^2 \left (4 C d^2-e (3 B d-A e)\right )\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^4}-\frac {e \left (a^2 C e^4+c^2 d^2 \left (3 C d^2-2 e (3 B d-5 A e)\right )-2 a c e^2 \left (4 C d^2-e (3 B d-A e)\right )\right ) \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^4} \]
-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
Time = 0.37 (sec) , antiderivative size = 466, normalized size of antiderivative = 0.89 \[ \int \frac {A+B x+C x^2}{(d+e x)^3 \left (a+c x^2\right )^2} \, dx=\frac {-\frac {e \left (c d^2+a e^2\right )^2 \left (C d^2+e (-B d+A e)\right )}{(d+e x)^2}-\frac {2 e \left (c d^2+a e^2\right ) \left (2 c C d^3+c d e (-3 B d+4 A e)+a e^2 (-2 C d+B e)\right )}{d+e x}+\frac {\left (c d^2+a e^2\right ) \left (a^3 C e^3+A c^3 d^3 x-a c^2 d \left (C d^2 x+B d (d-3 e x)+3 A e (-d+e x)\right )-a^2 c e (3 C d (d-e x)+e (-3 B d+A e+B e x))\right )}{a \left (a+c x^2\right )}+\frac {\sqrt {c} \left (A c d \left (c^2 d^4+10 a c d^2 e^2-15 a^2 e^4\right )+a \left (-2 a c d^2 e^2 (7 C d-9 B e)+c^2 d^4 (C d-3 B e)-3 a^2 e^4 (-3 C d+B e)\right )\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{3/2}}+2 \left (a^2 C e^5-2 a c e^3 \left (4 C d^2+e (-3 B d+A e)\right )+c^2 d^2 e \left (3 C d^2+2 e (-3 B d+5 A e)\right )\right ) \log (d+e x)-\left (a^2 C e^5-2 a c e^3 \left (4 C d^2+e (-3 B d+A e)\right )+c^2 d^2 e \left (3 C d^2+2 e (-3 B d+5 A e)\right )\right ) \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^4} \]
(-((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)
Time = 1.83 (sec) , antiderivative size = 537, normalized size of antiderivative = 1.02, 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.
| \(\displaystyle \int \frac {A+B x+C x^2}{\left (a+c x^2\right )^2 (d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 2178 |
| \(\displaystyle -\frac {\int -\frac {\frac {c^2 e^3 \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right ) x^3}{\left (c d^2+a e^2\right )^3}+\frac {c e^2 \left (A c \left (3 c^2 d^4-3 a c e^2 d^2-2 a^2 e^4\right )+a \left (2 a^2 C e^4+3 a c d (C d+B e) e^2-c^2 d^3 (3 C d-7 B e)\right )\right ) x^2}{\left (c d^2+a e^2\right )^3}+\frac {c e \left (A c^2 \left (3 c d^2+7 a e^2\right ) d^3+a \left (2 a^2 B e^5-a c d^2 (7 C d-9 B e) e^2-3 c^2 d^4 (C d-B e)\right )\right ) x}{\left (c d^2+a e^2\right )^3}+\frac {c \left (a c \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right ) d^3+A \left (c^3 d^6+9 a c^2 e^2 d^4+6 a^2 c e^4 d^2+2 a^3 e^6\right )\right )}{\left (c d^2+a e^2\right )^3}}{(d+e x)^3 \left (c x^2+a\right )}dx}{2 a c}-\frac {a \left (B c d \left (c d^2-3 a e^2\right )-e (A c-a C) \left (3 c d^2-a e^2\right )\right )-c x \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right )}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\frac {c^2 e^3 \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right ) x^3}{\left (c d^2+a e^2\right )^3}+\frac {c e^2 \left (A c \left (3 c^2 d^4-3 a c e^2 d^2-2 a^2 e^4\right )+a \left (2 a^2 C e^4+3 a c d (C d+B e) e^2-c^2 d^3 (3 C d-7 B e)\right )\right ) x^2}{\left (c d^2+a e^2\right )^3}+\frac {c e \left (A c^2 \left (3 c d^2+7 a e^2\right ) d^3+a \left (2 a^2 B e^5-a c d^2 (7 C d-9 B e) e^2-3 c^2 d^4 (C d-B e)\right )\right ) x}{\left (c d^2+a e^2\right )^3}+\frac {c \left (a c \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right ) d^3+A \left (c^3 d^6+9 a c^2 e^2 d^4+6 a^2 c e^4 d^2+2 a^3 e^6\right )\right )}{\left (c d^2+a e^2\right )^3}}{(d+e x)^3 \left (c x^2+a\right )}dx}{2 a c}-\frac {a \left (B c d \left (c d^2-3 a e^2\right )-e (A c-a C) \left (3 c d^2-a e^2\right )\right )-c x \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right )}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )^3}\) |
| \(\Big \downarrow \) 2160 |
| \(\displaystyle \frac {\int \left (\frac {\left (A c d \left (c^2 d^4+10 a c e^2 d^2-15 a^2 e^4\right )-a \left (-c^2 (C d-3 B e) d^4+2 a c e^2 (7 C d-9 B e) d^2-3 a^2 e^4 (3 C d-B e)\right )-2 a e \left (a^2 C e^4-2 a c \left (4 C d^2-e (3 B d-A e)\right ) e^2+c^2 \left (3 C d^4-2 d^2 e (3 B d-5 A e)\right )\right ) x\right ) c^2}{\left (c d^2+a e^2\right )^4 \left (c x^2+a\right )}+\frac {2 a e^2 \left (a^2 C e^4-2 a c \left (4 C d^2-e (3 B d-A e)\right ) e^2+c^2 \left (3 C d^4-2 d^2 e (3 B d-5 A e)\right )\right ) c}{\left (c d^2+a e^2\right )^4 (d+e x)}+\frac {2 a e^2 \left (2 c C d^3-c e (3 B d-4 A e) d-a e^2 (2 C d-B e)\right ) c}{\left (c d^2+a e^2\right )^3 (d+e x)^2}+\frac {2 a e^2 \left (C d^2-B e d+A e^2\right ) c}{\left (c d^2+a e^2\right )^2 (d+e x)^3}\right )dx}{2 a c}-\frac {a \left (B c d \left (c d^2-3 a e^2\right )-e (A c-a C) \left (3 c d^2-a e^2\right )\right )-c x \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right )}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {c^{3/2} \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (A c d \left (-15 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right )-a \left (-3 a^2 e^4 (3 C d-B e)+2 a c d^2 e^2 (7 C d-9 B e)-c^2 d^4 (C d-3 B e)\right )\right )}{\sqrt {a} \left (a e^2+c d^2\right )^4}-\frac {a c e \log \left (a+c x^2\right ) \left (a^2 C e^4-2 a c e^2 \left (4 C d^2-e (3 B d-A e)\right )+c^2 \left (3 C d^4-2 d^2 e (3 B d-5 A e)\right )\right )}{\left (a e^2+c d^2\right )^4}+\frac {2 a c e \log (d+e x) \left (a^2 C e^4-2 a c e^2 \left (4 C d^2-e (3 B d-A e)\right )+c^2 \left (3 C d^4-2 d^2 e (3 B d-5 A e)\right )\right )}{\left (a e^2+c d^2\right )^4}-\frac {a c e \left (A e^2-B d e+C d^2\right )}{(d+e x)^2 \left (a e^2+c d^2\right )^2}-\frac {2 a c e \left (-a e^2 (2 C d-B e)-c d e (3 B d-4 A e)+2 c C d^3\right )}{(d+e x) \left (a e^2+c d^2\right )^3}}{2 a c}-\frac {a \left (B c d \left (c d^2-3 a e^2\right )-e (A c-a C) \left (3 c d^2-a e^2\right )\right )-c x \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right )}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )^3}\) |
-1/2*(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) /(a*(c*d^2 + a*e^2)^3*(a + c*x^2)) + (-((a*c*e*(C*d^2 - B*d*e + A*e^2))/(( c*d^2 + a*e^2)^2*(d + e*x)^2)) - (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)))/((c*d^2 + a*e^2)^3*(d + e*x)) + (c^(3/2)*(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)/S qrt[a]])/(Sqrt[a]*(c*d^2 + a*e^2)^4) + (2*a*c*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 - (a*c*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])/(c* d^2 + a*e^2)^4)/(2*a*c)
3.1.56.3.1 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 = 0.68 (sec) , antiderivative size = 643, normalized size of antiderivative = 1.23
| method | result | size |
| default | \(-\frac {c \left (\frac {\frac {\left (3 A \,a^{2} c d \,e^{4}+2 A a \,c^{2} d^{3} e^{2}-d^{5} A \,c^{3}+B \,e^{5} a^{3}-2 B \,a^{2} c \,d^{2} e^{3}-3 B a \,c^{2} d^{4} e -3 C \,a^{3} d \,e^{4}-2 C \,a^{2} c \,d^{3} e^{2}+C \,c^{2} d^{5} a \right ) x}{2 a}+\frac {A \,a^{2} c \,e^{5}-2 A a \,c^{2} d^{2} e^{3}-3 A \,c^{3} d^{4} e -3 B \,a^{2} c d \,e^{4}-2 B a \,c^{2} d^{3} e^{2}+B \,c^{3} d^{5}-C \,a^{3} e^{5}+2 C \,a^{2} c \,d^{2} e^{3}+3 C a \,c^{2} d^{4} e}{2 c}}{c \,x^{2}+a}+\frac {\frac {\left (-4 A \,a^{2} c \,e^{5}+20 A a \,c^{2} d^{2} e^{3}+12 B \,a^{2} c d \,e^{4}-12 B a \,c^{2} d^{3} e^{2}+2 C \,a^{3} e^{5}-16 C \,a^{2} c \,d^{2} e^{3}+6 C a \,c^{2} d^{4} e \right ) \ln \left (c \,x^{2}+a \right )}{2 c}+\frac {\left (15 A \,a^{2} c d \,e^{4}-10 A a \,c^{2} d^{3} e^{2}-d^{5} A \,c^{3}+3 B \,e^{5} a^{3}-18 B \,a^{2} c \,d^{2} e^{3}+3 B a \,c^{2} d^{4} e -9 C \,a^{3} d \,e^{4}+14 C \,a^{2} c \,d^{3} e^{2}-C \,c^{2} d^{5} a \right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}}{2 a}\right )}{\left (e^{2} a +c \,d^{2}\right )^{4}}-\frac {e \left (4 A c d \,e^{2}+B \,e^{3} a -3 B c \,d^{2} e -2 C a d \,e^{2}+2 C c \,d^{3}\right )}{\left (e^{2} a +c \,d^{2}\right )^{3} \left (e x +d \right )}-\frac {e \left (A \,e^{2}-B d e +C \,d^{2}\right )}{2 \left (e^{2} a +c \,d^{2}\right )^{2} \left (e x +d \right )^{2}}-\frac {e \left (2 A a c \,e^{4}-10 A \,c^{2} d^{2} e^{2}-6 B a c d \,e^{3}+6 B \,c^{2} d^{3} e -a^{2} C \,e^{4}+8 C a c \,d^{2} e^{2}-3 C \,c^{2} d^{4}\right ) \ln \left (e x +d \right )}{\left (e^{2} a +c \,d^{2}\right )^{4}}\) | \(643\) |
| risch | \(\text {Expression too large to display}\) | \(64073\) |
-c/(a*e^2+c*d^2)^4*((1/2*(3*A*a^2*c*d*e^4+2*A*a*c^2*d^3*e^2-A*c^3*d^5+B*a^ 3*e^5-2*B*a^2*c*d^2*e^3-3*B*a*c^2*d^4*e-3*C*a^3*d*e^4-2*C*a^2*c*d^3*e^2+C* a*c^2*d^5)/a*x+1/2*(A*a^2*c*e^5-2*A*a*c^2*d^2*e^3-3*A*c^3*d^4*e-3*B*a^2*c* d*e^4-2*B*a*c^2*d^3*e^2+B*c^3*d^5-C*a^3*e^5+2*C*a^2*c*d^2*e^3+3*C*a*c^2*d^ 4*e)/c)/(c*x^2+a)+1/2/a*(1/2*(-4*A*a^2*c*e^5+20*A*a*c^2*d^2*e^3+12*B*a^2*c *d*e^4-12*B*a*c^2*d^3*e^2+2*C*a^3*e^5-16*C*a^2*c*d^2*e^3+6*C*a*c^2*d^4*e)/ c*ln(c*x^2+a)+(15*A*a^2*c*d*e^4-10*A*a*c^2*d^3*e^2-A*c^3*d^5+3*B*a^3*e^5-1 8*B*a^2*c*d^2*e^3+3*B*a*c^2*d^4*e-9*C*a^3*d*e^4+14*C*a^2*c*d^3*e^2-C*a*c^2 *d^5)/(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/(e*x+d)-1/2*e*(A*e^2-B*d*e+C*d ^2)/(a*e^2+c*d^2)^2/(e*x+d)^2-e*(2*A*a*c*e^4-10*A*c^2*d^2*e^2-6*B*a*c*d*e^ 3+6*B*c^2*d^3*e-C*a^2*e^4+8*C*a*c*d^2*e^2-3*C*c^2*d^4)/(a*e^2+c*d^2)^4*ln( e*x+d)
Timed out. \[ \int \frac {A+B x+C x^2}{(d+e x)^3 \left (a+c x^2\right )^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {A+B x+C x^2}{(d+e x)^3 \left (a+c x^2\right )^2} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 1030 vs. \(2 (505) = 1010\).
Time = 0.32 (sec) , antiderivative size = 1030, normalized size of antiderivative = 1.97 \[ \int \frac {A+B x+C x^2}{(d+e x)^3 \left (a+c x^2\right )^2} \, dx=-\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 )}} \]
-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)*sqr t(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 + ...
Leaf count of result is larger than twice the leaf count of optimal. 1013 vs. \(2 (505) = 1010\).
Time = 0.27 (sec) , antiderivative size = 1013, normalized size of antiderivative = 1.93 \[ \int \frac {A+B x+C x^2}{(d+e x)^3 \left (a+c x^2\right )^2} \, dx=-\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 | e x + 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 (e x + d\right )}^{2} a} \]
-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(e*x + 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 + 1 8*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 +...
Time = 27.67 (sec) , antiderivative size = 2828, normalized size of antiderivative = 5.40 \[ \int \frac {A+B x+C x^2}{(d+e x)^3 \left (a+c x^2\right )^2} \, dx=\text {Too large to display} \]
(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 + 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 - 14 5*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*...