Integrand size = 16, antiderivative size = 106 \[ \int \frac {A+B x}{x^5 (a+b x)} \, dx=-\frac {A}{4 a x^4}+\frac {A b-a B}{3 a^2 x^3}-\frac {b (A b-a B)}{2 a^3 x^2}+\frac {b^2 (A b-a B)}{a^4 x}+\frac {b^3 (A b-a B) \log (x)}{a^5}-\frac {b^3 (A b-a B) \log (a+b x)}{a^5} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {78} \[ \int \frac {A+B x}{x^5 (a+b x)} \, dx=\frac {b^3 \log (x) (A b-a B)}{a^5}-\frac {b^3 (A b-a B) \log (a+b x)}{a^5}+\frac {b^2 (A b-a B)}{a^4 x}-\frac {b (A b-a B)}{2 a^3 x^2}+\frac {A b-a B}{3 a^2 x^3}-\frac {A}{4 a x^4} \]
[In]
[Out]
Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {A}{a x^5}+\frac {-A b+a B}{a^2 x^4}-\frac {b (-A b+a B)}{a^3 x^3}+\frac {b^2 (-A b+a B)}{a^4 x^2}-\frac {b^3 (-A b+a B)}{a^5 x}+\frac {b^4 (-A b+a B)}{a^5 (a+b x)}\right ) \, dx \\ & = -\frac {A}{4 a x^4}+\frac {A b-a B}{3 a^2 x^3}-\frac {b (A b-a B)}{2 a^3 x^2}+\frac {b^2 (A b-a B)}{a^4 x}+\frac {b^3 (A b-a B) \log (x)}{a^5}-\frac {b^3 (A b-a B) \log (a+b x)}{a^5} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.94 \[ \int \frac {A+B x}{x^5 (a+b x)} \, dx=\frac {\frac {a \left (12 A b^3 x^3-6 a b^2 x^2 (A+2 B x)+2 a^2 b x (2 A+3 B x)-a^3 (3 A+4 B x)\right )}{x^4}+12 b^3 (A b-a B) \log (x)-12 b^3 (A b-a B) \log (a+b x)}{12 a^5} \]
[In]
[Out]
Time = 1.18 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.95
method | result | size |
default | \(-\frac {A}{4 a \,x^{4}}-\frac {-A b +B a}{3 x^{3} a^{2}}+\frac {b^{3} \left (A b -B a \right ) \ln \left (x \right )}{a^{5}}-\frac {b \left (A b -B a \right )}{2 a^{3} x^{2}}+\frac {b^{2} \left (A b -B a \right )}{a^{4} x}-\frac {b^{3} \left (A b -B a \right ) \ln \left (b x +a \right )}{a^{5}}\) | \(101\) |
norman | \(\frac {\frac {\left (A b -B a \right ) b^{2} x^{3}}{a^{4}}-\frac {A}{4 a}+\frac {\left (A b -B a \right ) x}{3 a^{2}}-\frac {\left (A b -B a \right ) b \,x^{2}}{2 a^{3}}}{x^{4}}+\frac {b^{3} \left (A b -B a \right ) \ln \left (x \right )}{a^{5}}-\frac {b^{3} \left (A b -B a \right ) \ln \left (b x +a \right )}{a^{5}}\) | \(101\) |
risch | \(\frac {\frac {\left (A b -B a \right ) b^{2} x^{3}}{a^{4}}-\frac {A}{4 a}+\frac {\left (A b -B a \right ) x}{3 a^{2}}-\frac {\left (A b -B a \right ) b \,x^{2}}{2 a^{3}}}{x^{4}}+\frac {b^{4} \ln \left (-x \right ) A}{a^{5}}-\frac {b^{3} \ln \left (-x \right ) B}{a^{4}}-\frac {b^{4} \ln \left (b x +a \right ) A}{a^{5}}+\frac {b^{3} \ln \left (b x +a \right ) B}{a^{4}}\) | \(116\) |
parallelrisch | \(\frac {12 A \ln \left (x \right ) x^{4} b^{4}-12 A \ln \left (b x +a \right ) x^{4} b^{4}-12 B \ln \left (x \right ) x^{4} a \,b^{3}+12 B \ln \left (b x +a \right ) x^{4} a \,b^{3}+12 A a \,b^{3} x^{3}-12 B \,a^{2} b^{2} x^{3}-6 A \,a^{2} b^{2} x^{2}+6 B \,a^{3} b \,x^{2}+4 A \,a^{3} b x -4 B \,a^{4} x -3 A \,a^{4}}{12 a^{5} x^{4}}\) | \(129\) |
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.10 \[ \int \frac {A+B x}{x^5 (a+b x)} \, dx=\frac {12 \, {\left (B a b^{3} - A b^{4}\right )} x^{4} \log \left (b x + a\right ) - 12 \, {\left (B a b^{3} - A b^{4}\right )} x^{4} \log \left (x\right ) - 3 \, A a^{4} - 12 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} x^{3} + 6 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} x^{2} - 4 \, {\left (B a^{4} - A a^{3} b\right )} x}{12 \, a^{5} x^{4}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (94) = 188\).
Time = 0.29 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.78 \[ \int \frac {A+B x}{x^5 (a+b x)} \, dx=\frac {- 3 A a^{3} + x^{3} \cdot \left (12 A b^{3} - 12 B a b^{2}\right ) + x^{2} \left (- 6 A a b^{2} + 6 B a^{2} b\right ) + x \left (4 A a^{2} b - 4 B a^{3}\right )}{12 a^{4} x^{4}} - \frac {b^{3} \left (- A b + B a\right ) \log {\left (x + \frac {- A a b^{4} + B a^{2} b^{3} - a b^{3} \left (- A b + B a\right )}{- 2 A b^{5} + 2 B a b^{4}} \right )}}{a^{5}} + \frac {b^{3} \left (- A b + B a\right ) \log {\left (x + \frac {- A a b^{4} + B a^{2} b^{3} + a b^{3} \left (- A b + B a\right )}{- 2 A b^{5} + 2 B a b^{4}} \right )}}{a^{5}} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.06 \[ \int \frac {A+B x}{x^5 (a+b x)} \, dx=\frac {{\left (B a b^{3} - A b^{4}\right )} \log \left (b x + a\right )}{a^{5}} - \frac {{\left (B a b^{3} - A b^{4}\right )} \log \left (x\right )}{a^{5}} - \frac {3 \, A a^{3} + 12 \, {\left (B a b^{2} - A b^{3}\right )} x^{3} - 6 \, {\left (B a^{2} b - A a b^{2}\right )} x^{2} + 4 \, {\left (B a^{3} - A a^{2} b\right )} x}{12 \, a^{4} x^{4}} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.15 \[ \int \frac {A+B x}{x^5 (a+b x)} \, dx=-\frac {{\left (B a b^{3} - A b^{4}\right )} \log \left ({\left | x \right |}\right )}{a^{5}} + \frac {{\left (B a b^{4} - A b^{5}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{5} b} - \frac {3 \, A a^{4} + 12 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} x^{3} - 6 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} x^{2} + 4 \, {\left (B a^{4} - A a^{3} b\right )} x}{12 \, a^{5} x^{4}} \]
[In]
[Out]
Time = 0.44 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.09 \[ \int \frac {A+B x}{x^5 (a+b x)} \, dx=-\frac {\frac {A}{4\,a}-\frac {x\,\left (A\,b-B\,a\right )}{3\,a^2}-\frac {b^2\,x^3\,\left (A\,b-B\,a\right )}{a^4}+\frac {b\,x^2\,\left (A\,b-B\,a\right )}{2\,a^3}}{x^4}-\frac {2\,b^3\,\mathrm {atanh}\left (\frac {b^3\,\left (A\,b-B\,a\right )\,\left (a+2\,b\,x\right )}{a\,\left (A\,b^4-B\,a\,b^3\right )}\right )\,\left (A\,b-B\,a\right )}{a^5} \]
[In]
[Out]