Integrand size = 27, antiderivative size = 571 \[ \int \frac {A+B x+C x^2}{(d+e x)^2 \left (a+c x^2\right )^3} \, dx=-\frac {e^3 \left (C d^2-B d e+A e^2\right )}{\left (c d^2+a e^2\right )^3 (d+e x)}-\frac {a \left (B c d^2-2 A c d e+2 a C d e-a B e^2\right )-\left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right ) x}{4 a \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}-\frac {4 a^2 e \left (a e^2 (2 C d-B e)-c d \left (2 C d^2-e (3 B d-4 A e)\right )\right )-\left (A c \left (3 c^2 d^4+12 a c d^2 e^2-7 a^2 e^4\right )+a \left (3 a^2 C e^4-2 a c d e^2 (6 C d-7 B e)+c^2 d^3 (C d-2 B e)\right )\right ) x}{8 a^2 \left (c d^2+a e^2\right )^3 \left (a+c x^2\right )}+\frac {\left (3 A c \left (c^3 d^6+5 a c^2 d^4 e^2+15 a^2 c d^2 e^4-5 a^3 e^6\right )+a \left (3 a^3 C e^6+a c^2 d^3 e^2 (13 C d-20 B e)-3 a^2 c d e^4 (11 C d-10 B e)+c^3 d^5 (C d-2 B e)\right )\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {c} \left (c d^2+a e^2\right )^4}-\frac {e^3 \left (a e^2 (2 C d-B e)-c d \left (4 C d^2-e (5 B d-6 A e)\right )\right ) \log (d+e x)}{\left (c d^2+a e^2\right )^4}+\frac {e^3 \left (a e^2 (2 C d-B e)-c d \left (4 C d^2-e (5 B d-6 A e)\right )\right ) \log \left (a+c x^2\right )}{2 \left (c d^2+a e^2\right )^4} \]
-e^3*(A*e^2-B*d*e+C*d^2)/(a*e^2+c*d^2)^3/(e*x+d)+1/4*(-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)^2+1/8*(-4*a^2*e*(a*e^2*(-B*e+2*C*d)-c*d*(2*C* d^2-e*(-4*A*e+3*B*d)))+(A*c*(-7*a^2*e^4+12*a*c*d^2*e^2+3*c^2*d^4)+a*(3*a^2 *C*e^4-2*a*c*d*e^2*(-7*B*e+6*C*d)+c^2*d^3*(-2*B*e+C*d)))*x)/a^2/(a*e^2+c*d ^2)^3/(c*x^2+a)-e^3*(a*e^2*(-B*e+2*C*d)-c*d*(4*C*d^2-e*(-6*A*e+5*B*d)))*ln (e*x+d)/(a*e^2+c*d^2)^4+1/2*e^3*(a*e^2*(-B*e+2*C*d)-c*d*(4*C*d^2-e*(-6*A*e +5*B*d)))*ln(c*x^2+a)/(a*e^2+c*d^2)^4+1/8*(3*A*c*(-5*a^3*e^6+15*a^2*c*d^2* e^4+5*a*c^2*d^4*e^2+c^3*d^6)+a*(3*a^3*C*e^6+a*c^2*d^3*e^2*(-20*B*e+13*C*d) -3*a^2*c*d*e^4*(-10*B*e+11*C*d)+c^3*d^5*(-2*B*e+C*d)))*arctan(x*c^(1/2)/a^ (1/2))/a^(5/2)/(a*e^2+c*d^2)^4/c^(1/2)
Time = 0.44 (sec) , antiderivative size = 498, normalized size of antiderivative = 0.87 \[ \int \frac {A+B x+C x^2}{(d+e x)^2 \left (a+c x^2\right )^3} \, dx=\frac {-\frac {8 e^3 \left (c d^2+a e^2\right ) \left (C d^2+e (-B d+A e)\right )}{d+e x}+\frac {2 \left (c d^2+a e^2\right )^2 \left (A c^2 d^2 x+a^2 e (-2 C d+B e+C e x)-a c \left (C d^2 x+B d (d-2 e x)+A e (-2 d+e x)\right )\right )}{a \left (a+c x^2\right )^2}+\frac {\left (c d^2+a e^2\right ) \left (3 A c^3 d^4 x+a c^2 d^2 \left (C d^2+2 e (-B d+6 A e)\right ) x+a^3 e^3 (-8 C d+4 B e+3 C e x)+a^2 c e \left (4 C d^2 (2 d-3 e x)+e (-2 B d (6 d-7 e x)+A e (16 d-7 e x))\right )\right )}{a^2 \left (a+c x^2\right )}+\frac {\left (3 A c \left (c^3 d^6+5 a c^2 d^4 e^2+15 a^2 c d^2 e^4-5 a^3 e^6\right )+a \left (3 a^3 C e^6+a c^2 d^3 e^2 (13 C d-20 B e)+c^3 d^5 (C d-2 B e)+3 a^2 c d e^4 (-11 C d+10 B e)\right )\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{5/2} \sqrt {c}}+8 e^3 \left (4 c C d^3+c d e (-5 B d+6 A e)+a e^2 (-2 C d+B e)\right ) \log (d+e x)-4 e^3 \left (4 c C d^3+c d e (-5 B d+6 A e)+a e^2 (-2 C d+B e)\right ) \log \left (a+c x^2\right )}{8 \left (c d^2+a e^2\right )^4} \]
((-8*e^3*(c*d^2 + a*e^2)*(C*d^2 + e*(-(B*d) + A*e)))/(d + e*x) + (2*(c*d^2 + a*e^2)^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)^2) + ((c*d^2 + a*e^2)* (3*A*c^3*d^4*x + a*c^2*d^2*(C*d^2 + 2*e*(-(B*d) + 6*A*e))*x + a^3*e^3*(-8* C*d + 4*B*e + 3*C*e*x) + a^2*c*e*(4*C*d^2*(2*d - 3*e*x) + e*(-2*B*d*(6*d - 7*e*x) + A*e*(16*d - 7*e*x)))))/(a^2*(a + c*x^2)) + ((3*A*c*(c^3*d^6 + 5* a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4 - 5*a^3*e^6) + a*(3*a^3*C*e^6 + a*c^2*d^3 *e^2*(13*C*d - 20*B*e) + c^3*d^5*(C*d - 2*B*e) + 3*a^2*c*d*e^4*(-11*C*d + 10*B*e)))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(a^(5/2)*Sqrt[c]) + 8*e^3*(4*c*C*d^ 3 + c*d*e*(-5*B*d + 6*A*e) + a*e^2*(-2*C*d + B*e))*Log[d + e*x] - 4*e^3*(4 *c*C*d^3 + c*d*e*(-5*B*d + 6*A*e) + a*e^2*(-2*C*d + B*e))*Log[a + c*x^2])/ (8*(c*d^2 + a*e^2)^4)
Time = 2.39 (sec) , antiderivative size = 603, normalized size of antiderivative = 1.06, 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.
| \(\displaystyle \int \frac {A+B x+C x^2}{\left (a+c x^2\right )^3 (d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 2178 |
| \(\displaystyle -\frac {\int -\frac {\frac {3 c e^2 \left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right ) x^2}{\left (c d^2+a e^2\right )^2}+\frac {2 c e \left (A c d \left (3 c d^2+a e^2\right )-a \left (c (3 C d-4 B e) d^2+a e^2 (C d-2 B e)\right )\right ) x}{\left (c d^2+a e^2\right )^2}+\frac {c \left (A \left (3 c^2 d^4+9 a c e^2 d^2+4 a^2 e^4\right )-a d^2 \left (a C e^2-c d (C d-2 B e)\right )\right )}{\left (c d^2+a e^2\right )^2}}{(d+e x)^2 \left (c x^2+a\right )^2}dx}{4 a c}-\frac {a \left (-a B e^2+2 a C d e-2 A c d e+B c d^2\right )-x \left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right )}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\frac {3 c e^2 \left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right ) x^2}{\left (c d^2+a e^2\right )^2}+\frac {2 c e \left (A c d \left (3 c d^2+a e^2\right )-a \left (c (3 C d-4 B e) d^2+a e^2 (C d-2 B e)\right )\right ) x}{\left (c d^2+a e^2\right )^2}+\frac {c \left (A \left (3 c^2 d^4+9 a c e^2 d^2+4 a^2 e^4\right )-a d^2 \left (a C e^2-c d (C d-2 B e)\right )\right )}{\left (c d^2+a e^2\right )^2}}{(d+e x)^2 \left (c x^2+a\right )^2}dx}{4 a c}-\frac {a \left (-a B e^2+2 a C d e-2 A c d e+B c d^2\right )-x \left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right )}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )^2}\) |
| \(\Big \downarrow \) 2178 |
| \(\displaystyle \frac {\frac {c \left (x \left (A c \left (-7 a^2 e^4+12 a c d^2 e^2+3 c^2 d^4\right )+a \left (3 a^2 C e^4-2 a c d e^2 (6 C d-7 B e)+c^2 d^3 (C d-2 B e)\right )\right )+4 a^2 e \left (-a e^2 (2 C d-B e)-c d e (3 B d-4 A e)+2 c C d^3\right )\right )}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )^3}-\frac {\int -\frac {\frac {e^2 \left (A c \left (3 c^2 d^4+12 a c e^2 d^2-7 a^2 e^4\right )+a \left (3 a^2 C e^4-2 a c d (6 C d-7 B e) e^2+c^2 d^3 (C d-2 B e)\right )\right ) x^2 c^2}{\left (c d^2+a e^2\right )^3}+\frac {\left (A \left (3 c^3 d^6+12 a c^2 e^2 d^4+33 a^2 c e^4 d^2+8 a^3 e^6\right )-a d^2 \left (5 a^2 C e^4-6 a c d (2 C d-3 B e) e^2-c^2 d^3 (C d-2 B e)\right )\right ) c^2}{\left (c d^2+a e^2\right )^3}+\frac {2 e \left (3 A c d \left (c d^2+3 a e^2\right )-a \left (a e^2 (5 C d-4 B e)-c d^2 (C d-2 B e)\right )\right ) x c^2}{\left (c d^2+a e^2\right )^2}}{(d+e x)^2 \left (c x^2+a\right )}dx}{2 a c}}{4 a c}-\frac {a \left (-a B e^2+2 a C d e-2 A c d e+B c d^2\right )-x \left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right )}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {\frac {e^2 \left (A c \left (3 c^2 d^4+12 a c e^2 d^2-7 a^2 e^4\right )+a \left (3 a^2 C e^4-2 a c d (6 C d-7 B e) e^2+c^2 d^3 (C d-2 B e)\right )\right ) x^2 c^2}{\left (c d^2+a e^2\right )^3}+\frac {\left (A \left (3 c^3 d^6+12 a c^2 e^2 d^4+33 a^2 c e^4 d^2+8 a^3 e^6\right )-a d^2 \left (5 a^2 C e^4-6 a c d (2 C d-3 B e) e^2-c^2 d^3 (C d-2 B e)\right )\right ) c^2}{\left (c d^2+a e^2\right )^3}+\frac {2 e \left (3 A c d \left (c d^2+3 a e^2\right )-a \left (a e^2 (5 C d-4 B e)-c d^2 (C d-2 B e)\right )\right ) x c^2}{\left (c d^2+a e^2\right )^2}}{(d+e x)^2 \left (c x^2+a\right )}dx}{2 a c}+\frac {c \left (x \left (A c \left (-7 a^2 e^4+12 a c d^2 e^2+3 c^2 d^4\right )+a \left (3 a^2 C e^4-2 a c d e^2 (6 C d-7 B e)+c^2 d^3 (C d-2 B e)\right )\right )+4 a^2 e \left (-a e^2 (2 C d-B e)-c d e (3 B d-4 A e)+2 c C d^3\right )\right )}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )^3}}{4 a c}-\frac {a \left (-a B e^2+2 a C d e-2 A c d e+B c d^2\right )-x \left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right )}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )^2}\) |
| \(\Big \downarrow \) 2160 |
| \(\displaystyle \frac {\frac {\int \left (\frac {8 a^2 c^2 \left (4 c C d^3-c e (5 B d-6 A e) d-a e^2 (2 C d-B e)\right ) e^4}{\left (c d^2+a e^2\right )^4 (d+e x)}+\frac {8 a^2 c^2 \left (C d^2-B e d+A e^2\right ) e^4}{\left (c d^2+a e^2\right )^3 (d+e x)^2}+\frac {c^2 \left (-8 a^2 c \left (4 c C d^3-c e (5 B d-6 A e) d-a e^2 (2 C d-B e)\right ) x e^3+3 A c \left (c^3 d^6+5 a c^2 e^2 d^4+15 a^2 c e^4 d^2-5 a^3 e^6\right )+a \left (3 a^3 C e^6-3 a^2 c d (11 C d-10 B e) e^4+a c^2 d^3 (13 C d-20 B e) e^2+c^3 d^5 (C d-2 B e)\right )\right )}{\left (c d^2+a e^2\right )^4 \left (c x^2+a\right )}\right )dx}{2 a c}+\frac {c \left (x \left (A c \left (-7 a^2 e^4+12 a c d^2 e^2+3 c^2 d^4\right )+a \left (3 a^2 C e^4-2 a c d e^2 (6 C d-7 B e)+c^2 d^3 (C d-2 B e)\right )\right )+4 a^2 e \left (-a e^2 (2 C d-B e)-c d e (3 B d-4 A e)+2 c C d^3\right )\right )}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )^3}}{4 a c}-\frac {a \left (-a B e^2+2 a C d e-2 A c d e+B c d^2\right )-x \left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right )}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {c \left (x \left (A c \left (-7 a^2 e^4+12 a c d^2 e^2+3 c^2 d^4\right )+a \left (3 a^2 C e^4-2 a c d e^2 (6 C d-7 B e)+c^2 d^3 (C d-2 B e)\right )\right )+4 a^2 e \left (-a e^2 (2 C d-B e)-c d e (3 B d-4 A e)+2 c C d^3\right )\right )}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )^3}+\frac {-\frac {8 a^2 c^2 e^3 \left (A e^2-B d e+C d^2\right )}{(d+e x) \left (a e^2+c d^2\right )^3}-\frac {4 a^2 c^2 e^3 \log \left (a+c x^2\right ) \left (-a e^2 (2 C d-B e)-c d e (5 B d-6 A e)+4 c C d^3\right )}{\left (a e^2+c d^2\right )^4}+\frac {8 a^2 c^2 e^3 \log (d+e x) \left (-a e^2 (2 C d-B e)-c d e (5 B d-6 A e)+4 c C d^3\right )}{\left (a e^2+c d^2\right )^4}+\frac {c^{3/2} \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (3 A c \left (-5 a^3 e^6+15 a^2 c d^2 e^4+5 a c^2 d^4 e^2+c^3 d^6\right )+a \left (3 a^3 C e^6-3 a^2 c d e^4 (11 C d-10 B e)+a c^2 d^3 e^2 (13 C d-20 B e)+c^3 d^5 (C d-2 B e)\right )\right )}{\sqrt {a} \left (a e^2+c d^2\right )^4}}{2 a c}}{4 a c}-\frac {a \left (-a B e^2+2 a C d e-2 A c d e+B c d^2\right )-x \left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d-2 B e)\right )\right )}{4 a \left (a+c x^2\right )^2 \left (a e^2+c d^2\right )^2}\) |
-1/4*(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) + ((c*(4*a^2*e*(2*c*C*d^3 - c*d*e*(3*B*d - 4*A*e) - a*e^2*(2*C*d - B*e)) + (A*c*(3*c^2*d^4 + 12*a*c*d^2*e^2 - 7*a^2*e^4) + a*(3*a^2*C*e^4 - 2*a*c*d *e^2*(6*C*d - 7*B*e) + c^2*d^3*(C*d - 2*B*e)))*x))/(2*a*(c*d^2 + a*e^2)^3* (a + c*x^2)) + ((-8*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)) + (c^(3/2)*(3*A*c*(c^3*d^6 + 5*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e ^4 - 5*a^3*e^6) + a*(3*a^3*C*e^6 + a*c^2*d^3*e^2*(13*C*d - 20*B*e) - 3*a^2 *c*d*e^4*(11*C*d - 10*B*e) + c^3*d^5*(C*d - 2*B*e)))*ArcTan[(Sqrt[c]*x)/Sq rt[a]])/(Sqrt[a]*(c*d^2 + a*e^2)^4) + (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))*Log[d + e*x])/(c*d^2 + a*e^2)^4 - (4* 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))*Log[ a + c*x^2])/(c*d^2 + a*e^2)^4)/(2*a*c))/(4*a*c)
3.1.62.3.1 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 = 0.74 (sec) , antiderivative size = 833, normalized size of antiderivative = 1.46
| method | result | size |
| default | \(-\frac {\frac {\frac {c \left (7 A \,a^{3} c \,e^{6}-5 A \,a^{2} c^{2} d^{2} e^{4}-15 A a \,c^{3} d^{4} e^{2}-3 A \,d^{6} c^{4}-14 B \,a^{3} c d \,e^{5}-12 B \,a^{2} c^{2} d^{3} e^{3}+2 B a \,c^{3} d^{5} e -3 C \,a^{4} e^{6}+9 C \,a^{3} c \,d^{2} e^{4}+11 C \,a^{2} c^{2} d^{4} e^{2}-C a \,c^{3} d^{6}\right ) x^{3}}{8 a^{2}}+\left (-2 A a \,c^{2} d \,e^{5}-2 A \,c^{3} d^{3} e^{3}-\frac {1}{2} B \,a^{2} c \,e^{6}+B a \,c^{2} d^{2} e^{4}+\frac {3}{2} B \,c^{3} d^{4} e^{2}+C \,a^{2} c d \,e^{5}-C \,c^{3} d^{5} e \right ) x^{2}+\frac {\left (9 A \,a^{3} c \,e^{6}-3 A \,a^{2} c^{2} d^{2} e^{4}-17 A a \,c^{3} d^{4} e^{2}-5 A \,d^{6} c^{4}-18 B \,a^{3} c d \,e^{5}-20 B \,a^{2} c^{2} d^{3} e^{3}-2 B a \,c^{3} d^{5} e -5 C \,a^{4} e^{6}+7 C \,a^{3} c \,d^{2} e^{4}+13 C \,a^{2} c^{2} d^{4} e^{2}+C a \,c^{3} d^{6}\right ) x}{8 a}-\frac {5 A \,a^{2} c d \,e^{5}}{2}-3 A a \,c^{2} d^{3} e^{3}-\frac {A \,c^{3} d^{5} e}{2}-\frac {3 B \,a^{3} e^{6}}{4}+\frac {3 B \,a^{2} c \,d^{2} e^{4}}{4}+\frac {7 B a \,c^{2} d^{4} e^{2}}{4}+\frac {B \,c^{3} d^{6}}{4}+\frac {3 C \,a^{3} d \,e^{5}}{2}+C \,a^{2} c \,d^{3} e^{3}-\frac {C a \,c^{2} d^{5} e}{2}}{\left (c \,x^{2}+a \right )^{2}}+\frac {\frac {\left (48 A \,a^{2} c^{2} d \,e^{5}+8 B \,a^{3} e^{6} c -40 B \,a^{2} c^{2} d^{2} e^{4}-16 C \,a^{3} c d \,e^{5}+32 C \,a^{2} c^{2} d^{3} e^{3}\right ) \ln \left (c \,x^{2}+a \right )}{2 c}+\frac {\left (15 A \,a^{3} c \,e^{6}-45 A \,a^{2} c^{2} d^{2} e^{4}-15 A a \,c^{3} d^{4} e^{2}-3 A \,d^{6} c^{4}-30 B \,a^{3} c d \,e^{5}+20 B \,a^{2} c^{2} d^{3} e^{3}+2 B a \,c^{3} d^{5} e -3 C \,a^{4} e^{6}+33 C \,a^{3} c \,d^{2} e^{4}-13 C \,a^{2} c^{2} d^{4} e^{2}-C a \,c^{3} d^{6}\right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}}{8 a^{2}}}{\left (e^{2} a +c \,d^{2}\right )^{4}}+\frac {e^{3} \left (6 A c d \,e^{2}+B \,e^{3} a -5 B c \,d^{2} e -2 C a d \,e^{2}+4 C c \,d^{3}\right ) \ln \left (e x +d \right )}{\left (e^{2} a +c \,d^{2}\right )^{4}}-\frac {e^{3} \left (A \,e^{2}-B d e +C \,d^{2}\right )}{\left (e^{2} a +c \,d^{2}\right )^{3} \left (e x +d \right )}\) | \(833\) |
| risch | \(\text {Expression too large to display}\) | \(65746\) |
-1/(a*e^2+c*d^2)^4*((1/8*c*(7*A*a^3*c*e^6-5*A*a^2*c^2*d^2*e^4-15*A*a*c^3*d ^4*e^2-3*A*c^4*d^6-14*B*a^3*c*d*e^5-12*B*a^2*c^2*d^3*e^3+2*B*a*c^3*d^5*e-3 *C*a^4*e^6+9*C*a^3*c*d^2*e^4+11*C*a^2*c^2*d^4*e^2-C*a*c^3*d^6)/a^2*x^3+(-2 *A*a*c^2*d*e^5-2*A*c^3*d^3*e^3-1/2*B*a^2*c*e^6+B*a*c^2*d^2*e^4+3/2*B*c^3*d ^4*e^2+C*a^2*c*d*e^5-C*c^3*d^5*e)*x^2+1/8*(9*A*a^3*c*e^6-3*A*a^2*c^2*d^2*e ^4-17*A*a*c^3*d^4*e^2-5*A*c^4*d^6-18*B*a^3*c*d*e^5-20*B*a^2*c^2*d^3*e^3-2* B*a*c^3*d^5*e-5*C*a^4*e^6+7*C*a^3*c*d^2*e^4+13*C*a^2*c^2*d^4*e^2+C*a*c^3*d ^6)/a*x-5/2*A*a^2*c*d*e^5-3*A*a*c^2*d^3*e^3-1/2*A*c^3*d^5*e-3/4*B*a^3*e^6+ 3/4*B*a^2*c*d^2*e^4+7/4*B*a*c^2*d^4*e^2+1/4*B*c^3*d^6+3/2*C*a^3*d*e^5+C*a^ 2*c*d^3*e^3-1/2*C*a*c^2*d^5*e)/(c*x^2+a)^2+1/8/a^2*(1/2*(48*A*a^2*c^2*d*e^ 5+8*B*a^3*c*e^6-40*B*a^2*c^2*d^2*e^4-16*C*a^3*c*d*e^5+32*C*a^2*c^2*d^3*e^3 )/c*ln(c*x^2+a)+(15*A*a^3*c*e^6-45*A*a^2*c^2*d^2*e^4-15*A*a*c^3*d^4*e^2-3* A*c^4*d^6-30*B*a^3*c*d*e^5+20*B*a^2*c^2*d^3*e^3+2*B*a*c^3*d^5*e-3*C*a^4*e^ 6+33*C*a^3*c*d^2*e^4-13*C*a^2*c^2*d^4*e^2-C*a*c^3*d^6)/(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*ln(e*x+d)-e^3*(A*e^2-B*d*e+C*d^2)/(a*e^2+c*d^2)^3/(e* x+d)
Timed out. \[ \int \frac {A+B x+C x^2}{(d+e x)^2 \left (a+c x^2\right )^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {A+B x+C x^2}{(d+e x)^2 \left (a+c x^2\right )^3} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 1196 vs. \(2 (555) = 1110\).
Time = 0.32 (sec) , antiderivative size = 1196, normalized size of antiderivative = 2.09 \[ \int \frac {A+B x+C x^2}{(d+e x)^2 \left (a+c x^2\right )^3} \, dx=\text {Too large to display} \]
-1/2*(4*C*c*d^3*e^3 - 5*B*c*d^2*e^4 + B*a*e^6 - 2*(C*a - 3*A*c)*d*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) + (4*C*c*d^3*e^3 - 5*B*c*d^2*e^4 + B*a*e^6 - 2*(C*a - 3*A*c) *d*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) - 1/8*(2*B*a*c^3*d^5*e + 20*B*a^2*c^2*d^3*e^3 - 30* B*a^3*c*d*e^5 - (C*a*c^3 + 3*A*c^4)*d^6 - (13*C*a^2*c^2 + 15*A*a*c^3)*d^4* e^2 + 3*(11*C*a^3*c - 15*A*a^2*c^2)*d^2*e^4 - 3*(C*a^4 - 5*A*a^3*c)*e^6)*a rctan(c*x/sqrt(a*c))/((a^2*c^4*d^8 + 4*a^3*c^3*d^6*e^2 + 6*a^4*c^2*d^4*e^4 + 4*a^5*c*d^2*e^6 + a^6*e^8)*sqrt(a*c)) - 1/8*(2*B*a^2*c^2*d^5 + 12*B*a^3 *c*d^3*e^2 - 14*B*a^4*d*e^4 + 8*A*a^4*e^5 - 4*(C*a^3*c + A*a^2*c^2)*d^4*e + 20*(C*a^4 - A*a^3*c)*d^2*e^3 + (2*B*a*c^3*d^3*e^2 - 22*B*a^2*c^2*d*e^4 - (C*a*c^3 + 3*A*c^4)*d^4*e + 4*(5*C*a^2*c^2 - 3*A*a*c^3)*d^2*e^3 - 3*(C*a^ 3*c - 5*A*a^2*c^2)*e^5)*x^4 + (2*B*a*c^3*d^4*e - 2*B*a^2*c^2*d^2*e^3 - 4*B *a^3*c*e^5 - (C*a*c^3 + 3*A*c^4)*d^5 + 4*(C*a^2*c^2 - 3*A*a*c^3)*d^3*e^2 + (5*C*a^3*c - 9*A*a^2*c^2)*d*e^4)*x^3 + (10*B*a^2*c^2*d^3*e^2 - 38*B*a^3*c *d*e^4 - (7*C*a^2*c^2 + 5*A*a*c^3)*d^4*e + 4*(9*C*a^3*c - 7*A*a^2*c^2)*d^2 *e^3 - 5*(C*a^4 - 5*A*a^3*c)*e^5)*x^2 - (6*B*a^3*c*d^2*e^3 + 6*B*a^4*e^5 - (C*a^2*c^2 - 5*A*a*c^3)*d^5 - 8*(C*a^3*c - 2*A*a^2*c^2)*d^3*e^2 - (7*C*a^ 4 - 11*A*a^3*c)*d*e^4)*x)/(a^4*c^3*d^7 + 3*a^5*c^2*d^5*e^2 + 3*a^6*c*d^3*e ^4 + a^7*d*e^6 + (a^2*c^5*d^6*e + 3*a^3*c^4*d^4*e^3 + 3*a^4*c^3*d^2*e^5...
Leaf count of result is larger than twice the leaf count of optimal. 1175 vs. \(2 (555) = 1110\).
Time = 0.30 (sec) , antiderivative size = 1175, normalized size of antiderivative = 2.06 \[ \int \frac {A+B x+C x^2}{(d+e x)^2 \left (a+c x^2\right )^3} \, dx=-\frac {{\left (4 \, C c d^{3} e^{3} - 5 \, B c d^{2} e^{4} - 2 \, C a d e^{5} + 6 \, A c d e^{5} + B a e^{6}\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^{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 {\frac {C d^{2} e^{9}}{e x + d} - \frac {B d e^{10}}{e x + d} + \frac {A e^{11}}{e x + d}}{c^{3} d^{6} e^{6} + 3 \, a c^{2} d^{4} e^{8} + 3 \, a^{2} c d^{2} e^{10} + a^{3} e^{12}} + \frac {{\left (C a c^{3} d^{6} e^{2} + 3 \, A c^{4} d^{6} e^{2} - 2 \, B a c^{3} d^{5} e^{3} + 13 \, C a^{2} c^{2} d^{4} e^{4} + 15 \, A a c^{3} d^{4} e^{4} - 20 \, B a^{2} c^{2} d^{3} e^{5} - 33 \, C a^{3} c d^{2} e^{6} + 45 \, A a^{2} c^{2} d^{2} e^{6} + 30 \, B a^{3} c d e^{7} + 3 \, C a^{4} e^{8} - 15 \, A a^{3} c e^{8}\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 )}{8 \, {\left (a^{2} c^{4} d^{8} + 4 \, a^{3} c^{3} d^{6} e^{2} + 6 \, a^{4} c^{2} d^{4} e^{4} + 4 \, a^{5} c d^{2} e^{6} + a^{6} e^{8}\right )} \sqrt {a c} e^{2}} + \frac {C a c^{4} d^{5} e + 3 \, A c^{5} d^{5} e - 2 \, B a c^{4} d^{4} e^{2} - 22 \, C a^{2} c^{3} d^{3} e^{3} + 14 \, A a c^{4} d^{3} e^{3} + 32 \, B a^{2} c^{3} d^{2} e^{4} + 17 \, C a^{3} c^{2} d e^{5} - 29 \, A a^{2} c^{3} d e^{5} - 6 \, B a^{3} c^{2} e^{6} - \frac {3 \, C a c^{4} d^{6} e^{2} + 9 \, A c^{5} d^{6} e^{2} - 6 \, B a c^{4} d^{5} e^{3} - 77 \, C a^{2} c^{3} d^{4} e^{4} + 41 \, A a c^{4} d^{4} e^{4} + 116 \, B a^{2} c^{3} d^{3} e^{5} + 77 \, C a^{3} c^{2} d^{2} e^{6} - 121 \, A a^{2} c^{3} d^{2} e^{6} - 38 \, B a^{3} c^{2} d e^{7} - 3 \, C a^{4} c e^{8} + 7 \, A a^{3} c^{2} e^{8}}{{\left (e x + d\right )} e} + \frac {3 \, C a c^{4} d^{7} e^{3} + 9 \, A c^{5} d^{7} e^{3} - 6 \, B a c^{4} d^{6} e^{4} - 89 \, C a^{2} c^{3} d^{5} e^{5} + 45 \, A a c^{4} d^{5} e^{5} + 140 \, B a^{2} c^{3} d^{4} e^{6} + 85 \, C a^{3} c^{2} d^{3} e^{7} - 145 \, A a^{2} c^{3} d^{3} e^{7} - 22 \, B a^{3} c^{2} d^{2} e^{8} + 17 \, C a^{4} c d e^{9} - 21 \, A a^{3} c^{2} d e^{9} - 8 \, B a^{4} c e^{10}}{{\left (e x + d\right )}^{2} e^{2}} - \frac {C a c^{4} d^{8} e^{4} + 3 \, A c^{5} d^{8} e^{4} - 2 \, B a c^{4} d^{7} e^{5} - 34 \, C a^{2} c^{3} d^{6} e^{6} + 18 \, A a c^{4} d^{6} e^{6} + 58 \, B a^{2} c^{3} d^{5} e^{7} + 20 \, C a^{3} c^{2} d^{4} e^{8} - 60 \, A a^{2} c^{3} d^{4} e^{8} + 26 \, B a^{3} c^{2} d^{3} e^{9} + 50 \, C a^{4} c d^{2} e^{10} - 66 \, A a^{3} c^{2} d^{2} e^{10} - 34 \, B a^{4} c d e^{11} - 5 \, C a^{5} e^{12} + 9 \, A a^{4} c e^{12}}{{\left (e x + d\right )}^{3} e^{3}}}{8 \, {\left (c d^{2} + a e^{2}\right )}^{4} a^{2} {\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}} \]
-1/2*(4*C*c*d^3*e^3 - 5*B*c*d^2*e^4 - 2*C*a*d*e^5 + 6*A*c*d*e^5 + B*a*e^6) *log(c - 2*c*d/(e*x + d) + c*d^2/(e*x + d)^2 + a*e^2/(e*x + d)^2)/(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) - (C*d ^2*e^9/(e*x + d) - B*d*e^10/(e*x + d) + A*e^11/(e*x + d))/(c^3*d^6*e^6 + 3 *a*c^2*d^4*e^8 + 3*a^2*c*d^2*e^10 + a^3*e^12) + 1/8*(C*a*c^3*d^6*e^2 + 3*A *c^4*d^6*e^2 - 2*B*a*c^3*d^5*e^3 + 13*C*a^2*c^2*d^4*e^4 + 15*A*a*c^3*d^4*e ^4 - 20*B*a^2*c^2*d^3*e^5 - 33*C*a^3*c*d^2*e^6 + 45*A*a^2*c^2*d^2*e^6 + 30 *B*a^3*c*d*e^7 + 3*C*a^4*e^8 - 15*A*a^3*c*e^8)*arctan((c*d - c*d^2/(e*x + d) - a*e^2/(e*x + d))/(sqrt(a*c)*e))/((a^2*c^4*d^8 + 4*a^3*c^3*d^6*e^2 + 6 *a^4*c^2*d^4*e^4 + 4*a^5*c*d^2*e^6 + a^6*e^8)*sqrt(a*c)*e^2) + 1/8*(C*a*c^ 4*d^5*e + 3*A*c^5*d^5*e - 2*B*a*c^4*d^4*e^2 - 22*C*a^2*c^3*d^3*e^3 + 14*A* a*c^4*d^3*e^3 + 32*B*a^2*c^3*d^2*e^4 + 17*C*a^3*c^2*d*e^5 - 29*A*a^2*c^3*d *e^5 - 6*B*a^3*c^2*e^6 - (3*C*a*c^4*d^6*e^2 + 9*A*c^5*d^6*e^2 - 6*B*a*c^4* d^5*e^3 - 77*C*a^2*c^3*d^4*e^4 + 41*A*a*c^4*d^4*e^4 + 116*B*a^2*c^3*d^3*e^ 5 + 77*C*a^3*c^2*d^2*e^6 - 121*A*a^2*c^3*d^2*e^6 - 38*B*a^3*c^2*d*e^7 - 3* C*a^4*c*e^8 + 7*A*a^3*c^2*e^8)/((e*x + d)*e) + (3*C*a*c^4*d^7*e^3 + 9*A*c^ 5*d^7*e^3 - 6*B*a*c^4*d^6*e^4 - 89*C*a^2*c^3*d^5*e^5 + 45*A*a*c^4*d^5*e^5 + 140*B*a^2*c^3*d^4*e^6 + 85*C*a^3*c^2*d^3*e^7 - 145*A*a^2*c^3*d^3*e^7 - 2 2*B*a^3*c^2*d^2*e^8 + 17*C*a^4*c*d*e^9 - 21*A*a^3*c^2*d*e^9 - 8*B*a^4*c*e^ 10)/((e*x + d)^2*e^2) - (C*a*c^4*d^8*e^4 + 3*A*c^5*d^8*e^4 - 2*B*a*c^4*...
Time = 16.66 (sec) , antiderivative size = 6848, normalized size of antiderivative = 11.99 \[ \int \frac {A+B x+C x^2}{(d+e x)^2 \left (a+c x^2\right )^3} \, dx=\text {Too large to display} \]
symsum(log(root(17920*a^9*c^5*d^8*e^8*z^3 + 14336*a^10*c^4*d^6*e^10*z^3 + 14336*a^8*c^6*d^10*e^6*z^3 + 7168*a^11*c^3*d^4*e^12*z^3 + 7168*a^7*c^7*d^1 2*e^4*z^3 + 2048*a^12*c^2*d^2*e^14*z^3 + 2048*a^6*c^8*d^14*e^2*z^3 + 256*a ^5*c^9*d^16*z^3 + 256*a^13*c*e^16*z^3 + 948*B*C*a^7*c*d*e^11*z - 12*A*B*a* c^7*d^11*e*z + 9768*B*C*a^5*c^3*d^5*e^7*z - 7476*B*C*a^6*c^2*d^3*e^9*z - 3 28*B*C*a^4*c^4*d^7*e^5*z - 92*B*C*a^3*c^5*d^9*e^3*z - 12486*A*C*a^5*c^3*d^ 4*e^8*z + 5868*A*C*a^6*c^2*d^2*e^10*z + 282*A*C*a^3*c^5*d^8*e^4*z + 168*A* C*a^4*c^4*d^6*e^6*z + 108*A*C*a^2*c^6*d^10*e^2*z + 14820*A*B*a^5*c^3*d^3*e ^9*z - 840*A*B*a^4*c^4*d^5*e^7*z - 600*A*B*a^3*c^5*d^7*e^5*z - 180*A*B*a^2 *c^6*d^9*e^3*z - 4*B*C*a^2*c^6*d^11*e*z - 3204*A*B*a^6*c^2*d*e^11*z + 4239 *C^2*a^6*c^2*d^4*e^8*z - 3924*C^2*a^5*c^3*d^6*e^6*z + 103*C^2*a^4*c^4*d^8* e^4*z + 26*C^2*a^3*c^5*d^10*e^2*z - 6000*B^2*a^5*c^3*d^4*e^8*z + 2820*B^2* a^6*c^2*d^2*e^10*z + 280*B^2*a^4*c^4*d^6*e^6*z + 80*B^2*a^3*c^5*d^8*e^4*z + 4*B^2*a^2*c^6*d^10*e^2*z - 8262*A^2*a^5*c^3*d^2*e^10*z + 1575*A^2*a^4*c^ 4*d^4*e^8*z + 1260*A^2*a^3*c^5*d^6*e^6*z + 495*A^2*a^2*c^6*d^8*e^4*z - 90* A*C*a^7*c*e^12*z + 6*A*C*a*c^7*d^12*z - 966*C^2*a^7*c*d^2*e^10*z + 90*A^2* a*c^7*d^10*e^2*z + C^2*a^2*c^6*d^12*z + 225*A^2*a^6*c^2*e^12*z - 192*B^2*a ^7*c*e^12*z + 9*A^2*c^8*d^12*z + 9*C^2*a^8*e^12*z + 78*A*B*C*a*c^4*d^6*e^4 + 942*A*B*C*a^2*c^3*d^4*e^6 - 342*A*B*C*a^3*c^2*d^2*e^8 - 129*B*C^2*a^4*c *d^2*e^8 + 990*A^2*C*a^3*c^2*d*e^9 - 234*A^2*C*a*c^4*d^5*e^5 - 24*A*C^2...