Integrand size = 16, antiderivative size = 140 \[ \int \frac {A+B x}{x^4 (a+b x)^3} \, dx=-\frac {A}{3 a^3 x^3}+\frac {3 A b-a B}{2 a^4 x^2}-\frac {3 b (2 A b-a B)}{a^5 x}-\frac {b^2 (A b-a B)}{2 a^4 (a+b x)^2}-\frac {b^2 (4 A b-3 a B)}{a^5 (a+b x)}-\frac {2 b^2 (5 A b-3 a B) \log (x)}{a^6}+\frac {2 b^2 (5 A b-3 a B) \log (a+b x)}{a^6} \]
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Time = 0.08 (sec) , antiderivative size = 140, 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} \[ \int \frac {A+B x}{x^4 (a+b x)^3} \, dx=-\frac {2 b^2 \log (x) (5 A b-3 a B)}{a^6}+\frac {2 b^2 (5 A b-3 a B) \log (a+b x)}{a^6}-\frac {b^2 (4 A b-3 a B)}{a^5 (a+b x)}-\frac {3 b (2 A b-a B)}{a^5 x}-\frac {b^2 (A b-a B)}{2 a^4 (a+b x)^2}+\frac {3 A b-a B}{2 a^4 x^2}-\frac {A}{3 a^3 x^3} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {A}{a^3 x^4}+\frac {-3 A b+a B}{a^4 x^3}-\frac {3 b (-2 A b+a B)}{a^5 x^2}+\frac {2 b^2 (-5 A b+3 a B)}{a^6 x}-\frac {b^3 (-A b+a B)}{a^4 (a+b x)^3}-\frac {b^3 (-4 A b+3 a B)}{a^5 (a+b x)^2}-\frac {2 b^3 (-5 A b+3 a B)}{a^6 (a+b x)}\right ) \, dx \\ & = -\frac {A}{3 a^3 x^3}+\frac {3 A b-a B}{2 a^4 x^2}-\frac {3 b (2 A b-a B)}{a^5 x}-\frac {b^2 (A b-a B)}{2 a^4 (a+b x)^2}-\frac {b^2 (4 A b-3 a B)}{a^5 (a+b x)}-\frac {2 b^2 (5 A b-3 a B) \log (x)}{a^6}+\frac {2 b^2 (5 A b-3 a B) \log (a+b x)}{a^6} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.92 \[ \int \frac {A+B x}{x^4 (a+b x)^3} \, dx=\frac {\frac {a \left (-60 A b^4 x^4+18 a b^3 x^3 (-5 A+2 B x)-a^4 (2 A+3 B x)+a^3 b x (5 A+12 B x)+2 a^2 b^2 x^2 (-10 A+27 B x)\right )}{x^3 (a+b x)^2}+12 b^2 (-5 A b+3 a B) \log (x)+12 b^2 (5 A b-3 a B) \log (a+b x)}{6 a^6} \]
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Time = 1.20 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.96
method | result | size |
default | \(-\frac {A}{3 a^{3} x^{3}}-\frac {-3 A b +B a}{2 x^{2} a^{4}}-\frac {3 b \left (2 A b -B a \right )}{a^{5} x}-\frac {2 b^{2} \left (5 A b -3 B a \right ) \ln \left (x \right )}{a^{6}}-\frac {b^{2} \left (4 A b -3 B a \right )}{a^{5} \left (b x +a \right )}-\frac {b^{2} \left (A b -B a \right )}{2 a^{4} \left (b x +a \right )^{2}}+\frac {2 b^{2} \left (5 A b -3 B a \right ) \ln \left (b x +a \right )}{a^{6}}\) | \(134\) |
norman | \(\frac {-\frac {A}{3 a}+\frac {\left (5 A b -3 B a \right ) x}{6 a^{2}}-\frac {2 b \left (5 A b -3 B a \right ) x^{2}}{3 a^{3}}-\frac {2 \left (5 b^{5} A -3 a \,b^{4} B \right ) x^{4}}{a^{5} b}-\frac {\left (15 b^{5} A -9 a \,b^{4} B \right ) x^{3}}{a^{4} b^{2}}}{x^{3} \left (b x +a \right )^{2}}-\frac {2 b^{2} \left (5 A b -3 B a \right ) \ln \left (x \right )}{a^{6}}+\frac {2 b^{2} \left (5 A b -3 B a \right ) \ln \left (b x +a \right )}{a^{6}}\) | \(145\) |
risch | \(\frac {-\frac {2 b^{3} \left (5 A b -3 B a \right ) x^{4}}{a^{5}}-\frac {3 b^{2} \left (5 A b -3 B a \right ) x^{3}}{a^{4}}-\frac {2 b \left (5 A b -3 B a \right ) x^{2}}{3 a^{3}}+\frac {\left (5 A b -3 B a \right ) x}{6 a^{2}}-\frac {A}{3 a}}{x^{3} \left (b x +a \right )^{2}}-\frac {10 b^{3} \ln \left (x \right ) A}{a^{6}}+\frac {6 b^{2} \ln \left (x \right ) B}{a^{5}}+\frac {10 b^{3} \ln \left (-b x -a \right ) A}{a^{6}}-\frac {6 b^{2} \ln \left (-b x -a \right ) B}{a^{5}}\) | \(151\) |
parallelrisch | \(-\frac {60 A \ln \left (x \right ) x^{5} b^{7}-60 A \ln \left (b x +a \right ) x^{5} b^{7}-36 B \ln \left (x \right ) x^{5} a \,b^{6}+36 B \ln \left (b x +a \right ) x^{5} a \,b^{6}+120 A \ln \left (x \right ) x^{4} a \,b^{6}-120 A \ln \left (b x +a \right ) x^{4} a \,b^{6}-72 B \ln \left (x \right ) x^{4} a^{2} b^{5}+72 B \ln \left (b x +a \right ) x^{4} a^{2} b^{5}+60 A \ln \left (x \right ) x^{3} a^{2} b^{5}-60 A \ln \left (b x +a \right ) x^{3} a^{2} b^{5}+60 A \,x^{4} a \,b^{6}-36 B \ln \left (x \right ) x^{3} a^{3} b^{4}+36 B \ln \left (b x +a \right ) x^{3} a^{3} b^{4}-36 B \,x^{4} a^{2} b^{5}+90 A \,x^{3} a^{2} b^{5}-54 B \,x^{3} a^{3} b^{4}+20 A \,x^{2} a^{3} b^{4}-12 B \,x^{2} a^{4} b^{3}-5 A x \,a^{4} b^{3}+3 B x \,a^{5} b^{2}+2 A \,a^{5} b^{2}}{6 a^{6} b^{2} x^{3} \left (b x +a \right )^{2}}\) | \(297\) |
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Time = 0.24 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.87 \[ \int \frac {A+B x}{x^4 (a+b x)^3} \, dx=-\frac {2 \, A a^{5} - 12 \, {\left (3 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} - 18 \, {\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{3} - 4 \, {\left (3 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} x^{2} + {\left (3 \, B a^{5} - 5 \, A a^{4} b\right )} x + 12 \, {\left ({\left (3 \, B a b^{4} - 5 \, A b^{5}\right )} x^{5} + 2 \, {\left (3 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} + {\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{3}\right )} \log \left (b x + a\right ) - 12 \, {\left ({\left (3 \, B a b^{4} - 5 \, A b^{5}\right )} x^{5} + 2 \, {\left (3 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} + {\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{3}\right )} \log \left (x\right )}{6 \, {\left (a^{6} b^{2} x^{5} + 2 \, a^{7} b x^{4} + a^{8} x^{3}\right )}} \]
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Time = 0.41 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.87 \[ \int \frac {A+B x}{x^4 (a+b x)^3} \, dx=\frac {- 2 A a^{4} + x^{4} \left (- 60 A b^{4} + 36 B a b^{3}\right ) + x^{3} \left (- 90 A a b^{3} + 54 B a^{2} b^{2}\right ) + x^{2} \left (- 20 A a^{2} b^{2} + 12 B a^{3} b\right ) + x \left (5 A a^{3} b - 3 B a^{4}\right )}{6 a^{7} x^{3} + 12 a^{6} b x^{4} + 6 a^{5} b^{2} x^{5}} + \frac {2 b^{2} \left (- 5 A b + 3 B a\right ) \log {\left (x + \frac {- 10 A a b^{3} + 6 B a^{2} b^{2} - 2 a b^{2} \left (- 5 A b + 3 B a\right )}{- 20 A b^{4} + 12 B a b^{3}} \right )}}{a^{6}} - \frac {2 b^{2} \left (- 5 A b + 3 B a\right ) \log {\left (x + \frac {- 10 A a b^{3} + 6 B a^{2} b^{2} + 2 a b^{2} \left (- 5 A b + 3 B a\right )}{- 20 A b^{4} + 12 B a b^{3}} \right )}}{a^{6}} \]
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Time = 0.19 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.17 \[ \int \frac {A+B x}{x^4 (a+b x)^3} \, dx=-\frac {2 \, A a^{4} - 12 \, {\left (3 \, B a b^{3} - 5 \, A b^{4}\right )} x^{4} - 18 \, {\left (3 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{3} - 4 \, {\left (3 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x^{2} + {\left (3 \, B a^{4} - 5 \, A a^{3} b\right )} x}{6 \, {\left (a^{5} b^{2} x^{5} + 2 \, a^{6} b x^{4} + a^{7} x^{3}\right )}} - \frac {2 \, {\left (3 \, B a b^{2} - 5 \, A b^{3}\right )} \log \left (b x + a\right )}{a^{6}} + \frac {2 \, {\left (3 \, B a b^{2} - 5 \, A b^{3}\right )} \log \left (x\right )}{a^{6}} \]
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Time = 0.28 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.13 \[ \int \frac {A+B x}{x^4 (a+b x)^3} \, dx=\frac {2 \, {\left (3 \, B a b^{2} - 5 \, A b^{3}\right )} \log \left ({\left | x \right |}\right )}{a^{6}} - \frac {2 \, {\left (3 \, B a b^{3} - 5 \, A b^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{6} b} - \frac {2 \, A a^{5} - 12 \, {\left (3 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{4} - 18 \, {\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{3} - 4 \, {\left (3 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} x^{2} + {\left (3 \, B a^{5} - 5 \, A a^{4} b\right )} x}{6 \, {\left (b x + a\right )}^{2} a^{6} x^{3}} \]
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Time = 0.41 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.17 \[ \int \frac {A+B x}{x^4 (a+b x)^3} \, dx=\frac {4\,b^2\,\mathrm {atanh}\left (\frac {2\,b^2\,\left (5\,A\,b-3\,B\,a\right )\,\left (a+2\,b\,x\right )}{a\,\left (10\,A\,b^3-6\,B\,a\,b^2\right )}\right )\,\left (5\,A\,b-3\,B\,a\right )}{a^6}-\frac {\frac {A}{3\,a}-\frac {x\,\left (5\,A\,b-3\,B\,a\right )}{6\,a^2}+\frac {3\,b^2\,x^3\,\left (5\,A\,b-3\,B\,a\right )}{a^4}+\frac {2\,b^3\,x^4\,\left (5\,A\,b-3\,B\,a\right )}{a^5}+\frac {2\,b\,x^2\,\left (5\,A\,b-3\,B\,a\right )}{3\,a^3}}{a^2\,x^3+2\,a\,b\,x^4+b^2\,x^5} \]
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