Integrand size = 11, antiderivative size = 90 \[ \int \frac {x^6}{(a+b x)^4} \, dx=\frac {10 a^2 x}{b^6}-\frac {2 a x^2}{b^5}+\frac {x^3}{3 b^4}-\frac {a^6}{3 b^7 (a+b x)^3}+\frac {3 a^5}{b^7 (a+b x)^2}-\frac {15 a^4}{b^7 (a+b x)}-\frac {20 a^3 \log (a+b x)}{b^7} \]
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Time = 0.04 (sec) , antiderivative size = 90, 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} \[ \int \frac {x^6}{(a+b x)^4} \, dx=-\frac {a^6}{3 b^7 (a+b x)^3}+\frac {3 a^5}{b^7 (a+b x)^2}-\frac {15 a^4}{b^7 (a+b x)}-\frac {20 a^3 \log (a+b x)}{b^7}+\frac {10 a^2 x}{b^6}-\frac {2 a x^2}{b^5}+\frac {x^3}{3 b^4} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {10 a^2}{b^6}-\frac {4 a x}{b^5}+\frac {x^2}{b^4}+\frac {a^6}{b^6 (a+b x)^4}-\frac {6 a^5}{b^6 (a+b x)^3}+\frac {15 a^4}{b^6 (a+b x)^2}-\frac {20 a^3}{b^6 (a+b x)}\right ) \, dx \\ & = \frac {10 a^2 x}{b^6}-\frac {2 a x^2}{b^5}+\frac {x^3}{3 b^4}-\frac {a^6}{3 b^7 (a+b x)^3}+\frac {3 a^5}{b^7 (a+b x)^2}-\frac {15 a^4}{b^7 (a+b x)}-\frac {20 a^3 \log (a+b x)}{b^7} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00 \[ \int \frac {x^6}{(a+b x)^4} \, dx=\frac {10 a^2 x}{b^6}-\frac {2 a x^2}{b^5}+\frac {x^3}{3 b^4}-\frac {a^6}{3 b^7 (a+b x)^3}+\frac {3 a^5}{b^7 (a+b x)^2}-\frac {15 a^4}{b^7 (a+b x)}-\frac {20 a^3 \log (a+b x)}{b^7} \]
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Time = 0.17 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.86
method | result | size |
risch | \(\frac {x^{3}}{3 b^{4}}-\frac {2 a \,x^{2}}{b^{5}}+\frac {10 a^{2} x}{b^{6}}+\frac {-15 a^{4} b \,x^{2}-27 a^{5} x -\frac {37 a^{6}}{3 b}}{b^{6} \left (b x +a \right )^{3}}-\frac {20 a^{3} \ln \left (b x +a \right )}{b^{7}}\) | \(77\) |
norman | \(\frac {\frac {x^{6}}{3 b}-\frac {a \,x^{5}}{b^{2}}+\frac {5 a^{2} x^{4}}{b^{3}}-\frac {110 a^{6}}{3 b^{7}}-\frac {60 a^{4} x^{2}}{b^{5}}-\frac {90 a^{5} x}{b^{6}}}{\left (b x +a \right )^{3}}-\frac {20 a^{3} \ln \left (b x +a \right )}{b^{7}}\) | \(81\) |
default | \(\frac {\frac {1}{3} b^{2} x^{3}-2 a b \,x^{2}+10 a^{2} x}{b^{6}}-\frac {20 a^{3} \ln \left (b x +a \right )}{b^{7}}-\frac {a^{6}}{3 b^{7} \left (b x +a \right )^{3}}+\frac {3 a^{5}}{b^{7} \left (b x +a \right )^{2}}-\frac {15 a^{4}}{b^{7} \left (b x +a \right )}\) | \(87\) |
parallelrisch | \(-\frac {-b^{6} x^{6}+3 a \,x^{5} b^{5}+60 \ln \left (b x +a \right ) x^{3} a^{3} b^{3}-15 a^{2} x^{4} b^{4}+180 \ln \left (b x +a \right ) x^{2} a^{4} b^{2}+180 \ln \left (b x +a \right ) x \,a^{5} b +180 a^{4} x^{2} b^{2}+60 \ln \left (b x +a \right ) a^{6}+270 a^{5} x b +110 a^{6}}{3 b^{7} \left (b x +a \right )^{3}}\) | \(123\) |
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Time = 0.22 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.54 \[ \int \frac {x^6}{(a+b x)^4} \, dx=\frac {b^{6} x^{6} - 3 \, a b^{5} x^{5} + 15 \, a^{2} b^{4} x^{4} + 73 \, a^{3} b^{3} x^{3} + 39 \, a^{4} b^{2} x^{2} - 51 \, a^{5} b x - 37 \, a^{6} - 60 \, {\left (a^{3} b^{3} x^{3} + 3 \, a^{4} b^{2} x^{2} + 3 \, a^{5} b x + a^{6}\right )} \log \left (b x + a\right )}{3 \, {\left (b^{10} x^{3} + 3 \, a b^{9} x^{2} + 3 \, a^{2} b^{8} x + a^{3} b^{7}\right )}} \]
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Time = 0.29 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.19 \[ \int \frac {x^6}{(a+b x)^4} \, dx=- \frac {20 a^{3} \log {\left (a + b x \right )}}{b^{7}} + \frac {10 a^{2} x}{b^{6}} - \frac {2 a x^{2}}{b^{5}} + \frac {- 37 a^{6} - 81 a^{5} b x - 45 a^{4} b^{2} x^{2}}{3 a^{3} b^{7} + 9 a^{2} b^{8} x + 9 a b^{9} x^{2} + 3 b^{10} x^{3}} + \frac {x^{3}}{3 b^{4}} \]
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Time = 0.20 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.13 \[ \int \frac {x^6}{(a+b x)^4} \, dx=-\frac {45 \, a^{4} b^{2} x^{2} + 81 \, a^{5} b x + 37 \, a^{6}}{3 \, {\left (b^{10} x^{3} + 3 \, a b^{9} x^{2} + 3 \, a^{2} b^{8} x + a^{3} b^{7}\right )}} - \frac {20 \, a^{3} \log \left (b x + a\right )}{b^{7}} + \frac {b^{2} x^{3} - 6 \, a b x^{2} + 30 \, a^{2} x}{3 \, b^{6}} \]
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Time = 0.29 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.92 \[ \int \frac {x^6}{(a+b x)^4} \, dx=-\frac {20 \, a^{3} \log \left ({\left | b x + a \right |}\right )}{b^{7}} - \frac {45 \, a^{4} b^{2} x^{2} + 81 \, a^{5} b x + 37 \, a^{6}}{3 \, {\left (b x + a\right )}^{3} b^{7}} + \frac {b^{8} x^{3} - 6 \, a b^{7} x^{2} + 30 \, a^{2} b^{6} x}{3 \, b^{12}} \]
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Time = 0.17 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.88 \[ \int \frac {x^6}{(a+b x)^4} \, dx=-\frac {3\,a\,{\left (a+b\,x\right )}^2-\frac {{\left (a+b\,x\right )}^3}{3}+\frac {15\,a^4}{a+b\,x}-\frac {3\,a^5}{{\left (a+b\,x\right )}^2}+\frac {a^6}{3\,{\left (a+b\,x\right )}^3}+20\,a^3\,\ln \left (a+b\,x\right )-15\,a^2\,b\,x}{b^7} \]
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