Integrand size = 27, antiderivative size = 753 \[ \int \frac {A+B x+C x^2}{(d+e x)^3 \left (a+c x^2\right )^3} \, dx=-\frac {e^3 \left (C d^2-B d e+A e^2\right )}{2 \left (c d^2+a e^2\right )^3 (d+e x)^2}+\frac {e^3 \left (a e^2 (2 C d-B e)-c d \left (4 C d^2-e (5 B d-6 A e)\right )\right )}{\left (c d^2+a e^2\right )^4 (d+e x)}-\frac {a \left (B c d \left (c d^2-3 a e^2\right )-(A c-a C) e \left (3 c d^2-a e^2\right )\right )-c \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right ) x}{4 a \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )^2}+\frac {4 a^2 e \left (a^2 C e^4+c^2 d^2 \left (3 C d^2-2 e (3 B d-5 A e)\right )-2 a c e^2 \left (4 C d^2-e (3 B d-A e)\right )\right )+c \left (3 A c d \left (c^2 d^4+6 a c d^2 e^2-11 a^2 e^4\right )-a \left (2 a c d^2 e^2 (13 C d-19 B e)-c^2 d^4 (C d-3 B e)-7 a^2 e^4 (3 C d-B e)\right )\right ) x}{8 a^2 \left (c d^2+a e^2\right )^4 \left (a+c x^2\right )}+\frac {\sqrt {c} \left (3 A c d \left (c^3 d^6+7 a c^2 d^4 e^2+35 a^2 c d^2 e^4-35 a^3 e^6\right )+a \left (a c^2 d^4 e^2 (23 C d-45 B e)-5 a^2 c d^2 e^4 (25 C d-27 B e)+c^3 d^6 (C d-3 B e)+15 a^3 e^6 (3 C d-B e)\right )\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} \left (c d^2+a e^2\right )^5}+\frac {e^3 \left (a^2 C e^4-a c e^2 \left (13 C d^2-9 B d e+3 A e^2\right )+c^2 d^2 \left (10 C d^2-3 e (5 B d-7 A e)\right )\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^5}-\frac {e^3 \left (a^2 C e^4-a c e^2 \left (13 C d^2-9 B d e+3 A e^2\right )+c^2 d^2 \left (10 C d^2-3 e (5 B d-7 A e)\right )\right ) \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^5} \]
-1/2*e^3*(A*e^2-B*d*e+C*d^2)/(a*e^2+c*d^2)^3/(e*x+d)^2+e^3*(a*e^2*(-B*e+2* C*d)-c*d*(4*C*d^2-e*(-6*A*e+5*B*d)))/(a*e^2+c*d^2)^4/(e*x+d)+1/4*(-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) ^2+1/8*(4*a^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)))+c*(3*A*c*d*(-11*a^2*e^4+6*a*c*d^2*e^2+c^2*d^4)-a *(2*a*c*d^2*e^2*(-19*B*e+13*C*d)-c^2*d^4*(-3*B*e+C*d)-7*a^2*e^4*(-B*e+3*C* d)))*x)/a^2/(a*e^2+c*d^2)^4/(c*x^2+a)+e^3*(a^2*C*e^4-a*c*e^2*(3*A*e^2-9*B* d*e+13*C*d^2)+c^2*d^2*(10*C*d^2-3*e*(-7*A*e+5*B*d)))*ln(e*x+d)/(a*e^2+c*d^ 2)^5-1/2*e^3*(a^2*C*e^4-a*c*e^2*(3*A*e^2-9*B*d*e+13*C*d^2)+c^2*d^2*(10*C*d ^2-3*e*(-7*A*e+5*B*d)))*ln(c*x^2+a)/(a*e^2+c*d^2)^5+1/8*(3*A*c*d*(-35*a^3* e^6+35*a^2*c*d^2*e^4+7*a*c^2*d^4*e^2+c^3*d^6)+a*(a*c^2*d^4*e^2*(-45*B*e+23 *C*d)-5*a^2*c*d^2*e^4*(-27*B*e+25*C*d)+c^3*d^6*(-3*B*e+C*d)+15*a^3*e^6*(-B *e+3*C*d)))*arctan(x*c^(1/2)/a^(1/2))*c^(1/2)/a^(5/2)/(a*e^2+c*d^2)^5
Time = 0.58 (sec) , antiderivative size = 672, normalized size of antiderivative = 0.89 \[ \int \frac {A+B x+C x^2}{(d+e x)^3 \left (a+c x^2\right )^3} \, dx=\frac {-\frac {4 e^3 \left (c d^2+a e^2\right )^2 \left (C d^2+e (-B d+A e)\right )}{(d+e x)^2}-\frac {8 e^3 \left (c d^2+a e^2\right ) \left (4 c C d^3+c d e (-5 B d+6 A e)+a e^2 (-2 C d+B e)\right )}{d+e x}+\frac {2 \left (c d^2+a e^2\right )^2 \left (a^3 C e^3+A c^3 d^3 x-a c^2 d \left (C d^2 x+B d (d-3 e x)+3 A e (-d+e x)\right )-a^2 c e (3 C d (d-e x)+e (-3 B d+A e+B e x))\right )}{a \left (a+c x^2\right )^2}+\frac {\left (c d^2+a e^2\right ) \left (4 a^4 C e^5+3 A c^4 d^5 x+a c^3 d^3 \left (C d^2+3 e (-B d+6 A e)\right ) x+a^3 c e^3 (C d (-32 d+21 e x)+e (24 B d-8 A e-7 B e x))+a^2 c^2 d e \left (2 C d^2 (6 d-13 e x)+e \left (-24 B d^2+40 A d e+38 B d e x-33 A e^2 x\right )\right )\right )}{a^2 \left (a+c x^2\right )}+\frac {\sqrt {c} \left (3 A c d \left (c^3 d^6+7 a c^2 d^4 e^2+35 a^2 c d^2 e^4-35 a^3 e^6\right )+a \left (a c^2 d^4 e^2 (23 C d-45 B e)-5 a^2 c d^2 e^4 (25 C d-27 B e)+c^3 d^6 (C d-3 B e)-15 a^3 e^6 (-3 C d+B e)\right )\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{5/2}}+8 \left (a^2 C e^7+a c e^5 \left (-13 C d^2+9 B d e-3 A e^2\right )+c^2 d^2 e^3 \left (10 C d^2+3 e (-5 B d+7 A e)\right )\right ) \log (d+e x)-4 \left (a^2 C e^7+a c e^5 \left (-13 C d^2+9 B d e-3 A e^2\right )+c^2 d^2 e^3 \left (10 C d^2+3 e (-5 B d+7 A e)\right )\right ) \log \left (a+c x^2\right )}{8 \left (c d^2+a e^2\right )^5} \]
((-4*e^3*(c*d^2 + a*e^2)^2*(C*d^2 + e*(-(B*d) + A*e)))/(d + e*x)^2 - (8*e^ 3*(c*d^2 + a*e^2)*(4*c*C*d^3 + c*d*e*(-5*B*d + 6*A*e) + a*e^2*(-2*C*d + B* e)))/(d + e*x) + (2*(c*d^2 + a*e^2)^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)^2) + ((c*d^2 + a*e^2)*(4*a^4*C *e^5 + 3*A*c^4*d^5*x + a*c^3*d^3*(C*d^2 + 3*e*(-(B*d) + 6*A*e))*x + a^3*c* e^3*(C*d*(-32*d + 21*e*x) + e*(24*B*d - 8*A*e - 7*B*e*x)) + a^2*c^2*d*e*(2 *C*d^2*(6*d - 13*e*x) + e*(-24*B*d^2 + 40*A*d*e + 38*B*d*e*x - 33*A*e^2*x) )))/(a^2*(a + c*x^2)) + (Sqrt[c]*(3*A*c*d*(c^3*d^6 + 7*a*c^2*d^4*e^2 + 35* a^2*c*d^2*e^4 - 35*a^3*e^6) + a*(a*c^2*d^4*e^2*(23*C*d - 45*B*e) - 5*a^2*c *d^2*e^4*(25*C*d - 27*B*e) + c^3*d^6*(C*d - 3*B*e) - 15*a^3*e^6*(-3*C*d + B*e)))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/a^(5/2) + 8*(a^2*C*e^7 + a*c*e^5*(-13* C*d^2 + 9*B*d*e - 3*A*e^2) + c^2*d^2*e^3*(10*C*d^2 + 3*e*(-5*B*d + 7*A*e)) )*Log[d + e*x] - 4*(a^2*C*e^7 + a*c*e^5*(-13*C*d^2 + 9*B*d*e - 3*A*e^2) + c^2*d^2*e^3*(10*C*d^2 + 3*e*(-5*B*d + 7*A*e)))*Log[a + c*x^2])/(8*(c*d^2 + a*e^2)^5)
Time = 3.75 (sec) , antiderivative size = 794, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2178, 25, 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.
| \(\displaystyle \int \frac {A+B x+C x^2}{\left (a+c x^2\right )^3 (d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 2178 |
| \(\displaystyle -\frac {\int -\frac {\frac {3 c^2 e^3 \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right ) x^3}{\left (c d^2+a e^2\right )^3}+\frac {c e^2 \left (A c \left (9 c^2 d^4-15 a c e^2 d^2-4 a^2 e^4\right )+a \left (4 a^2 C e^4+3 a c d (5 C d+B e) e^2-c^2 d^3 (9 C d-23 B e)\right )\right ) x^2}{\left (c d^2+a e^2\right )^3}+\frac {c e \left (A c^2 \left (9 c d^2+5 a e^2\right ) d^3+a \left (4 a^2 B e^5-5 a c d^2 (C d-3 B e) e^2-3 c^2 d^4 (3 C d-5 B e)\right )\right ) x}{\left (c d^2+a e^2\right )^3}+\frac {c \left (a c \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right ) d^3+A \left (3 c^3 d^6+15 a c^2 e^2 d^4+12 a^2 c e^4 d^2+4 a^3 e^6\right )\right )}{\left (c d^2+a e^2\right )^3}}{(d+e x)^3 \left (c x^2+a\right )^2}dx}{4 a c}-\frac {a \left (B c d \left (c d^2-3 a e^2\right )-e (A c-a C) \left (3 c d^2-a e^2\right )\right )-c x \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right )}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\frac {3 c^2 e^3 \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right ) x^3}{\left (c d^2+a e^2\right )^3}+\frac {c e^2 \left (A c \left (9 c^2 d^4-15 a c e^2 d^2-4 a^2 e^4\right )+a \left (4 a^2 C e^4+3 a c d (5 C d+B e) e^2-c^2 d^3 (9 C d-23 B e)\right )\right ) x^2}{\left (c d^2+a e^2\right )^3}+\frac {c e \left (A c^2 \left (9 c d^2+5 a e^2\right ) d^3+a \left (4 a^2 B e^5-5 a c d^2 (C d-3 B e) e^2-3 c^2 d^4 (3 C d-5 B e)\right )\right ) x}{\left (c d^2+a e^2\right )^3}+\frac {c \left (a c \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right ) d^3+A \left (3 c^3 d^6+15 a c^2 e^2 d^4+12 a^2 c e^4 d^2+4 a^3 e^6\right )\right )}{\left (c d^2+a e^2\right )^3}}{(d+e x)^3 \left (c x^2+a\right )^2}dx}{4 a c}-\frac {a \left (B c d \left (c d^2-3 a e^2\right )-e (A c-a C) \left (3 c d^2-a e^2\right )\right )-c x \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right )}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )^3}\) |
| \(\Big \downarrow \) 2178 |
| \(\displaystyle \frac {\frac {c \left (4 e \left (a^2 C e^4-2 a c \left (4 C d^2-e (3 B d-A e)\right ) e^2+c^2 \left (3 C d^4-2 d^2 e (3 B d-5 A e)\right )\right ) a^2+c \left (3 A c d \left (c^2 d^4+6 a c e^2 d^2-11 a^2 e^4\right )-a \left (-c^2 (C d-3 B e) d^4+2 a c e^2 (13 C d-19 B e) d^2-7 a^2 e^4 (3 C d-B e)\right )\right ) x\right )}{2 a \left (c d^2+a e^2\right )^4 \left (c x^2+a\right )}-\frac {\int -\frac {\frac {c^3 e^3 \left (3 A c d \left (c^2 d^4+6 a c e^2 d^2-11 a^2 e^4\right )-a \left (-c^2 (C d-3 B e) d^4+2 a c e^2 (13 C d-19 B e) d^2-7 a^2 e^4 (3 C d-B e)\right )\right ) x^3}{\left (c d^2+a e^2\right )^4}+\frac {c^2 e^2 \left (A c \left (9 c^3 d^6+54 a c^2 e^2 d^4-19 a^2 c e^4 d^2-16 a^3 e^6\right )+a \left (8 a^3 C e^6-a^2 c d (C d-27 B e) e^4-6 a c^2 d^3 (9 C d-11 B e) e^2+3 c^3 d^5 (C d-3 B e)\right )\right ) x^2}{\left (c d^2+a e^2\right )^4}+\frac {c^2 e \left (3 A c^2 \left (3 c^2 d^4+18 a c e^2 d^2+31 a^2 e^4\right ) d^3+a \left (8 a^3 B e^7-a^2 c d^2 (65 C d-59 B e) e^4-2 a c^2 d^4 (7 C d+3 B e) e^2+3 c^3 d^6 (C d-3 B e)\right )\right ) x}{\left (c d^2+a e^2\right )^4}+\frac {c^2 \left (a c \left (c^2 (C d-3 B e) d^4+2 a c e^2 (11 C d-21 B e) d^2-9 a^2 e^4 (3 C d-B e)\right ) d^3+A \left (3 c^4 d^8+18 a c^3 e^2 d^6+87 a^2 c^2 e^4 d^4+32 a^3 c e^6 d^2+8 a^4 e^8\right )\right )}{\left (c d^2+a e^2\right )^4}}{(d+e x)^3 \left (c x^2+a\right )}dx}{2 a c}}{4 a c}-\frac {a \left (B c d \left (c d^2-3 a e^2\right )-(A c-a C) e \left (3 c d^2-a e^2\right )\right )-c \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right ) x}{4 a \left (c d^2+a e^2\right )^3 \left (c x^2+a\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {c \left (4 e \left (a^2 C e^4-2 a c \left (4 C d^2-e (3 B d-A e)\right ) e^2+c^2 \left (3 C d^4-2 d^2 e (3 B d-5 A e)\right )\right ) a^2+c \left (3 A c d \left (c^2 d^4+6 a c e^2 d^2-11 a^2 e^4\right )-a \left (-c^2 (C d-3 B e) d^4+2 a c e^2 (13 C d-19 B e) d^2-7 a^2 e^4 (3 C d-B e)\right )\right ) x\right )}{2 a \left (c d^2+a e^2\right )^4 \left (c x^2+a\right )}+\frac {\int \frac {\frac {c^3 e^3 \left (3 A c d \left (c^2 d^4+6 a c e^2 d^2-11 a^2 e^4\right )-a \left (-c^2 (C d-3 B e) d^4+2 a c e^2 (13 C d-19 B e) d^2-7 a^2 e^4 (3 C d-B e)\right )\right ) x^3}{\left (c d^2+a e^2\right )^4}+\frac {c^2 e^2 \left (A c \left (9 c^3 d^6+54 a c^2 e^2 d^4-19 a^2 c e^4 d^2-16 a^3 e^6\right )+a \left (8 a^3 C e^6-a^2 c d (C d-27 B e) e^4-6 a c^2 d^3 (9 C d-11 B e) e^2+3 c^3 d^5 (C d-3 B e)\right )\right ) x^2}{\left (c d^2+a e^2\right )^4}+\frac {c^2 e \left (3 A c^2 \left (3 c^2 d^4+18 a c e^2 d^2+31 a^2 e^4\right ) d^3+a \left (8 a^3 B e^7-a^2 c d^2 (65 C d-59 B e) e^4-2 a c^2 d^4 (7 C d+3 B e) e^2+3 c^3 d^6 (C d-3 B e)\right )\right ) x}{\left (c d^2+a e^2\right )^4}+\frac {c^2 \left (a c \left (c^2 (C d-3 B e) d^4+2 a c e^2 (11 C d-21 B e) d^2-9 a^2 e^4 (3 C d-B e)\right ) d^3+A \left (3 c^4 d^8+18 a c^3 e^2 d^6+87 a^2 c^2 e^4 d^4+32 a^3 c e^6 d^2+8 a^4 e^8\right )\right )}{\left (c d^2+a e^2\right )^4}}{(d+e x)^3 \left (c x^2+a\right )}dx}{2 a c}}{4 a c}-\frac {a \left (B c d \left (c d^2-3 a e^2\right )-(A c-a C) e \left (3 c d^2-a e^2\right )\right )-c \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right ) x}{4 a \left (c d^2+a e^2\right )^3 \left (c x^2+a\right )^2}\) |
| \(\Big \downarrow \) 2160 |
| \(\displaystyle \frac {\frac {\int \left (\frac {8 a^2 c^2 \left (a^2 C e^4-a c \left (13 C d^2-9 B e d+3 A e^2\right ) e^2+c^2 \left (10 C d^4-3 d^2 e (5 B d-7 A e)\right )\right ) e^4}{\left (c d^2+a e^2\right )^5 (d+e x)}+\frac {8 a^2 c^2 \left (4 c C d^3-c e (5 B d-6 A e) d-a e^2 (2 C d-B e)\right ) e^4}{\left (c d^2+a e^2\right )^4 (d+e x)^2}+\frac {8 a^2 c^2 \left (C d^2-B e d+A e^2\right ) e^4}{\left (c d^2+a e^2\right )^3 (d+e x)^3}+\frac {c^3 \left (-8 a^2 \left (a^2 C e^4-a c \left (13 C d^2-9 B e d+3 A e^2\right ) e^2+c^2 \left (10 C d^4-3 d^2 e (5 B d-7 A e)\right )\right ) x e^3+3 A c d \left (c^3 d^6+7 a c^2 e^2 d^4+35 a^2 c e^4 d^2-35 a^3 e^6\right )+a \left (c^3 (C d-3 B e) d^6+a c^2 e^2 (23 C d-45 B e) d^4-5 a^2 c e^4 (25 C d-27 B e) d^2+15 a^3 e^6 (3 C d-B e)\right )\right )}{\left (c d^2+a e^2\right )^5 \left (c x^2+a\right )}\right )dx}{2 a c}+\frac {c \left (c x \left (3 A c d \left (-11 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right )-a \left (-7 a^2 e^4 (3 C d-B e)+2 a c d^2 e^2 (13 C d-19 B e)-c^2 d^4 (C d-3 B e)\right )\right )+4 a^2 e \left (a^2 C e^4-2 a c e^2 \left (4 C d^2-e (3 B d-A e)\right )+c^2 \left (3 C d^4-2 d^2 e (3 B d-5 A e)\right )\right )\right )}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )^4}}{4 a c}-\frac {a \left (B c d \left (c d^2-3 a e^2\right )-e (A c-a C) \left (3 c d^2-a e^2\right )\right )-c x \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right )}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {c \left (c x \left (3 A c d \left (-11 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right )-a \left (-7 a^2 e^4 (3 C d-B e)+2 a c d^2 e^2 (13 C d-19 B e)-c^2 d^4 (C d-3 B e)\right )\right )+4 a^2 e \left (a^2 C e^4-2 a c e^2 \left (4 C d^2-e (3 B d-A e)\right )+c^2 \left (3 C d^4-2 d^2 e (3 B d-5 A e)\right )\right )\right )}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )^4}+\frac {-\frac {4 a^2 c^2 e^3 \left (A e^2-B d e+C d^2\right )}{(d+e x)^2 \left (a e^2+c d^2\right )^3}-\frac {4 a^2 c^2 e^3 \log \left (a+c x^2\right ) \left (a^2 C e^4-a c e^2 \left (3 A e^2-9 B d e+13 C d^2\right )+c^2 \left (10 C d^4-3 d^2 e (5 B d-7 A e)\right )\right )}{\left (a e^2+c d^2\right )^5}+\frac {8 a^2 c^2 e^3 \log (d+e x) \left (a^2 C e^4-a c e^2 \left (3 A e^2-9 B d e+13 C d^2\right )+c^2 \left (10 C d^4-3 d^2 e (5 B d-7 A e)\right )\right )}{\left (a e^2+c d^2\right )^5}-\frac {8 a^2 c^2 e^3 \left (-a e^2 (2 C d-B e)-c d e (5 B d-6 A e)+4 c C d^3\right )}{(d+e x) \left (a e^2+c d^2\right )^4}+\frac {c^{5/2} \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (3 A c d \left (-35 a^3 e^6+35 a^2 c d^2 e^4+7 a c^2 d^4 e^2+c^3 d^6\right )+a \left (15 a^3 e^6 (3 C d-B e)-5 a^2 c d^2 e^4 (25 C d-27 B e)+a c^2 d^4 e^2 (23 C d-45 B e)+c^3 d^6 (C d-3 B e)\right )\right )}{\sqrt {a} \left (a e^2+c d^2\right )^5}}{2 a c}}{4 a c}-\frac {a \left (B c d \left (c d^2-3 a e^2\right )-e (A c-a C) \left (3 c d^2-a e^2\right )\right )-c x \left (A c d \left (c d^2-3 a e^2\right )-a \left (c d^2 (C d-3 B e)-a e^2 (3 C d-B e)\right )\right )}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )^3}\) |
-1/4*(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)^2) + ((c*(4*a^2*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))) + c*(3*A*c*d*(c^2*d^4 + 6*a*c*d^2*e^2 - 11*a^2*e^4) - a*(2*a*c*d^2*e^2*(13*C *d - 19*B*e) - c^2*d^4*(C*d - 3*B*e) - 7*a^2*e^4*(3*C*d - B*e)))*x))/(2*a* (c*d^2 + a*e^2)^4*(a + c*x^2)) + ((-4*a^2*c^2*e^3*(C*d^2 - B*d*e + A*e^2)) /((c*d^2 + a*e^2)^3*(d + e*x)^2) - (8*a^2*c^2*e^3*(4*c*C*d^3 - c*d*e*(5*B* d - 6*A*e) - a*e^2*(2*C*d - B*e)))/((c*d^2 + a*e^2)^4*(d + e*x)) + (c^(5/2 )*(3*A*c*d*(c^3*d^6 + 7*a*c^2*d^4*e^2 + 35*a^2*c*d^2*e^4 - 35*a^3*e^6) + a *(a*c^2*d^4*e^2*(23*C*d - 45*B*e) - 5*a^2*c*d^2*e^4*(25*C*d - 27*B*e) + c^ 3*d^6*(C*d - 3*B*e) + 15*a^3*e^6*(3*C*d - B*e)))*ArcTan[(Sqrt[c]*x)/Sqrt[a ]])/(Sqrt[a]*(c*d^2 + a*e^2)^5) + (8*a^2*c^2*e^3*(a^2*C*e^4 - a*c*e^2*(13* C*d^2 - 9*B*d*e + 3*A*e^2) + c^2*(10*C*d^4 - 3*d^2*e*(5*B*d - 7*A*e)))*Log [d + e*x])/(c*d^2 + a*e^2)^5 - (4*a^2*c^2*e^3*(a^2*C*e^4 - a*c*e^2*(13*C*d ^2 - 9*B*d*e + 3*A*e^2) + c^2*(10*C*d^4 - 3*d^2*e*(5*B*d - 7*A*e)))*Log[a + c*x^2])/(c*d^2 + a*e^2)^5)/(2*a*c))/(4*a*c)
3.1.63.3.1 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 = 0.82 (sec) , antiderivative size = 1055, normalized size of antiderivative = 1.40
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1055\) |
| risch | \(\text {Expression too large to display}\) | \(104197\) |
-c/(a*e^2+c*d^2)^5*((1/8*c*(33*A*a^3*c*d*e^6+15*A*a^2*c^2*d^3*e^4-21*A*a*c ^3*d^5*e^2-3*A*c^4*d^7+7*B*a^4*e^7-31*B*a^3*c*d^2*e^5-35*B*a^2*c^2*d^4*e^3 +3*B*a*c^3*d^6*e-21*C*a^4*d*e^6+5*C*a^3*c*d^3*e^4+25*C*a^2*c^2*d^5*e^2-C*a *c^3*d^7)/a^2*x^3+(A*a^2*c*e^7-4*A*a*c^2*d^2*e^5-5*A*c^3*d^4*e^3-3*B*a^2*c *d*e^6+3*B*c^3*d^5*e^2-1/2*C*a^3*e^7+7/2*C*a^2*c*d^2*e^5+5/2*C*a*c^2*d^4*e ^3-3/2*C*c^3*d^6*e)*x^2+1/8*(39*A*a^3*c*d*e^6+25*A*a^2*c^2*d^3*e^4-19*A*a* c^3*d^5*e^2-5*A*c^4*d^7+9*B*a^4*e^7-33*B*a^3*c*d^2*e^5-45*B*a^2*c^2*d^4*e^ 3-3*B*a*c^3*d^6*e-27*C*a^4*d*e^6-5*C*a^3*c*d^3*e^4+23*C*a^2*c^2*d^5*e^2+C* a*c^3*d^7)/a*x+1/4*(5*A*a^3*c*e^7-17*A*a^2*c^2*d^2*e^5-25*A*a*c^3*d^4*e^3- 3*A*c^4*d^6*e-15*B*a^3*c*d*e^6-5*B*a^2*c^2*d^3*e^4+11*B*a*c^3*d^5*e^2+B*c^ 4*d^7-3*C*a^4*e^7+15*C*a^3*c*d^2*e^5+15*C*a^2*c^2*d^4*e^3-3*C*a*c^3*d^6*e) /c)/(c*x^2+a)^2+1/8/a^2*(1/2*(-24*A*a^3*c*e^7+168*A*a^2*c^2*d^2*e^5+72*B*a ^3*c*d*e^6-120*B*a^2*c^2*d^3*e^4+8*C*a^4*e^7-104*C*a^3*c*d^2*e^5+80*C*a^2* c^2*d^4*e^3)/c*ln(c*x^2+a)+(105*A*a^3*c*d*e^6-105*A*a^2*c^2*d^3*e^4-21*A*a *c^3*d^5*e^2-3*A*c^4*d^7+15*B*a^4*e^7-135*B*a^3*c*d^2*e^5+45*B*a^2*c^2*d^4 *e^3+3*B*a*c^3*d^6*e-45*C*a^4*d*e^6+125*C*a^3*c*d^3*e^4-23*C*a^2*c^2*d^5*e ^2-C*a*c^3*d^7)/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))))-e^3*(6*A*c*d*e^2+B*a *e^3-5*B*c*d^2*e-2*C*a*d*e^2+4*C*c*d^3)/(a*e^2+c*d^2)^4/(e*x+d)-1/2*e^3*(A *e^2-B*d*e+C*d^2)/(a*e^2+c*d^2)^3/(e*x+d)^2-e^3*(3*A*a*c*e^4-21*A*c^2*d^2* e^2-9*B*a*c*d*e^3+15*B*c^2*d^3*e-C*a^2*e^4+13*C*a*c*d^2*e^2-10*C*c^2*d^...
Timed out. \[ \int \frac {A+B x+C x^2}{(d+e x)^3 \left (a+c x^2\right )^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {A+B x+C x^2}{(d+e x)^3 \left (a+c x^2\right )^3} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 1835 vs. \(2 (732) = 1464\).
Time = 0.33 (sec) , antiderivative size = 1835, normalized size of antiderivative = 2.44 \[ \int \frac {A+B x+C x^2}{(d+e x)^3 \left (a+c x^2\right )^3} \, dx=\text {Too large to display} \]
-1/2*(10*C*c^2*d^4*e^3 - 15*B*c^2*d^3*e^4 + 9*B*a*c*d*e^6 - (13*C*a*c - 21 *A*c^2)*d^2*e^5 + (C*a^2 - 3*A*a*c)*e^7)*log(c*x^2 + a)/(c^5*d^10 + 5*a*c^ 4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 + 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 + a^ 5*e^10) + (10*C*c^2*d^4*e^3 - 15*B*c^2*d^3*e^4 + 9*B*a*c*d*e^6 - (13*C*a*c - 21*A*c^2)*d^2*e^5 + (C*a^2 - 3*A*a*c)*e^7)*log(e*x + d)/(c^5*d^10 + 5*a *c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 + 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 + a^5*e^10) - 1/8*(3*B*a*c^4*d^6*e + 45*B*a^2*c^3*d^4*e^3 - 135*B*a^3*c^2*d ^2*e^5 + 15*B*a^4*c*e^7 - (C*a*c^4 + 3*A*c^5)*d^7 - (23*C*a^2*c^3 + 21*A*a *c^4)*d^5*e^2 + 5*(25*C*a^3*c^2 - 21*A*a^2*c^3)*d^3*e^4 - 15*(3*C*a^4*c - 7*A*a^3*c^2)*d*e^6)*arctan(c*x/sqrt(a*c))/((a^2*c^5*d^10 + 5*a^3*c^4*d^8*e ^2 + 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6 + 5*a^6*c*d^2*e^8 + a^7*e^10) *sqrt(a*c)) - 1/8*(2*B*a^2*c^3*d^7 + 20*B*a^3*c^2*d^5*e^2 - 74*B*a^4*c*d^3 *e^4 + 4*B*a^5*d*e^6 + 4*A*a^5*e^7 - 6*(C*a^3*c^2 + A*a^2*c^3)*d^6*e + 4*( 18*C*a^4*c - 11*A*a^3*c^2)*d^4*e^3 - 2*(9*C*a^5 - 31*A*a^4*c)*d^2*e^5 + (3 *B*a*c^4*d^4*e^3 - 78*B*a^2*c^3*d^2*e^5 + 15*B*a^3*c^2*e^7 - (C*a*c^4 + 3* A*c^5)*d^5*e^2 + 2*(29*C*a^2*c^3 - 9*A*a*c^4)*d^3*e^4 - (37*C*a^3*c^2 - 81 *A*a^2*c^3)*d*e^6)*x^5 + 2*(3*B*a*c^4*d^5*e^2 - 48*B*a^2*c^3*d^3*e^4 - 3*B *a^3*c^2*d*e^6 - (C*a*c^4 + 3*A*c^5)*d^6*e + 2*(19*C*a^2*c^3 - 9*A*a*c^4)* d^4*e^3 - (11*C*a^3*c^2 - 39*A*a^2*c^3)*d^2*e^5 - 2*(C*a^4*c - 3*A*a^3*c^2 )*e^7)*x^4 + (3*B*a*c^4*d^6*e + 7*B*a^2*c^3*d^4*e^3 - 163*B*a^3*c^2*d^2...
Leaf count of result is larger than twice the leaf count of optimal. 1614 vs. \(2 (732) = 1464\).
Time = 0.28 (sec) , antiderivative size = 1614, normalized size of antiderivative = 2.14 \[ \int \frac {A+B x+C x^2}{(d+e x)^3 \left (a+c x^2\right )^3} \, dx=\text {Too large to display} \]
-1/2*(10*C*c^2*d^4*e^3 - 15*B*c^2*d^3*e^4 - 13*C*a*c*d^2*e^5 + 21*A*c^2*d^ 2*e^5 + 9*B*a*c*d*e^6 + C*a^2*e^7 - 3*A*a*c*e^7)*log(c*x^2 + a)/(c^5*d^10 + 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 + 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2* e^8 + a^5*e^10) + (10*C*c^2*d^4*e^4 - 15*B*c^2*d^3*e^5 - 13*C*a*c*d^2*e^6 + 21*A*c^2*d^2*e^6 + 9*B*a*c*d*e^7 + C*a^2*e^8 - 3*A*a*c*e^8)*log(abs(e*x + d))/(c^5*d^10*e + 5*a*c^4*d^8*e^3 + 10*a^2*c^3*d^6*e^5 + 10*a^3*c^2*d^4* e^7 + 5*a^4*c*d^2*e^9 + a^5*e^11) + 1/8*(C*a*c^4*d^7 + 3*A*c^5*d^7 - 3*B*a *c^4*d^6*e + 23*C*a^2*c^3*d^5*e^2 + 21*A*a*c^4*d^5*e^2 - 45*B*a^2*c^3*d^4* e^3 - 125*C*a^3*c^2*d^3*e^4 + 105*A*a^2*c^3*d^3*e^4 + 135*B*a^3*c^2*d^2*e^ 5 + 45*C*a^4*c*d*e^6 - 105*A*a^3*c^2*d*e^6 - 15*B*a^4*c*e^7)*arctan(c*x/sq rt(a*c))/((a^2*c^5*d^10 + 5*a^3*c^4*d^8*e^2 + 10*a^4*c^3*d^6*e^4 + 10*a^5* c^2*d^4*e^6 + 5*a^6*c*d^2*e^8 + a^7*e^10)*sqrt(a*c)) + 1/8*(C*a*c^4*d^5*e^ 2*x^5 + 3*A*c^5*d^5*e^2*x^5 - 3*B*a*c^4*d^4*e^3*x^5 - 58*C*a^2*c^3*d^3*e^4 *x^5 + 18*A*a*c^4*d^3*e^4*x^5 + 78*B*a^2*c^3*d^2*e^5*x^5 + 37*C*a^3*c^2*d* e^6*x^5 - 81*A*a^2*c^3*d*e^6*x^5 - 15*B*a^3*c^2*e^7*x^5 + 2*C*a*c^4*d^6*e* x^4 + 6*A*c^5*d^6*e*x^4 - 6*B*a*c^4*d^5*e^2*x^4 - 76*C*a^2*c^3*d^4*e^3*x^4 + 36*A*a*c^4*d^4*e^3*x^4 + 96*B*a^2*c^3*d^3*e^4*x^4 + 22*C*a^3*c^2*d^2*e^ 5*x^4 - 78*A*a^2*c^3*d^2*e^5*x^4 + 6*B*a^3*c^2*d*e^6*x^4 + 4*C*a^4*c*e^7*x ^4 - 12*A*a^3*c^2*e^7*x^4 + C*a*c^4*d^7*x^3 + 3*A*c^5*d^7*x^3 - 3*B*a*c^4* d^6*e*x^3 - 3*C*a^2*c^3*d^5*e^2*x^3 + 23*A*a*c^4*d^5*e^2*x^3 - 7*B*a^2*...
Time = 17.49 (sec) , antiderivative size = 8774, normalized size of antiderivative = 11.65 \[ \int \frac {A+B x+C x^2}{(d+e x)^3 \left (a+c x^2\right )^3} \, dx=\text {Too large to display} \]
((x^5*(3*A*c^5*d^5*e^2 - 15*B*a^3*c^2*e^7 + 18*A*a*c^4*d^3*e^4 - 81*A*a^2* c^3*d*e^6 - 3*B*a*c^4*d^4*e^3 + C*a*c^4*d^5*e^2 + 37*C*a^3*c^2*d*e^6 + 78* B*a^2*c^3*d^2*e^5 - 58*C*a^2*c^3*d^3*e^4))/(8*a^2*(a^4*e^8 + c^4*d^8 + 4*a *c^3*d^6*e^2 + 4*a^3*c*d^2*e^6 + 6*a^2*c^2*d^4*e^4)) - (2*A*a^3*e^7 + B*c^ 3*d^7 + 2*B*a^3*d*e^6 - 3*A*c^3*d^6*e - 9*C*a^3*d^2*e^5 - 22*A*a*c^2*d^4*e ^3 + 31*A*a^2*c*d^2*e^5 + 10*B*a*c^2*d^5*e^2 - 37*B*a^2*c*d^3*e^4 + 36*C*a ^2*c*d^4*e^3 - 3*C*a*c^2*d^6*e)/(4*(a^4*e^8 + c^4*d^8 + 4*a*c^3*d^6*e^2 + 4*a^3*c*d^2*e^6 + 6*a^2*c^2*d^4*e^4)) + (x*(5*A*c^4*d^7 - 8*B*a^4*e^7 - C* a*c^3*d^7 + 28*C*a^4*d*e^6 + 26*A*a*c^3*d^5*e^2 + 91*B*a^3*c*d^2*e^5 - 77* C*a^3*c*d^3*e^4 + 49*A*a^2*c^2*d^3*e^4 + 2*B*a^2*c^2*d^4*e^3 - 10*C*a^2*c^ 2*d^5*e^2 - 68*A*a^3*c*d*e^6 - B*a*c^3*d^6*e))/(8*a*(a^4*e^8 + c^4*d^8 + 4 *a*c^3*d^6*e^2 + 4*a^3*c*d^2*e^6 + 6*a^2*c^2*d^4*e^4)) + (x^2*(3*C*a^4*e^7 - 9*A*a^3*c*e^7 + 5*A*c^4*d^6*e + 37*A*a*c^3*d^4*e^3 - 10*B*a*c^3*d^5*e^2 + 23*C*a^3*c*d^2*e^5 - 73*A*a^2*c^2*d^2*e^5 + 88*B*a^2*c^2*d^3*e^4 - 71*C *a^2*c^2*d^4*e^3 + 2*B*a^3*c*d*e^6 + 5*C*a*c^3*d^6*e))/(4*a*(a^4*e^8 + c^4 *d^8 + 4*a*c^3*d^6*e^2 + 4*a^3*c*d^2*e^6 + 6*a^2*c^2*d^4*e^4)) + (x^3*(3*A *c^5*d^7 - 25*B*a^4*c*e^7 + C*a*c^4*d^7 + 23*A*a*c^4*d^5*e^2 - 151*A*a^3*c ^2*d*e^6 + 61*A*a^2*c^3*d^3*e^4 - 7*B*a^2*c^3*d^4*e^3 + 163*B*a^3*c^2*d^2* e^5 - 3*C*a^2*c^3*d^5*e^2 - 129*C*a^3*c^2*d^3*e^4 - 3*B*a*c^4*d^6*e + 67*C *a^4*c*d*e^6))/(8*a^2*(a^4*e^8 + c^4*d^8 + 4*a*c^3*d^6*e^2 + 4*a^3*c*d^...