Integrand size = 11, antiderivative size = 81 \[ \int \frac {x^6}{(a+b x)^2} \, dx=\frac {5 a^4 x}{b^6}-\frac {2 a^3 x^2}{b^5}+\frac {a^2 x^3}{b^4}-\frac {a x^4}{2 b^3}+\frac {x^5}{5 b^2}-\frac {a^6}{b^7 (a+b x)}-\frac {6 a^5 \log (a+b x)}{b^7} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 81, 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} \[ \int \frac {x^6}{(a+b x)^2} \, dx=-\frac {a^6}{b^7 (a+b x)}-\frac {6 a^5 \log (a+b x)}{b^7}+\frac {5 a^4 x}{b^6}-\frac {2 a^3 x^2}{b^5}+\frac {a^2 x^3}{b^4}-\frac {a x^4}{2 b^3}+\frac {x^5}{5 b^2} \]
[In]
[Out]
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {5 a^4}{b^6}-\frac {4 a^3 x}{b^5}+\frac {3 a^2 x^2}{b^4}-\frac {2 a x^3}{b^3}+\frac {x^4}{b^2}+\frac {a^6}{b^6 (a+b x)^2}-\frac {6 a^5}{b^6 (a+b x)}\right ) \, dx \\ & = \frac {5 a^4 x}{b^6}-\frac {2 a^3 x^2}{b^5}+\frac {a^2 x^3}{b^4}-\frac {a x^4}{2 b^3}+\frac {x^5}{5 b^2}-\frac {a^6}{b^7 (a+b x)}-\frac {6 a^5 \log (a+b x)}{b^7} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.95 \[ \int \frac {x^6}{(a+b x)^2} \, dx=\frac {50 a^4 b x-20 a^3 b^2 x^2+10 a^2 b^3 x^3-5 a b^4 x^4+2 b^5 x^5-\frac {10 a^6}{a+b x}-60 a^5 \log (a+b x)}{10 b^7} \]
[In]
[Out]
Time = 0.18 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.96
method | result | size |
default | \(\frac {\frac {1}{5} b^{4} x^{5}-\frac {1}{2} a \,b^{3} x^{4}+a^{2} b^{2} x^{3}-2 a^{3} b \,x^{2}+5 a^{4} x}{b^{6}}-\frac {6 a^{5} \ln \left (b x +a \right )}{b^{7}}-\frac {a^{6}}{b^{7} \left (b x +a \right )}\) | \(78\) |
risch | \(\frac {5 a^{4} x}{b^{6}}-\frac {2 a^{3} x^{2}}{b^{5}}+\frac {a^{2} x^{3}}{b^{4}}-\frac {a \,x^{4}}{2 b^{3}}+\frac {x^{5}}{5 b^{2}}-\frac {a^{6}}{b^{7} \left (b x +a \right )}-\frac {6 a^{5} \ln \left (b x +a \right )}{b^{7}}\) | \(78\) |
norman | \(\frac {\frac {x^{6}}{5 b}-\frac {3 a \,x^{5}}{10 b^{2}}-\frac {6 a^{6}}{b^{7}}+\frac {a^{2} x^{4}}{2 b^{3}}-\frac {a^{3} x^{3}}{b^{4}}+\frac {3 a^{4} x^{2}}{b^{5}}}{b x +a}-\frac {6 a^{5} \ln \left (b x +a \right )}{b^{7}}\) | \(83\) |
parallelrisch | \(-\frac {-2 b^{6} x^{6}+3 a \,x^{5} b^{5}-5 a^{2} x^{4} b^{4}+10 a^{3} x^{3} b^{3}+60 \ln \left (b x +a \right ) x \,a^{5} b -30 a^{4} x^{2} b^{2}+60 \ln \left (b x +a \right ) a^{6}+60 a^{6}}{10 b^{7} \left (b x +a \right )}\) | \(93\) |
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.19 \[ \int \frac {x^6}{(a+b x)^2} \, dx=\frac {2 \, b^{6} x^{6} - 3 \, a b^{5} x^{5} + 5 \, a^{2} b^{4} x^{4} - 10 \, a^{3} b^{3} x^{3} + 30 \, a^{4} b^{2} x^{2} + 50 \, a^{5} b x - 10 \, a^{6} - 60 \, {\left (a^{5} b x + a^{6}\right )} \log \left (b x + a\right )}{10 \, {\left (b^{8} x + a b^{7}\right )}} \]
[In]
[Out]
Time = 0.12 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.96 \[ \int \frac {x^6}{(a+b x)^2} \, dx=- \frac {a^{6}}{a b^{7} + b^{8} x} - \frac {6 a^{5} \log {\left (a + b x \right )}}{b^{7}} + \frac {5 a^{4} x}{b^{6}} - \frac {2 a^{3} x^{2}}{b^{5}} + \frac {a^{2} x^{3}}{b^{4}} - \frac {a x^{4}}{2 b^{3}} + \frac {x^{5}}{5 b^{2}} \]
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.01 \[ \int \frac {x^6}{(a+b x)^2} \, dx=-\frac {a^{6}}{b^{8} x + a b^{7}} - \frac {6 \, a^{5} \log \left (b x + a\right )}{b^{7}} + \frac {2 \, b^{4} x^{5} - 5 \, a b^{3} x^{4} + 10 \, a^{2} b^{2} x^{3} - 20 \, a^{3} b x^{2} + 50 \, a^{4} x}{10 \, b^{6}} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.27 \[ \int \frac {x^6}{(a+b x)^2} \, dx=-\frac {{\left (b x + a\right )}^{5} {\left (\frac {15 \, a}{b x + a} - \frac {50 \, a^{2}}{{\left (b x + a\right )}^{2}} + \frac {100 \, a^{3}}{{\left (b x + a\right )}^{3}} - \frac {150 \, a^{4}}{{\left (b x + a\right )}^{4}} - 2\right )}}{10 \, b^{7}} + \frac {6 \, a^{5} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{7}} - \frac {a^{6}}{{\left (b x + a\right )} b^{7}} \]
[In]
[Out]
Time = 0.14 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.02 \[ \int \frac {x^6}{(a+b x)^2} \, dx=\frac {x^5}{5\,b^2}-\frac {6\,a^5\,\ln \left (a+b\,x\right )}{b^7}-\frac {a\,x^4}{2\,b^3}+\frac {5\,a^4\,x}{b^6}+\frac {a^2\,x^3}{b^4}-\frac {2\,a^3\,x^2}{b^5}-\frac {a^6}{b\,\left (x\,b^7+a\,b^6\right )} \]
[In]
[Out]