Integrand size = 15, antiderivative size = 122 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^6} \, dx=-\frac {\left (a+b \sqrt [3]{x}\right )^{11}}{5 a x^5}+\frac {2 b \left (a+b \sqrt [3]{x}\right )^{11}}{35 a^2 x^{14/3}}-\frac {6 b^2 \left (a+b \sqrt [3]{x}\right )^{11}}{455 a^3 x^{13/3}}+\frac {b^3 \left (a+b \sqrt [3]{x}\right )^{11}}{455 a^4 x^4}-\frac {b^4 \left (a+b \sqrt [3]{x}\right )^{11}}{5005 a^5 x^{11/3}} \]
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Time = 0.03 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {272, 47, 37} \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^6} \, dx=-\frac {b^4 \left (a+b \sqrt [3]{x}\right )^{11}}{5005 a^5 x^{11/3}}+\frac {b^3 \left (a+b \sqrt [3]{x}\right )^{11}}{455 a^4 x^4}-\frac {6 b^2 \left (a+b \sqrt [3]{x}\right )^{11}}{455 a^3 x^{13/3}}+\frac {2 b \left (a+b \sqrt [3]{x}\right )^{11}}{35 a^2 x^{14/3}}-\frac {\left (a+b \sqrt [3]{x}\right )^{11}}{5 a x^5} \]
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Rule 37
Rule 47
Rule 272
Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int \frac {(a+b x)^{10}}{x^{16}} \, dx,x,\sqrt [3]{x}\right ) \\ & = -\frac {\left (a+b \sqrt [3]{x}\right )^{11}}{5 a x^5}-\frac {(4 b) \text {Subst}\left (\int \frac {(a+b x)^{10}}{x^{15}} \, dx,x,\sqrt [3]{x}\right )}{5 a} \\ & = -\frac {\left (a+b \sqrt [3]{x}\right )^{11}}{5 a x^5}+\frac {2 b \left (a+b \sqrt [3]{x}\right )^{11}}{35 a^2 x^{14/3}}+\frac {\left (6 b^2\right ) \text {Subst}\left (\int \frac {(a+b x)^{10}}{x^{14}} \, dx,x,\sqrt [3]{x}\right )}{35 a^2} \\ & = -\frac {\left (a+b \sqrt [3]{x}\right )^{11}}{5 a x^5}+\frac {2 b \left (a+b \sqrt [3]{x}\right )^{11}}{35 a^2 x^{14/3}}-\frac {6 b^2 \left (a+b \sqrt [3]{x}\right )^{11}}{455 a^3 x^{13/3}}-\frac {\left (12 b^3\right ) \text {Subst}\left (\int \frac {(a+b x)^{10}}{x^{13}} \, dx,x,\sqrt [3]{x}\right )}{455 a^3} \\ & = -\frac {\left (a+b \sqrt [3]{x}\right )^{11}}{5 a x^5}+\frac {2 b \left (a+b \sqrt [3]{x}\right )^{11}}{35 a^2 x^{14/3}}-\frac {6 b^2 \left (a+b \sqrt [3]{x}\right )^{11}}{455 a^3 x^{13/3}}+\frac {b^3 \left (a+b \sqrt [3]{x}\right )^{11}}{455 a^4 x^4}+\frac {b^4 \text {Subst}\left (\int \frac {(a+b x)^{10}}{x^{12}} \, dx,x,\sqrt [3]{x}\right )}{455 a^4} \\ & = -\frac {\left (a+b \sqrt [3]{x}\right )^{11}}{5 a x^5}+\frac {2 b \left (a+b \sqrt [3]{x}\right )^{11}}{35 a^2 x^{14/3}}-\frac {6 b^2 \left (a+b \sqrt [3]{x}\right )^{11}}{455 a^3 x^{13/3}}+\frac {b^3 \left (a+b \sqrt [3]{x}\right )^{11}}{455 a^4 x^4}-\frac {b^4 \left (a+b \sqrt [3]{x}\right )^{11}}{5005 a^5 x^{11/3}} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^6} \, dx=\frac {-1001 a^{10}-10725 a^9 b \sqrt [3]{x}-51975 a^8 b^2 x^{2/3}-150150 a^7 b^3 x-286650 a^6 b^4 x^{4/3}-378378 a^5 b^5 x^{5/3}-350350 a^4 b^6 x^2-225225 a^3 b^7 x^{7/3}-96525 a^2 b^8 x^{8/3}-25025 a b^9 x^3-3003 b^{10} x^{10/3}}{5005 x^5} \]
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Time = 3.61 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.93
method | result | size |
derivativedivides | \(-\frac {3 b^{10}}{5 x^{\frac {5}{3}}}-\frac {378 a^{5} b^{5}}{5 x^{\frac {10}{3}}}-\frac {45 a^{3} b^{7}}{x^{\frac {8}{3}}}-\frac {30 a^{7} b^{3}}{x^{4}}-\frac {5 a \,b^{9}}{x^{2}}-\frac {135 a^{8} b^{2}}{13 x^{\frac {13}{3}}}-\frac {135 a^{2} b^{8}}{7 x^{\frac {7}{3}}}-\frac {a^{10}}{5 x^{5}}-\frac {15 a^{9} b}{7 x^{\frac {14}{3}}}-\frac {70 a^{4} b^{6}}{x^{3}}-\frac {630 a^{6} b^{4}}{11 x^{\frac {11}{3}}}\) | \(113\) |
default | \(-\frac {3 b^{10}}{5 x^{\frac {5}{3}}}-\frac {378 a^{5} b^{5}}{5 x^{\frac {10}{3}}}-\frac {45 a^{3} b^{7}}{x^{\frac {8}{3}}}-\frac {30 a^{7} b^{3}}{x^{4}}-\frac {5 a \,b^{9}}{x^{2}}-\frac {135 a^{8} b^{2}}{13 x^{\frac {13}{3}}}-\frac {135 a^{2} b^{8}}{7 x^{\frac {7}{3}}}-\frac {a^{10}}{5 x^{5}}-\frac {15 a^{9} b}{7 x^{\frac {14}{3}}}-\frac {70 a^{4} b^{6}}{x^{3}}-\frac {630 a^{6} b^{4}}{11 x^{\frac {11}{3}}}\) | \(113\) |
trager | \(\frac {\left (-1+x \right ) \left (a^{9} x^{4}+150 a^{6} b^{3} x^{4}+350 a^{3} b^{6} x^{4}+25 b^{9} x^{4}+a^{9} x^{3}+150 a^{6} b^{3} x^{3}+350 a^{3} b^{6} x^{3}+25 b^{9} x^{3}+a^{9} x^{2}+150 a^{6} b^{3} x^{2}+350 a^{3} b^{6} x^{2}+a^{9} x +150 x \,a^{6} b^{3}+a^{9}\right ) a}{5 x^{5}}-\frac {3 \left (77 b^{9} x^{3}+5775 a^{3} b^{6} x^{2}+7350 x \,a^{6} b^{3}+275 a^{9}\right ) b}{385 x^{\frac {14}{3}}}-\frac {27 \left (325 b^{6} x^{2}+1274 a^{3} b^{3} x +175 a^{6}\right ) a^{2} b^{2}}{455 x^{\frac {13}{3}}}\) | \(206\) |
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Time = 0.28 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^6} \, dx=-\frac {25025 \, a b^{9} x^{3} + 350350 \, a^{4} b^{6} x^{2} + 150150 \, a^{7} b^{3} x + 1001 \, a^{10} + 297 \, {\left (325 \, a^{2} b^{8} x^{2} + 1274 \, a^{5} b^{5} x + 175 \, a^{8} b^{2}\right )} x^{\frac {2}{3}} + 39 \, {\left (77 \, b^{10} x^{3} + 5775 \, a^{3} b^{7} x^{2} + 7350 \, a^{6} b^{4} x + 275 \, a^{9} b\right )} x^{\frac {1}{3}}}{5005 \, x^{5}} \]
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Time = 0.68 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.17 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^6} \, dx=- \frac {a^{10}}{5 x^{5}} - \frac {15 a^{9} b}{7 x^{\frac {14}{3}}} - \frac {135 a^{8} b^{2}}{13 x^{\frac {13}{3}}} - \frac {30 a^{7} b^{3}}{x^{4}} - \frac {630 a^{6} b^{4}}{11 x^{\frac {11}{3}}} - \frac {378 a^{5} b^{5}}{5 x^{\frac {10}{3}}} - \frac {70 a^{4} b^{6}}{x^{3}} - \frac {45 a^{3} b^{7}}{x^{\frac {8}{3}}} - \frac {135 a^{2} b^{8}}{7 x^{\frac {7}{3}}} - \frac {5 a b^{9}}{x^{2}} - \frac {3 b^{10}}{5 x^{\frac {5}{3}}} \]
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Time = 0.19 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^6} \, dx=-\frac {3003 \, b^{10} x^{\frac {10}{3}} + 25025 \, a b^{9} x^{3} + 96525 \, a^{2} b^{8} x^{\frac {8}{3}} + 225225 \, a^{3} b^{7} x^{\frac {7}{3}} + 350350 \, a^{4} b^{6} x^{2} + 378378 \, a^{5} b^{5} x^{\frac {5}{3}} + 286650 \, a^{6} b^{4} x^{\frac {4}{3}} + 150150 \, a^{7} b^{3} x + 51975 \, a^{8} b^{2} x^{\frac {2}{3}} + 10725 \, a^{9} b x^{\frac {1}{3}} + 1001 \, a^{10}}{5005 \, x^{5}} \]
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Time = 0.29 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^6} \, dx=-\frac {3003 \, b^{10} x^{\frac {10}{3}} + 25025 \, a b^{9} x^{3} + 96525 \, a^{2} b^{8} x^{\frac {8}{3}} + 225225 \, a^{3} b^{7} x^{\frac {7}{3}} + 350350 \, a^{4} b^{6} x^{2} + 378378 \, a^{5} b^{5} x^{\frac {5}{3}} + 286650 \, a^{6} b^{4} x^{\frac {4}{3}} + 150150 \, a^{7} b^{3} x + 51975 \, a^{8} b^{2} x^{\frac {2}{3}} + 10725 \, a^{9} b x^{\frac {1}{3}} + 1001 \, a^{10}}{5005 \, x^{5}} \]
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Time = 5.58 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b \sqrt [3]{x}\right )^{10}}{x^6} \, dx=-\frac {\frac {a^{10}}{5}+\frac {3\,b^{10}\,x^{10/3}}{5}+30\,a^7\,b^3\,x+5\,a\,b^9\,x^3+\frac {15\,a^9\,b\,x^{1/3}}{7}+70\,a^4\,b^6\,x^2+\frac {135\,a^8\,b^2\,x^{2/3}}{13}+\frac {630\,a^6\,b^4\,x^{4/3}}{11}+\frac {378\,a^5\,b^5\,x^{5/3}}{5}+45\,a^3\,b^7\,x^{7/3}+\frac {135\,a^2\,b^8\,x^{8/3}}{7}}{x^5} \]
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