Optimal. Leaf size=217 \[ -\frac {a^{15}}{10 x^{10}}-\frac {45 a^{14} b}{29 x^{29/3}}-\frac {45 a^{13} b^2}{4 x^{28/3}}-\frac {455 a^{12} b^3}{9 x^9}-\frac {315 a^{11} b^4}{2 x^{26/3}}-\frac {9009 a^{10} b^5}{25 x^{25/3}}-\frac {5005 a^9 b^6}{8 x^8}-\frac {19305 a^8 b^7}{23 x^{23/3}}-\frac {1755 a^7 b^8}{2 x^{22/3}}-\frac {715 a^6 b^9}{x^7}-\frac {9009 a^5 b^{10}}{20 x^{20/3}}-\frac {4095 a^4 b^{11}}{19 x^{19/3}}-\frac {455 a^3 b^{12}}{6 x^6}-\frac {315 a^2 b^{13}}{17 x^{17/3}}-\frac {45 a b^{14}}{16 x^{16/3}}-\frac {b^{15}}{5 x^5} \]
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Rubi [A]
time = 0.08, 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 = {272, 45}
\begin {gather*} -\frac {a^{15}}{10 x^{10}}-\frac {45 a^{14} b}{29 x^{29/3}}-\frac {45 a^{13} b^2}{4 x^{28/3}}-\frac {455 a^{12} b^3}{9 x^9}-\frac {315 a^{11} b^4}{2 x^{26/3}}-\frac {9009 a^{10} b^5}{25 x^{25/3}}-\frac {5005 a^9 b^6}{8 x^8}-\frac {19305 a^8 b^7}{23 x^{23/3}}-\frac {1755 a^7 b^8}{2 x^{22/3}}-\frac {715 a^6 b^9}{x^7}-\frac {9009 a^5 b^{10}}{20 x^{20/3}}-\frac {4095 a^4 b^{11}}{19 x^{19/3}}-\frac {455 a^3 b^{12}}{6 x^6}-\frac {315 a^2 b^{13}}{17 x^{17/3}}-\frac {45 a b^{14}}{16 x^{16/3}}-\frac {b^{15}}{5 x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {\left (a+b \sqrt [3]{x}\right )^{15}}{x^{11}} \, dx &=3 \text {Subst}\left (\int \frac {(a+b x)^{15}}{x^{31}} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \text {Subst}\left (\int \left (\frac {a^{15}}{x^{31}}+\frac {15 a^{14} b}{x^{30}}+\frac {105 a^{13} b^2}{x^{29}}+\frac {455 a^{12} b^3}{x^{28}}+\frac {1365 a^{11} b^4}{x^{27}}+\frac {3003 a^{10} b^5}{x^{26}}+\frac {5005 a^9 b^6}{x^{25}}+\frac {6435 a^8 b^7}{x^{24}}+\frac {6435 a^7 b^8}{x^{23}}+\frac {5005 a^6 b^9}{x^{22}}+\frac {3003 a^5 b^{10}}{x^{21}}+\frac {1365 a^4 b^{11}}{x^{20}}+\frac {455 a^3 b^{12}}{x^{19}}+\frac {105 a^2 b^{13}}{x^{18}}+\frac {15 a b^{14}}{x^{17}}+\frac {b^{15}}{x^{16}}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {a^{15}}{10 x^{10}}-\frac {45 a^{14} b}{29 x^{29/3}}-\frac {45 a^{13} b^2}{4 x^{28/3}}-\frac {455 a^{12} b^3}{9 x^9}-\frac {315 a^{11} b^4}{2 x^{26/3}}-\frac {9009 a^{10} b^5}{25 x^{25/3}}-\frac {5005 a^9 b^6}{8 x^8}-\frac {19305 a^8 b^7}{23 x^{23/3}}-\frac {1755 a^7 b^8}{2 x^{22/3}}-\frac {715 a^6 b^9}{x^7}-\frac {9009 a^5 b^{10}}{20 x^{20/3}}-\frac {4095 a^4 b^{11}}{19 x^{19/3}}-\frac {455 a^3 b^{12}}{6 x^6}-\frac {315 a^2 b^{13}}{17 x^{17/3}}-\frac {45 a b^{14}}{16 x^{16/3}}-\frac {b^{15}}{5 x^5}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 189, normalized size = 0.87 \begin {gather*} \frac {-77558760 a^{15}-1203498000 a^{14} b \sqrt [3]{x}-8725360500 a^{13} b^2 x^{2/3}-39210262000 a^{12} b^3 x-122155047000 a^{11} b^4 x^{4/3}-279490747536 a^{10} b^5 x^{5/3}-485226992250 a^9 b^6 x^2-650987766000 a^8 b^7 x^{7/3}-680578119000 a^7 b^8 x^{8/3}-554545134000 a^6 b^9 x^3-349363434420 a^5 b^{10} x^{10/3}-167159538000 a^4 b^{11} x^{11/3}-58815393000 a^3 b^{12} x^4-14371182000 a^2 b^{13} x^{13/3}-2181340125 a b^{14} x^{14/3}-155117520 b^{15} x^5}{775587600 x^{10}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 168, normalized size = 0.77
method | result | size |
derivativedivides | \(-\frac {a^{15}}{10 x^{10}}-\frac {45 a^{14} b}{29 x^{\frac {29}{3}}}-\frac {45 a^{13} b^{2}}{4 x^{\frac {28}{3}}}-\frac {455 a^{12} b^{3}}{9 x^{9}}-\frac {315 a^{11} b^{4}}{2 x^{\frac {26}{3}}}-\frac {9009 a^{10} b^{5}}{25 x^{\frac {25}{3}}}-\frac {5005 a^{9} b^{6}}{8 x^{8}}-\frac {19305 a^{8} b^{7}}{23 x^{\frac {23}{3}}}-\frac {1755 a^{7} b^{8}}{2 x^{\frac {22}{3}}}-\frac {715 a^{6} b^{9}}{x^{7}}-\frac {9009 a^{5} b^{10}}{20 x^{\frac {20}{3}}}-\frac {4095 a^{4} b^{11}}{19 x^{\frac {19}{3}}}-\frac {455 a^{3} b^{12}}{6 x^{6}}-\frac {315 a^{2} b^{13}}{17 x^{\frac {17}{3}}}-\frac {45 a \,b^{14}}{16 x^{\frac {16}{3}}}-\frac {b^{15}}{5 x^{5}}\) | \(168\) |
default | \(-\frac {a^{15}}{10 x^{10}}-\frac {45 a^{14} b}{29 x^{\frac {29}{3}}}-\frac {45 a^{13} b^{2}}{4 x^{\frac {28}{3}}}-\frac {455 a^{12} b^{3}}{9 x^{9}}-\frac {315 a^{11} b^{4}}{2 x^{\frac {26}{3}}}-\frac {9009 a^{10} b^{5}}{25 x^{\frac {25}{3}}}-\frac {5005 a^{9} b^{6}}{8 x^{8}}-\frac {19305 a^{8} b^{7}}{23 x^{\frac {23}{3}}}-\frac {1755 a^{7} b^{8}}{2 x^{\frac {22}{3}}}-\frac {715 a^{6} b^{9}}{x^{7}}-\frac {9009 a^{5} b^{10}}{20 x^{\frac {20}{3}}}-\frac {4095 a^{4} b^{11}}{19 x^{\frac {19}{3}}}-\frac {455 a^{3} b^{12}}{6 x^{6}}-\frac {315 a^{2} b^{13}}{17 x^{\frac {17}{3}}}-\frac {45 a \,b^{14}}{16 x^{\frac {16}{3}}}-\frac {b^{15}}{5 x^{5}}\) | \(168\) |
trager | \(\frac {\left (x -1\right ) \left (36 a^{15} x^{9}+18200 a^{12} b^{3} x^{9}+225225 a^{9} b^{6} x^{9}+257400 a^{6} b^{9} x^{9}+27300 a^{3} b^{12} x^{9}+72 b^{15} x^{9}+36 a^{15} x^{8}+18200 a^{12} b^{3} x^{8}+225225 a^{9} b^{6} x^{8}+257400 a^{6} b^{9} x^{8}+27300 a^{3} b^{12} x^{8}+72 b^{15} x^{8}+36 a^{15} x^{7}+18200 a^{12} b^{3} x^{7}+225225 a^{9} b^{6} x^{7}+257400 a^{6} b^{9} x^{7}+27300 a^{3} b^{12} x^{7}+72 b^{15} x^{7}+36 a^{15} x^{6}+18200 a^{12} b^{3} x^{6}+225225 a^{9} b^{6} x^{6}+257400 a^{6} b^{9} x^{6}+27300 a^{3} b^{12} x^{6}+72 b^{15} x^{6}+36 a^{15} x^{5}+18200 a^{12} b^{3} x^{5}+225225 a^{9} b^{6} x^{5}+257400 a^{6} b^{9} x^{5}+27300 a^{3} b^{12} x^{5}+72 b^{15} x^{5}+36 a^{15} x^{4}+18200 a^{12} b^{3} x^{4}+225225 a^{9} b^{6} x^{4}+257400 a^{6} b^{9} x^{4}+27300 a^{3} b^{12} x^{4}+36 a^{15} x^{3}+18200 a^{12} b^{3} x^{3}+225225 a^{9} b^{6} x^{3}+257400 a^{6} b^{9} x^{3}+36 a^{15} x^{2}+18200 a^{12} b^{3} x^{2}+225225 a^{9} b^{6} x^{2}+36 a^{15} x +18200 a^{12} b^{3} x +36 a^{15}\right )}{360 x^{10}}-\frac {9 \left (466900 b^{12} x^{4}+11350339 a^{3} b^{9} x^{3}+21149700 a^{6} b^{6} x^{2}+3968650 a^{9} b^{3} x +39100 a^{12}\right ) a^{2} b}{226780 x^{\frac {29}{3}}}-\frac {9 \left (2375 b^{12} x^{4}+182000 a^{3} b^{9} x^{3}+741000 a^{6} b^{6} x^{2}+304304 a^{9} b^{3} x +9500 a^{12}\right ) a \,b^{2}}{7600 x^{\frac {28}{3}}}\) | \(562\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 167, normalized size = 0.77 \begin {gather*} -\frac {155117520 \, b^{15} x^{5} + 2181340125 \, a b^{14} x^{\frac {14}{3}} + 14371182000 \, a^{2} b^{13} x^{\frac {13}{3}} + 58815393000 \, a^{3} b^{12} x^{4} + 167159538000 \, a^{4} b^{11} x^{\frac {11}{3}} + 349363434420 \, a^{5} b^{10} x^{\frac {10}{3}} + 554545134000 \, a^{6} b^{9} x^{3} + 680578119000 \, a^{7} b^{8} x^{\frac {8}{3}} + 650987766000 \, a^{8} b^{7} x^{\frac {7}{3}} + 485226992250 \, a^{9} b^{6} x^{2} + 279490747536 \, a^{10} b^{5} x^{\frac {5}{3}} + 122155047000 \, a^{11} b^{4} x^{\frac {4}{3}} + 39210262000 \, a^{12} b^{3} x + 8725360500 \, a^{13} b^{2} x^{\frac {2}{3}} + 1203498000 \, a^{14} b x^{\frac {1}{3}} + 77558760 \, a^{15}}{775587600 \, x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 169, normalized size = 0.78 \begin {gather*} -\frac {155117520 \, b^{15} x^{5} + 58815393000 \, a^{3} b^{12} x^{4} + 554545134000 \, a^{6} b^{9} x^{3} + 485226992250 \, a^{9} b^{6} x^{2} + 39210262000 \, a^{12} b^{3} x + 77558760 \, a^{15} + 918459 \, {\left (2375 \, a b^{14} x^{4} + 182000 \, a^{4} b^{11} x^{3} + 741000 \, a^{7} b^{8} x^{2} + 304304 \, a^{10} b^{5} x + 9500 \, a^{13} b^{2}\right )} x^{\frac {2}{3}} + 30780 \, {\left (466900 \, a^{2} b^{13} x^{4} + 11350339 \, a^{5} b^{10} x^{3} + 21149700 \, a^{8} b^{7} x^{2} + 3968650 \, a^{11} b^{4} x + 39100 \, a^{14} b\right )} x^{\frac {1}{3}}}{775587600 \, x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.89, size = 219, normalized size = 1.01 \begin {gather*} - \frac {a^{15}}{10 x^{10}} - \frac {45 a^{14} b}{29 x^{\frac {29}{3}}} - \frac {45 a^{13} b^{2}}{4 x^{\frac {28}{3}}} - \frac {455 a^{12} b^{3}}{9 x^{9}} - \frac {315 a^{11} b^{4}}{2 x^{\frac {26}{3}}} - \frac {9009 a^{10} b^{5}}{25 x^{\frac {25}{3}}} - \frac {5005 a^{9} b^{6}}{8 x^{8}} - \frac {19305 a^{8} b^{7}}{23 x^{\frac {23}{3}}} - \frac {1755 a^{7} b^{8}}{2 x^{\frac {22}{3}}} - \frac {715 a^{6} b^{9}}{x^{7}} - \frac {9009 a^{5} b^{10}}{20 x^{\frac {20}{3}}} - \frac {4095 a^{4} b^{11}}{19 x^{\frac {19}{3}}} - \frac {455 a^{3} b^{12}}{6 x^{6}} - \frac {315 a^{2} b^{13}}{17 x^{\frac {17}{3}}} - \frac {45 a b^{14}}{16 x^{\frac {16}{3}}} - \frac {b^{15}}{5 x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.30, size = 167, normalized size = 0.77 \begin {gather*} -\frac {155117520 \, b^{15} x^{5} + 2181340125 \, a b^{14} x^{\frac {14}{3}} + 14371182000 \, a^{2} b^{13} x^{\frac {13}{3}} + 58815393000 \, a^{3} b^{12} x^{4} + 167159538000 \, a^{4} b^{11} x^{\frac {11}{3}} + 349363434420 \, a^{5} b^{10} x^{\frac {10}{3}} + 554545134000 \, a^{6} b^{9} x^{3} + 680578119000 \, a^{7} b^{8} x^{\frac {8}{3}} + 650987766000 \, a^{8} b^{7} x^{\frac {7}{3}} + 485226992250 \, a^{9} b^{6} x^{2} + 279490747536 \, a^{10} b^{5} x^{\frac {5}{3}} + 122155047000 \, a^{11} b^{4} x^{\frac {4}{3}} + 39210262000 \, a^{12} b^{3} x + 8725360500 \, a^{13} b^{2} x^{\frac {2}{3}} + 1203498000 \, a^{14} b x^{\frac {1}{3}} + 77558760 \, a^{15}}{775587600 \, x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.19, size = 167, normalized size = 0.77 \begin {gather*} -\frac {\frac {a^{15}}{10}+\frac {b^{15}\,x^5}{5}+\frac {455\,a^{12}\,b^3\,x}{9}+\frac {45\,a^{14}\,b\,x^{1/3}}{29}+\frac {45\,a\,b^{14}\,x^{14/3}}{16}+\frac {5005\,a^9\,b^6\,x^2}{8}+715\,a^6\,b^9\,x^3+\frac {455\,a^3\,b^{12}\,x^4}{6}+\frac {45\,a^{13}\,b^2\,x^{2/3}}{4}+\frac {315\,a^{11}\,b^4\,x^{4/3}}{2}+\frac {9009\,a^{10}\,b^5\,x^{5/3}}{25}+\frac {19305\,a^8\,b^7\,x^{7/3}}{23}+\frac {1755\,a^7\,b^8\,x^{8/3}}{2}+\frac {9009\,a^5\,b^{10}\,x^{10/3}}{20}+\frac {4095\,a^4\,b^{11}\,x^{11/3}}{19}+\frac {315\,a^2\,b^{13}\,x^{13/3}}{17}}{x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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