Optimal. Leaf size=217 \[ \frac {a^{15} x^5}{5}+\frac {45}{16} a^{14} b x^{16/3}+\frac {315}{17} a^{13} b^2 x^{17/3}+\frac {455}{6} a^{12} b^3 x^6+\frac {4095}{19} a^{11} b^4 x^{19/3}+\frac {9009}{20} a^{10} b^5 x^{20/3}+715 a^9 b^6 x^7+\frac {1755}{2} a^8 b^7 x^{22/3}+\frac {19305}{23} a^7 b^8 x^{23/3}+\frac {5005}{8} a^6 b^9 x^8+\frac {9009}{25} a^5 b^{10} x^{25/3}+\frac {315}{2} a^4 b^{11} x^{26/3}+\frac {455}{9} a^3 b^{12} x^9+\frac {45}{4} a^2 b^{13} x^{28/3}+\frac {45}{29} a b^{14} x^{29/3}+\frac {b^{15} x^{10}}{10} \]
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Rubi [A] time = 0.13, antiderivative size = 217, normalized size of antiderivative = 1.00, 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 43
Rule 266
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*}
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Mathematica [A] time = 0.10, size = 217, normalized size = 1.00 \[ \frac {a^{15} x^5}{5}+\frac {45}{16} a^{14} b x^{16/3}+\frac {315}{17} a^{13} b^2 x^{17/3}+\frac {455}{6} a^{12} b^3 x^6+\frac {4095}{19} a^{11} b^4 x^{19/3}+\frac {9009}{20} a^{10} b^5 x^{20/3}+715 a^9 b^6 x^7+\frac {1755}{2} a^8 b^7 x^{22/3}+\frac {19305}{23} a^7 b^8 x^{23/3}+\frac {5005}{8} a^6 b^9 x^8+\frac {9009}{25} a^5 b^{10} x^{25/3}+\frac {315}{2} a^4 b^{11} x^{26/3}+\frac {455}{9} a^3 b^{12} x^9+\frac {45}{4} a^2 b^{13} x^{28/3}+\frac {45}{29} a b^{14} x^{29/3}+\frac {b^{15} x^{10}}{10} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 179, normalized size = 0.82 \[ \frac {1}{10} \, b^{15} x^{10} + \frac {455}{9} \, a^{3} b^{12} x^{9} + \frac {5005}{8} \, a^{6} b^{9} x^{8} + 715 \, a^{9} b^{6} x^{7} + \frac {455}{6} \, a^{12} b^{3} x^{6} + \frac {1}{5} \, a^{15} x^{5} + \frac {9}{226780} \, {\left (39100 \, a b^{14} x^{9} + 3968650 \, a^{4} b^{11} x^{8} + 21149700 \, a^{7} b^{8} x^{7} + 11350339 \, a^{10} b^{5} x^{6} + 466900 \, a^{13} b^{2} x^{5}\right )} x^{\frac {2}{3}} + \frac {9}{7600} \, {\left (9500 \, a^{2} b^{13} x^{9} + 304304 \, a^{5} b^{10} x^{8} + 741000 \, a^{8} b^{7} x^{7} + 182000 \, a^{11} b^{4} x^{6} + 2375 \, a^{14} b x^{5}\right )} x^{\frac {1}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.46, size = 167, normalized size = 0.77 \[ \frac {1}{10} \, b^{15} x^{10} + \frac {45}{29} \, a b^{14} x^{\frac {29}{3}} + \frac {45}{4} \, a^{2} b^{13} x^{\frac {28}{3}} + \frac {455}{9} \, a^{3} b^{12} x^{9} + \frac {315}{2} \, a^{4} b^{11} x^{\frac {26}{3}} + \frac {9009}{25} \, a^{5} b^{10} x^{\frac {25}{3}} + \frac {5005}{8} \, a^{6} b^{9} x^{8} + \frac {19305}{23} \, a^{7} b^{8} x^{\frac {23}{3}} + \frac {1755}{2} \, a^{8} b^{7} x^{\frac {22}{3}} + 715 \, a^{9} b^{6} x^{7} + \frac {9009}{20} \, a^{10} b^{5} x^{\frac {20}{3}} + \frac {4095}{19} \, a^{11} b^{4} x^{\frac {19}{3}} + \frac {455}{6} \, a^{12} b^{3} x^{6} + \frac {315}{17} \, a^{13} b^{2} x^{\frac {17}{3}} + \frac {45}{16} \, a^{14} b x^{\frac {16}{3}} + \frac {1}{5} \, a^{15} x^{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 168, normalized size = 0.77 \[ \frac {b^{15} x^{10}}{10}+\frac {45 a \,b^{14} x^{\frac {29}{3}}}{29}+\frac {45 a^{2} b^{13} x^{\frac {28}{3}}}{4}+\frac {455 a^{3} b^{12} x^{9}}{9}+\frac {315 a^{4} b^{11} x^{\frac {26}{3}}}{2}+\frac {9009 a^{5} b^{10} x^{\frac {25}{3}}}{25}+\frac {5005 a^{6} b^{9} x^{8}}{8}+\frac {19305 a^{7} b^{8} x^{\frac {23}{3}}}{23}+\frac {1755 a^{8} b^{7} x^{\frac {22}{3}}}{2}+715 a^{9} b^{6} x^{7}+\frac {9009 a^{10} b^{5} x^{\frac {20}{3}}}{20}+\frac {4095 a^{11} b^{4} x^{\frac {19}{3}}}{19}+\frac {455 a^{12} b^{3} x^{6}}{6}+\frac {315 a^{13} b^{2} x^{\frac {17}{3}}}{17}+\frac {45 a^{14} b \,x^{\frac {16}{3}}}{16}+\frac {a^{15} x^{5}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.91, size = 251, normalized size = 1.16 \[ \frac {{\left (b x^{\frac {1}{3}} + a\right )}^{30}}{10 \, b^{15}} - \frac {42 \, {\left (b x^{\frac {1}{3}} + a\right )}^{29} a}{29 \, b^{15}} + \frac {39 \, {\left (b x^{\frac {1}{3}} + a\right )}^{28} a^{2}}{4 \, b^{15}} - \frac {364 \, {\left (b x^{\frac {1}{3}} + a\right )}^{27} a^{3}}{9 \, b^{15}} + \frac {231 \, {\left (b x^{\frac {1}{3}} + a\right )}^{26} a^{4}}{2 \, b^{15}} - \frac {6006 \, {\left (b x^{\frac {1}{3}} + a\right )}^{25} a^{5}}{25 \, b^{15}} + \frac {3003 \, {\left (b x^{\frac {1}{3}} + a\right )}^{24} a^{6}}{8 \, b^{15}} - \frac {10296 \, {\left (b x^{\frac {1}{3}} + a\right )}^{23} a^{7}}{23 \, b^{15}} + \frac {819 \, {\left (b x^{\frac {1}{3}} + a\right )}^{22} a^{8}}{2 \, b^{15}} - \frac {286 \, {\left (b x^{\frac {1}{3}} + a\right )}^{21} a^{9}}{b^{15}} + \frac {3003 \, {\left (b x^{\frac {1}{3}} + a\right )}^{20} a^{10}}{20 \, b^{15}} - \frac {1092 \, {\left (b x^{\frac {1}{3}} + a\right )}^{19} a^{11}}{19 \, b^{15}} + \frac {91 \, {\left (b x^{\frac {1}{3}} + a\right )}^{18} a^{12}}{6 \, b^{15}} - \frac {42 \, {\left (b x^{\frac {1}{3}} + a\right )}^{17} a^{13}}{17 \, b^{15}} + \frac {3 \, {\left (b x^{\frac {1}{3}} + a\right )}^{16} a^{14}}{16 \, b^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 167, normalized size = 0.77 \[ \frac {a^{15}\,x^5}{5}+\frac {b^{15}\,x^{10}}{10}+\frac {45\,a^{14}\,b\,x^{16/3}}{16}+\frac {45\,a\,b^{14}\,x^{29/3}}{29}+\frac {455\,a^{12}\,b^3\,x^6}{6}+715\,a^9\,b^6\,x^7+\frac {5005\,a^6\,b^9\,x^8}{8}+\frac {455\,a^3\,b^{12}\,x^9}{9}+\frac {315\,a^{13}\,b^2\,x^{17/3}}{17}+\frac {4095\,a^{11}\,b^4\,x^{19/3}}{19}+\frac {9009\,a^{10}\,b^5\,x^{20/3}}{20}+\frac {1755\,a^8\,b^7\,x^{22/3}}{2}+\frac {19305\,a^7\,b^8\,x^{23/3}}{23}+\frac {9009\,a^5\,b^{10}\,x^{25/3}}{25}+\frac {315\,a^4\,b^{11}\,x^{26/3}}{2}+\frac {45\,a^2\,b^{13}\,x^{28/3}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.80, size = 218, normalized size = 1.00 \[ \frac {a^{15} x^{5}}{5} + \frac {45 a^{14} b x^{\frac {16}{3}}}{16} + \frac {315 a^{13} b^{2} x^{\frac {17}{3}}}{17} + \frac {455 a^{12} b^{3} x^{6}}{6} + \frac {4095 a^{11} b^{4} x^{\frac {19}{3}}}{19} + \frac {9009 a^{10} b^{5} x^{\frac {20}{3}}}{20} + 715 a^{9} b^{6} x^{7} + \frac {1755 a^{8} b^{7} x^{\frac {22}{3}}}{2} + \frac {19305 a^{7} b^{8} x^{\frac {23}{3}}}{23} + \frac {5005 a^{6} b^{9} x^{8}}{8} + \frac {9009 a^{5} b^{10} x^{\frac {25}{3}}}{25} + \frac {315 a^{4} b^{11} x^{\frac {26}{3}}}{2} + \frac {455 a^{3} b^{12} x^{9}}{9} + \frac {45 a^{2} b^{13} x^{\frac {28}{3}}}{4} + \frac {45 a b^{14} x^{\frac {29}{3}}}{29} + \frac {b^{15} x^{10}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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