Integrand size = 20, antiderivative size = 200 \[ \int \frac {A+B x}{(a+b x)^2 (d+e x)^4} \, dx=-\frac {b^2 (A b-a B)}{(b d-a e)^4 (a+b x)}+\frac {B d-A e}{3 (b d-a e)^2 (d+e x)^3}+\frac {b B d-2 A b e+a B e}{2 (b d-a e)^3 (d+e x)^2}+\frac {b (b B d-3 A b e+2 a B e)}{(b d-a e)^4 (d+e x)}+\frac {b^2 (b B d-4 A b e+3 a B e) \log (a+b x)}{(b d-a e)^5}-\frac {b^2 (b B d-4 A b e+3 a B e) \log (d+e x)}{(b d-a e)^5} \]
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Time = 0.14 (sec) , antiderivative size = 200, 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} \[ \int \frac {A+B x}{(a+b x)^2 (d+e x)^4} \, dx=-\frac {b^2 (A b-a B)}{(a+b x) (b d-a e)^4}+\frac {b^2 \log (a+b x) (3 a B e-4 A b e+b B d)}{(b d-a e)^5}-\frac {b^2 \log (d+e x) (3 a B e-4 A b e+b B d)}{(b d-a e)^5}+\frac {b (2 a B e-3 A b e+b B d)}{(d+e x) (b d-a e)^4}+\frac {a B e-2 A b e+b B d}{2 (d+e x)^2 (b d-a e)^3}+\frac {B d-A e}{3 (d+e x)^3 (b d-a e)^2} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {b^3 (A b-a B)}{(b d-a e)^4 (a+b x)^2}+\frac {b^3 (b B d-4 A b e+3 a B e)}{(b d-a e)^5 (a+b x)}+\frac {e (-B d+A e)}{(b d-a e)^2 (d+e x)^4}+\frac {e (-b B d+2 A b e-a B e)}{(b d-a e)^3 (d+e x)^3}+\frac {b e (-b B d+3 A b e-2 a B e)}{(b d-a e)^4 (d+e x)^2}+\frac {b^2 e (-b B d+4 A b e-3 a B e)}{(b d-a e)^5 (d+e x)}\right ) \, dx \\ & = -\frac {b^2 (A b-a B)}{(b d-a e)^4 (a+b x)}+\frac {B d-A e}{3 (b d-a e)^2 (d+e x)^3}+\frac {b B d-2 A b e+a B e}{2 (b d-a e)^3 (d+e x)^2}+\frac {b (b B d-3 A b e+2 a B e)}{(b d-a e)^4 (d+e x)}+\frac {b^2 (b B d-4 A b e+3 a B e) \log (a+b x)}{(b d-a e)^5}-\frac {b^2 (b B d-4 A b e+3 a B e) \log (d+e x)}{(b d-a e)^5} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.94 \[ \int \frac {A+B x}{(a+b x)^2 (d+e x)^4} \, dx=\frac {-\frac {6 b^2 (A b-a B) (b d-a e)}{a+b x}+\frac {2 (b d-a e)^3 (B d-A e)}{(d+e x)^3}+\frac {3 (b d-a e)^2 (b B d-2 A b e+a B e)}{(d+e x)^2}+\frac {6 b (b d-a e) (b B d-3 A b e+2 a B e)}{d+e x}+6 b^2 (b B d-4 A b e+3 a B e) \log (a+b x)-6 b^2 (b B d-4 A b e+3 a B e) \log (d+e x)}{6 (b d-a e)^5} \]
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Time = 0.80 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.02
method | result | size |
default | \(\frac {b^{2} \left (4 A b e -3 B a e -B b d \right ) \ln \left (b x +a \right )}{\left (a e -b d \right )^{5}}-\frac {\left (A b -B a \right ) b^{2}}{\left (a e -b d \right )^{4} \left (b x +a \right )}-\frac {A e -B d}{3 \left (a e -b d \right )^{2} \left (e x +d \right )^{3}}-\frac {b \left (3 A b e -2 B a e -B b d \right )}{\left (a e -b d \right )^{4} \left (e x +d \right )}+\frac {2 A b e -B a e -B b d}{2 \left (a e -b d \right )^{3} \left (e x +d \right )^{2}}-\frac {b^{2} \left (4 A b e -3 B a e -B b d \right ) \ln \left (e x +d \right )}{\left (a e -b d \right )^{5}}\) | \(203\) |
norman | \(\frac {-\frac {2 A \,a^{3} b \,e^{6}-10 A \,a^{2} b^{2} d \,e^{5}+26 A a \,b^{3} d^{2} e^{4}+6 A \,b^{4} d^{3} e^{3}+B \,a^{3} b d \,e^{5}-8 B \,a^{2} b^{2} d^{2} e^{4}-17 B a \,b^{3} d^{3} e^{3}}{6 b \,e^{3} \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}-\frac {\left (4 A \,b^{4} e^{4}-3 B a \,b^{3} e^{4}-B \,b^{4} d \,e^{3}\right ) x^{3}}{e b \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}-\frac {\left (4 A a \,b^{3} e^{5}+20 A \,b^{4} d \,e^{4}-3 B \,a^{2} b^{2} e^{5}-16 B a \,b^{3} d \,e^{4}-5 B \,b^{4} d^{2} e^{3}\right ) x^{2}}{2 e^{2} b \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}+\frac {\left (4 A \,a^{2} b^{2} e^{6}-32 A a \,b^{3} d \,e^{5}-44 A \,b^{4} d^{2} e^{4}-3 B \,a^{3} b \,e^{6}+23 B \,a^{2} b^{2} d \,e^{5}+41 B a \,b^{3} d^{2} e^{4}+11 B \,b^{4} d^{3} e^{3}\right ) x}{6 b \,e^{3} \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}}{\left (b x +a \right ) \left (e x +d \right )^{3}}+\frac {b^{2} \left (4 A b e -3 B a e -B b d \right ) \ln \left (b x +a \right )}{a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}}-\frac {b^{2} \left (4 A b e -3 B a e -B b d \right ) \ln \left (e x +d \right )}{a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}}\) | \(703\) |
risch | \(\frac {-\frac {b^{2} e^{2} \left (4 A b e -3 B a e -B b d \right ) x^{3}}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}-\frac {e \left (a e +5 b d \right ) b \left (4 A b e -3 B a e -B b d \right ) x^{2}}{2 \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}+\frac {\left (4 A \,a^{2} b \,e^{3}-32 A a \,b^{2} d \,e^{2}-44 A \,b^{3} d^{2} e -3 B \,a^{3} e^{3}+23 B \,a^{2} b d \,e^{2}+41 B a \,b^{2} d^{2} e +11 b^{3} B \,d^{3}\right ) x}{6 a^{4} e^{4}-24 a^{3} b d \,e^{3}+36 a^{2} b^{2} d^{2} e^{2}-24 a \,b^{3} d^{3} e +6 b^{4} d^{4}}-\frac {2 a^{3} A \,e^{3}-10 A \,a^{2} b d \,e^{2}+26 A a \,b^{2} d^{2} e +6 A \,b^{3} d^{3}+B \,a^{3} d \,e^{2}-8 B \,a^{2} b \,d^{2} e -17 B a \,b^{2} d^{3}}{6 \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}}{\left (b x +a \right ) \left (e x +d \right )^{3}}+\frac {4 b^{3} \ln \left (-b x -a \right ) A e}{a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}}-\frac {3 b^{2} \ln \left (-b x -a \right ) B a e}{a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}}-\frac {b^{3} \ln \left (-b x -a \right ) B d}{a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}}-\frac {4 b^{3} \ln \left (e x +d \right ) A e}{a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}}+\frac {3 b^{2} \ln \left (e x +d \right ) B a e}{a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}}+\frac {b^{3} \ln \left (e x +d \right ) B d}{a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}}\) | \(912\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1347\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1219 vs. \(2 (195) = 390\).
Time = 0.26 (sec) , antiderivative size = 1219, normalized size of antiderivative = 6.10 \[ \int \frac {A+B x}{(a+b x)^2 (d+e x)^4} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1445 vs. \(2 (190) = 380\).
Time = 2.95 (sec) , antiderivative size = 1445, normalized size of antiderivative = 7.22 \[ \int \frac {A+B x}{(a+b x)^2 (d+e x)^4} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 761 vs. \(2 (195) = 390\).
Time = 0.24 (sec) , antiderivative size = 761, normalized size of antiderivative = 3.80 \[ \int \frac {A+B x}{(a+b x)^2 (d+e x)^4} \, dx=\frac {{\left (B b^{3} d + {\left (3 \, B a b^{2} - 4 \, A b^{3}\right )} e\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 {{\left (B b^{3} d + {\left (3 \, B a b^{2} - 4 \, A b^{3}\right )} e\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 {2 \, A a^{3} e^{3} - {\left (17 \, B a b^{2} - 6 \, A b^{3}\right )} d^{3} - 2 \, {\left (4 \, B a^{2} b - 13 \, A a b^{2}\right )} d^{2} e + {\left (B a^{3} - 10 \, A a^{2} b\right )} d e^{2} - 6 \, {\left (B b^{3} d e^{2} + {\left (3 \, B a b^{2} - 4 \, A b^{3}\right )} e^{3}\right )} x^{3} - 3 \, {\left (5 \, B b^{3} d^{2} e + 4 \, {\left (4 \, B a b^{2} - 5 \, A b^{3}\right )} d e^{2} + {\left (3 \, B a^{2} b - 4 \, A a b^{2}\right )} e^{3}\right )} x^{2} - {\left (11 \, B b^{3} d^{3} + {\left (41 \, B a b^{2} - 44 \, A b^{3}\right )} d^{2} e + {\left (23 \, B a^{2} b - 32 \, A a b^{2}\right )} d e^{2} - {\left (3 \, B a^{3} - 4 \, A a^{2} b\right )} e^{3}\right )} x}{6 \, {\left (a b^{4} d^{7} - 4 \, a^{2} b^{3} d^{6} e + 6 \, a^{3} b^{2} d^{5} e^{2} - 4 \, a^{4} b d^{4} e^{3} + a^{5} d^{3} e^{4} + {\left (b^{5} d^{4} e^{3} - 4 \, a b^{4} d^{3} e^{4} + 6 \, a^{2} b^{3} d^{2} e^{5} - 4 \, a^{3} b^{2} d e^{6} + a^{4} b e^{7}\right )} x^{4} + {\left (3 \, b^{5} d^{5} e^{2} - 11 \, a b^{4} d^{4} e^{3} + 14 \, a^{2} b^{3} d^{3} e^{4} - 6 \, a^{3} b^{2} d^{2} e^{5} - a^{4} b d e^{6} + a^{5} e^{7}\right )} x^{3} + 3 \, {\left (b^{5} d^{6} e - 3 \, a b^{4} d^{5} e^{2} + 2 \, a^{2} b^{3} d^{4} e^{3} + 2 \, a^{3} b^{2} d^{3} e^{4} - 3 \, a^{4} b d^{2} e^{5} + a^{5} d e^{6}\right )} x^{2} + {\left (b^{5} d^{7} - a b^{4} d^{6} e - 6 \, a^{2} b^{3} d^{5} e^{2} + 14 \, a^{3} b^{2} d^{4} e^{3} - 11 \, a^{4} b d^{3} e^{4} + 3 \, a^{5} d^{2} e^{5}\right )} x\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 432 vs. \(2 (195) = 390\).
Time = 0.30 (sec) , antiderivative size = 432, normalized size of antiderivative = 2.16 \[ \int \frac {A+B x}{(a+b x)^2 (d+e x)^4} \, dx=-\frac {{\left (B b^{4} d + 3 \, B a b^{3} e - 4 \, A b^{4} e\right )} \log \left ({\left | \frac {b d}{b x + a} - \frac {a e}{b x + a} + e \right |}\right )}{b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}} + \frac {\frac {B a b^{6}}{b x + a} - \frac {A b^{7}}{b x + a}}{b^{8} d^{4} - 4 \, a b^{7} d^{3} e + 6 \, a^{2} b^{6} d^{2} e^{2} - 4 \, a^{3} b^{5} d e^{3} + a^{4} b^{4} e^{4}} - \frac {11 \, B b^{3} d e^{3} + 15 \, B a b^{2} e^{4} - 26 \, A b^{3} e^{4} + \frac {3 \, {\left (9 \, B b^{5} d^{2} e^{2} + 2 \, B a b^{4} d e^{3} - 20 \, A b^{5} d e^{3} - 11 \, B a^{2} b^{3} e^{4} + 20 \, A a b^{4} e^{4}\right )}}{{\left (b x + a\right )} b} + \frac {18 \, {\left (B b^{7} d^{3} e - B a b^{6} d^{2} e^{2} - 2 \, A b^{7} d^{2} e^{2} - B a^{2} b^{5} d e^{3} + 4 \, A a b^{6} d e^{3} + B a^{3} b^{4} e^{4} - 2 \, A a^{2} b^{5} e^{4}\right )}}{{\left (b x + a\right )}^{2} b^{2}}}{6 \, {\left (b d - a e\right )}^{5} {\left (\frac {b d}{b x + a} - \frac {a e}{b x + a} + e\right )}^{3}} \]
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Time = 1.94 (sec) , antiderivative size = 527, normalized size of antiderivative = 2.64 \[ \int \frac {A+B x}{(a+b x)^2 (d+e x)^4} \, dx=\frac {\frac {x\,\left (-a^2\,e^2+8\,a\,b\,d\,e+11\,b^2\,d^2\right )\,\left (3\,B\,a\,e-4\,A\,b\,e+B\,b\,d\right )}{6\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}-\frac {B\,a^3\,d\,e^2+2\,A\,a^3\,e^3-8\,B\,a^2\,b\,d^2\,e-10\,A\,a^2\,b\,d\,e^2-17\,B\,a\,b^2\,d^3+26\,A\,a\,b^2\,d^2\,e+6\,A\,b^3\,d^3}{6\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}+\frac {b^2\,e^2\,x^3\,\left (3\,B\,a\,e-4\,A\,b\,e+B\,b\,d\right )}{a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4}+\frac {b\,x^2\,\left (a\,e^2+5\,b\,d\,e\right )\,\left (3\,B\,a\,e-4\,A\,b\,e+B\,b\,d\right )}{2\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}}{x^3\,\left (a\,e^3+3\,b\,d\,e^2\right )+x^2\,\left (3\,b\,d^2\,e+3\,a\,d\,e^2\right )+a\,d^3+x\,\left (b\,d^3+3\,a\,e\,d^2\right )+b\,e^3\,x^4}+\frac {\ln \left (a+b\,x\right )\,\left (b^3\,\left (4\,A\,e-B\,d\right )-3\,B\,a\,b^2\,e\right )}{{\left (a\,e-b\,d\right )}^5}+\frac {\ln \left (d+e\,x\right )\,\left (B\,b^3\,d-b^2\,e\,\left (4\,A\,b-3\,B\,a\right )\right )}{{\left (a\,e-b\,d\right )}^5} \]
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