Integrand size = 24, antiderivative size = 283 \[ \int \frac {A+B x}{(d+e x)^3 \left (b x+c x^2\right )^2} \, dx=-\frac {A}{b^2 d^3 x}+\frac {c^3 (b B-A c)}{b^2 (c d-b e)^3 (b+c x)}+\frac {e^2 (B d-A e)}{2 d^2 (c d-b e)^2 (d+e x)^2}-\frac {e^2 (2 A e (2 c d-b e)-B d (3 c d-b e))}{d^3 (c d-b e)^3 (d+e x)}+\frac {(b B d-2 A c d-3 A b e) \log (x)}{b^3 d^4}+\frac {c^3 \left (2 A c^2 d+4 b^2 B e-b c (B d+5 A e)\right ) \log (b+c x)}{b^3 (c d-b e)^4}-\frac {e^2 \left (B d \left (6 c^2 d^2-4 b c d e+b^2 e^2\right )-A e \left (10 c^2 d^2-10 b c d e+3 b^2 e^2\right )\right ) \log (d+e x)}{d^4 (c d-b e)^4} \]
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Time = 0.30 (sec) , antiderivative size = 283, 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} \[ \int \frac {A+B x}{(d+e x)^3 \left (b x+c x^2\right )^2} \, dx=\frac {\log (x) (-3 A b e-2 A c d+b B d)}{b^3 d^4}+\frac {c^3 (b B-A c)}{b^2 (b+c x) (c d-b e)^3}-\frac {e^2 \log (d+e x) \left (B d \left (b^2 e^2-4 b c d e+6 c^2 d^2\right )-A e \left (3 b^2 e^2-10 b c d e+10 c^2 d^2\right )\right )}{d^4 (c d-b e)^4}-\frac {A}{b^2 d^3 x}+\frac {c^3 \log (b+c x) \left (-b c (5 A e+B d)+2 A c^2 d+4 b^2 B e\right )}{b^3 (c d-b e)^4}-\frac {e^2 (2 A e (2 c d-b e)-B d (3 c d-b e))}{d^3 (d+e x) (c d-b e)^3}+\frac {e^2 (B d-A e)}{2 d^2 (d+e x)^2 (c d-b e)^2} \]
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Rule 785
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {A}{b^2 d^3 x^2}+\frac {b B d-2 A c d-3 A b e}{b^3 d^4 x}+\frac {c^4 (b B-A c)}{b^2 (-c d+b e)^3 (b+c x)^2}+\frac {c^4 \left (2 A c^2 d+4 b^2 B e-b c (B d+5 A e)\right )}{b^3 (c d-b e)^4 (b+c x)}-\frac {e^3 (B d-A e)}{d^2 (c d-b e)^2 (d+e x)^3}+\frac {e^3 (2 A e (2 c d-b e)-B d (3 c d-b e))}{d^3 (c d-b e)^3 (d+e x)^2}+\frac {e^3 \left (-B d \left (6 c^2 d^2-4 b c d e+b^2 e^2\right )+A e \left (10 c^2 d^2-10 b c d e+3 b^2 e^2\right )\right )}{d^4 (c d-b e)^4 (d+e x)}\right ) \, dx \\ & = -\frac {A}{b^2 d^3 x}+\frac {c^3 (b B-A c)}{b^2 (c d-b e)^3 (b+c x)}+\frac {e^2 (B d-A e)}{2 d^2 (c d-b e)^2 (d+e x)^2}-\frac {e^2 (2 A e (2 c d-b e)-B d (3 c d-b e))}{d^3 (c d-b e)^3 (d+e x)}+\frac {(b B d-2 A c d-3 A b e) \log (x)}{b^3 d^4}+\frac {c^3 \left (2 A c^2 d+4 b^2 B e-b c (B d+5 A e)\right ) \log (b+c x)}{b^3 (c d-b e)^4}-\frac {e^2 \left (B d \left (6 c^2 d^2-4 b c d e+b^2 e^2\right )-A e \left (10 c^2 d^2-10 b c d e+3 b^2 e^2\right )\right ) \log (d+e x)}{d^4 (c d-b e)^4} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.99 \[ \int \frac {A+B x}{(d+e x)^3 \left (b x+c x^2\right )^2} \, dx=-\frac {A}{b^2 d^3 x}+\frac {c^3 (-b B+A c)}{b^2 (-c d+b e)^3 (b+c x)}+\frac {e^2 (B d-A e)}{2 d^2 (c d-b e)^2 (d+e x)^2}+\frac {e^2 (B d (3 c d-b e)+2 A e (-2 c d+b e))}{d^3 (c d-b e)^3 (d+e x)}+\frac {(b B d-2 A c d-3 A b e) \log (x)}{b^3 d^4}+\frac {c^3 \left (2 A c^2 d+4 b^2 B e-b c (B d+5 A e)\right ) \log (b+c x)}{b^3 (c d-b e)^4}+\frac {e^2 \left (-B d \left (6 c^2 d^2-4 b c d e+b^2 e^2\right )+A e \left (10 c^2 d^2-10 b c d e+3 b^2 e^2\right )\right ) \log (d+e x)}{d^4 (c d-b e)^4} \]
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Time = 0.40 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.01
method | result | size |
default | \(-\frac {A}{b^{2} d^{3} x}+\frac {\left (-3 A b e -2 A c d +B b d \right ) \ln \left (x \right )}{b^{3} d^{4}}-\frac {c^{3} \left (5 A b c e -2 A \,c^{2} d -4 b^{2} B e +B b c d \right ) \ln \left (c x +b \right )}{b^{3} \left (b e -c d \right )^{4}}+\frac {\left (A c -B b \right ) c^{3}}{b^{2} \left (b e -c d \right )^{3} \left (c x +b \right )}-\frac {e^{2} \left (2 A b \,e^{2}-4 A c d e -B b d e +3 B c \,d^{2}\right )}{d^{3} \left (b e -c d \right )^{3} \left (e x +d \right )}+\frac {e^{2} \left (3 A \,b^{2} e^{3}-10 A b c d \,e^{2}+10 A \,c^{2} d^{2} e -B \,b^{2} d \,e^{2}+4 B b c \,d^{2} e -6 B \,c^{2} d^{3}\right ) \ln \left (e x +d \right )}{d^{4} \left (b e -c d \right )^{4}}-\frac {\left (A e -B d \right ) e^{2}}{2 d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{2}}\) | \(285\) |
norman | \(\frac {\frac {\left (6 A \,b^{5} e^{5}-12 A \,b^{4} c d \,e^{4}+4 A \,b^{3} c^{2} d^{2} e^{3}+A b \,c^{4} d^{4} e -2 A \,c^{5} d^{5}-2 B \,b^{5} d \,e^{4}+4 B \,b^{4} c \,d^{2} e^{3}+B b \,c^{4} d^{5}\right ) x^{2}}{d^{3} b^{3} \left (b^{3} e^{3}-3 b^{2} c d \,e^{2}+3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}-\frac {A}{d b}+\frac {e \left (9 A \,b^{5} e^{5}-7 A \,b^{4} c d \,e^{4}-18 A \,b^{3} c^{2} d^{2} e^{3}+8 A \,b^{2} c^{3} d^{3} e^{2}+4 A b \,c^{4} d^{4} e -8 A \,c^{5} d^{5}-3 B \,b^{5} d \,e^{4}+3 B \,b^{4} c \,d^{2} e^{3}+8 B \,b^{3} c^{2} d^{3} e^{2}+4 B b \,c^{4} d^{5}\right ) x^{3}}{2 d^{4} b^{3} \left (b^{3} e^{3}-3 b^{2} c d \,e^{2}+3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}+\frac {e^{2} c \left (9 A \,b^{4} e^{4}-19 A \,b^{3} c d \,e^{3}+6 A \,b^{2} c^{2} d^{2} e^{2}+2 A b \,c^{3} d^{3} e -4 A \,c^{4} d^{4}-3 B \,b^{4} d \,e^{3}+7 B \,b^{3} c \,d^{2} e^{2}+2 B b \,c^{3} d^{4}\right ) x^{4}}{2 d^{4} b^{3} \left (b^{3} e^{3}-3 b^{2} c d \,e^{2}+3 b \,c^{2} d^{2} e -c^{3} d^{3}\right )}}{\left (e x +d \right )^{2} x \left (c x +b \right )}+\frac {e^{2} \left (3 A \,b^{2} e^{3}-10 A b c d \,e^{2}+10 A \,c^{2} d^{2} e -B \,b^{2} d \,e^{2}+4 B b c \,d^{2} e -6 B \,c^{2} d^{3}\right ) \ln \left (e x +d \right )}{d^{4} \left (b^{4} e^{4}-4 b^{3} c d \,e^{3}+6 b^{2} c^{2} d^{2} e^{2}-4 b \,c^{3} d^{3} e +c^{4} d^{4}\right )}-\frac {\left (3 A b e +2 A c d -B b d \right ) \ln \left (x \right )}{b^{3} d^{4}}-\frac {c^{3} \left (5 A b c e -2 A \,c^{2} d -4 b^{2} B e +B b c d \right ) \ln \left (c x +b \right )}{\left (b^{4} e^{4}-4 b^{3} c d \,e^{3}+6 b^{2} c^{2} d^{2} e^{2}-4 b \,c^{3} d^{3} e +c^{4} d^{4}\right ) b^{3}}\) | \(714\) |
risch | \(\text {Expression too large to display}\) | \(1180\) |
parallelrisch | \(\text {Expression too large to display}\) | \(2554\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1924 vs. \(2 (279) = 558\).
Time = 264.40 (sec) , antiderivative size = 1924, normalized size of antiderivative = 6.80 \[ \int \frac {A+B x}{(d+e x)^3 \left (b x+c x^2\right )^2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {A+B x}{(d+e x)^3 \left (b x+c x^2\right )^2} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 813 vs. \(2 (279) = 558\).
Time = 0.23 (sec) , antiderivative size = 813, normalized size of antiderivative = 2.87 \[ \int \frac {A+B x}{(d+e x)^3 \left (b x+c x^2\right )^2} \, dx=-\frac {{\left ({\left (B b c^{4} - 2 \, A c^{5}\right )} d - {\left (4 \, B b^{2} c^{3} - 5 \, A b c^{4}\right )} e\right )} \log \left (c x + b\right )}{b^{3} c^{4} d^{4} - 4 \, b^{4} c^{3} d^{3} e + 6 \, b^{5} c^{2} d^{2} e^{2} - 4 \, b^{6} c d e^{3} + b^{7} e^{4}} - \frac {{\left (6 \, B c^{2} d^{3} e^{2} - 3 \, A b^{2} e^{5} - 2 \, {\left (2 \, B b c + 5 \, A c^{2}\right )} d^{2} e^{3} + {\left (B b^{2} + 10 \, A b c\right )} d e^{4}\right )} \log \left (e x + d\right )}{c^{4} d^{8} - 4 \, b c^{3} d^{7} e + 6 \, b^{2} c^{2} d^{6} e^{2} - 4 \, b^{3} c d^{5} e^{3} + b^{4} d^{4} e^{4}} - \frac {2 \, A b c^{3} d^{5} - 6 \, A b^{2} c^{2} d^{4} e + 6 \, A b^{3} c d^{3} e^{2} - 2 \, A b^{4} d^{2} e^{3} - 2 \, {\left (3 \, A b^{3} c e^{5} + {\left (B b c^{3} - 2 \, A c^{4}\right )} d^{3} e^{2} + 3 \, {\left (B b^{2} c^{2} + A b c^{3}\right )} d^{2} e^{3} - {\left (B b^{3} c + 7 \, A b^{2} c^{2}\right )} d e^{4}\right )} x^{3} - {\left (6 \, A b^{4} e^{5} + 4 \, {\left (B b c^{3} - 2 \, A c^{4}\right )} d^{4} e + {\left (7 \, B b^{2} c^{2} + 10 \, A b c^{3}\right )} d^{3} e^{2} + 3 \, {\left (B b^{3} c - 5 \, A b^{2} c^{2}\right )} d^{2} e^{3} - {\left (2 \, B b^{4} + 5 \, A b^{3} c\right )} d e^{4}\right )} x^{2} - {\left (2 \, A b c^{3} d^{4} e + 9 \, A b^{4} d e^{4} + 2 \, {\left (B b c^{3} - 2 \, A c^{4}\right )} d^{5} + {\left (7 \, B b^{3} c + 6 \, A b^{2} c^{2}\right )} d^{3} e^{2} - {\left (3 \, B b^{4} + 19 \, A b^{3} c\right )} d^{2} e^{3}\right )} x}{2 \, {\left ({\left (b^{2} c^{4} d^{6} e^{2} - 3 \, b^{3} c^{3} d^{5} e^{3} + 3 \, b^{4} c^{2} d^{4} e^{4} - b^{5} c d^{3} e^{5}\right )} x^{4} + {\left (2 \, b^{2} c^{4} d^{7} e - 5 \, b^{3} c^{3} d^{6} e^{2} + 3 \, b^{4} c^{2} d^{5} e^{3} + b^{5} c d^{4} e^{4} - b^{6} d^{3} e^{5}\right )} x^{3} + {\left (b^{2} c^{4} d^{8} - b^{3} c^{3} d^{7} e - 3 \, b^{4} c^{2} d^{6} e^{2} + 5 \, b^{5} c d^{5} e^{3} - 2 \, b^{6} d^{4} e^{4}\right )} x^{2} + {\left (b^{3} c^{3} d^{8} - 3 \, b^{4} c^{2} d^{7} e + 3 \, b^{5} c d^{6} e^{2} - b^{6} d^{5} e^{3}\right )} x\right )}} - \frac {{\left (3 \, A b e - {\left (B b - 2 \, A c\right )} d\right )} \log \left (x\right )}{b^{3} d^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 773 vs. \(2 (279) = 558\).
Time = 0.28 (sec) , antiderivative size = 773, normalized size of antiderivative = 2.73 \[ \int \frac {A+B x}{(d+e x)^3 \left (b x+c x^2\right )^2} \, dx=-\frac {{\left (B b c^{5} d - 2 \, A c^{6} d - 4 \, B b^{2} c^{4} e + 5 \, A b c^{5} e\right )} \log \left ({\left | c x + b \right |}\right )}{b^{3} c^{5} d^{4} - 4 \, b^{4} c^{4} d^{3} e + 6 \, b^{5} c^{3} d^{2} e^{2} - 4 \, b^{6} c^{2} d e^{3} + b^{7} c e^{4}} - \frac {{\left (6 \, B c^{2} d^{3} e^{3} - 4 \, B b c d^{2} e^{4} - 10 \, A c^{2} d^{2} e^{4} + B b^{2} d e^{5} + 10 \, A b c d e^{5} - 3 \, A b^{2} e^{6}\right )} \log \left ({\left | e x + d \right |}\right )}{c^{4} d^{8} e - 4 \, b c^{3} d^{7} e^{2} + 6 \, b^{2} c^{2} d^{6} e^{3} - 4 \, b^{3} c d^{5} e^{4} + b^{4} d^{4} e^{5}} + \frac {{\left (B b d - 2 \, A c d - 3 \, A b e\right )} \log \left ({\left | x \right |}\right )}{b^{3} d^{4}} - \frac {2 \, A b c^{4} d^{7} - 8 \, A b^{2} c^{3} d^{6} e + 12 \, A b^{3} c^{2} d^{5} e^{2} - 8 \, A b^{4} c d^{4} e^{3} + 2 \, A b^{5} d^{3} e^{4} - 2 \, {\left (B b c^{4} d^{5} e^{2} - 2 \, A c^{5} d^{5} e^{2} + 2 \, B b^{2} c^{3} d^{4} e^{3} + 5 \, A b c^{4} d^{4} e^{3} - 4 \, B b^{3} c^{2} d^{3} e^{4} - 10 \, A b^{2} c^{3} d^{3} e^{4} + B b^{4} c d^{2} e^{5} + 10 \, A b^{3} c^{2} d^{2} e^{5} - 3 \, A b^{4} c d e^{6}\right )} x^{3} - {\left (4 \, B b c^{4} d^{6} e - 8 \, A c^{5} d^{6} e + 3 \, B b^{2} c^{3} d^{5} e^{2} + 18 \, A b c^{4} d^{5} e^{2} - 4 \, B b^{3} c^{2} d^{4} e^{3} - 25 \, A b^{2} c^{3} d^{4} e^{3} - 5 \, B b^{4} c d^{3} e^{4} + 10 \, A b^{3} c^{2} d^{3} e^{4} + 2 \, B b^{5} d^{2} e^{5} + 11 \, A b^{4} c d^{2} e^{5} - 6 \, A b^{5} d e^{6}\right )} x^{2} - {\left (2 \, B b c^{4} d^{7} - 4 \, A c^{5} d^{7} - 2 \, B b^{2} c^{3} d^{6} e + 6 \, A b c^{4} d^{6} e + 7 \, B b^{3} c^{2} d^{5} e^{2} + 4 \, A b^{2} c^{3} d^{5} e^{2} - 10 \, B b^{4} c d^{4} e^{3} - 25 \, A b^{3} c^{2} d^{4} e^{3} + 3 \, B b^{5} d^{3} e^{4} + 28 \, A b^{4} c d^{3} e^{4} - 9 \, A b^{5} d^{2} e^{5}\right )} x}{2 \, {\left (c d - b e\right )}^{4} {\left (c x + b\right )} {\left (e x + d\right )}^{2} b^{2} d^{4} x} \]
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Time = 12.03 (sec) , antiderivative size = 726, normalized size of antiderivative = 2.57 \[ \int \frac {A+B x}{(d+e x)^3 \left (b x+c x^2\right )^2} \, dx=\frac {\ln \left (d+e\,x\right )\,\left (\left (3\,A\,e^5-B\,d\,e^4\right )\,b^2+\left (4\,B\,d^2\,e^3-10\,A\,d\,e^4\right )\,b\,c+\left (10\,A\,d^2\,e^3-6\,B\,d^3\,e^2\right )\,c^2\right )}{b^4\,d^4\,e^4-4\,b^3\,c\,d^5\,e^3+6\,b^2\,c^2\,d^6\,e^2-4\,b\,c^3\,d^7\,e+c^4\,d^8}-\frac {\frac {A}{b\,d}+\frac {x^2\,\left (-2\,B\,b^4\,d\,e^4+6\,A\,b^4\,e^5+3\,B\,b^3\,c\,d^2\,e^3-5\,A\,b^3\,c\,d\,e^4+7\,B\,b^2\,c^2\,d^3\,e^2-15\,A\,b^2\,c^2\,d^2\,e^3+4\,B\,b\,c^3\,d^4\,e+10\,A\,b\,c^3\,d^3\,e^2-8\,A\,c^4\,d^4\,e\right )}{2\,b^2\,d^3\,\left (b^3\,e^3-3\,b^2\,c\,d\,e^2+3\,b\,c^2\,d^2\,e-c^3\,d^3\right )}+\frac {x\,\left (-3\,B\,b^4\,d\,e^3+9\,A\,b^4\,e^4+7\,B\,b^3\,c\,d^2\,e^2-19\,A\,b^3\,c\,d\,e^3+6\,A\,b^2\,c^2\,d^2\,e^2+2\,B\,b\,c^3\,d^4+2\,A\,b\,c^3\,d^3\,e-4\,A\,c^4\,d^4\right )}{2\,b^2\,d^2\,\left (b^3\,e^3-3\,b^2\,c\,d\,e^2+3\,b\,c^2\,d^2\,e-c^3\,d^3\right )}+\frac {c\,e^2\,x^3\,\left (-B\,b^3\,d\,e^2+3\,A\,b^3\,e^3+3\,B\,b^2\,c\,d^2\,e-7\,A\,b^2\,c\,d\,e^2+B\,b\,c^2\,d^3+3\,A\,b\,c^2\,d^2\,e-2\,A\,c^3\,d^3\right )}{b^2\,d^3\,\left (b^3\,e^3-3\,b^2\,c\,d\,e^2+3\,b\,c^2\,d^2\,e-c^3\,d^3\right )}}{x^2\,\left (c\,d^2+2\,b\,e\,d\right )+x^3\,\left (b\,e^2+2\,c\,d\,e\right )+c\,e^2\,x^4+b\,d^2\,x}+\frac {\ln \left (b+c\,x\right )\,\left (e\,\left (4\,B\,b^2\,c^3-5\,A\,b\,c^4\right )+d\,\left (2\,A\,c^5-B\,b\,c^4\right )\right )}{b^7\,e^4-4\,b^6\,c\,d\,e^3+6\,b^5\,c^2\,d^2\,e^2-4\,b^4\,c^3\,d^3\,e+b^3\,c^4\,d^4}-\frac {\ln \left (x\right )\,\left (d\,\left (2\,A\,c-B\,b\right )+3\,A\,b\,e\right )}{b^3\,d^4} \]
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