Integrand size = 15, antiderivative size = 179 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right ) x^5} \, dx=-\frac {3}{11 b x^{11/3}}+\frac {3 a}{10 b^2 x^{10/3}}-\frac {a^2}{3 b^3 x^3}+\frac {3 a^3}{8 b^4 x^{8/3}}-\frac {3 a^4}{7 b^5 x^{7/3}}+\frac {a^5}{2 b^6 x^2}-\frac {3 a^6}{5 b^7 x^{5/3}}+\frac {3 a^7}{4 b^8 x^{4/3}}-\frac {a^8}{b^9 x}+\frac {3 a^9}{2 b^{10} x^{2/3}}-\frac {3 a^{10}}{b^{11} \sqrt [3]{x}}+\frac {3 a^{11} \log \left (b+a \sqrt [3]{x}\right )}{b^{12}}-\frac {a^{11} \log (x)}{b^{12}} \]
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Time = 0.07 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {269, 272, 46} \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right ) x^5} \, dx=\frac {3 a^{11} \log \left (a \sqrt [3]{x}+b\right )}{b^{12}}-\frac {a^{11} \log (x)}{b^{12}}-\frac {3 a^{10}}{b^{11} \sqrt [3]{x}}+\frac {3 a^9}{2 b^{10} x^{2/3}}-\frac {a^8}{b^9 x}+\frac {3 a^7}{4 b^8 x^{4/3}}-\frac {3 a^6}{5 b^7 x^{5/3}}+\frac {a^5}{2 b^6 x^2}-\frac {3 a^4}{7 b^5 x^{7/3}}+\frac {3 a^3}{8 b^4 x^{8/3}}-\frac {a^2}{3 b^3 x^3}+\frac {3 a}{10 b^2 x^{10/3}}-\frac {3}{11 b x^{11/3}} \]
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Rule 46
Rule 269
Rule 272
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\left (b+a \sqrt [3]{x}\right ) x^{14/3}} \, dx \\ & = 3 \text {Subst}\left (\int \frac {1}{x^{12} (b+a x)} \, dx,x,\sqrt [3]{x}\right ) \\ & = 3 \text {Subst}\left (\int \left (\frac {1}{b x^{12}}-\frac {a}{b^2 x^{11}}+\frac {a^2}{b^3 x^{10}}-\frac {a^3}{b^4 x^9}+\frac {a^4}{b^5 x^8}-\frac {a^5}{b^6 x^7}+\frac {a^6}{b^7 x^6}-\frac {a^7}{b^8 x^5}+\frac {a^8}{b^9 x^4}-\frac {a^9}{b^{10} x^3}+\frac {a^{10}}{b^{11} x^2}-\frac {a^{11}}{b^{12} x}+\frac {a^{12}}{b^{12} (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right ) \\ & = -\frac {3}{11 b x^{11/3}}+\frac {3 a}{10 b^2 x^{10/3}}-\frac {a^2}{3 b^3 x^3}+\frac {3 a^3}{8 b^4 x^{8/3}}-\frac {3 a^4}{7 b^5 x^{7/3}}+\frac {a^5}{2 b^6 x^2}-\frac {3 a^6}{5 b^7 x^{5/3}}+\frac {3 a^7}{4 b^8 x^{4/3}}-\frac {a^8}{b^9 x}+\frac {3 a^9}{2 b^{10} x^{2/3}}-\frac {3 a^{10}}{b^{11} \sqrt [3]{x}}+\frac {3 a^{11} \log \left (b+a \sqrt [3]{x}\right )}{b^{12}}-\frac {a^{11} \log (x)}{b^{12}} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right ) x^5} \, dx=\frac {\frac {b \left (-2520 b^{10}+2772 a b^9 \sqrt [3]{x}-3080 a^2 b^8 x^{2/3}+3465 a^3 b^7 x-3960 a^4 b^6 x^{4/3}+4620 a^5 b^5 x^{5/3}-5544 a^6 b^4 x^2+6930 a^7 b^3 x^{7/3}-9240 a^8 b^2 x^{8/3}+13860 a^9 b x^3-27720 a^{10} x^{10/3}\right )}{x^{11/3}}+27720 a^{11} \log \left (b+a \sqrt [3]{x}\right )-9240 a^{11} \log (x)}{9240 b^{12}} \]
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Time = 3.75 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(-\frac {3}{11 b \,x^{\frac {11}{3}}}+\frac {3 a}{10 b^{2} x^{\frac {10}{3}}}-\frac {a^{2}}{3 b^{3} x^{3}}+\frac {3 a^{3}}{8 b^{4} x^{\frac {8}{3}}}-\frac {3 a^{4}}{7 b^{5} x^{\frac {7}{3}}}+\frac {a^{5}}{2 b^{6} x^{2}}-\frac {3 a^{6}}{5 b^{7} x^{\frac {5}{3}}}+\frac {3 a^{7}}{4 b^{8} x^{\frac {4}{3}}}-\frac {a^{8}}{b^{9} x}+\frac {3 a^{9}}{2 b^{10} x^{\frac {2}{3}}}-\frac {3 a^{10}}{b^{11} x^{\frac {1}{3}}}+\frac {3 a^{11} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{b^{12}}-\frac {a^{11} \ln \left (x \right )}{b^{12}}\) | \(144\) |
default | \(-\frac {3}{11 b \,x^{\frac {11}{3}}}+\frac {3 a}{10 b^{2} x^{\frac {10}{3}}}-\frac {a^{2}}{3 b^{3} x^{3}}+\frac {3 a^{3}}{8 b^{4} x^{\frac {8}{3}}}-\frac {3 a^{4}}{7 b^{5} x^{\frac {7}{3}}}+\frac {a^{5}}{2 b^{6} x^{2}}-\frac {3 a^{6}}{5 b^{7} x^{\frac {5}{3}}}+\frac {3 a^{7}}{4 b^{8} x^{\frac {4}{3}}}-\frac {a^{8}}{b^{9} x}+\frac {3 a^{9}}{2 b^{10} x^{\frac {2}{3}}}-\frac {3 a^{10}}{b^{11} x^{\frac {1}{3}}}+\frac {3 a^{11} \ln \left (b +a \,x^{\frac {1}{3}}\right )}{b^{12}}-\frac {a^{11} \ln \left (x \right )}{b^{12}}\) | \(144\) |
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Time = 0.27 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right ) x^5} \, dx=\frac {27720 \, a^{11} x^{4} \log \left (a x^{\frac {1}{3}} + b\right ) - 27720 \, a^{11} x^{4} \log \left (x^{\frac {1}{3}}\right ) - 9240 \, a^{8} b^{3} x^{3} + 4620 \, a^{5} b^{6} x^{2} - 3080 \, a^{2} b^{9} x - 198 \, {\left (140 \, a^{10} b x^{3} - 35 \, a^{7} b^{4} x^{2} + 20 \, a^{4} b^{7} x - 14 \, a b^{10}\right )} x^{\frac {2}{3}} + 63 \, {\left (220 \, a^{9} b^{2} x^{3} - 88 \, a^{6} b^{5} x^{2} + 55 \, a^{3} b^{8} x - 40 \, b^{11}\right )} x^{\frac {1}{3}}}{9240 \, b^{12} x^{4}} \]
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Time = 3.09 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right ) x^5} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {11}{3}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {3}{11 b x^{\frac {11}{3}}} & \text {for}\: a = 0 \\- \frac {1}{4 a x^{4}} & \text {for}\: b = 0 \\- \frac {a^{11} \log {\left (x \right )}}{b^{12}} + \frac {3 a^{11} \log {\left (\sqrt [3]{x} + \frac {b}{a} \right )}}{b^{12}} - \frac {3 a^{10}}{b^{11} \sqrt [3]{x}} + \frac {3 a^{9}}{2 b^{10} x^{\frac {2}{3}}} - \frac {a^{8}}{b^{9} x} + \frac {3 a^{7}}{4 b^{8} x^{\frac {4}{3}}} - \frac {3 a^{6}}{5 b^{7} x^{\frac {5}{3}}} + \frac {a^{5}}{2 b^{6} x^{2}} - \frac {3 a^{4}}{7 b^{5} x^{\frac {7}{3}}} + \frac {3 a^{3}}{8 b^{4} x^{\frac {8}{3}}} - \frac {a^{2}}{3 b^{3} x^{3}} + \frac {3 a}{10 b^{2} x^{\frac {10}{3}}} - \frac {3}{11 b x^{\frac {11}{3}}} & \text {otherwise} \end {cases} \]
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Time = 0.23 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right ) x^5} \, dx=\frac {3 \, a^{11} \log \left (a + \frac {b}{x^{\frac {1}{3}}}\right )}{b^{12}} - \frac {3 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{11}}{11 \, b^{12}} + \frac {33 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{10} a}{10 \, b^{12}} - \frac {55 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{9} a^{2}}{3 \, b^{12}} + \frac {495 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{8} a^{3}}{8 \, b^{12}} - \frac {990 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{7} a^{4}}{7 \, b^{12}} + \frac {231 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{6} a^{5}}{b^{12}} - \frac {1386 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{5} a^{6}}{5 \, b^{12}} + \frac {495 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{4} a^{7}}{2 \, b^{12}} - \frac {165 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{3} a^{8}}{b^{12}} + \frac {165 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )}^{2} a^{9}}{2 \, b^{12}} - \frac {33 \, {\left (a + \frac {b}{x^{\frac {1}{3}}}\right )} a^{10}}{b^{12}} \]
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Time = 0.28 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.82 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right ) x^5} \, dx=\frac {3 \, a^{11} \log \left ({\left | a x^{\frac {1}{3}} + b \right |}\right )}{b^{12}} - \frac {a^{11} \log \left ({\left | x \right |}\right )}{b^{12}} - \frac {27720 \, a^{10} b x^{\frac {10}{3}} - 13860 \, a^{9} b^{2} x^{3} + 9240 \, a^{8} b^{3} x^{\frac {8}{3}} - 6930 \, a^{7} b^{4} x^{\frac {7}{3}} + 5544 \, a^{6} b^{5} x^{2} - 4620 \, a^{5} b^{6} x^{\frac {5}{3}} + 3960 \, a^{4} b^{7} x^{\frac {4}{3}} - 3465 \, a^{3} b^{8} x + 3080 \, a^{2} b^{9} x^{\frac {2}{3}} - 2772 \, a b^{10} x^{\frac {1}{3}} + 2520 \, b^{11}}{9240 \, b^{12} x^{\frac {11}{3}}} \]
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Time = 0.08 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.77 \[ \int \frac {1}{\left (a+\frac {b}{\sqrt [3]{x}}\right ) x^5} \, dx=-\frac {2520\,b^{11}-55440\,a^{11}\,x^{11/3}\,\mathrm {atanh}\left (\frac {2\,a\,x^{1/3}}{b}+1\right )-3465\,a^3\,b^8\,x-2772\,a\,b^{10}\,x^{1/3}+27720\,a^{10}\,b\,x^{10/3}+5544\,a^6\,b^5\,x^2-13860\,a^9\,b^2\,x^3+3080\,a^2\,b^9\,x^{2/3}+3960\,a^4\,b^7\,x^{4/3}-4620\,a^5\,b^6\,x^{5/3}-6930\,a^7\,b^4\,x^{7/3}+9240\,a^8\,b^3\,x^{8/3}}{9240\,b^{12}\,x^{11/3}} \]
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