Integrand size = 17, antiderivative size = 104 \[ \int \frac {1}{x^6 \left (a+b x^{3/2}\right )^{2/3}} \, dx=-\frac {\sqrt [3]{a+b x^{3/2}}}{5 a x^5}+\frac {9 b \sqrt [3]{a+b x^{3/2}}}{35 a^2 x^{7/2}}-\frac {27 b^2 \sqrt [3]{a+b x^{3/2}}}{70 a^3 x^2}+\frac {81 b^3 \sqrt [3]{a+b x^{3/2}}}{70 a^4 \sqrt {x}} \]
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Time = 0.03 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {277, 270} \[ \int \frac {1}{x^6 \left (a+b x^{3/2}\right )^{2/3}} \, dx=\frac {81 b^3 \sqrt [3]{a+b x^{3/2}}}{70 a^4 \sqrt {x}}-\frac {27 b^2 \sqrt [3]{a+b x^{3/2}}}{70 a^3 x^2}+\frac {9 b \sqrt [3]{a+b x^{3/2}}}{35 a^2 x^{7/2}}-\frac {\sqrt [3]{a+b x^{3/2}}}{5 a x^5} \]
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Rule 270
Rule 277
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt [3]{a+b x^{3/2}}}{5 a x^5}-\frac {(9 b) \int \frac {1}{x^{9/2} \left (a+b x^{3/2}\right )^{2/3}} \, dx}{10 a} \\ & = -\frac {\sqrt [3]{a+b x^{3/2}}}{5 a x^5}+\frac {9 b \sqrt [3]{a+b x^{3/2}}}{35 a^2 x^{7/2}}+\frac {\left (27 b^2\right ) \int \frac {1}{x^3 \left (a+b x^{3/2}\right )^{2/3}} \, dx}{35 a^2} \\ & = -\frac {\sqrt [3]{a+b x^{3/2}}}{5 a x^5}+\frac {9 b \sqrt [3]{a+b x^{3/2}}}{35 a^2 x^{7/2}}-\frac {27 b^2 \sqrt [3]{a+b x^{3/2}}}{70 a^3 x^2}-\frac {\left (81 b^3\right ) \int \frac {1}{x^{3/2} \left (a+b x^{3/2}\right )^{2/3}} \, dx}{140 a^3} \\ & = -\frac {\sqrt [3]{a+b x^{3/2}}}{5 a x^5}+\frac {9 b \sqrt [3]{a+b x^{3/2}}}{35 a^2 x^{7/2}}-\frac {27 b^2 \sqrt [3]{a+b x^{3/2}}}{70 a^3 x^2}+\frac {81 b^3 \sqrt [3]{a+b x^{3/2}}}{70 a^4 \sqrt {x}} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.57 \[ \int \frac {1}{x^6 \left (a+b x^{3/2}\right )^{2/3}} \, dx=\frac {\sqrt [3]{a+b x^{3/2}} \left (-14 a^3+18 a^2 b x^{3/2}-27 a b^2 x^3+81 b^3 x^{9/2}\right )}{70 a^4 x^5} \]
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\[\int \frac {1}{x^{6} \left (a +b \,x^{\frac {3}{2}}\right )^{\frac {2}{3}}}d x\]
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none
Time = 0.53 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.51 \[ \int \frac {1}{x^6 \left (a+b x^{3/2}\right )^{2/3}} \, dx=-\frac {{\left (27 \, a b^{2} x^{3} + 14 \, a^{3} - 9 \, {\left (9 \, b^{3} x^{4} + 2 \, a^{2} b x\right )} \sqrt {x}\right )} {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}}}{70 \, a^{4} x^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 736 vs. \(2 (95) = 190\).
Time = 6.88 (sec) , antiderivative size = 736, normalized size of antiderivative = 7.08 \[ \int \frac {1}{x^6 \left (a+b x^{3/2}\right )^{2/3}} \, dx=- \frac {56 a^{6} b^{\frac {28}{3}} x^{9} \sqrt [3]{\frac {a}{b x^{\frac {3}{2}}} + 1} \Gamma \left (- \frac {10}{3}\right )}{81 a^{7} b^{9} x^{\frac {27}{2}} \Gamma \left (\frac {2}{3}\right ) + 243 a^{6} b^{10} x^{15} \Gamma \left (\frac {2}{3}\right ) + 243 a^{5} b^{11} x^{\frac {33}{2}} \Gamma \left (\frac {2}{3}\right ) + 81 a^{4} b^{12} x^{18} \Gamma \left (\frac {2}{3}\right )} - \frac {96 a^{5} b^{\frac {31}{3}} x^{\frac {21}{2}} \sqrt [3]{\frac {a}{b x^{\frac {3}{2}}} + 1} \Gamma \left (- \frac {10}{3}\right )}{81 a^{7} b^{9} x^{\frac {27}{2}} \Gamma \left (\frac {2}{3}\right ) + 243 a^{6} b^{10} x^{15} \Gamma \left (\frac {2}{3}\right ) + 243 a^{5} b^{11} x^{\frac {33}{2}} \Gamma \left (\frac {2}{3}\right ) + 81 a^{4} b^{12} x^{18} \Gamma \left (\frac {2}{3}\right )} - \frac {60 a^{4} b^{\frac {34}{3}} x^{12} \sqrt [3]{\frac {a}{b x^{\frac {3}{2}}} + 1} \Gamma \left (- \frac {10}{3}\right )}{81 a^{7} b^{9} x^{\frac {27}{2}} \Gamma \left (\frac {2}{3}\right ) + 243 a^{6} b^{10} x^{15} \Gamma \left (\frac {2}{3}\right ) + 243 a^{5} b^{11} x^{\frac {33}{2}} \Gamma \left (\frac {2}{3}\right ) + 81 a^{4} b^{12} x^{18} \Gamma \left (\frac {2}{3}\right )} + \frac {160 a^{3} b^{\frac {37}{3}} x^{\frac {27}{2}} \sqrt [3]{\frac {a}{b x^{\frac {3}{2}}} + 1} \Gamma \left (- \frac {10}{3}\right )}{81 a^{7} b^{9} x^{\frac {27}{2}} \Gamma \left (\frac {2}{3}\right ) + 243 a^{6} b^{10} x^{15} \Gamma \left (\frac {2}{3}\right ) + 243 a^{5} b^{11} x^{\frac {33}{2}} \Gamma \left (\frac {2}{3}\right ) + 81 a^{4} b^{12} x^{18} \Gamma \left (\frac {2}{3}\right )} + \frac {720 a^{2} b^{\frac {40}{3}} x^{15} \sqrt [3]{\frac {a}{b x^{\frac {3}{2}}} + 1} \Gamma \left (- \frac {10}{3}\right )}{81 a^{7} b^{9} x^{\frac {27}{2}} \Gamma \left (\frac {2}{3}\right ) + 243 a^{6} b^{10} x^{15} \Gamma \left (\frac {2}{3}\right ) + 243 a^{5} b^{11} x^{\frac {33}{2}} \Gamma \left (\frac {2}{3}\right ) + 81 a^{4} b^{12} x^{18} \Gamma \left (\frac {2}{3}\right )} + \frac {864 a b^{\frac {43}{3}} x^{\frac {33}{2}} \sqrt [3]{\frac {a}{b x^{\frac {3}{2}}} + 1} \Gamma \left (- \frac {10}{3}\right )}{81 a^{7} b^{9} x^{\frac {27}{2}} \Gamma \left (\frac {2}{3}\right ) + 243 a^{6} b^{10} x^{15} \Gamma \left (\frac {2}{3}\right ) + 243 a^{5} b^{11} x^{\frac {33}{2}} \Gamma \left (\frac {2}{3}\right ) + 81 a^{4} b^{12} x^{18} \Gamma \left (\frac {2}{3}\right )} + \frac {324 b^{\frac {46}{3}} x^{18} \sqrt [3]{\frac {a}{b x^{\frac {3}{2}}} + 1} \Gamma \left (- \frac {10}{3}\right )}{81 a^{7} b^{9} x^{\frac {27}{2}} \Gamma \left (\frac {2}{3}\right ) + 243 a^{6} b^{10} x^{15} \Gamma \left (\frac {2}{3}\right ) + 243 a^{5} b^{11} x^{\frac {33}{2}} \Gamma \left (\frac {2}{3}\right ) + 81 a^{4} b^{12} x^{18} \Gamma \left (\frac {2}{3}\right )} \]
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Time = 0.20 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.66 \[ \int \frac {1}{x^6 \left (a+b x^{3/2}\right )^{2/3}} \, dx=\frac {\frac {140 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}} b^{3}}{\sqrt {x}} - \frac {105 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {4}{3}} b^{2}}{x^{2}} + \frac {60 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {7}{3}} b}{x^{\frac {7}{2}}} - \frac {14 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {10}{3}}}{x^{5}}}{70 \, a^{4}} \]
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\[ \int \frac {1}{x^6 \left (a+b x^{3/2}\right )^{2/3}} \, dx=\int { \frac {1}{{\left (b x^{\frac {3}{2}} + a\right )}^{\frac {2}{3}} x^{6}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^6 \left (a+b x^{3/2}\right )^{2/3}} \, dx=\int \frac {1}{x^6\,{\left (a+b\,x^{3/2}\right )}^{2/3}} \,d x \]
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