Integrand size = 13, antiderivative size = 69 \[ \int \frac {\left (a+b x^3\right )^5}{x^{31}} \, dx=-\frac {a^5}{30 x^{30}}-\frac {5 a^4 b}{27 x^{27}}-\frac {5 a^3 b^2}{12 x^{24}}-\frac {10 a^2 b^3}{21 x^{21}}-\frac {5 a b^4}{18 x^{18}}-\frac {b^5}{15 x^{15}} \]
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Time = 0.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \[ \int \frac {\left (a+b x^3\right )^5}{x^{31}} \, dx=-\frac {a^5}{30 x^{30}}-\frac {5 a^4 b}{27 x^{27}}-\frac {5 a^3 b^2}{12 x^{24}}-\frac {10 a^2 b^3}{21 x^{21}}-\frac {5 a b^4}{18 x^{18}}-\frac {b^5}{15 x^{15}} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {(a+b x)^5}{x^{11}} \, dx,x,x^3\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \left (\frac {a^5}{x^{11}}+\frac {5 a^4 b}{x^{10}}+\frac {10 a^3 b^2}{x^9}+\frac {10 a^2 b^3}{x^8}+\frac {5 a b^4}{x^7}+\frac {b^5}{x^6}\right ) \, dx,x,x^3\right ) \\ & = -\frac {a^5}{30 x^{30}}-\frac {5 a^4 b}{27 x^{27}}-\frac {5 a^3 b^2}{12 x^{24}}-\frac {10 a^2 b^3}{21 x^{21}}-\frac {5 a b^4}{18 x^{18}}-\frac {b^5}{15 x^{15}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^3\right )^5}{x^{31}} \, dx=-\frac {a^5}{30 x^{30}}-\frac {5 a^4 b}{27 x^{27}}-\frac {5 a^3 b^2}{12 x^{24}}-\frac {10 a^2 b^3}{21 x^{21}}-\frac {5 a b^4}{18 x^{18}}-\frac {b^5}{15 x^{15}} \]
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Time = 3.59 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.84
method | result | size |
default | \(-\frac {a^{5}}{30 x^{30}}-\frac {5 a^{4} b}{27 x^{27}}-\frac {5 a^{3} b^{2}}{12 x^{24}}-\frac {10 a^{2} b^{3}}{21 x^{21}}-\frac {5 a \,b^{4}}{18 x^{18}}-\frac {b^{5}}{15 x^{15}}\) | \(58\) |
norman | \(\frac {-\frac {1}{15} b^{5} x^{15}-\frac {1}{30} a^{5}-\frac {5}{27} a^{4} b \,x^{3}-\frac {5}{12} a^{3} b^{2} x^{6}-\frac {10}{21} a^{2} b^{3} x^{9}-\frac {5}{18} a \,b^{4} x^{12}}{x^{30}}\) | \(59\) |
risch | \(\frac {-\frac {1}{15} b^{5} x^{15}-\frac {1}{30} a^{5}-\frac {5}{27} a^{4} b \,x^{3}-\frac {5}{12} a^{3} b^{2} x^{6}-\frac {10}{21} a^{2} b^{3} x^{9}-\frac {5}{18} a \,b^{4} x^{12}}{x^{30}}\) | \(59\) |
gosper | \(-\frac {252 b^{5} x^{15}+1050 a \,b^{4} x^{12}+1800 a^{2} b^{3} x^{9}+1575 a^{3} b^{2} x^{6}+700 a^{4} b \,x^{3}+126 a^{5}}{3780 x^{30}}\) | \(60\) |
parallelrisch | \(\frac {-252 b^{5} x^{15}-1050 a \,b^{4} x^{12}-1800 a^{2} b^{3} x^{9}-1575 a^{3} b^{2} x^{6}-700 a^{4} b \,x^{3}-126 a^{5}}{3780 x^{30}}\) | \(60\) |
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Time = 0.28 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b x^3\right )^5}{x^{31}} \, dx=-\frac {252 \, b^{5} x^{15} + 1050 \, a b^{4} x^{12} + 1800 \, a^{2} b^{3} x^{9} + 1575 \, a^{3} b^{2} x^{6} + 700 \, a^{4} b x^{3} + 126 \, a^{5}}{3780 \, x^{30}} \]
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Time = 0.38 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b x^3\right )^5}{x^{31}} \, dx=\frac {- 126 a^{5} - 700 a^{4} b x^{3} - 1575 a^{3} b^{2} x^{6} - 1800 a^{2} b^{3} x^{9} - 1050 a b^{4} x^{12} - 252 b^{5} x^{15}}{3780 x^{30}} \]
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Time = 0.20 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b x^3\right )^5}{x^{31}} \, dx=-\frac {252 \, b^{5} x^{15} + 1050 \, a b^{4} x^{12} + 1800 \, a^{2} b^{3} x^{9} + 1575 \, a^{3} b^{2} x^{6} + 700 \, a^{4} b x^{3} + 126 \, a^{5}}{3780 \, x^{30}} \]
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Time = 0.27 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b x^3\right )^5}{x^{31}} \, dx=-\frac {252 \, b^{5} x^{15} + 1050 \, a b^{4} x^{12} + 1800 \, a^{2} b^{3} x^{9} + 1575 \, a^{3} b^{2} x^{6} + 700 \, a^{4} b x^{3} + 126 \, a^{5}}{3780 \, x^{30}} \]
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Time = 0.05 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b x^3\right )^5}{x^{31}} \, dx=-\frac {\frac {a^5}{30}+\frac {5\,a^4\,b\,x^3}{27}+\frac {5\,a^3\,b^2\,x^6}{12}+\frac {10\,a^2\,b^3\,x^9}{21}+\frac {5\,a\,b^4\,x^{12}}{18}+\frac {b^5\,x^{15}}{15}}{x^{30}} \]
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