Integrand size = 15, antiderivative size = 68 \[ \int \frac {\sqrt [3]{a+b x^3}}{x^{11}} \, dx=-\frac {\left (a+b x^3\right )^{4/3}}{10 a x^{10}}+\frac {3 b \left (a+b x^3\right )^{4/3}}{35 a^2 x^7}-\frac {9 b^2 \left (a+b x^3\right )^{4/3}}{140 a^3 x^4} \]
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Time = 0.01 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {277, 270} \[ \int \frac {\sqrt [3]{a+b x^3}}{x^{11}} \, dx=-\frac {9 b^2 \left (a+b x^3\right )^{4/3}}{140 a^3 x^4}+\frac {3 b \left (a+b x^3\right )^{4/3}}{35 a^2 x^7}-\frac {\left (a+b x^3\right )^{4/3}}{10 a x^{10}} \]
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Rule 270
Rule 277
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a+b x^3\right )^{4/3}}{10 a x^{10}}-\frac {(3 b) \int \frac {\sqrt [3]{a+b x^3}}{x^8} \, dx}{5 a} \\ & = -\frac {\left (a+b x^3\right )^{4/3}}{10 a x^{10}}+\frac {3 b \left (a+b x^3\right )^{4/3}}{35 a^2 x^7}+\frac {\left (9 b^2\right ) \int \frac {\sqrt [3]{a+b x^3}}{x^5} \, dx}{35 a^2} \\ & = -\frac {\left (a+b x^3\right )^{4/3}}{10 a x^{10}}+\frac {3 b \left (a+b x^3\right )^{4/3}}{35 a^2 x^7}-\frac {9 b^2 \left (a+b x^3\right )^{4/3}}{140 a^3 x^4} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.78 \[ \int \frac {\sqrt [3]{a+b x^3}}{x^{11}} \, dx=\frac {\sqrt [3]{a+b x^3} \left (-14 a^3-2 a^2 b x^3+3 a b^2 x^6-9 b^3 x^9\right )}{140 a^3 x^{10}} \]
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Time = 4.04 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.57
method | result | size |
gosper | \(-\frac {\left (b \,x^{3}+a \right )^{\frac {4}{3}} \left (9 b^{2} x^{6}-12 a b \,x^{3}+14 a^{2}\right )}{140 x^{10} a^{3}}\) | \(39\) |
pseudoelliptic | \(-\frac {\left (b \,x^{3}+a \right )^{\frac {4}{3}} \left (9 b^{2} x^{6}-12 a b \,x^{3}+14 a^{2}\right )}{140 x^{10} a^{3}}\) | \(39\) |
trager | \(-\frac {\left (9 b^{3} x^{9}-3 a \,b^{2} x^{6}+2 a^{2} b \,x^{3}+14 a^{3}\right ) \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{140 x^{10} a^{3}}\) | \(50\) |
risch | \(-\frac {\left (9 b^{3} x^{9}-3 a \,b^{2} x^{6}+2 a^{2} b \,x^{3}+14 a^{3}\right ) \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{140 x^{10} a^{3}}\) | \(50\) |
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Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt [3]{a+b x^3}}{x^{11}} \, dx=-\frac {{\left (9 \, b^{3} x^{9} - 3 \, a b^{2} x^{6} + 2 \, a^{2} b x^{3} + 14 \, a^{3}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{140 \, a^{3} x^{10}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 520 vs. \(2 (61) = 122\).
Time = 0.86 (sec) , antiderivative size = 520, normalized size of antiderivative = 7.65 \[ \int \frac {\sqrt [3]{a+b x^3}}{x^{11}} \, dx=\frac {28 a^{5} b^{\frac {13}{3}} \sqrt [3]{\frac {a}{b x^{3}} + 1} \Gamma \left (- \frac {10}{3}\right )}{27 a^{5} b^{4} x^{9} \Gamma \left (- \frac {1}{3}\right ) + 54 a^{4} b^{5} x^{12} \Gamma \left (- \frac {1}{3}\right ) + 27 a^{3} b^{6} x^{15} \Gamma \left (- \frac {1}{3}\right )} + \frac {60 a^{4} b^{\frac {16}{3}} x^{3} \sqrt [3]{\frac {a}{b x^{3}} + 1} \Gamma \left (- \frac {10}{3}\right )}{27 a^{5} b^{4} x^{9} \Gamma \left (- \frac {1}{3}\right ) + 54 a^{4} b^{5} x^{12} \Gamma \left (- \frac {1}{3}\right ) + 27 a^{3} b^{6} x^{15} \Gamma \left (- \frac {1}{3}\right )} + \frac {30 a^{3} b^{\frac {19}{3}} x^{6} \sqrt [3]{\frac {a}{b x^{3}} + 1} \Gamma \left (- \frac {10}{3}\right )}{27 a^{5} b^{4} x^{9} \Gamma \left (- \frac {1}{3}\right ) + 54 a^{4} b^{5} x^{12} \Gamma \left (- \frac {1}{3}\right ) + 27 a^{3} b^{6} x^{15} \Gamma \left (- \frac {1}{3}\right )} + \frac {10 a^{2} b^{\frac {22}{3}} x^{9} \sqrt [3]{\frac {a}{b x^{3}} + 1} \Gamma \left (- \frac {10}{3}\right )}{27 a^{5} b^{4} x^{9} \Gamma \left (- \frac {1}{3}\right ) + 54 a^{4} b^{5} x^{12} \Gamma \left (- \frac {1}{3}\right ) + 27 a^{3} b^{6} x^{15} \Gamma \left (- \frac {1}{3}\right )} + \frac {30 a b^{\frac {25}{3}} x^{12} \sqrt [3]{\frac {a}{b x^{3}} + 1} \Gamma \left (- \frac {10}{3}\right )}{27 a^{5} b^{4} x^{9} \Gamma \left (- \frac {1}{3}\right ) + 54 a^{4} b^{5} x^{12} \Gamma \left (- \frac {1}{3}\right ) + 27 a^{3} b^{6} x^{15} \Gamma \left (- \frac {1}{3}\right )} + \frac {18 b^{\frac {28}{3}} x^{15} \sqrt [3]{\frac {a}{b x^{3}} + 1} \Gamma \left (- \frac {10}{3}\right )}{27 a^{5} b^{4} x^{9} \Gamma \left (- \frac {1}{3}\right ) + 54 a^{4} b^{5} x^{12} \Gamma \left (- \frac {1}{3}\right ) + 27 a^{3} b^{6} x^{15} \Gamma \left (- \frac {1}{3}\right )} \]
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Time = 0.20 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt [3]{a+b x^3}}{x^{11}} \, dx=-\frac {\frac {35 \, {\left (b x^{3} + a\right )}^{\frac {4}{3}} b^{2}}{x^{4}} - \frac {40 \, {\left (b x^{3} + a\right )}^{\frac {7}{3}} b}{x^{7}} + \frac {14 \, {\left (b x^{3} + a\right )}^{\frac {10}{3}}}{x^{10}}}{140 \, a^{3}} \]
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\[ \int \frac {\sqrt [3]{a+b x^3}}{x^{11}} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x^{11}} \,d x } \]
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Time = 6.39 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt [3]{a+b x^3}}{x^{11}} \, dx=\frac {3\,b^2\,{\left (b\,x^3+a\right )}^{1/3}}{140\,a^2\,x^4}-\frac {b\,{\left (b\,x^3+a\right )}^{1/3}}{70\,a\,x^7}-\frac {9\,b^3\,{\left (b\,x^3+a\right )}^{1/3}}{140\,a^3\,x}-\frac {{\left (b\,x^3+a\right )}^{1/3}}{10\,x^{10}} \]
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