Integrand size = 11, antiderivative size = 65 \[ \int \frac {x^4}{(a+b x)^4} \, dx=\frac {x}{b^4}-\frac {a^4}{3 b^5 (a+b x)^3}+\frac {2 a^3}{b^5 (a+b x)^2}-\frac {6 a^2}{b^5 (a+b x)}-\frac {4 a \log (a+b x)}{b^5} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int \frac {x^4}{(a+b x)^4} \, dx=-\frac {a^4}{3 b^5 (a+b x)^3}+\frac {2 a^3}{b^5 (a+b x)^2}-\frac {6 a^2}{b^5 (a+b x)}-\frac {4 a \log (a+b x)}{b^5}+\frac {x}{b^4} \]
[In]
[Out]
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{b^4}+\frac {a^4}{b^4 (a+b x)^4}-\frac {4 a^3}{b^4 (a+b x)^3}+\frac {6 a^2}{b^4 (a+b x)^2}-\frac {4 a}{b^4 (a+b x)}\right ) \, dx \\ & = \frac {x}{b^4}-\frac {a^4}{3 b^5 (a+b x)^3}+\frac {2 a^3}{b^5 (a+b x)^2}-\frac {6 a^2}{b^5 (a+b x)}-\frac {4 a \log (a+b x)}{b^5} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.78 \[ \int \frac {x^4}{(a+b x)^4} \, dx=-\frac {-3 b x+\frac {a^2 \left (13 a^2+30 a b x+18 b^2 x^2\right )}{(a+b x)^3}+12 a \log (a+b x)}{3 b^5} \]
[In]
[Out]
Time = 0.18 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.83
method | result | size |
risch | \(\frac {x}{b^{4}}+\frac {-6 a^{2} b \,x^{2}-10 a^{3} x -\frac {13 a^{4}}{3 b}}{b^{4} \left (b x +a \right )^{3}}-\frac {4 a \ln \left (b x +a \right )}{b^{5}}\) | \(54\) |
norman | \(\frac {\frac {x^{4}}{b}-\frac {22 a^{4}}{3 b^{5}}-\frac {12 a^{2} x^{2}}{b^{3}}-\frac {18 a^{3} x}{b^{4}}}{\left (b x +a \right )^{3}}-\frac {4 a \ln \left (b x +a \right )}{b^{5}}\) | \(58\) |
default | \(\frac {x}{b^{4}}-\frac {a^{4}}{3 b^{5} \left (b x +a \right )^{3}}+\frac {2 a^{3}}{b^{5} \left (b x +a \right )^{2}}-\frac {6 a^{2}}{b^{5} \left (b x +a \right )}-\frac {4 a \ln \left (b x +a \right )}{b^{5}}\) | \(64\) |
parallelrisch | \(-\frac {12 \ln \left (b x +a \right ) x^{3} a \,b^{3}-3 b^{4} x^{4}+36 \ln \left (b x +a \right ) x^{2} a^{2} b^{2}+36 \ln \left (b x +a \right ) x \,a^{3} b +36 a^{2} b^{2} x^{2}+12 a^{4} \ln \left (b x +a \right )+54 a^{3} b x +22 a^{4}}{3 b^{5} \left (b x +a \right )^{3}}\) | \(101\) |
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.78 \[ \int \frac {x^4}{(a+b x)^4} \, dx=\frac {3 \, b^{4} x^{4} + 9 \, a b^{3} x^{3} - 9 \, a^{2} b^{2} x^{2} - 27 \, a^{3} b x - 13 \, a^{4} - 12 \, {\left (a b^{3} x^{3} + 3 \, a^{2} b^{2} x^{2} + 3 \, a^{3} b x + a^{4}\right )} \log \left (b x + a\right )}{3 \, {\left (b^{8} x^{3} + 3 \, a b^{7} x^{2} + 3 \, a^{2} b^{6} x + a^{3} b^{5}\right )}} \]
[In]
[Out]
Time = 0.20 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.26 \[ \int \frac {x^4}{(a+b x)^4} \, dx=- \frac {4 a \log {\left (a + b x \right )}}{b^{5}} + \frac {- 13 a^{4} - 30 a^{3} b x - 18 a^{2} b^{2} x^{2}}{3 a^{3} b^{5} + 9 a^{2} b^{6} x + 9 a b^{7} x^{2} + 3 b^{8} x^{3}} + \frac {x}{b^{4}} \]
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.22 \[ \int \frac {x^4}{(a+b x)^4} \, dx=-\frac {18 \, a^{2} b^{2} x^{2} + 30 \, a^{3} b x + 13 \, a^{4}}{3 \, {\left (b^{8} x^{3} + 3 \, a b^{7} x^{2} + 3 \, a^{2} b^{6} x + a^{3} b^{5}\right )}} + \frac {x}{b^{4}} - \frac {4 \, a \log \left (b x + a\right )}{b^{5}} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.85 \[ \int \frac {x^4}{(a+b x)^4} \, dx=\frac {x}{b^{4}} - \frac {4 \, a \log \left ({\left | b x + a \right |}\right )}{b^{5}} - \frac {18 \, a^{2} b^{2} x^{2} + 30 \, a^{3} b x + 13 \, a^{4}}{3 \, {\left (b x + a\right )}^{3} b^{5}} \]
[In]
[Out]
Time = 0.19 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.85 \[ \int \frac {x^4}{(a+b x)^4} \, dx=-\frac {4\,a\,\ln \left (a+b\,x\right )-b\,x+\frac {6\,a^2}{a+b\,x}-\frac {2\,a^3}{{\left (a+b\,x\right )}^2}+\frac {a^4}{3\,{\left (a+b\,x\right )}^3}}{b^5} \]
[In]
[Out]