Integrand size = 11, antiderivative size = 86 \[ \int \frac {x^6}{(a+b x)^3} \, dx=-\frac {10 a^3 x}{b^6}+\frac {3 a^2 x^2}{b^5}-\frac {a x^3}{b^4}+\frac {x^4}{4 b^3}-\frac {a^6}{2 b^7 (a+b x)^2}+\frac {6 a^5}{b^7 (a+b x)}+\frac {15 a^4 \log (a+b x)}{b^7} \]
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Time = 0.04 (sec) , antiderivative size = 86, 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} \[ \int \frac {x^6}{(a+b x)^3} \, dx=-\frac {a^6}{2 b^7 (a+b x)^2}+\frac {6 a^5}{b^7 (a+b x)}+\frac {15 a^4 \log (a+b x)}{b^7}-\frac {10 a^3 x}{b^6}+\frac {3 a^2 x^2}{b^5}-\frac {a x^3}{b^4}+\frac {x^4}{4 b^3} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {10 a^3}{b^6}+\frac {6 a^2 x}{b^5}-\frac {3 a x^2}{b^4}+\frac {x^3}{b^3}+\frac {a^6}{b^6 (a+b x)^3}-\frac {6 a^5}{b^6 (a+b x)^2}+\frac {15 a^4}{b^6 (a+b x)}\right ) \, dx \\ & = -\frac {10 a^3 x}{b^6}+\frac {3 a^2 x^2}{b^5}-\frac {a x^3}{b^4}+\frac {x^4}{4 b^3}-\frac {a^6}{2 b^7 (a+b x)^2}+\frac {6 a^5}{b^7 (a+b x)}+\frac {15 a^4 \log (a+b x)}{b^7} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.90 \[ \int \frac {x^6}{(a+b x)^3} \, dx=\frac {-40 a^3 b x+12 a^2 b^2 x^2-4 a b^3 x^3+b^4 x^4-\frac {2 a^6}{(a+b x)^2}+\frac {24 a^5}{a+b x}+60 a^4 \log (a+b x)}{4 b^7} \]
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Time = 0.18 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.92
method | result | size |
risch | \(\frac {x^{4}}{4 b^{3}}-\frac {a \,x^{3}}{b^{4}}+\frac {3 a^{2} x^{2}}{b^{5}}-\frac {10 a^{3} x}{b^{6}}+\frac {6 a^{5} x +\frac {11 a^{6}}{2 b}}{b^{6} \left (b x +a \right )^{2}}+\frac {15 a^{4} \ln \left (b x +a \right )}{b^{7}}\) | \(79\) |
norman | \(\frac {\frac {x^{6}}{4 b}-\frac {a \,x^{5}}{2 b^{2}}+\frac {5 a^{2} x^{4}}{4 b^{3}}+\frac {45 a^{6}}{2 b^{7}}-\frac {5 a^{3} x^{3}}{b^{4}}+\frac {30 a^{5} x}{b^{6}}}{\left (b x +a \right )^{2}}+\frac {15 a^{4} \ln \left (b x +a \right )}{b^{7}}\) | \(81\) |
default | \(-\frac {-\frac {1}{4} b^{3} x^{4}+a \,b^{2} x^{3}-3 a^{2} b \,x^{2}+10 a^{3} x}{b^{6}}+\frac {15 a^{4} \ln \left (b x +a \right )}{b^{7}}-\frac {a^{6}}{2 b^{7} \left (b x +a \right )^{2}}+\frac {6 a^{5}}{b^{7} \left (b x +a \right )}\) | \(83\) |
parallelrisch | \(\frac {b^{6} x^{6}-2 a \,x^{5} b^{5}+5 a^{2} x^{4} b^{4}+60 \ln \left (b x +a \right ) x^{2} a^{4} b^{2}-20 a^{3} x^{3} b^{3}+120 \ln \left (b x +a \right ) x \,a^{5} b +60 \ln \left (b x +a \right ) a^{6}+120 a^{5} x b +90 a^{6}}{4 b^{7} \left (b x +a \right )^{2}}\) | \(105\) |
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none
Time = 0.21 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.36 \[ \int \frac {x^6}{(a+b x)^3} \, dx=\frac {b^{6} x^{6} - 2 \, a b^{5} x^{5} + 5 \, a^{2} b^{4} x^{4} - 20 \, a^{3} b^{3} x^{3} - 68 \, a^{4} b^{2} x^{2} - 16 \, a^{5} b x + 22 \, a^{6} + 60 \, {\left (a^{4} b^{2} x^{2} + 2 \, a^{5} b x + a^{6}\right )} \log \left (b x + a\right )}{4 \, {\left (b^{9} x^{2} + 2 \, a b^{8} x + a^{2} b^{7}\right )}} \]
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Time = 0.20 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.07 \[ \int \frac {x^6}{(a+b x)^3} \, dx=\frac {15 a^{4} \log {\left (a + b x \right )}}{b^{7}} - \frac {10 a^{3} x}{b^{6}} + \frac {3 a^{2} x^{2}}{b^{5}} - \frac {a x^{3}}{b^{4}} + \frac {11 a^{6} + 12 a^{5} b x}{2 a^{2} b^{7} + 4 a b^{8} x + 2 b^{9} x^{2}} + \frac {x^{4}}{4 b^{3}} \]
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none
Time = 0.21 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.06 \[ \int \frac {x^6}{(a+b x)^3} \, dx=\frac {12 \, a^{5} b x + 11 \, a^{6}}{2 \, {\left (b^{9} x^{2} + 2 \, a b^{8} x + a^{2} b^{7}\right )}} + \frac {15 \, a^{4} \log \left (b x + a\right )}{b^{7}} + \frac {b^{3} x^{4} - 4 \, a b^{2} x^{3} + 12 \, a^{2} b x^{2} - 40 \, a^{3} x}{4 \, b^{6}} \]
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none
Time = 0.31 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.97 \[ \int \frac {x^6}{(a+b x)^3} \, dx=\frac {15 \, a^{4} \log \left ({\left | b x + a \right |}\right )}{b^{7}} + \frac {12 \, a^{5} b x + 11 \, a^{6}}{2 \, {\left (b x + a\right )}^{2} b^{7}} + \frac {b^{9} x^{4} - 4 \, a b^{8} x^{3} + 12 \, a^{2} b^{7} x^{2} - 40 \, a^{3} b^{6} x}{4 \, b^{12}} \]
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Time = 0.18 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.91 \[ \int \frac {x^6}{(a+b x)^3} \, dx=\frac {\frac {{\left (a+b\,x\right )}^4}{4}-2\,a\,{\left (a+b\,x\right )}^3+\frac {15\,a^2\,{\left (a+b\,x\right )}^2}{2}+\frac {6\,a^5}{a+b\,x}-\frac {a^6}{2\,{\left (a+b\,x\right )}^2}+15\,a^4\,\ln \left (a+b\,x\right )-20\,a^3\,b\,x}{b^7} \]
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