Integrand size = 26, antiderivative size = 137 \[ \int \frac {1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^{11/2}} \, dx=-\frac {3 a^2}{10 b^3 \left (a+b \sqrt [3]{x}\right )^9 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac {2 a}{3 b^3 \left (a+b \sqrt [3]{x}\right )^8 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}-\frac {3}{8 b^3 \left (a+b \sqrt [3]{x}\right )^7 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \]
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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} \[ \int \frac {1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^{11/2}} \, dx=-\frac {3 a^2}{10 b^3 \left (a+b \sqrt [3]{x}\right )^9 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac {2 a}{3 b^3 \left (a+b \sqrt [3]{x}\right )^8 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}-\frac {3}{8 b^3 \left (a+b \sqrt [3]{x}\right )^7 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \]
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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 )^{11/2}} \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {\left (3 b^{11} \left (a+b \sqrt [3]{x}\right )\right ) \text {Subst}\left (\int \frac {x^2}{\left (a b+b^2 x\right )^{11}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \\ & = \frac {\left (3 b^{11} \left (a+b \sqrt [3]{x}\right )\right ) \text {Subst}\left (\int \left (\frac {a^2}{b^{13} (a+b x)^{11}}-\frac {2 a}{b^{13} (a+b x)^{10}}+\frac {1}{b^{13} (a+b x)^9}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \\ & = -\frac {3 a^2}{10 b^3 \left (a+b \sqrt [3]{x}\right )^9 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac {2 a}{3 b^3 \left (a+b \sqrt [3]{x}\right )^8 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}-\frac {3}{8 b^3 \left (a+b \sqrt [3]{x}\right )^7 \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 \[ \int \frac {1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^{11/2}} \, dx=\frac {\left (a+b \sqrt [3]{x}\right ) \left (-a^2-10 a b \sqrt [3]{x}-45 b^2 x^{2/3}\right )}{120 b^3 \left (\left (a+b \sqrt [3]{x}\right )^2\right )^{11/2}} \]
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Time = 1.18 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.31
method | result | size |
derivativedivides | \(-\frac {\left (45 b^{2} x^{\frac {2}{3}}+10 a b \,x^{\frac {1}{3}}+a^{2}\right ) \left (a +b \,x^{\frac {1}{3}}\right )}{120 b^{3} {\left (\left (a +b \,x^{\frac {1}{3}}\right )^{2}\right )}^{\frac {11}{2}}}\) | \(43\) |
default | \(-\frac {\left (-5834520 x^{\frac {34}{3}} a^{18} b^{34}+2207250 x^{\frac {38}{3}} a^{14} b^{38}-13023000 x^{\frac {31}{3}} a^{21} b^{31}+6925500 x^{\frac {35}{3}} a^{17} b^{35}-12561075 x^{\frac {28}{3}} a^{24} b^{28}+8650665 x^{\frac {32}{3}} a^{20} b^{32}-3059244 x^{\frac {25}{3}} a^{27} b^{25}+1739880 x^{\frac {29}{3}} a^{23} b^{29}-8850060 x^{\frac {26}{3}} a^{26} b^{26}-12242880 x^{\frac {23}{3}} a^{29} b^{23}+a^{52}-2290640 a^{13} b^{39} x^{13}-3926065 a^{16} b^{36} x^{12}-378620 a^{19} b^{33} x^{11}+9124622 a^{22} b^{30} x^{10}+15708340 a^{25} b^{27} x^{9}+11830105 a^{28} b^{24} x^{8}+2682800 a^{31} b^{21} x^{7}-2306600 a^{34} b^{18} x^{6}-1957000 a^{37} b^{15} x^{5}-506615 a^{40} b^{12} x^{4}+1220 a^{43} b^{9} x^{3}+14950 a^{46} b^{6} x^{2}-100 a^{49} b^{3} x +45 x^{\frac {52}{3}} b^{52}+5519790 x^{\frac {22}{3}} a^{30} b^{22}-7044003 x^{\frac {20}{3}} a^{32} b^{20}+6159780 x^{\frac {19}{3}} a^{33} b^{19}-1200420 x^{\frac {17}{3}} a^{35} b^{17}+2548755 x^{\frac {16}{3}} a^{36} b^{16}+686610 x^{\frac {14}{3}} a^{38} b^{14}+233640 x^{\frac {13}{3}} a^{39} b^{13}+367740 x^{\frac {11}{3}} a^{41} b^{11}-141372 x^{\frac {10}{3}} a^{42} b^{10}+40095 x^{\frac {8}{3}} a^{44} b^{8}-31680 x^{\frac {7}{3}} a^{45} b^{7}-4752 x^{\frac {5}{3}} a^{47} b^{5}+990 x^{\frac {4}{3}} a^{48} b^{4}-440 x^{17} a \,b^{51}+21230 x^{16} a^{4} b^{48}+50248 x^{15} a^{7} b^{45}-8460 x^{\frac {49}{3}} a^{3} b^{49}+27270 x^{\frac {46}{3}} a^{6} b^{46}+2376 x^{\frac {50}{3}} a^{2} b^{50}+435060 x^{\frac {43}{3}} a^{9} b^{43}-35640 x^{\frac {47}{3}} a^{5} b^{47}+1187253 x^{\frac {40}{3}} a^{12} b^{40}-227205 x^{\frac {44}{3}} a^{8} b^{44}+41040 x^{\frac {37}{3}} a^{15} b^{37}-83700 x^{\frac {41}{3}} a^{11} b^{41}-435820 a^{10} b^{42} x^{14}\right ) \left (a +b \,x^{\frac {1}{3}}\right )}{120 b^{3} \left (b^{2} x^{\frac {2}{3}}-a b \,x^{\frac {1}{3}}+a^{2}\right )^{10} \left (b^{3} x +a^{3}\right )^{10} \left (a^{2}+2 a b \,x^{\frac {1}{3}}+b^{2} x^{\frac {2}{3}}\right )^{\frac {11}{2}}}\) | \(611\) |
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Leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (107) = 214\).
Time = 0.58 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.50 \[ \int \frac {1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^{11/2}} \, dx=\frac {440 \, a b^{21} x^{7} - 25630 \, a^{4} b^{18} x^{6} + 186252 \, a^{7} b^{15} x^{5} - 326150 \, a^{10} b^{12} x^{4} + 154000 \, a^{13} b^{9} x^{3} - 16005 \, a^{16} b^{6} x^{2} + 110 \, a^{19} b^{3} x - a^{22} - 27 \, {\left (88 \, a^{2} b^{20} x^{6} - 2200 \, a^{5} b^{17} x^{5} + 9625 \, a^{8} b^{14} x^{4} - 10910 \, a^{11} b^{11} x^{3} + 3245 \, a^{14} b^{8} x^{2} - 176 \, a^{17} b^{5} x\right )} x^{\frac {2}{3}} - 9 \, {\left (5 \, b^{22} x^{7} - 990 \, a^{3} b^{19} x^{6} + 12705 \, a^{6} b^{16} x^{5} - 34760 \, a^{9} b^{13} x^{4} + 25542 \, a^{12} b^{10} x^{3} - 4620 \, a^{15} b^{7} x^{2} + 110 \, a^{18} b^{4} x\right )} x^{\frac {1}{3}}}{120 \, {\left (b^{33} x^{10} + 10 \, a^{3} b^{30} x^{9} + 45 \, a^{6} b^{27} x^{8} + 120 \, a^{9} b^{24} x^{7} + 210 \, a^{12} b^{21} x^{6} + 252 \, a^{15} b^{18} x^{5} + 210 \, a^{18} b^{15} x^{4} + 120 \, a^{21} b^{12} x^{3} + 45 \, a^{24} b^{9} x^{2} + 10 \, a^{27} b^{6} x + a^{30} b^{3}\right )}} \]
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\[ \int \frac {1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^{11/2}} \, dx=\int \frac {1}{\left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac {2}{3}}\right )^{\frac {11}{2}}}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.39 \[ \int \frac {1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^{11/2}} \, dx=-\frac {3}{8 \, b^{11} {\left (x^{\frac {1}{3}} + \frac {a}{b}\right )}^{8}} + \frac {2 \, a}{3 \, b^{12} {\left (x^{\frac {1}{3}} + \frac {a}{b}\right )}^{9}} - \frac {3 \, a^{2}}{10 \, b^{13} {\left (x^{\frac {1}{3}} + \frac {a}{b}\right )}^{10}} \]
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Time = 0.29 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.31 \[ \int \frac {1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^{11/2}} \, dx=-\frac {45 \, b^{2} x^{\frac {2}{3}} + 10 \, a b x^{\frac {1}{3}} + a^{2}}{120 \, {\left (b x^{\frac {1}{3}} + a\right )}^{10} b^{3} \mathrm {sgn}\left (b x^{\frac {1}{3}} + a\right )} \]
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Time = 10.06 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.39 \[ \int \frac {1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^{11/2}} \, dx=-\frac {\sqrt {a^2+b^2\,x^{2/3}+2\,a\,b\,x^{1/3}}\,\left (a^2+45\,b^2\,x^{2/3}+10\,a\,b\,x^{1/3}\right )}{120\,b^3\,{\left (a+b\,x^{1/3}\right )}^{11}} \]
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