Optimal. Leaf size=217 \[ \frac{315}{17} a^{13} b^2 x^{17/3}+\frac{455}{6} a^{12} b^3 x^6+\frac{4095}{19} a^{11} b^4 x^{19/3}+\frac{9009}{20} a^{10} b^5 x^{20/3}+715 a^9 b^6 x^7+\frac{1755}{2} a^8 b^7 x^{22/3}+\frac{19305}{23} a^7 b^8 x^{23/3}+\frac{5005}{8} a^6 b^9 x^8+\frac{9009}{25} a^5 b^{10} x^{25/3}+\frac{315}{2} a^4 b^{11} x^{26/3}+\frac{455}{9} a^3 b^{12} x^9+\frac{45}{4} a^2 b^{13} x^{28/3}+\frac{45}{16} a^{14} b x^{16/3}+\frac{a^{15} x^5}{5}+\frac{45}{29} a b^{14} x^{29/3}+\frac{b^{15} x^{10}}{10} \]
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Rubi [A] time = 0.133412, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{315}{17} a^{13} b^2 x^{17/3}+\frac{455}{6} a^{12} b^3 x^6+\frac{4095}{19} a^{11} b^4 x^{19/3}+\frac{9009}{20} a^{10} b^5 x^{20/3}+715 a^9 b^6 x^7+\frac{1755}{2} a^8 b^7 x^{22/3}+\frac{19305}{23} a^7 b^8 x^{23/3}+\frac{5005}{8} a^6 b^9 x^8+\frac{9009}{25} a^5 b^{10} x^{25/3}+\frac{315}{2} a^4 b^{11} x^{26/3}+\frac{455}{9} a^3 b^{12} x^9+\frac{45}{4} a^2 b^{13} x^{28/3}+\frac{45}{16} a^{14} b x^{16/3}+\frac{a^{15} x^5}{5}+\frac{45}{29} a b^{14} x^{29/3}+\frac{b^{15} x^{10}}{10} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \left (a+b \sqrt [3]{x}\right )^{15} x^4 \, dx &=3 \operatorname{Subst}\left (\int x^{14} (a+b x)^{15} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname{Subst}\left (\int \left (a^{15} x^{14}+15 a^{14} b x^{15}+105 a^{13} b^2 x^{16}+455 a^{12} b^3 x^{17}+1365 a^{11} b^4 x^{18}+3003 a^{10} b^5 x^{19}+5005 a^9 b^6 x^{20}+6435 a^8 b^7 x^{21}+6435 a^7 b^8 x^{22}+5005 a^6 b^9 x^{23}+3003 a^5 b^{10} x^{24}+1365 a^4 b^{11} x^{25}+455 a^3 b^{12} x^{26}+105 a^2 b^{13} x^{27}+15 a b^{14} x^{28}+b^{15} x^{29}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{a^{15} x^5}{5}+\frac{45}{16} a^{14} b x^{16/3}+\frac{315}{17} a^{13} b^2 x^{17/3}+\frac{455}{6} a^{12} b^3 x^6+\frac{4095}{19} a^{11} b^4 x^{19/3}+\frac{9009}{20} a^{10} b^5 x^{20/3}+715 a^9 b^6 x^7+\frac{1755}{2} a^8 b^7 x^{22/3}+\frac{19305}{23} a^7 b^8 x^{23/3}+\frac{5005}{8} a^6 b^9 x^8+\frac{9009}{25} a^5 b^{10} x^{25/3}+\frac{315}{2} a^4 b^{11} x^{26/3}+\frac{455}{9} a^3 b^{12} x^9+\frac{45}{4} a^2 b^{13} x^{28/3}+\frac{45}{29} a b^{14} x^{29/3}+\frac{b^{15} x^{10}}{10}\\ \end{align*}
Mathematica [A] time = 0.0943194, size = 217, normalized size = 1. \[ \frac{315}{17} a^{13} b^2 x^{17/3}+\frac{455}{6} a^{12} b^3 x^6+\frac{4095}{19} a^{11} b^4 x^{19/3}+\frac{9009}{20} a^{10} b^5 x^{20/3}+715 a^9 b^6 x^7+\frac{1755}{2} a^8 b^7 x^{22/3}+\frac{19305}{23} a^7 b^8 x^{23/3}+\frac{5005}{8} a^6 b^9 x^8+\frac{9009}{25} a^5 b^{10} x^{25/3}+\frac{315}{2} a^4 b^{11} x^{26/3}+\frac{455}{9} a^3 b^{12} x^9+\frac{45}{4} a^2 b^{13} x^{28/3}+\frac{45}{16} a^{14} b x^{16/3}+\frac{a^{15} x^5}{5}+\frac{45}{29} a b^{14} x^{29/3}+\frac{b^{15} x^{10}}{10} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 168, normalized size = 0.8 \begin{align*}{\frac{{a}^{15}{x}^{5}}{5}}+{\frac{45\,{a}^{14}b}{16}{x}^{{\frac{16}{3}}}}+{\frac{315\,{a}^{13}{b}^{2}}{17}{x}^{{\frac{17}{3}}}}+{\frac{455\,{a}^{12}{b}^{3}{x}^{6}}{6}}+{\frac{4095\,{a}^{11}{b}^{4}}{19}{x}^{{\frac{19}{3}}}}+{\frac{9009\,{a}^{10}{b}^{5}}{20}{x}^{{\frac{20}{3}}}}+715\,{a}^{9}{b}^{6}{x}^{7}+{\frac{1755\,{a}^{8}{b}^{7}}{2}{x}^{{\frac{22}{3}}}}+{\frac{19305\,{a}^{7}{b}^{8}}{23}{x}^{{\frac{23}{3}}}}+{\frac{5005\,{a}^{6}{b}^{9}{x}^{8}}{8}}+{\frac{9009\,{a}^{5}{b}^{10}}{25}{x}^{{\frac{25}{3}}}}+{\frac{315\,{a}^{4}{b}^{11}}{2}{x}^{{\frac{26}{3}}}}+{\frac{455\,{a}^{3}{b}^{12}{x}^{9}}{9}}+{\frac{45\,{a}^{2}{b}^{13}}{4}{x}^{{\frac{28}{3}}}}+{\frac{45\,a{b}^{14}}{29}{x}^{{\frac{29}{3}}}}+{\frac{{b}^{15}{x}^{10}}{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.97053, size = 339, normalized size = 1.56 \begin{align*} \frac{{\left (b x^{\frac{1}{3}} + a\right )}^{30}}{10 \, b^{15}} - \frac{42 \,{\left (b x^{\frac{1}{3}} + a\right )}^{29} a}{29 \, b^{15}} + \frac{39 \,{\left (b x^{\frac{1}{3}} + a\right )}^{28} a^{2}}{4 \, b^{15}} - \frac{364 \,{\left (b x^{\frac{1}{3}} + a\right )}^{27} a^{3}}{9 \, b^{15}} + \frac{231 \,{\left (b x^{\frac{1}{3}} + a\right )}^{26} a^{4}}{2 \, b^{15}} - \frac{6006 \,{\left (b x^{\frac{1}{3}} + a\right )}^{25} a^{5}}{25 \, b^{15}} + \frac{3003 \,{\left (b x^{\frac{1}{3}} + a\right )}^{24} a^{6}}{8 \, b^{15}} - \frac{10296 \,{\left (b x^{\frac{1}{3}} + a\right )}^{23} a^{7}}{23 \, b^{15}} + \frac{819 \,{\left (b x^{\frac{1}{3}} + a\right )}^{22} a^{8}}{2 \, b^{15}} - \frac{286 \,{\left (b x^{\frac{1}{3}} + a\right )}^{21} a^{9}}{b^{15}} + \frac{3003 \,{\left (b x^{\frac{1}{3}} + a\right )}^{20} a^{10}}{20 \, b^{15}} - \frac{1092 \,{\left (b x^{\frac{1}{3}} + a\right )}^{19} a^{11}}{19 \, b^{15}} + \frac{91 \,{\left (b x^{\frac{1}{3}} + a\right )}^{18} a^{12}}{6 \, b^{15}} - \frac{42 \,{\left (b x^{\frac{1}{3}} + a\right )}^{17} a^{13}}{17 \, b^{15}} + \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{16} a^{14}}{16 \, b^{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4921, size = 489, normalized size = 2.25 \begin{align*} \frac{1}{10} \, b^{15} x^{10} + \frac{455}{9} \, a^{3} b^{12} x^{9} + \frac{5005}{8} \, a^{6} b^{9} x^{8} + 715 \, a^{9} b^{6} x^{7} + \frac{455}{6} \, a^{12} b^{3} x^{6} + \frac{1}{5} \, a^{15} x^{5} + \frac{9}{226780} \,{\left (39100 \, a b^{14} x^{9} + 3968650 \, a^{4} b^{11} x^{8} + 21149700 \, a^{7} b^{8} x^{7} + 11350339 \, a^{10} b^{5} x^{6} + 466900 \, a^{13} b^{2} x^{5}\right )} x^{\frac{2}{3}} + \frac{9}{7600} \,{\left (9500 \, a^{2} b^{13} x^{9} + 304304 \, a^{5} b^{10} x^{8} + 741000 \, a^{8} b^{7} x^{7} + 182000 \, a^{11} b^{4} x^{6} + 2375 \, a^{14} b x^{5}\right )} x^{\frac{1}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 13.211, size = 218, normalized size = 1. \begin{align*} \frac{a^{15} x^{5}}{5} + \frac{45 a^{14} b x^{\frac{16}{3}}}{16} + \frac{315 a^{13} b^{2} x^{\frac{17}{3}}}{17} + \frac{455 a^{12} b^{3} x^{6}}{6} + \frac{4095 a^{11} b^{4} x^{\frac{19}{3}}}{19} + \frac{9009 a^{10} b^{5} x^{\frac{20}{3}}}{20} + 715 a^{9} b^{6} x^{7} + \frac{1755 a^{8} b^{7} x^{\frac{22}{3}}}{2} + \frac{19305 a^{7} b^{8} x^{\frac{23}{3}}}{23} + \frac{5005 a^{6} b^{9} x^{8}}{8} + \frac{9009 a^{5} b^{10} x^{\frac{25}{3}}}{25} + \frac{315 a^{4} b^{11} x^{\frac{26}{3}}}{2} + \frac{455 a^{3} b^{12} x^{9}}{9} + \frac{45 a^{2} b^{13} x^{\frac{28}{3}}}{4} + \frac{45 a b^{14} x^{\frac{29}{3}}}{29} + \frac{b^{15} x^{10}}{10} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25463, size = 225, normalized size = 1.04 \begin{align*} \frac{1}{10} \, b^{15} x^{10} + \frac{45}{29} \, a b^{14} x^{\frac{29}{3}} + \frac{45}{4} \, a^{2} b^{13} x^{\frac{28}{3}} + \frac{455}{9} \, a^{3} b^{12} x^{9} + \frac{315}{2} \, a^{4} b^{11} x^{\frac{26}{3}} + \frac{9009}{25} \, a^{5} b^{10} x^{\frac{25}{3}} + \frac{5005}{8} \, a^{6} b^{9} x^{8} + \frac{19305}{23} \, a^{7} b^{8} x^{\frac{23}{3}} + \frac{1755}{2} \, a^{8} b^{7} x^{\frac{22}{3}} + 715 \, a^{9} b^{6} x^{7} + \frac{9009}{20} \, a^{10} b^{5} x^{\frac{20}{3}} + \frac{4095}{19} \, a^{11} b^{4} x^{\frac{19}{3}} + \frac{455}{6} \, a^{12} b^{3} x^{6} + \frac{315}{17} \, a^{13} b^{2} x^{\frac{17}{3}} + \frac{45}{16} \, a^{14} b x^{\frac{16}{3}} + \frac{1}{5} \, a^{15} x^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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