Integrand size = 20, antiderivative size = 199 \[ \int \frac {A+B x}{(a+b x)^3 (d+e x)^3} \, dx=-\frac {b (A b-a B)}{2 (b d-a e)^3 (a+b x)^2}-\frac {b (b B d-3 A b e+2 a B e)}{(b d-a e)^4 (a+b x)}-\frac {e (B d-A e)}{2 (b d-a e)^3 (d+e x)^2}-\frac {e (2 b B d-3 A b e+a B e)}{(b d-a e)^4 (d+e x)}-\frac {3 b e (b B d-2 A b e+a B e) \log (a+b x)}{(b d-a e)^5}+\frac {3 b e (b B d-2 A b e+a B e) \log (d+e x)}{(b d-a e)^5} \]
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Time = 0.16 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \[ \int \frac {A+B x}{(a+b x)^3 (d+e x)^3} \, dx=-\frac {b (A b-a B)}{2 (a+b x)^2 (b d-a e)^3}-\frac {b (2 a B e-3 A b e+b B d)}{(a+b x) (b d-a e)^4}-\frac {e (a B e-3 A b e+2 b B d)}{(d+e x) (b d-a e)^4}-\frac {e (B d-A e)}{2 (d+e x)^2 (b d-a e)^3}-\frac {3 b e \log (a+b x) (a B e-2 A b e+b B d)}{(b d-a e)^5}+\frac {3 b e \log (d+e x) (a B e-2 A b e+b B d)}{(b d-a e)^5} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {b^2 (A b-a B)}{(b d-a e)^3 (a+b x)^3}+\frac {b^2 (b B d-3 A b e+2 a B e)}{(b d-a e)^4 (a+b x)^2}+\frac {3 b^2 e (-b B d+2 A b e-a B e)}{(b d-a e)^5 (a+b x)}-\frac {e^2 (-B d+A e)}{(b d-a e)^3 (d+e x)^3}-\frac {e^2 (-2 b B d+3 A b e-a B e)}{(b d-a e)^4 (d+e x)^2}-\frac {3 b e^2 (-b B d+2 A b e-a B e)}{(b d-a e)^5 (d+e x)}\right ) \, dx \\ & = -\frac {b (A b-a B)}{2 (b d-a e)^3 (a+b x)^2}-\frac {b (b B d-3 A b e+2 a B e)}{(b d-a e)^4 (a+b x)}-\frac {e (B d-A e)}{2 (b d-a e)^3 (d+e x)^2}-\frac {e (2 b B d-3 A b e+a B e)}{(b d-a e)^4 (d+e x)}-\frac {3 b e (b B d-2 A b e+a B e) \log (a+b x)}{(b d-a e)^5}+\frac {3 b e (b B d-2 A b e+a B e) \log (d+e x)}{(b d-a e)^5} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.93 \[ \int \frac {A+B x}{(a+b x)^3 (d+e x)^3} \, dx=\frac {-\frac {b (A b-a B) (b d-a e)^2}{(a+b x)^2}-\frac {2 b (b d-a e) (b B d-3 A b e+2 a B e)}{a+b x}+\frac {e (b d-a e)^2 (-B d+A e)}{(d+e x)^2}+\frac {2 e (b d-a e) (-2 b B d+3 A b e-a B e)}{d+e x}-6 b e (b B d-2 A b e+a B e) \log (a+b x)+6 b e (b B d-2 A b e+a B e) \log (d+e x)}{2 (b d-a e)^5} \]
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Time = 0.80 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.01
method | result | size |
default | \(\frac {b \left (3 A b e -2 B a e -B b d \right )}{\left (a e -b d \right )^{4} \left (b x +a \right )}+\frac {\left (A b -B a \right ) b}{2 \left (a e -b d \right )^{3} \left (b x +a \right )^{2}}-\frac {3 b e \left (2 A b e -B a e -B b d \right ) \ln \left (b x +a \right )}{\left (a e -b d \right )^{5}}-\frac {\left (A e -B d \right ) e}{2 \left (a e -b d \right )^{3} \left (e x +d \right )^{2}}+\frac {e \left (3 A b e -B a e -2 B b d \right )}{\left (a e -b d \right )^{4} \left (e x +d \right )}+\frac {3 b e \left (2 A b e -B a e -B b d \right ) \ln \left (e x +d \right )}{\left (a e -b d \right )^{5}}\) | \(200\) |
norman | \(\frac {\frac {\left (6 A \,b^{4} e^{4}-3 B a \,b^{3} e^{4}-3 B \,b^{4} d \,e^{3}\right ) x^{3}}{\left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) b e}+\frac {\left (2 A \,a^{2} b^{3} e^{5}+14 A a \,b^{4} d \,e^{4}+2 A \,b^{5} d^{2} e^{3}-B \,a^{3} b^{2} e^{5}-8 B \,a^{2} b^{3} d \,e^{4}-8 B a \,b^{4} d^{2} e^{3}-B \,b^{5} d^{3} e^{2}\right ) x}{e^{2} b^{2} \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}-\frac {A \,a^{3} b^{2} e^{5}-7 A \,a^{2} b^{3} d \,e^{4}-7 A a \,b^{4} d^{2} e^{3}+A \,b^{5} d^{3} e^{2}+B \,a^{3} b^{2} d \,e^{4}+10 B \,a^{2} b^{3} d^{2} e^{3}+B a \,b^{4} d^{3} e^{2}}{2 e^{2} b^{2} \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}+\frac {\left (18 A a \,b^{4} e^{5}+18 A \,b^{5} d \,e^{4}-9 B \,a^{2} b^{3} e^{5}-18 B a \,b^{4} d \,e^{4}-9 B \,b^{5} d^{2} e^{3}\right ) x^{2}}{2 e^{2} b^{2} \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}}{\left (b x +a \right )^{2} \left (e x +d \right )^{2}}-\frac {3 b e \left (2 A b e -B a e -B b d \right ) \ln \left (b x +a \right )}{a^{5} e^{5}-5 a^{4} d \,e^{4} b +10 a^{3} d^{2} e^{3} b^{2}-10 a^{2} d^{3} e^{2} b^{3}+5 a \,d^{4} e \,b^{4}-b^{5} d^{5}}+\frac {3 b e \left (2 A b e -B a e -B b d \right ) \ln \left (e x +d \right )}{a^{5} e^{5}-5 a^{4} d \,e^{4} b +10 a^{3} d^{2} e^{3} b^{2}-10 a^{2} d^{3} e^{2} b^{3}+5 a \,d^{4} e \,b^{4}-b^{5} d^{5}}\) | \(703\) |
risch | \(\frac {\frac {3 b^{2} e^{2} \left (2 A b e -B a e -B b d \right ) x^{3}}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}+\frac {9 b e \left (a e +b d \right ) \left (2 A b e -B a e -B b d \right ) x^{2}}{2 \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}+\frac {\left (2 A \,a^{2} b \,e^{3}+14 A a \,b^{2} d \,e^{2}+2 A \,b^{3} d^{2} e -B \,a^{3} e^{3}-8 B \,a^{2} b d \,e^{2}-8 B a \,b^{2} d^{2} e -b^{3} B \,d^{3}\right ) x}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}-\frac {a^{3} A \,e^{3}-7 A \,a^{2} b d \,e^{2}-7 A a \,b^{2} d^{2} e +A \,b^{3} d^{3}+B \,a^{3} d \,e^{2}+10 B \,a^{2} b \,d^{2} e +B a \,b^{2} d^{3}}{2 \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}}{\left (b x +a \right )^{2} \left (e x +d \right )^{2}}-\frac {6 b^{2} e^{2} \ln \left (b x +a \right ) A}{a^{5} e^{5}-5 a^{4} d \,e^{4} b +10 a^{3} d^{2} e^{3} b^{2}-10 a^{2} d^{3} e^{2} b^{3}+5 a \,d^{4} e \,b^{4}-b^{5} d^{5}}+\frac {3 b \,e^{2} \ln \left (b x +a \right ) B a}{a^{5} e^{5}-5 a^{4} d \,e^{4} b +10 a^{3} d^{2} e^{3} b^{2}-10 a^{2} d^{3} e^{2} b^{3}+5 a \,d^{4} e \,b^{4}-b^{5} d^{5}}+\frac {3 b^{2} e \ln \left (b x +a \right ) B d}{a^{5} e^{5}-5 a^{4} d \,e^{4} b +10 a^{3} d^{2} e^{3} b^{2}-10 a^{2} d^{3} e^{2} b^{3}+5 a \,d^{4} e \,b^{4}-b^{5} d^{5}}+\frac {6 b^{2} e^{2} \ln \left (-e x -d \right ) A}{a^{5} e^{5}-5 a^{4} d \,e^{4} b +10 a^{3} d^{2} e^{3} b^{2}-10 a^{2} d^{3} e^{2} b^{3}+5 a \,d^{4} e \,b^{4}-b^{5} d^{5}}-\frac {3 b \,e^{2} \ln \left (-e x -d \right ) B a}{a^{5} e^{5}-5 a^{4} d \,e^{4} b +10 a^{3} d^{2} e^{3} b^{2}-10 a^{2} d^{3} e^{2} b^{3}+5 a \,d^{4} e \,b^{4}-b^{5} d^{5}}-\frac {3 b^{2} e \ln \left (-e x -d \right ) B d}{a^{5} e^{5}-5 a^{4} d \,e^{4} b +10 a^{3} d^{2} e^{3} b^{2}-10 a^{2} d^{3} e^{2} b^{3}+5 a \,d^{4} e \,b^{4}-b^{5} d^{5}}\) | \(914\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1388\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1215 vs. \(2 (195) = 390\).
Time = 0.26 (sec) , antiderivative size = 1215, normalized size of antiderivative = 6.11 \[ \int \frac {A+B x}{(a+b x)^3 (d+e x)^3} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1431 vs. \(2 (192) = 384\).
Time = 2.94 (sec) , antiderivative size = 1431, normalized size of antiderivative = 7.19 \[ \int \frac {A+B x}{(a+b x)^3 (d+e x)^3} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 745 vs. \(2 (195) = 390\).
Time = 0.21 (sec) , antiderivative size = 745, normalized size of antiderivative = 3.74 \[ \int \frac {A+B x}{(a+b x)^3 (d+e x)^3} \, dx=-\frac {3 \, {\left (B b^{2} d e + {\left (B a b - 2 \, A b^{2}\right )} e^{2}\right )} \log \left (b x + a\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} + \frac {3 \, {\left (B b^{2} d e + {\left (B a b - 2 \, A b^{2}\right )} e^{2}\right )} \log \left (e x + d\right )}{b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}} - \frac {A a^{3} e^{3} + {\left (B a b^{2} + A b^{3}\right )} d^{3} + {\left (10 \, B a^{2} b - 7 \, A a b^{2}\right )} d^{2} e + {\left (B a^{3} - 7 \, A a^{2} b\right )} d e^{2} + 6 \, {\left (B b^{3} d e^{2} + {\left (B a b^{2} - 2 \, A b^{3}\right )} e^{3}\right )} x^{3} + 9 \, {\left (B b^{3} d^{2} e + 2 \, {\left (B a b^{2} - A b^{3}\right )} d e^{2} + {\left (B a^{2} b - 2 \, A a b^{2}\right )} e^{3}\right )} x^{2} + 2 \, {\left (B b^{3} d^{3} + 2 \, {\left (4 \, B a b^{2} - A b^{3}\right )} d^{2} e + 2 \, {\left (4 \, B a^{2} b - 7 \, A a b^{2}\right )} d e^{2} + {\left (B a^{3} - 2 \, A a^{2} b\right )} e^{3}\right )} x}{2 \, {\left (a^{2} b^{4} d^{6} - 4 \, a^{3} b^{3} d^{5} e + 6 \, a^{4} b^{2} d^{4} e^{2} - 4 \, a^{5} b d^{3} e^{3} + a^{6} d^{2} e^{4} + {\left (b^{6} d^{4} e^{2} - 4 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{4} + 2 \, {\left (b^{6} d^{5} e - 3 \, a b^{5} d^{4} e^{2} + 2 \, a^{2} b^{4} d^{3} e^{3} + 2 \, a^{3} b^{3} d^{2} e^{4} - 3 \, a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x^{3} + {\left (b^{6} d^{6} - 9 \, a^{2} b^{4} d^{4} e^{2} + 16 \, a^{3} b^{3} d^{3} e^{3} - 9 \, a^{4} b^{2} d^{2} e^{4} + a^{6} e^{6}\right )} x^{2} + 2 \, {\left (a b^{5} d^{6} - 3 \, a^{2} b^{4} d^{5} e + 2 \, a^{3} b^{3} d^{4} e^{2} + 2 \, a^{4} b^{2} d^{3} e^{3} - 3 \, a^{5} b d^{2} e^{4} + a^{6} d e^{5}\right )} x\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 532 vs. \(2 (195) = 390\).
Time = 0.28 (sec) , antiderivative size = 532, normalized size of antiderivative = 2.67 \[ \int \frac {A+B x}{(a+b x)^3 (d+e x)^3} \, dx=-\frac {3 \, {\left (B b^{3} d e + B a b^{2} e^{2} - 2 \, A b^{3} e^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}} + \frac {3 \, {\left (B b^{2} d e^{2} + B a b e^{3} - 2 \, A b^{2} e^{3}\right )} \log \left ({\left | e x + d \right |}\right )}{b^{5} d^{5} e - 5 \, a b^{4} d^{4} e^{2} + 10 \, a^{2} b^{3} d^{3} e^{3} - 10 \, a^{3} b^{2} d^{2} e^{4} + 5 \, a^{4} b d e^{5} - a^{5} e^{6}} - \frac {6 \, B b^{3} d e^{2} x^{3} + 6 \, B a b^{2} e^{3} x^{3} - 12 \, A b^{3} e^{3} x^{3} + 9 \, B b^{3} d^{2} e x^{2} + 18 \, B a b^{2} d e^{2} x^{2} - 18 \, A b^{3} d e^{2} x^{2} + 9 \, B a^{2} b e^{3} x^{2} - 18 \, A a b^{2} e^{3} x^{2} + 2 \, B b^{3} d^{3} x + 16 \, B a b^{2} d^{2} e x - 4 \, A b^{3} d^{2} e x + 16 \, B a^{2} b d e^{2} x - 28 \, A a b^{2} d e^{2} x + 2 \, B a^{3} e^{3} x - 4 \, A a^{2} b e^{3} x + B a b^{2} d^{3} + A b^{3} d^{3} + 10 \, B a^{2} b d^{2} e - 7 \, A a b^{2} d^{2} e + B a^{3} d e^{2} - 7 \, A a^{2} b d e^{2} + A a^{3} e^{3}}{2 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} {\left (b e x^{2} + b d x + a e x + a d\right )}^{2}} \]
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Time = 1.78 (sec) , antiderivative size = 726, normalized size of antiderivative = 3.65 \[ \int \frac {A+B x}{(a+b x)^3 (d+e x)^3} \, dx=\frac {2\,\mathrm {atanh}\left (\frac {\left (3\,b\,e^2\,\left (2\,A\,b-B\,a\right )-3\,B\,b^2\,d\,e\right )\,\left (\frac {a^5\,e^5-3\,a^4\,b\,d\,e^4+2\,a^3\,b^2\,d^2\,e^3+2\,a^2\,b^3\,d^3\,e^2-3\,a\,b^4\,d^4\,e+b^5\,d^5}{a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4}+2\,b\,e\,x\right )\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{{\left (a\,e-b\,d\right )}^5\,\left (-6\,A\,b^2\,e^2+3\,B\,d\,b^2\,e+3\,B\,a\,b\,e^2\right )}\right )\,\left (3\,b\,e^2\,\left (2\,A\,b-B\,a\right )-3\,B\,b^2\,d\,e\right )}{{\left (a\,e-b\,d\right )}^5}-\frac {\frac {B\,a^3\,d\,e^2+A\,a^3\,e^3+10\,B\,a^2\,b\,d^2\,e-7\,A\,a^2\,b\,d\,e^2+B\,a\,b^2\,d^3-7\,A\,a\,b^2\,d^2\,e+A\,b^3\,d^3}{2\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}+\frac {9\,x^2\,\left (d\,b^2\,e+a\,b\,e^2\right )\,\left (B\,a\,e-2\,A\,b\,e+B\,b\,d\right )}{2\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}+\frac {x\,\left (a^2\,e^2+7\,a\,b\,d\,e+b^2\,d^2\right )\,\left (B\,a\,e-2\,A\,b\,e+B\,b\,d\right )}{a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4}+\frac {3\,b^2\,e^2\,x^3\,\left (B\,a\,e-2\,A\,b\,e+B\,b\,d\right )}{a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4}}{x\,\left (2\,e\,a^2\,d+2\,b\,a\,d^2\right )+x^2\,\left (a^2\,e^2+4\,a\,b\,d\,e+b^2\,d^2\right )+x^3\,\left (2\,d\,b^2\,e+2\,a\,b\,e^2\right )+a^2\,d^2+b^2\,e^2\,x^4} \]
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