Integrand size = 24, antiderivative size = 147 \[ \int \frac {A+B x}{(d+e x) \left (b x+c x^2\right )^2} \, dx=-\frac {A}{b^2 d x}+\frac {c (b B-A c)}{b^2 (c d-b e) (b+c x)}+\frac {(b B d-2 A c d-A b e) \log (x)}{b^3 d^2}+\frac {c \left (2 A c^2 d+2 b^2 B e-b c (B d+3 A e)\right ) \log (b+c x)}{b^3 (c d-b e)^2}-\frac {e^2 (B d-A e) \log (d+e x)}{d^2 (c d-b e)^2} \]
[Out]
Time = 0.14 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {785} \[ \int \frac {A+B x}{(d+e x) \left (b x+c x^2\right )^2} \, dx=\frac {\log (x) (-A b e-2 A c d+b B d)}{b^3 d^2}+\frac {c (b B-A c)}{b^2 (b+c x) (c d-b e)}-\frac {A}{b^2 d x}+\frac {c \log (b+c x) \left (-b c (3 A e+B d)+2 A c^2 d+2 b^2 B e\right )}{b^3 (c d-b e)^2}-\frac {e^2 (B d-A e) \log (d+e x)}{d^2 (c d-b e)^2} \]
[In]
[Out]
Rule 785
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {A}{b^2 d x^2}+\frac {b B d-2 A c d-A b e}{b^3 d^2 x}+\frac {c^2 (b B-A c)}{b^2 (-c d+b e) (b+c x)^2}+\frac {c^2 \left (2 A c^2 d+2 b^2 B e-b c (B d+3 A e)\right )}{b^3 (c d-b e)^2 (b+c x)}-\frac {e^3 (B d-A e)}{d^2 (c d-b e)^2 (d+e x)}\right ) \, dx \\ & = -\frac {A}{b^2 d x}+\frac {c (b B-A c)}{b^2 (c d-b e) (b+c x)}+\frac {(b B d-2 A c d-A b e) \log (x)}{b^3 d^2}+\frac {c \left (2 A c^2 d+2 b^2 B e-b c (B d+3 A e)\right ) \log (b+c x)}{b^3 (c d-b e)^2}-\frac {e^2 (B d-A e) \log (d+e x)}{d^2 (c d-b e)^2} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.99 \[ \int \frac {A+B x}{(d+e x) \left (b x+c x^2\right )^2} \, dx=-\frac {A}{b^2 d x}+\frac {c (-b B+A c)}{b^2 (-c d+b e) (b+c x)}+\frac {(b B d-2 A c d-A b e) \log (x)}{b^3 d^2}+\frac {c \left (2 A c^2 d+2 b^2 B e-b c (B d+3 A e)\right ) \log (b+c x)}{b^3 (c d-b e)^2}+\frac {e^2 (-B d+A e) \log (d+e x)}{d^2 (c d-b e)^2} \]
[In]
[Out]
Time = 0.32 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00
method | result | size |
default | \(-\frac {A}{b^{2} d x}+\frac {\left (-A b e -2 A c d +B b d \right ) \ln \left (x \right )}{b^{3} d^{2}}-\frac {c \left (3 A b c e -2 A \,c^{2} d -2 b^{2} B e +B b c d \right ) \ln \left (c x +b \right )}{b^{3} \left (b e -c d \right )^{2}}+\frac {\left (A c -B b \right ) c}{b^{2} \left (b e -c d \right ) \left (c x +b \right )}+\frac {\left (A e -B d \right ) e^{2} \ln \left (e x +d \right )}{d^{2} \left (b e -c d \right )^{2}}\) | \(147\) |
norman | \(\frac {\frac {\left (A b c e -2 A \,c^{2} d +B b c d \right ) c \,x^{2}}{d \,b^{3} \left (b e -c d \right )}-\frac {A}{d b}}{x \left (c x +b \right )}+\frac {e^{2} \left (A e -B d \right ) \ln \left (e x +d \right )}{d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}-\frac {\left (A b e +2 A c d -B b d \right ) \ln \left (x \right )}{d^{2} b^{3}}-\frac {c \left (3 A b c e -2 A \,c^{2} d -2 b^{2} B e +B b c d \right ) \ln \left (c x +b \right )}{\left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) b^{3}}\) | \(192\) |
risch | \(\frac {-\frac {c \left (A b e -2 A c d +B b d \right ) x}{b^{2} d \left (b e -c d \right )}-\frac {A}{d b}}{x \left (c x +b \right )}+\frac {e^{3} \ln \left (-e x -d \right ) A}{d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}-\frac {e^{2} \ln \left (-e x -d \right ) B}{d \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right )}-\frac {3 c^{2} \ln \left (c x +b \right ) A e}{\left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) b^{2}}+\frac {2 c^{3} \ln \left (c x +b \right ) A d}{\left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) b^{3}}+\frac {2 c \ln \left (c x +b \right ) B e}{\left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) b}-\frac {c^{2} \ln \left (c x +b \right ) B d}{\left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) b^{2}}-\frac {\ln \left (-x \right ) A e}{d^{2} b^{2}}-\frac {2 \ln \left (-x \right ) A c}{d \,b^{3}}+\frac {\ln \left (-x \right ) B}{d \,b^{2}}\) | \(332\) |
parallelrisch | \(-\frac {B x \,b^{3} c^{2} d^{2} e +A \ln \left (x \right ) x^{2} b^{3} c^{2} e^{3}-A \ln \left (e x +d \right ) x^{2} b^{3} c^{2} e^{3}+A \ln \left (x \right ) x \,b^{4} c \,e^{3}-A \ln \left (e x +d \right ) x \,b^{4} c \,e^{3}+A x \,b^{3} c^{2} d \,e^{2}-3 A x \,b^{2} c^{3} d^{2} e -2 A \ln \left (c x +b \right ) x b \,c^{4} d^{3}-B \ln \left (x \right ) x \,b^{2} c^{3} d^{3}+B \ln \left (c x +b \right ) x \,b^{2} c^{3} d^{3}-B \ln \left (x \right ) x^{2} b \,c^{4} d^{3}+B \ln \left (c x +b \right ) x^{2} b \,c^{4} d^{3}+2 A \ln \left (x \right ) x b \,c^{4} d^{3}+A \,b^{2} c^{3} d^{3}+2 A \ln \left (x \right ) x^{2} c^{5} d^{3}-2 A \ln \left (c x +b \right ) x^{2} c^{5} d^{3}+2 A x b \,c^{4} d^{3}-B x \,b^{2} c^{3} d^{3}-3 A \ln \left (x \right ) x^{2} b \,c^{4} d^{2} e +3 A \ln \left (c x +b \right ) x^{2} b \,c^{4} d^{2} e -3 A \ln \left (x \right ) x \,b^{2} c^{3} d^{2} e +3 A \ln \left (c x +b \right ) x \,b^{2} c^{3} d^{2} e +A \,b^{4} c d \,e^{2}-2 A \,b^{3} c^{2} d^{2} e -B \ln \left (x \right ) x^{2} b^{3} c^{2} d \,e^{2}+2 B \ln \left (x \right ) x^{2} b^{2} c^{3} d^{2} e -2 B \ln \left (c x +b \right ) x^{2} b^{2} c^{3} d^{2} e +B \ln \left (e x +d \right ) x^{2} b^{3} c^{2} d \,e^{2}-B \ln \left (x \right ) x \,b^{4} c d \,e^{2}+2 B \ln \left (x \right ) x \,b^{3} c^{2} d^{2} e -2 B \ln \left (c x +b \right ) x \,b^{3} c^{2} d^{2} e +B \ln \left (e x +d \right ) x \,b^{4} c d \,e^{2}}{\left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) \left (c x +b \right ) x \,b^{3} c \,d^{2}}\) | \(556\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 450 vs. \(2 (147) = 294\).
Time = 23.08 (sec) , antiderivative size = 450, normalized size of antiderivative = 3.06 \[ \int \frac {A+B x}{(d+e x) \left (b x+c x^2\right )^2} \, dx=-\frac {A b^{2} c^{2} d^{3} - 2 \, A b^{3} c d^{2} e + A b^{4} d e^{2} + {\left (A b^{3} c d e^{2} - {\left (B b^{2} c^{2} - 2 \, A b c^{3}\right )} d^{3} + {\left (B b^{3} c - 3 \, A b^{2} c^{2}\right )} d^{2} e\right )} x + {\left ({\left ({\left (B b c^{3} - 2 \, A c^{4}\right )} d^{3} - {\left (2 \, B b^{2} c^{2} - 3 \, A b c^{3}\right )} d^{2} e\right )} x^{2} + {\left ({\left (B b^{2} c^{2} - 2 \, A b c^{3}\right )} d^{3} - {\left (2 \, B b^{3} c - 3 \, A b^{2} c^{2}\right )} d^{2} e\right )} x\right )} \log \left (c x + b\right ) + {\left ({\left (B b^{3} c d e^{2} - A b^{3} c e^{3}\right )} x^{2} + {\left (B b^{4} d e^{2} - A b^{4} e^{3}\right )} x\right )} \log \left (e x + d\right ) - {\left ({\left (B b^{3} c d e^{2} - A b^{3} c e^{3} + {\left (B b c^{3} - 2 \, A c^{4}\right )} d^{3} - {\left (2 \, B b^{2} c^{2} - 3 \, A b c^{3}\right )} d^{2} e\right )} x^{2} + {\left (B b^{4} d e^{2} - A b^{4} e^{3} + {\left (B b^{2} c^{2} - 2 \, A b c^{3}\right )} d^{3} - {\left (2 \, B b^{3} c - 3 \, A b^{2} c^{2}\right )} d^{2} e\right )} x\right )} \log \left (x\right )}{{\left (b^{3} c^{3} d^{4} - 2 \, b^{4} c^{2} d^{3} e + b^{5} c d^{2} e^{2}\right )} x^{2} + {\left (b^{4} c^{2} d^{4} - 2 \, b^{5} c d^{3} e + b^{6} d^{2} e^{2}\right )} x} \]
[In]
[Out]
Timed out. \[ \int \frac {A+B x}{(d+e x) \left (b x+c x^2\right )^2} \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.54 \[ \int \frac {A+B x}{(d+e x) \left (b x+c x^2\right )^2} \, dx=-\frac {{\left ({\left (B b c^{2} - 2 \, A c^{3}\right )} d - {\left (2 \, B b^{2} c - 3 \, A b c^{2}\right )} e\right )} \log \left (c x + b\right )}{b^{3} c^{2} d^{2} - 2 \, b^{4} c d e + b^{5} e^{2}} - \frac {{\left (B d e^{2} - A e^{3}\right )} \log \left (e x + d\right )}{c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}} - \frac {A b c d - A b^{2} e - {\left (A b c e + {\left (B b c - 2 \, A c^{2}\right )} d\right )} x}{{\left (b^{2} c^{2} d^{2} - b^{3} c d e\right )} x^{2} + {\left (b^{3} c d^{2} - b^{4} d e\right )} x} - \frac {{\left (A b e - {\left (B b - 2 \, A c\right )} d\right )} \log \left (x\right )}{b^{3} d^{2}} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.80 \[ \int \frac {A+B x}{(d+e x) \left (b x+c x^2\right )^2} \, dx=-\frac {{\left (B b c^{3} d - 2 \, A c^{4} d - 2 \, B b^{2} c^{2} e + 3 \, A b c^{3} e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{3} c^{3} d^{2} - 2 \, b^{4} c^{2} d e + b^{5} c e^{2}} - \frac {{\left (B d e^{3} - A e^{4}\right )} \log \left ({\left | e x + d \right |}\right )}{c^{2} d^{4} e - 2 \, b c d^{3} e^{2} + b^{2} d^{2} e^{3}} + \frac {{\left (B b d - 2 \, A c d - A b e\right )} \log \left ({\left | x \right |}\right )}{b^{3} d^{2}} - \frac {A b c^{2} d^{3} - 2 \, A b^{2} c d^{2} e + A b^{3} d e^{2} - {\left (B b c^{2} d^{3} - 2 \, A c^{3} d^{3} - B b^{2} c d^{2} e + 3 \, A b c^{2} d^{2} e - A b^{2} c d e^{2}\right )} x}{{\left (c d - b e\right )}^{2} {\left (c x + b\right )} b^{2} d^{2} x} \]
[In]
[Out]
Time = 10.84 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.37 \[ \int \frac {A+B x}{(d+e x) \left (b x+c x^2\right )^2} \, dx=\frac {\ln \left (b+c\,x\right )\,\left (d\,\left (2\,A\,c^3-B\,b\,c^2\right )-3\,A\,b\,c^2\,e+2\,B\,b^2\,c\,e\right )}{b^5\,e^2-2\,b^4\,c\,d\,e+b^3\,c^2\,d^2}-\frac {\frac {A}{b\,d}+\frac {x\,\left (A\,b\,c\,e-2\,A\,c^2\,d+B\,b\,c\,d\right )}{b^2\,d\,\left (b\,e-c\,d\right )}}{c\,x^2+b\,x}+\frac {\ln \left (d+e\,x\right )\,\left (A\,e^3-B\,d\,e^2\right )}{b^2\,d^2\,e^2-2\,b\,c\,d^3\,e+c^2\,d^4}-\frac {\ln \left (x\right )\,\left (b\,\left (A\,e-B\,d\right )+2\,A\,c\,d\right )}{b^3\,d^2} \]
[In]
[Out]