Integrand size = 27, antiderivative size = 596 \[ \int \frac {A+B x}{\left (a+b x+c x^2\right )^2 \left (d+f x^2\right )} \, dx=\frac {A b c (c d+a f)-(A b-a B) \left (2 c^2 d+b^2 f-2 a c f\right )-c \left (A b^2 f+2 A c (c d-a f)-b B (c d+a f)\right ) x}{\left (b^2-4 a c\right ) \left (b^2 d f+(c d-a f)^2\right ) \left (a+b x+c x^2\right )}-\frac {f^{3/2} \left (A b^2 d f+2 b B d (c d-a f)-A (c d-a f)^2\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {d}}\right )}{\sqrt {d} \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )^2}-\frac {\left (b^5 B d f^2-2 A b^4 f^2 (c d-a f)-4 A c^2 (c d-3 a f) (c d-a f)^2+b^3 B f \left (5 c^2 d^2-4 a c d f-a^2 f^2\right )-4 A b^2 c f \left (2 c^2 d^2-3 a c d f+3 a^2 f^2\right )+2 b B c \left (c^3 d^3-7 a c^2 d^2 f+3 a^2 c d f^2+3 a^3 f^3\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )^2}-\frac {f \left (2 A b f (c d-a f)+B \left (c^2 d^2-2 a c d f-f \left (b^2 d-a^2 f\right )\right )\right ) \log \left (a+b x+c x^2\right )}{2 \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )^2}+\frac {f \left (2 A b f (c d-a f)+B \left (c^2 d^2-2 a c d f-f \left (b^2 d-a^2 f\right )\right )\right ) \log \left (d+f x^2\right )}{2 \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )^2} \]
(A*b*c*(a*f+c*d)-(A*b-B*a)*(-2*a*c*f+b^2*f+2*c^2*d)-c*(A*b^2*f+2*A*c*(-a*f +c*d)-b*B*(a*f+c*d))*x)/(-4*a*c+b^2)/(b^2*d*f+(-a*f+c*d)^2)/(c*x^2+b*x+a)- (b^5*B*d*f^2-2*A*b^4*f^2*(-a*f+c*d)-4*A*c^2*(-3*a*f+c*d)*(-a*f+c*d)^2+b^3* B*f*(-a^2*f^2-4*a*c*d*f+5*c^2*d^2)-4*A*b^2*c*f*(3*a^2*f^2-3*a*c*d*f+2*c^2* d^2)+2*b*B*c*(3*a^3*f^3+3*a^2*c*d*f^2-7*a*c^2*d^2*f+c^3*d^3))*arctanh((2*c *x+b)/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(3/2)/(c^2*d^2-2*a*c*d*f+f*(a^2*f+b ^2*d))^2-1/2*f*(2*A*b*f*(-a*f+c*d)+B*(c^2*d^2-2*a*c*d*f-f*(-a^2*f+b^2*d))) *ln(c*x^2+b*x+a)/(c^2*d^2-2*a*c*d*f+f*(a^2*f+b^2*d))^2+1/2*f*(2*A*b*f*(-a* f+c*d)+B*(c^2*d^2-2*a*c*d*f-f*(-a^2*f+b^2*d)))*ln(f*x^2+d)/(c^2*d^2-2*a*c* d*f+f*(a^2*f+b^2*d))^2-f^(3/2)*(A*b^2*d*f+2*b*B*d*(-a*f+c*d)-A*(-a*f+c*d)^ 2)*arctan(x*f^(1/2)/d^(1/2))/(c^2*d^2-2*a*c*d*f+f*(a^2*f+b^2*d))^2/d^(1/2)
Time = 1.10 (sec) , antiderivative size = 523, normalized size of antiderivative = 0.88 \[ \int \frac {A+B x}{\left (a+b x+c x^2\right )^2 \left (d+f x^2\right )} \, dx=\frac {-\frac {2 \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right ) \left (A \left (b^3 f+b c (c d-3 a f)+b^2 c f x+2 c^2 (c d-a f) x\right )+B \left (2 a^2 c f-b c^2 d x-a \left (2 c^2 d+b^2 f+b c f x\right )\right )\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\frac {2 f^{3/2} \left (-A b^2 d f+A (c d-a f)^2+2 b B d (-c d+a f)\right ) \arctan \left (\frac {\sqrt {f} x}{\sqrt {d}}\right )}{\sqrt {d}}-\frac {2 \left (b^5 B d f^2-4 A c^2 (c d-3 a f) (c d-a f)^2+2 A b^4 f^2 (-c d+a f)-b^3 B f \left (-5 c^2 d^2+4 a c d f+a^2 f^2\right )-4 A b^2 c f \left (2 c^2 d^2-3 a c d f+3 a^2 f^2\right )+2 b B c \left (c^3 d^3-7 a c^2 d^2 f+3 a^2 c d f^2+3 a^3 f^3\right )\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}}+f \left (2 A b f (c d-a f)+B \left (c^2 d^2-2 a c d f+f \left (-b^2 d+a^2 f\right )\right )\right ) \log \left (d+f x^2\right )+f \left (2 A b f (-c d+a f)+B \left (-c^2 d^2+2 a c d f+f \left (b^2 d-a^2 f\right )\right )\right ) \log (a+x (b+c x))}{2 \left (c^2 d^2-2 a c d f+f \left (b^2 d+a^2 f\right )\right )^2} \]
((-2*(c^2*d^2 - 2*a*c*d*f + f*(b^2*d + a^2*f))*(A*(b^3*f + b*c*(c*d - 3*a* f) + b^2*c*f*x + 2*c^2*(c*d - a*f)*x) + B*(2*a^2*c*f - b*c^2*d*x - a*(2*c^ 2*d + b^2*f + b*c*f*x))))/((b^2 - 4*a*c)*(a + x*(b + c*x))) + (2*f^(3/2)*( -(A*b^2*d*f) + A*(c*d - a*f)^2 + 2*b*B*d*(-(c*d) + a*f))*ArcTan[(Sqrt[f]*x )/Sqrt[d]])/Sqrt[d] - (2*(b^5*B*d*f^2 - 4*A*c^2*(c*d - 3*a*f)*(c*d - a*f)^ 2 + 2*A*b^4*f^2*(-(c*d) + a*f) - b^3*B*f*(-5*c^2*d^2 + 4*a*c*d*f + a^2*f^2 ) - 4*A*b^2*c*f*(2*c^2*d^2 - 3*a*c*d*f + 3*a^2*f^2) + 2*b*B*c*(c^3*d^3 - 7 *a*c^2*d^2*f + 3*a^2*c*d*f^2 + 3*a^3*f^3))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + f*(2*A*b*f*(c*d - a*f) + B*(c^2*d^2 - 2*a* c*d*f + f*(-(b^2*d) + a^2*f)))*Log[d + f*x^2] + f*(2*A*b*f*(-(c*d) + a*f) + B*(-(c^2*d^2) + 2*a*c*d*f + f*(b^2*d - a^2*f)))*Log[a + x*(b + c*x)])/(2 *(c^2*d^2 - 2*a*c*d*f + f*(b^2*d + a^2*f))^2)
Time = 1.42 (sec) , antiderivative size = 594, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {1351, 25, 2142, 25, 27, 452, 218, 240, 1142, 1083, 219, 1103}
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.
| \(\displaystyle \int \frac {A+B x}{\left (d+f x^2\right ) \left (a+b x+c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1351 |
| \(\displaystyle \frac {-(A b-a B) \left (-2 a c f+b^2 f+2 c^2 d\right )-c x \left (2 A c (c d-a f)-b B (a f+c d)+A b^2 f\right )+A b c (a f+c d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left ((c d-a f)^2+b^2 d f\right )}-\frac {\int -\frac {-c f \left (A f b^2-B (c d+a f) b+2 A c (c d-a f)\right ) x^2-\left (b^2-4 a c\right ) f (B c d+A b f-a B f) x+(b B-2 A c) \left (d f b^2+(c d-a f)^2\right )+a f \left (A f b^2-B (c d+a f) b+2 A c (c d-a f)\right )}{\left (c x^2+b x+a\right ) \left (f x^2+d\right )}dx}{\left (b^2-4 a c\right ) \left ((c d-a f)^2+b^2 d f\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {-c f \left (A f b^2-B (c d+a f) b+2 A c (c d-a f)\right ) x^2-\left (b^2-4 a c\right ) f (B c d+A b f-a B f) x+(b B-2 A c) \left (d f b^2+(c d-a f)^2\right )+a f \left (A f b^2-B (c d+a f) b+2 A c (c d-a f)\right )}{\left (c x^2+b x+a\right ) \left (f x^2+d\right )}dx}{\left (b^2-4 a c\right ) \left ((c d-a f)^2+b^2 d f\right )}+\frac {-(A b-a B) \left (-2 a c f+b^2 f+2 c^2 d\right )-c x \left (2 A c (c d-a f)-b B (a f+c d)+A b^2 f\right )+A b c (a f+c d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left ((c d-a f)^2+b^2 d f\right )}\) |
| \(\Big \downarrow \) 2142 |
| \(\displaystyle \frac {\frac {\int -\frac {\left (b^2-4 a c\right ) f^2 \left (A d f b^2+2 B d (c d-a f) b-A (c d-a f)^2-\left (2 A b f (c d-a f)+B \left (c^2 d^2-2 a c f d-f \left (b^2 d-a^2 f\right )\right )\right ) x\right )}{f x^2+d}dx}{f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2}+\frac {\int \frac {B d f^2 b^5-2 A f^2 (c d-a f) b^4+B f \left (2 c^2 d^2-3 a c f d-a^2 f^2\right ) b^3-2 A c f \left (2 c^2 d^2-5 a c f d+5 a^2 f^2\right ) b^2+B c \left (c^3 d^3-5 a c^2 f d^2-a^2 c f^2 d+5 a^3 f^3\right ) b-2 A c^2 (c d-3 a f) (c d-a f)^2-c \left (b^2-4 a c\right ) f \left (2 A b f (c d-a f)+B \left (c^2 d^2-2 a c f d-f \left (b^2 d-a^2 f\right )\right )\right ) x}{c x^2+b x+a}dx}{f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2}}{\left (b^2-4 a c\right ) \left ((c d-a f)^2+b^2 d f\right )}+\frac {-(A b-a B) \left (-2 a c f+b^2 f+2 c^2 d\right )-c x \left (2 A c (c d-a f)-b B (a f+c d)+A b^2 f\right )+A b c (a f+c d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left ((c d-a f)^2+b^2 d f\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {B d f^2 b^5-2 A f^2 (c d-a f) b^4+B f \left (2 c^2 d^2-3 a c f d-a^2 f^2\right ) b^3-2 A c f \left (2 c^2 d^2-5 a c f d+5 a^2 f^2\right ) b^2+B c \left (c^3 d^3-5 a c^2 f d^2-a^2 c f^2 d+5 a^3 f^3\right ) b-2 A c^2 (c d-3 a f) (c d-a f)^2-c \left (b^2-4 a c\right ) f \left (2 A b f (c d-a f)+B \left (c^2 d^2-2 a c f d-f \left (b^2 d-a^2 f\right )\right )\right ) x}{c x^2+b x+a}dx}{f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2}-\frac {\int \frac {\left (b^2-4 a c\right ) f^2 \left (A d f b^2+2 B d (c d-a f) b-A (c d-a f)^2-\left (2 A b f (c d-a f)+B \left (c^2 d^2-2 a c f d-f \left (b^2 d-a^2 f\right )\right )\right ) x\right )}{f x^2+d}dx}{f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2}}{\left (b^2-4 a c\right ) \left ((c d-a f)^2+b^2 d f\right )}+\frac {-(A b-a B) \left (-2 a c f+b^2 f+2 c^2 d\right )-c x \left (2 A c (c d-a f)-b B (a f+c d)+A b^2 f\right )+A b c (a f+c d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left ((c d-a f)^2+b^2 d f\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {B d f^2 b^5-2 A f^2 (c d-a f) b^4+B f \left (2 c^2 d^2-3 a c f d-a^2 f^2\right ) b^3-2 A c f \left (2 c^2 d^2-5 a c f d+5 a^2 f^2\right ) b^2+B c \left (c^3 d^3-5 a c^2 f d^2-a^2 c f^2 d+5 a^3 f^3\right ) b-2 A c^2 (c d-3 a f) (c d-a f)^2-c \left (b^2-4 a c\right ) f \left (2 A b f (c d-a f)+B \left (c^2 d^2-2 a c f d-f \left (b^2 d-a^2 f\right )\right )\right ) x}{c x^2+b x+a}dx}{f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2}-\frac {f^2 \left (b^2-4 a c\right ) \int \frac {A d f b^2+2 B d (c d-a f) b-A (c d-a f)^2-\left (2 A b f (c d-a f)+B \left (c^2 d^2-2 a c f d-f \left (b^2 d-a^2 f\right )\right )\right ) x}{f x^2+d}dx}{f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2}}{\left (b^2-4 a c\right ) \left ((c d-a f)^2+b^2 d f\right )}+\frac {-(A b-a B) \left (-2 a c f+b^2 f+2 c^2 d\right )-c x \left (2 A c (c d-a f)-b B (a f+c d)+A b^2 f\right )+A b c (a f+c d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left ((c d-a f)^2+b^2 d f\right )}\) |
| \(\Big \downarrow \) 452 |
| \(\displaystyle \frac {\frac {\int \frac {B d f^2 b^5-2 A f^2 (c d-a f) b^4+B f \left (2 c^2 d^2-3 a c f d-a^2 f^2\right ) b^3-2 A c f \left (2 c^2 d^2-5 a c f d+5 a^2 f^2\right ) b^2+B c \left (c^3 d^3-5 a c^2 f d^2-a^2 c f^2 d+5 a^3 f^3\right ) b-2 A c^2 (c d-3 a f) (c d-a f)^2-c \left (b^2-4 a c\right ) f \left (2 A b f (c d-a f)+B \left (c^2 d^2-2 a c f d-f \left (b^2 d-a^2 f\right )\right )\right ) x}{c x^2+b x+a}dx}{f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2}-\frac {f^2 \left (b^2-4 a c\right ) \left (\left (-A (c d-a f)^2+2 b B d (c d-a f)+A b^2 d f\right ) \int \frac {1}{f x^2+d}dx-\left (B \left (-f \left (b^2 d-a^2 f\right )-2 a c d f+c^2 d^2\right )+2 A b f (c d-a f)\right ) \int \frac {x}{f x^2+d}dx\right )}{f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2}}{\left (b^2-4 a c\right ) \left ((c d-a f)^2+b^2 d f\right )}+\frac {-(A b-a B) \left (-2 a c f+b^2 f+2 c^2 d\right )-c x \left (2 A c (c d-a f)-b B (a f+c d)+A b^2 f\right )+A b c (a f+c d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left ((c d-a f)^2+b^2 d f\right )}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\int \frac {B d f^2 b^5-2 A f^2 (c d-a f) b^4+B f \left (2 c^2 d^2-3 a c f d-a^2 f^2\right ) b^3-2 A c f \left (2 c^2 d^2-5 a c f d+5 a^2 f^2\right ) b^2+B c \left (c^3 d^3-5 a c^2 f d^2-a^2 c f^2 d+5 a^3 f^3\right ) b-2 A c^2 (c d-3 a f) (c d-a f)^2-c \left (b^2-4 a c\right ) f \left (2 A b f (c d-a f)+B \left (c^2 d^2-2 a c f d-f \left (b^2 d-a^2 f\right )\right )\right ) x}{c x^2+b x+a}dx}{f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2}-\frac {f^2 \left (b^2-4 a c\right ) \left (\frac {\arctan \left (\frac {\sqrt {f} x}{\sqrt {d}}\right ) \left (-A (c d-a f)^2+2 b B d (c d-a f)+A b^2 d f\right )}{\sqrt {d} \sqrt {f}}-\left (B \left (-f \left (b^2 d-a^2 f\right )-2 a c d f+c^2 d^2\right )+2 A b f (c d-a f)\right ) \int \frac {x}{f x^2+d}dx\right )}{f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2}}{\left (b^2-4 a c\right ) \left ((c d-a f)^2+b^2 d f\right )}+\frac {-(A b-a B) \left (-2 a c f+b^2 f+2 c^2 d\right )-c x \left (2 A c (c d-a f)-b B (a f+c d)+A b^2 f\right )+A b c (a f+c d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left ((c d-a f)^2+b^2 d f\right )}\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle \frac {\frac {\int \frac {B d f^2 b^5-2 A f^2 (c d-a f) b^4+B f \left (2 c^2 d^2-3 a c f d-a^2 f^2\right ) b^3-2 A c f \left (2 c^2 d^2-5 a c f d+5 a^2 f^2\right ) b^2+B c \left (c^3 d^3-5 a c^2 f d^2-a^2 c f^2 d+5 a^3 f^3\right ) b-2 A c^2 (c d-3 a f) (c d-a f)^2-c \left (b^2-4 a c\right ) f \left (2 A b f (c d-a f)+B \left (c^2 d^2-2 a c f d-f \left (b^2 d-a^2 f\right )\right )\right ) x}{c x^2+b x+a}dx}{f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2}-\frac {f^2 \left (b^2-4 a c\right ) \left (\frac {\arctan \left (\frac {\sqrt {f} x}{\sqrt {d}}\right ) \left (-A (c d-a f)^2+2 b B d (c d-a f)+A b^2 d f\right )}{\sqrt {d} \sqrt {f}}-\frac {\log \left (d+f x^2\right ) \left (B \left (-f \left (b^2 d-a^2 f\right )-2 a c d f+c^2 d^2\right )+2 A b f (c d-a f)\right )}{2 f}\right )}{f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2}}{\left (b^2-4 a c\right ) \left ((c d-a f)^2+b^2 d f\right )}+\frac {-(A b-a B) \left (-2 a c f+b^2 f+2 c^2 d\right )-c x \left (2 A c (c d-a f)-b B (a f+c d)+A b^2 f\right )+A b c (a f+c d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left ((c d-a f)^2+b^2 d f\right )}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {\frac {1}{2} \left (-4 A b^2 c f \left (3 a^2 f^2-3 a c d f+2 c^2 d^2\right )+b^3 B f \left (-a^2 f^2-4 a c d f+5 c^2 d^2\right )+2 b B c \left (3 a^3 f^3+3 a^2 c d f^2-7 a c^2 d^2 f+c^3 d^3\right )-2 A b^4 f^2 (c d-a f)-4 A c^2 (c d-3 a f) (c d-a f)^2+b^5 B d f^2\right ) \int \frac {1}{c x^2+b x+a}dx-\frac {1}{2} f \left (b^2-4 a c\right ) \left (B \left (-f \left (b^2 d-a^2 f\right )-2 a c d f+c^2 d^2\right )+2 A b f (c d-a f)\right ) \int \frac {b+2 c x}{c x^2+b x+a}dx}{f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2}-\frac {f^2 \left (b^2-4 a c\right ) \left (\frac {\arctan \left (\frac {\sqrt {f} x}{\sqrt {d}}\right ) \left (-A (c d-a f)^2+2 b B d (c d-a f)+A b^2 d f\right )}{\sqrt {d} \sqrt {f}}-\frac {\log \left (d+f x^2\right ) \left (B \left (-f \left (b^2 d-a^2 f\right )-2 a c d f+c^2 d^2\right )+2 A b f (c d-a f)\right )}{2 f}\right )}{f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2}}{\left (b^2-4 a c\right ) \left ((c d-a f)^2+b^2 d f\right )}+\frac {-(A b-a B) \left (-2 a c f+b^2 f+2 c^2 d\right )-c x \left (2 A c (c d-a f)-b B (a f+c d)+A b^2 f\right )+A b c (a f+c d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left ((c d-a f)^2+b^2 d f\right )}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {-\frac {1}{2} f \left (b^2-4 a c\right ) \left (B \left (-f \left (b^2 d-a^2 f\right )-2 a c d f+c^2 d^2\right )+2 A b f (c d-a f)\right ) \int \frac {b+2 c x}{c x^2+b x+a}dx-\left (-4 A b^2 c f \left (3 a^2 f^2-3 a c d f+2 c^2 d^2\right )+b^3 B f \left (-a^2 f^2-4 a c d f+5 c^2 d^2\right )+2 b B c \left (3 a^3 f^3+3 a^2 c d f^2-7 a c^2 d^2 f+c^3 d^3\right )-2 A b^4 f^2 (c d-a f)-4 A c^2 (c d-3 a f) (c d-a f)^2+b^5 B d f^2\right ) \int \frac {1}{b^2-(b+2 c x)^2-4 a c}d(b+2 c x)}{f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2}-\frac {f^2 \left (b^2-4 a c\right ) \left (\frac {\arctan \left (\frac {\sqrt {f} x}{\sqrt {d}}\right ) \left (-A (c d-a f)^2+2 b B d (c d-a f)+A b^2 d f\right )}{\sqrt {d} \sqrt {f}}-\frac {\log \left (d+f x^2\right ) \left (B \left (-f \left (b^2 d-a^2 f\right )-2 a c d f+c^2 d^2\right )+2 A b f (c d-a f)\right )}{2 f}\right )}{f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2}}{\left (b^2-4 a c\right ) \left ((c d-a f)^2+b^2 d f\right )}+\frac {-(A b-a B) \left (-2 a c f+b^2 f+2 c^2 d\right )-c x \left (2 A c (c d-a f)-b B (a f+c d)+A b^2 f\right )+A b c (a f+c d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left ((c d-a f)^2+b^2 d f\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {-\frac {1}{2} f \left (b^2-4 a c\right ) \left (B \left (-f \left (b^2 d-a^2 f\right )-2 a c d f+c^2 d^2\right )+2 A b f (c d-a f)\right ) \int \frac {b+2 c x}{c x^2+b x+a}dx-\frac {\text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (-4 A b^2 c f \left (3 a^2 f^2-3 a c d f+2 c^2 d^2\right )+b^3 B f \left (-a^2 f^2-4 a c d f+5 c^2 d^2\right )+2 b B c \left (3 a^3 f^3+3 a^2 c d f^2-7 a c^2 d^2 f+c^3 d^3\right )-2 A b^4 f^2 (c d-a f)-4 A c^2 (c d-3 a f) (c d-a f)^2+b^5 B d f^2\right )}{\sqrt {b^2-4 a c}}}{f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2}-\frac {f^2 \left (b^2-4 a c\right ) \left (\frac {\arctan \left (\frac {\sqrt {f} x}{\sqrt {d}}\right ) \left (-A (c d-a f)^2+2 b B d (c d-a f)+A b^2 d f\right )}{\sqrt {d} \sqrt {f}}-\frac {\log \left (d+f x^2\right ) \left (B \left (-f \left (b^2 d-a^2 f\right )-2 a c d f+c^2 d^2\right )+2 A b f (c d-a f)\right )}{2 f}\right )}{f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2}}{\left (b^2-4 a c\right ) \left ((c d-a f)^2+b^2 d f\right )}+\frac {-(A b-a B) \left (-2 a c f+b^2 f+2 c^2 d\right )-c x \left (2 A c (c d-a f)-b B (a f+c d)+A b^2 f\right )+A b c (a f+c d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left ((c d-a f)^2+b^2 d f\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {-\frac {1}{2} f \left (b^2-4 a c\right ) \log \left (a+b x+c x^2\right ) \left (B \left (-f \left (b^2 d-a^2 f\right )-2 a c d f+c^2 d^2\right )+2 A b f (c d-a f)\right )-\frac {\text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (-4 A b^2 c f \left (3 a^2 f^2-3 a c d f+2 c^2 d^2\right )+b^3 B f \left (-a^2 f^2-4 a c d f+5 c^2 d^2\right )+2 b B c \left (3 a^3 f^3+3 a^2 c d f^2-7 a c^2 d^2 f+c^3 d^3\right )-2 A b^4 f^2 (c d-a f)-4 A c^2 (c d-3 a f) (c d-a f)^2+b^5 B d f^2\right )}{\sqrt {b^2-4 a c}}}{f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2}-\frac {f^2 \left (b^2-4 a c\right ) \left (\frac {\arctan \left (\frac {\sqrt {f} x}{\sqrt {d}}\right ) \left (-A (c d-a f)^2+2 b B d (c d-a f)+A b^2 d f\right )}{\sqrt {d} \sqrt {f}}-\frac {\log \left (d+f x^2\right ) \left (B \left (-f \left (b^2 d-a^2 f\right )-2 a c d f+c^2 d^2\right )+2 A b f (c d-a f)\right )}{2 f}\right )}{f \left (a^2 f+b^2 d\right )-2 a c d f+c^2 d^2}}{\left (b^2-4 a c\right ) \left ((c d-a f)^2+b^2 d f\right )}+\frac {-(A b-a B) \left (-2 a c f+b^2 f+2 c^2 d\right )-c x \left (2 A c (c d-a f)-b B (a f+c d)+A b^2 f\right )+A b c (a f+c d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left ((c d-a f)^2+b^2 d f\right )}\) |
(A*b*c*(c*d + a*f) - (A*b - a*B)*(2*c^2*d + b^2*f - 2*a*c*f) - c*(A*b^2*f + 2*A*c*(c*d - a*f) - b*B*(c*d + a*f))*x)/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(a + b*x + c*x^2)) + ((-(((b^5*B*d*f^2 - 2*A*b^4*f^2*(c*d - a*f) - 4*A*c^2*(c*d - 3*a*f)*(c*d - a*f)^2 + b^3*B*f*(5*c^2*d^2 - 4*a*c*d*f - a ^2*f^2) - 4*A*b^2*c*f*(2*c^2*d^2 - 3*a*c*d*f + 3*a^2*f^2) + 2*b*B*c*(c^3*d ^3 - 7*a*c^2*d^2*f + 3*a^2*c*d*f^2 + 3*a^3*f^3))*ArcTanh[(b + 2*c*x)/Sqrt[ b^2 - 4*a*c]])/Sqrt[b^2 - 4*a*c]) - ((b^2 - 4*a*c)*f*(2*A*b*f*(c*d - a*f) + B*(c^2*d^2 - 2*a*c*d*f - f*(b^2*d - a^2*f)))*Log[a + b*x + c*x^2])/2)/(c ^2*d^2 - 2*a*c*d*f + f*(b^2*d + a^2*f)) - ((b^2 - 4*a*c)*f^2*(((A*b^2*d*f + 2*b*B*d*(c*d - a*f) - A*(c*d - a*f)^2)*ArcTan[(Sqrt[f]*x)/Sqrt[d]])/(Sqr t[d]*Sqrt[f]) - ((2*A*b*f*(c*d - a*f) + B*(c^2*d^2 - 2*a*c*d*f - f*(b^2*d - a^2*f)))*Log[d + f*x^2])/(2*f)))/(c^2*d^2 - 2*a*c*d*f + f*(b^2*d + a^2*f )))/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2))
3.1.5.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[((c_) + (d_.)*(x_))/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[c Int[1/ (a + b*x^2), x], x] + Simp[d Int[x/(a + b*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c^2 + a*d^2, 0]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (f _.)*(x_)^2)^(q_), x_Symbol] :> Simp[(a + b*x + c*x^2)^(p + 1)*((d + f*x^2)^ (q + 1)/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 1)))*((g*c)*((-b)*(c* d + a*f)) + (g*b - a*h)*(2*c^2*d + b^2*f - c*(2*a*f)) + c*(g*(2*c^2*d + b^2 *f - c*(2*a*f)) - h*(b*c*d + a*b*f))*x), x] + Simp[1/((b^2 - 4*a*c)*(b^2*d* f + (c*d - a*f)^2)*(p + 1)) Int[(a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^q*S imp[(b*h - 2*g*c)*((c*d - a*f)^2 - (b*d)*((-b)*f))*(p + 1) + (b^2*(g*f) - b *(h*c*d + a*h*f) + 2*(g*c*(c*d - a*f)))*(a*f*(p + 1) - c*d*(p + 2)) - (2*f* ((g*c)*((-b)*(c*d + a*f)) + (g*b - a*h)*(2*c^2*d + b^2*f - c*(2*a*f)))*(p + q + 2) - (b^2*(g*f) - b*(h*c*d + a*h*f) + 2*(g*c*(c*d - a*f)))*(b*f*(p + 1 )))*x - c*f*(b^2*(g*f) - b*(h*c*d + a*h*f) + 2*(g*c*(c*d - a*f)))*(2*p + 2* q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, f, g, h, q}, x] && NeQ[b^2 - 4 *a*c, 0] && LtQ[p, -1] && NeQ[b^2*d*f + (c*d - a*f)^2, 0] && !( !IntegerQ[ p] && ILtQ[q, -1])
Int[(Px_)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_) + (f_.)*(x_)^2)), x_Sym bol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2] , q = c^2*d^2 + b^2*d*f - 2*a*c*d*f + a^2*f^2}, Simp[1/q Int[(A*c^2*d - a *c*C*d + A*b^2*f - a*b*B*f - a*A*c*f + a^2*C*f + c*(B*c*d - b*C*d + A*b*f - a*B*f)*x)/(a + b*x + c*x^2), x], x] + Simp[1/q Int[(c*C*d^2 + b*B*d*f - A*c*d*f - a*C*d*f + a*A*f^2 - f*(B*c*d - b*C*d + A*b*f - a*B*f)*x)/(d + f*x ^2), x], x] /; NeQ[q, 0]] /; FreeQ[{a, b, c, d, f}, x] && PolyQ[Px, x, 2]
Leaf count of result is larger than twice the leaf count of optimal. \(1253\) vs. \(2(580)=1160\).
Time = 1.55 (sec) , antiderivative size = 1254, normalized size of antiderivative = 2.10
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1254\) |
| risch | \(\text {Expression too large to display}\) | \(3364134\) |
f^2/(a^4*f^4-4*a^3*c*d*f^3+2*a^2*b^2*d*f^3+6*a^2*c^2*d^2*f^2-4*a*b^2*c*d^2 *f^2-4*a*c^3*d^3*f+b^4*d^2*f^2+2*b^2*c^2*d^3*f+c^4*d^4)*(1/2*(-2*A*a*b*f^2 +2*A*b*c*d*f+B*a^2*f^2-2*B*a*c*d*f-B*b^2*d*f+B*c^2*d^2)/f*ln(f*x^2+d)+(A*a ^2*f^2-2*A*a*c*d*f-A*b^2*d*f+A*c^2*d^2+2*B*a*b*d*f-2*B*b*c*d^2)/(d*f)^(1/2 )*arctan(f*x/(d*f)^(1/2)))-1/(a^4*f^4-4*a^3*c*d*f^3+2*a^2*b^2*d*f^3+6*a^2* c^2*d^2*f^2-4*a*b^2*c*d^2*f^2-4*a*c^3*d^3*f+b^4*d^2*f^2+2*b^2*c^2*d^3*f+c^ 4*d^4)*((c*(2*A*a^3*c*f^3-A*a^2*b^2*f^3-6*A*a^2*c^2*d*f^2+4*A*a*b^2*c*d*f^ 2+6*A*a*c^3*d^2*f-A*b^4*d*f^2-3*A*b^2*c^2*d^2*f-2*A*c^4*d^3+B*a^3*b*f^3-B* a^2*b*c*d*f^2+B*a*b^3*d*f^2-B*a*b*c^2*d^2*f+B*b^3*c*d^2*f+B*b*c^3*d^3)/(4* a*c-b^2)*x+(3*A*a^3*b*c*f^3-A*a^2*b^3*f^3-7*A*a^2*b*c^2*d*f^2+5*A*a*b^3*c* d*f^2+5*A*a*b*c^3*d^2*f-A*b^5*d*f^2-2*A*b^3*c^2*d^2*f-A*b*c^4*d^3-2*B*a^4* c*f^3+B*a^3*b^2*f^3+6*B*a^3*c^2*d*f^2-4*B*a^2*b^2*c*d*f^2-6*B*a^2*c^3*d^2* f+B*a*b^4*d*f^2+3*B*a*b^2*c^2*d^2*f+2*B*a*c^4*d^3)/(4*a*c-b^2))/(c*x^2+b*x +a)+1/(4*a*c-b^2)*(1/2*(-8*A*a^2*b*c^2*f^3+2*A*a*b^3*c*f^3+8*A*a*b*c^3*d*f ^2-2*A*b^3*c^2*d*f^2+4*B*a^3*c^2*f^3-B*a^2*b^2*c*f^3-8*B*a^2*c^3*d*f^2-2*B *a*b^2*c^2*d*f^2+4*B*a*c^4*d^2*f+B*b^4*c*d*f^2-B*b^2*c^3*d^2*f)/c*ln(c*x^2 +b*x+a)+2*(6*A*a^3*c^2*f^3-10*A*a^2*b^2*c*f^3-14*A*a^2*c^3*d*f^2+2*A*a*b^4 *f^3+10*A*a*b^2*c^2*d*f^2+10*A*a*c^4*d^2*f-2*A*b^4*c*d*f^2-4*A*b^2*c^3*d^2 *f-2*A*c^5*d^3+5*B*a^3*b*c*f^3-B*a^2*b^3*f^3-B*a^2*b*c^2*d*f^2-3*B*a*b^3*c *d*f^2-5*B*a*b*c^3*d^2*f+b^5*B*d*f^2+2*B*b^3*c^2*d^2*f+B*b*c^4*d^3-1/2*...
Timed out. \[ \int \frac {A+B x}{\left (a+b x+c x^2\right )^2 \left (d+f x^2\right )} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {A+B x}{\left (a+b x+c x^2\right )^2 \left (d+f x^2\right )} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {A+B x}{\left (a+b x+c x^2\right )^2 \left (d+f x^2\right )} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Leaf count of result is larger than twice the leaf count of optimal. 1313 vs. \(2 (579) = 1158\).
Time = 0.28 (sec) , antiderivative size = 1313, normalized size of antiderivative = 2.20 \[ \int \frac {A+B x}{\left (a+b x+c x^2\right )^2 \left (d+f x^2\right )} \, dx=\text {Too large to display} \]
-1/2*(B*c^2*d^2*f - B*b^2*d*f^2 - 2*B*a*c*d*f^2 + 2*A*b*c*d*f^2 + B*a^2*f^ 3 - 2*A*a*b*f^3)*log(c*x^2 + b*x + a)/(c^4*d^4 + 2*b^2*c^2*d^3*f - 4*a*c^3 *d^3*f + b^4*d^2*f^2 - 4*a*b^2*c*d^2*f^2 + 6*a^2*c^2*d^2*f^2 + 2*a^2*b^2*d *f^3 - 4*a^3*c*d*f^3 + a^4*f^4) + 1/2*(B*c^2*d^2*f - B*b^2*d*f^2 - 2*B*a*c *d*f^2 + 2*A*b*c*d*f^2 + B*a^2*f^3 - 2*A*a*b*f^3)*log(f*x^2 + d)/(c^4*d^4 + 2*b^2*c^2*d^3*f - 4*a*c^3*d^3*f + b^4*d^2*f^2 - 4*a*b^2*c*d^2*f^2 + 6*a^ 2*c^2*d^2*f^2 + 2*a^2*b^2*d*f^3 - 4*a^3*c*d*f^3 + a^4*f^4) - (2*B*b*c*d^2* f^2 - A*c^2*d^2*f^2 - 2*B*a*b*d*f^3 + A*b^2*d*f^3 + 2*A*a*c*d*f^3 - A*a^2* f^4)*arctan(f*x/sqrt(d*f))/((c^4*d^4 + 2*b^2*c^2*d^3*f - 4*a*c^3*d^3*f + b ^4*d^2*f^2 - 4*a*b^2*c*d^2*f^2 + 6*a^2*c^2*d^2*f^2 + 2*a^2*b^2*d*f^3 - 4*a ^3*c*d*f^3 + a^4*f^4)*sqrt(d*f)) + (2*B*b*c^4*d^3 - 4*A*c^5*d^3 + 5*B*b^3* c^2*d^2*f - 14*B*a*b*c^3*d^2*f - 8*A*b^2*c^3*d^2*f + 20*A*a*c^4*d^2*f + B* b^5*d*f^2 - 4*B*a*b^3*c*d*f^2 - 2*A*b^4*c*d*f^2 + 6*B*a^2*b*c^2*d*f^2 + 12 *A*a*b^2*c^2*d*f^2 - 28*A*a^2*c^3*d*f^2 - B*a^2*b^3*f^3 + 2*A*a*b^4*f^3 + 6*B*a^3*b*c*f^3 - 12*A*a^2*b^2*c*f^3 + 12*A*a^3*c^2*f^3)*arctan((2*c*x + b )/sqrt(-b^2 + 4*a*c))/((b^2*c^4*d^4 - 4*a*c^5*d^4 + 2*b^4*c^2*d^3*f - 12*a *b^2*c^3*d^3*f + 16*a^2*c^4*d^3*f + b^6*d^2*f^2 - 8*a*b^4*c*d^2*f^2 + 22*a ^2*b^2*c^2*d^2*f^2 - 24*a^3*c^3*d^2*f^2 + 2*a^2*b^4*d*f^3 - 12*a^3*b^2*c*d *f^3 + 16*a^4*c^2*d*f^3 + a^4*b^2*f^4 - 4*a^5*c*f^4)*sqrt(-b^2 + 4*a*c)) + (2*B*a*c^4*d^3 - A*b*c^4*d^3 + 3*B*a*b^2*c^2*d^2*f - 2*A*b^3*c^2*d^2*f...
Time = 18.33 (sec) , antiderivative size = 23006, normalized size of antiderivative = 38.60 \[ \int \frac {A+B x}{\left (a+b x+c x^2\right )^2 \left (d+f x^2\right )} \, dx=\text {Too large to display} \]
((A*b^3*f + A*b*c^2*d - 2*B*a*c^2*d - B*a*b^2*f + 2*B*a^2*c*f - 3*A*a*b*c* f)/((4*a*c - b^2)*(a^2*f^2 + c^2*d^2 + b^2*d*f - 2*a*c*d*f)) - (x*(2*A*a*c ^2*f - 2*A*c^3*d + B*b*c^2*d - A*b^2*c*f + B*a*b*c*f))/((4*a*c - b^2)*(a^2 *f^2 + c^2*d^2 + b^2*d*f - 2*a*c*d*f)))/(a + b*x + c*x^2) + symsum(log((x* (4*A^3*b^3*c^4*f^6 + 16*B^3*a^3*c^4*f^6 - 3*B^3*a^2*b^2*c^3*f^6 + B^3*b^2* c^5*d^2*f^4 - 16*A^3*a*b*c^5*f^6 + 20*A^2*B*a^2*c^5*f^6 - 3*A^2*B*b^4*c^3* f^6 + 4*A^2*B*c^7*d^2*f^4 - 16*B^3*a^2*c^5*d*f^5 + 6*B^3*a*b^2*c^4*d*f^5 - 24*A^2*B*a*c^6*d*f^5 + 6*A*B^2*a*b^3*c^3*f^6 - 28*A*B^2*a^2*b*c^4*f^6 + 8 *A^2*B*a*b^2*c^4*f^6 - 4*A*B^2*b*c^6*d^2*f^4 - 6*A*B^2*b^3*c^4*d*f^5 + 8*A ^2*B*b^2*c^5*d*f^5 + 16*A*B^2*a*b*c^5*d*f^5))/(16*a^2*c^6*d^4 + a^4*b^4*f^ 4 + b^4*c^4*d^4 + 16*a^6*c^2*f^4 + b^8*d^2*f^2 - 8*a*b^2*c^5*d^4 - 8*a^5*b ^2*c*f^4 + 2*a^2*b^6*d*f^3 - 64*a^3*c^5*d^3*f - 64*a^5*c^3*d*f^3 + 2*b^6*c ^2*d^3*f + 96*a^4*c^4*d^2*f^2 + 54*a^2*b^4*c^2*d^2*f^2 - 112*a^3*b^2*c^3*d ^2*f^2 - 20*a*b^4*c^3*d^3*f - 12*a*b^6*c*d^2*f^2 - 20*a^3*b^4*c*d*f^3 + 64 *a^2*b^2*c^4*d^3*f + 64*a^4*b^2*c^2*d*f^3) - root(2560*a^3*b^2*c^9*d^8*f*z ^4 - 1152*a^2*b^4*c^8*d^8*f*z^4 + 384*a^5*b^8*c*d^3*f^6*z^4 + 384*a*b^8*c^ 5*d^7*f^2*z^4 + 288*a^3*b^10*c*d^4*f^5*z^4 + 288*a*b^10*c^3*d^6*f^3*z^4 + 224*a^7*b^6*c*d^2*f^7*z^4 - 192*a^10*b^2*c^2*d*f^8*z^4 + 224*a*b^6*c^7*d^8 *f*z^4 + 80*a*b^12*c*d^5*f^4*z^4 + 48*a^9*b^4*c*d*f^8*z^4 - 33920*a^6*b^2* c^6*d^5*f^4*z^4 + 27936*a^5*b^4*c^5*d^5*f^4*z^4 + 26112*a^7*b^2*c^5*d^4...