Integrand size = 26, antiderivative size = 137 \[ \int \frac {1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^{7/2}} \, dx=-\frac {a^2}{2 b^3 \left (a+b \sqrt [3]{x}\right )^5 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac {6 a}{5 b^3 \left (a+b \sqrt [3]{x}\right )^4 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}-\frac {3}{4 b^3 \left (a+b \sqrt [3]{x}\right )^3 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \]
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Time = 0.05 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1355, 660, 45} \[ \int \frac {1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^{7/2}} \, dx=-\frac {a^2}{2 b^3 \left (a+b \sqrt [3]{x}\right )^5 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac {6 a}{5 b^3 \left (a+b \sqrt [3]{x}\right )^4 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}-\frac {3}{4 b^3 \left (a+b \sqrt [3]{x}\right )^3 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \]
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Rule 45
Rule 660
Rule 1355
Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int \frac {x^2}{\left (a^2+2 a b x+b^2 x^2\right )^{7/2}} \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {\left (3 b^7 \left (a+b \sqrt [3]{x}\right )\right ) \text {Subst}\left (\int \frac {x^2}{\left (a b+b^2 x\right )^7} \, dx,x,\sqrt [3]{x}\right )}{\sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \\ & = \frac {\left (3 b^7 \left (a+b \sqrt [3]{x}\right )\right ) \text {Subst}\left (\int \left (\frac {a^2}{b^9 (a+b x)^7}-\frac {2 a}{b^9 (a+b x)^6}+\frac {1}{b^9 (a+b x)^5}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \\ & = -\frac {a^2}{2 b^3 \left (a+b \sqrt [3]{x}\right )^5 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac {6 a}{5 b^3 \left (a+b \sqrt [3]{x}\right )^4 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}-\frac {3}{4 b^3 \left (a+b \sqrt [3]{x}\right )^3 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.41 \[ \int \frac {1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^{7/2}} \, dx=\frac {\left (a+b \sqrt [3]{x}\right ) \left (-a^2-6 a b \sqrt [3]{x}-15 b^2 x^{2/3}\right )}{20 b^3 \left (\left (a+b \sqrt [3]{x}\right )^2\right )^{7/2}} \]
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Time = 0.49 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.31
method | result | size |
derivativedivides | \(-\frac {\left (15 b^{2} x^{\frac {2}{3}}+6 a b \,x^{\frac {1}{3}}+a^{2}\right ) \left (a +b \,x^{\frac {1}{3}}\right )}{20 b^{3} {\left (\left (a +b \,x^{\frac {1}{3}}\right )^{2}\right )}^{\frac {7}{2}}}\) | \(43\) |
default | \(-\frac {\left (280 x^{9} a^{5} b^{27}-540 x^{\frac {29}{3}} a^{3} b^{29}-84 x^{\frac {31}{3}} a \,b^{31}-2106 x^{\frac {26}{3}} a^{6} b^{26}+567 x^{\frac {28}{3}} a^{4} b^{28}-792 x^{\frac {23}{3}} a^{9} b^{23}+3996 x^{\frac {25}{3}} a^{7} b^{25}+7344 x^{\frac {20}{3}} a^{12} b^{20}+7470 x^{\frac {22}{3}} a^{10} b^{22}+14580 x^{\frac {17}{3}} a^{15} b^{17}+3240 x^{\frac {19}{3}} a^{13} b^{19}-6264 x^{\frac {16}{3}} a^{16} b^{16}+280 x^{10} a^{2} b^{30}+a^{32}+2640 a^{20} b^{12} x^{4}+2820 a^{23} b^{9} x^{3}+666 a^{26} b^{6} x^{2}-8 a^{29} b^{3} x -3465 a^{8} b^{24} x^{8}-11004 a^{11} b^{21} x^{7}-12858 a^{14} b^{18} x^{6}-4824 a^{17} b^{15} x^{5}+15 x^{\frac {32}{3}} b^{32}+11250 x^{\frac {14}{3}} a^{18} b^{14}-9108 x^{\frac {13}{3}} a^{19} b^{13}+3024 x^{\frac {11}{3}} a^{21} b^{11}-4374 x^{\frac {10}{3}} a^{22} b^{10}-567 x^{\frac {8}{3}} a^{24} b^{8}-540 x^{\frac {7}{3}} a^{25} b^{7}-336 x^{\frac {5}{3}} a^{27} b^{5}+105 x^{\frac {4}{3}} a^{28} b^{4}\right ) \left (a +b \,x^{\frac {1}{3}}\right )}{20 b^{3} \left (b^{2} x^{\frac {2}{3}}-a b \,x^{\frac {1}{3}}+a^{2}\right )^{6} \left (b^{3} x +a^{3}\right )^{6} \left (a^{2}+2 a b \,x^{\frac {1}{3}}+b^{2} x^{\frac {2}{3}}\right )^{\frac {7}{2}}}\) | \(391\) |
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Time = 0.34 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.53 \[ \int \frac {1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^{7/2}} \, dx=-\frac {280 \, a^{2} b^{12} x^{4} - 1400 \, a^{5} b^{9} x^{3} + 735 \, a^{8} b^{6} x^{2} - 14 \, a^{11} b^{3} x + a^{14} + 3 \, {\left (5 \, b^{14} x^{4} - 210 \, a^{3} b^{11} x^{3} + 483 \, a^{6} b^{8} x^{2} - 112 \, a^{9} b^{5} x\right )} x^{\frac {2}{3}} - 3 \, {\left (28 \, a b^{13} x^{4} - 357 \, a^{4} b^{10} x^{3} + 390 \, a^{7} b^{7} x^{2} - 35 \, a^{10} b^{4} x\right )} x^{\frac {1}{3}}}{20 \, {\left (b^{21} x^{6} + 6 \, a^{3} b^{18} x^{5} + 15 \, a^{6} b^{15} x^{4} + 20 \, a^{9} b^{12} x^{3} + 15 \, a^{12} b^{9} x^{2} + 6 \, a^{15} b^{6} x + a^{18} b^{3}\right )}} \]
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\[ \int \frac {1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^{7/2}} \, dx=\int \frac {1}{\left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac {2}{3}}\right )^{\frac {7}{2}}}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.39 \[ \int \frac {1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^{7/2}} \, dx=-\frac {3}{4 \, b^{7} {\left (x^{\frac {1}{3}} + \frac {a}{b}\right )}^{4}} + \frac {6 \, a}{5 \, b^{8} {\left (x^{\frac {1}{3}} + \frac {a}{b}\right )}^{5}} - \frac {a^{2}}{2 \, b^{9} {\left (x^{\frac {1}{3}} + \frac {a}{b}\right )}^{6}} \]
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Time = 0.29 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.31 \[ \int \frac {1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^{7/2}} \, dx=-\frac {15 \, b^{2} x^{\frac {2}{3}} + 6 \, a b x^{\frac {1}{3}} + a^{2}}{20 \, {\left (b x^{\frac {1}{3}} + a\right )}^{6} b^{3} \mathrm {sgn}\left (b x^{\frac {1}{3}} + a\right )} \]
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Time = 9.49 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.39 \[ \int \frac {1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^{7/2}} \, dx=-\frac {\sqrt {a^2+b^2\,x^{2/3}+2\,a\,b\,x^{1/3}}\,\left (a^2+15\,b^2\,x^{2/3}+6\,a\,b\,x^{1/3}\right )}{20\,b^3\,{\left (a+b\,x^{1/3}\right )}^7} \]
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