Optimal. Leaf size=224 \[ -\frac{b^8 \left (a+b \sqrt [3]{x}\right )^{16}}{3922512 a^9 x^{16/3}}+\frac{b^7 \left (a+b \sqrt [3]{x}\right )^{16}}{245157 a^8 x^{17/3}}-\frac{b^6 \left (a+b \sqrt [3]{x}\right )^{16}}{28842 a^7 x^6}+\frac{b^5 \left (a+b \sqrt [3]{x}\right )^{16}}{4807 a^6 x^{19/3}}-\frac{b^4 \left (a+b \sqrt [3]{x}\right )^{16}}{1012 a^5 x^{20/3}}+\frac{b^3 \left (a+b \sqrt [3]{x}\right )^{16}}{253 a^4 x^7}-\frac{7 b^2 \left (a+b \sqrt [3]{x}\right )^{16}}{506 a^3 x^{22/3}}+\frac{b \left (a+b \sqrt [3]{x}\right )^{16}}{23 a^2 x^{23/3}}-\frac{\left (a+b \sqrt [3]{x}\right )^{16}}{8 a x^8} \]
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Rubi [A] time = 0.110531, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {266, 45, 37} \[ -\frac{b^8 \left (a+b \sqrt [3]{x}\right )^{16}}{3922512 a^9 x^{16/3}}+\frac{b^7 \left (a+b \sqrt [3]{x}\right )^{16}}{245157 a^8 x^{17/3}}-\frac{b^6 \left (a+b \sqrt [3]{x}\right )^{16}}{28842 a^7 x^6}+\frac{b^5 \left (a+b \sqrt [3]{x}\right )^{16}}{4807 a^6 x^{19/3}}-\frac{b^4 \left (a+b \sqrt [3]{x}\right )^{16}}{1012 a^5 x^{20/3}}+\frac{b^3 \left (a+b \sqrt [3]{x}\right )^{16}}{253 a^4 x^7}-\frac{7 b^2 \left (a+b \sqrt [3]{x}\right )^{16}}{506 a^3 x^{22/3}}+\frac{b \left (a+b \sqrt [3]{x}\right )^{16}}{23 a^2 x^{23/3}}-\frac{\left (a+b \sqrt [3]{x}\right )^{16}}{8 a x^8} \]
Antiderivative was successfully verified.
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Rule 266
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{\left (a+b \sqrt [3]{x}\right )^{15}}{x^9} \, dx &=3 \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{25}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{\left (a+b \sqrt [3]{x}\right )^{16}}{8 a x^8}-\frac{b \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{24}} \, dx,x,\sqrt [3]{x}\right )}{a}\\ &=-\frac{\left (a+b \sqrt [3]{x}\right )^{16}}{8 a x^8}+\frac{b \left (a+b \sqrt [3]{x}\right )^{16}}{23 a^2 x^{23/3}}+\frac{\left (7 b^2\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{23}} \, dx,x,\sqrt [3]{x}\right )}{23 a^2}\\ &=-\frac{\left (a+b \sqrt [3]{x}\right )^{16}}{8 a x^8}+\frac{b \left (a+b \sqrt [3]{x}\right )^{16}}{23 a^2 x^{23/3}}-\frac{7 b^2 \left (a+b \sqrt [3]{x}\right )^{16}}{506 a^3 x^{22/3}}-\frac{\left (21 b^3\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{22}} \, dx,x,\sqrt [3]{x}\right )}{253 a^3}\\ &=-\frac{\left (a+b \sqrt [3]{x}\right )^{16}}{8 a x^8}+\frac{b \left (a+b \sqrt [3]{x}\right )^{16}}{23 a^2 x^{23/3}}-\frac{7 b^2 \left (a+b \sqrt [3]{x}\right )^{16}}{506 a^3 x^{22/3}}+\frac{b^3 \left (a+b \sqrt [3]{x}\right )^{16}}{253 a^4 x^7}+\frac{\left (5 b^4\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{21}} \, dx,x,\sqrt [3]{x}\right )}{253 a^4}\\ &=-\frac{\left (a+b \sqrt [3]{x}\right )^{16}}{8 a x^8}+\frac{b \left (a+b \sqrt [3]{x}\right )^{16}}{23 a^2 x^{23/3}}-\frac{7 b^2 \left (a+b \sqrt [3]{x}\right )^{16}}{506 a^3 x^{22/3}}+\frac{b^3 \left (a+b \sqrt [3]{x}\right )^{16}}{253 a^4 x^7}-\frac{b^4 \left (a+b \sqrt [3]{x}\right )^{16}}{1012 a^5 x^{20/3}}-\frac{b^5 \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{20}} \, dx,x,\sqrt [3]{x}\right )}{253 a^5}\\ &=-\frac{\left (a+b \sqrt [3]{x}\right )^{16}}{8 a x^8}+\frac{b \left (a+b \sqrt [3]{x}\right )^{16}}{23 a^2 x^{23/3}}-\frac{7 b^2 \left (a+b \sqrt [3]{x}\right )^{16}}{506 a^3 x^{22/3}}+\frac{b^3 \left (a+b \sqrt [3]{x}\right )^{16}}{253 a^4 x^7}-\frac{b^4 \left (a+b \sqrt [3]{x}\right )^{16}}{1012 a^5 x^{20/3}}+\frac{b^5 \left (a+b \sqrt [3]{x}\right )^{16}}{4807 a^6 x^{19/3}}+\frac{\left (3 b^6\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{19}} \, dx,x,\sqrt [3]{x}\right )}{4807 a^6}\\ &=-\frac{\left (a+b \sqrt [3]{x}\right )^{16}}{8 a x^8}+\frac{b \left (a+b \sqrt [3]{x}\right )^{16}}{23 a^2 x^{23/3}}-\frac{7 b^2 \left (a+b \sqrt [3]{x}\right )^{16}}{506 a^3 x^{22/3}}+\frac{b^3 \left (a+b \sqrt [3]{x}\right )^{16}}{253 a^4 x^7}-\frac{b^4 \left (a+b \sqrt [3]{x}\right )^{16}}{1012 a^5 x^{20/3}}+\frac{b^5 \left (a+b \sqrt [3]{x}\right )^{16}}{4807 a^6 x^{19/3}}-\frac{b^6 \left (a+b \sqrt [3]{x}\right )^{16}}{28842 a^7 x^6}-\frac{b^7 \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{18}} \, dx,x,\sqrt [3]{x}\right )}{14421 a^7}\\ &=-\frac{\left (a+b \sqrt [3]{x}\right )^{16}}{8 a x^8}+\frac{b \left (a+b \sqrt [3]{x}\right )^{16}}{23 a^2 x^{23/3}}-\frac{7 b^2 \left (a+b \sqrt [3]{x}\right )^{16}}{506 a^3 x^{22/3}}+\frac{b^3 \left (a+b \sqrt [3]{x}\right )^{16}}{253 a^4 x^7}-\frac{b^4 \left (a+b \sqrt [3]{x}\right )^{16}}{1012 a^5 x^{20/3}}+\frac{b^5 \left (a+b \sqrt [3]{x}\right )^{16}}{4807 a^6 x^{19/3}}-\frac{b^6 \left (a+b \sqrt [3]{x}\right )^{16}}{28842 a^7 x^6}+\frac{b^7 \left (a+b \sqrt [3]{x}\right )^{16}}{245157 a^8 x^{17/3}}+\frac{b^8 \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{17}} \, dx,x,\sqrt [3]{x}\right )}{245157 a^8}\\ &=-\frac{\left (a+b \sqrt [3]{x}\right )^{16}}{8 a x^8}+\frac{b \left (a+b \sqrt [3]{x}\right )^{16}}{23 a^2 x^{23/3}}-\frac{7 b^2 \left (a+b \sqrt [3]{x}\right )^{16}}{506 a^3 x^{22/3}}+\frac{b^3 \left (a+b \sqrt [3]{x}\right )^{16}}{253 a^4 x^7}-\frac{b^4 \left (a+b \sqrt [3]{x}\right )^{16}}{1012 a^5 x^{20/3}}+\frac{b^5 \left (a+b \sqrt [3]{x}\right )^{16}}{4807 a^6 x^{19/3}}-\frac{b^6 \left (a+b \sqrt [3]{x}\right )^{16}}{28842 a^7 x^6}+\frac{b^7 \left (a+b \sqrt [3]{x}\right )^{16}}{245157 a^8 x^{17/3}}-\frac{b^8 \left (a+b \sqrt [3]{x}\right )^{16}}{3922512 a^9 x^{16/3}}\\ \end{align*}
Mathematica [A] time = 0.0440574, size = 213, normalized size = 0.95 \[ -\frac{315 a^{13} b^2}{22 x^{22/3}}-\frac{65 a^{12} b^3}{x^7}-\frac{819 a^{11} b^4}{4 x^{20/3}}-\frac{9009 a^{10} b^5}{19 x^{19/3}}-\frac{5005 a^9 b^6}{6 x^6}-\frac{19305 a^8 b^7}{17 x^{17/3}}-\frac{19305 a^7 b^8}{16 x^{16/3}}-\frac{1001 a^6 b^9}{x^5}-\frac{1287 a^5 b^{10}}{2 x^{14/3}}-\frac{315 a^4 b^{11}}{x^{13/3}}-\frac{455 a^3 b^{12}}{4 x^4}-\frac{315 a^2 b^{13}}{11 x^{11/3}}-\frac{45 a^{14} b}{23 x^{23/3}}-\frac{a^{15}}{8 x^8}-\frac{9 a b^{14}}{2 x^{10/3}}-\frac{b^{15}}{3 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 168, normalized size = 0.8 \begin{align*} -{\frac{315\,{a}^{13}{b}^{2}}{22}{x}^{-{\frac{22}{3}}}}-{\frac{{a}^{15}}{8\,{x}^{8}}}-65\,{\frac{{a}^{12}{b}^{3}}{{x}^{7}}}-{\frac{455\,{a}^{3}{b}^{12}}{4\,{x}^{4}}}-1001\,{\frac{{a}^{6}{b}^{9}}{{x}^{5}}}-{\frac{315\,{a}^{2}{b}^{13}}{11}{x}^{-{\frac{11}{3}}}}-{\frac{819\,{a}^{11}{b}^{4}}{4}{x}^{-{\frac{20}{3}}}}-{\frac{19305\,{a}^{8}{b}^{7}}{17}{x}^{-{\frac{17}{3}}}}-{\frac{45\,{a}^{14}b}{23}{x}^{-{\frac{23}{3}}}}-{\frac{9009\,{a}^{10}{b}^{5}}{19}{x}^{-{\frac{19}{3}}}}-{\frac{1287\,{a}^{5}{b}^{10}}{2}{x}^{-{\frac{14}{3}}}}-{\frac{5005\,{a}^{9}{b}^{6}}{6\,{x}^{6}}}-{\frac{9\,a{b}^{14}}{2}{x}^{-{\frac{10}{3}}}}-315\,{\frac{{a}^{4}{b}^{11}}{{x}^{13/3}}}-{\frac{{b}^{15}}{3\,{x}^{3}}}-{\frac{19305\,{a}^{7}{b}^{8}}{16}{x}^{-{\frac{16}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.967952, size = 225, normalized size = 1. \begin{align*} -\frac{1307504 \, b^{15} x^{5} + 17651304 \, a b^{14} x^{\frac{14}{3}} + 112326480 \, a^{2} b^{13} x^{\frac{13}{3}} + 446185740 \, a^{3} b^{12} x^{4} + 1235591280 \, a^{4} b^{11} x^{\frac{11}{3}} + 2524136472 \, a^{5} b^{10} x^{\frac{10}{3}} + 3926434512 \, a^{6} b^{9} x^{3} + 4732755885 \, a^{7} b^{8} x^{\frac{8}{3}} + 4454358480 \, a^{8} b^{7} x^{\frac{7}{3}} + 3272028760 \, a^{9} b^{6} x^{2} + 1859890032 \, a^{10} b^{5} x^{\frac{5}{3}} + 803134332 \, a^{11} b^{4} x^{\frac{4}{3}} + 254963280 \, a^{12} b^{3} x + 56163240 \, a^{13} b^{2} x^{\frac{2}{3}} + 7674480 \, a^{14} b x^{\frac{1}{3}} + 490314 \, a^{15}}{3922512 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71947, size = 506, normalized size = 2.26 \begin{align*} -\frac{1307504 \, b^{15} x^{5} + 446185740 \, a^{3} b^{12} x^{4} + 3926434512 \, a^{6} b^{9} x^{3} + 3272028760 \, a^{9} b^{6} x^{2} + 254963280 \, a^{12} b^{3} x + 490314 \, a^{15} + 10557 \,{\left (1672 \, a b^{14} x^{4} + 117040 \, a^{4} b^{11} x^{3} + 448305 \, a^{7} b^{8} x^{2} + 176176 \, a^{10} b^{5} x + 5320 \, a^{13} b^{2}\right )} x^{\frac{2}{3}} + 2052 \,{\left (54740 \, a^{2} b^{13} x^{4} + 1230086 \, a^{5} b^{10} x^{3} + 2170740 \, a^{8} b^{7} x^{2} + 391391 \, a^{11} b^{4} x + 3740 \, a^{14} b\right )} x^{\frac{1}{3}}}{3922512 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 32.2652, size = 216, normalized size = 0.96 \begin{align*} - \frac{a^{15}}{8 x^{8}} - \frac{45 a^{14} b}{23 x^{\frac{23}{3}}} - \frac{315 a^{13} b^{2}}{22 x^{\frac{22}{3}}} - \frac{65 a^{12} b^{3}}{x^{7}} - \frac{819 a^{11} b^{4}}{4 x^{\frac{20}{3}}} - \frac{9009 a^{10} b^{5}}{19 x^{\frac{19}{3}}} - \frac{5005 a^{9} b^{6}}{6 x^{6}} - \frac{19305 a^{8} b^{7}}{17 x^{\frac{17}{3}}} - \frac{19305 a^{7} b^{8}}{16 x^{\frac{16}{3}}} - \frac{1001 a^{6} b^{9}}{x^{5}} - \frac{1287 a^{5} b^{10}}{2 x^{\frac{14}{3}}} - \frac{315 a^{4} b^{11}}{x^{\frac{13}{3}}} - \frac{455 a^{3} b^{12}}{4 x^{4}} - \frac{315 a^{2} b^{13}}{11 x^{\frac{11}{3}}} - \frac{9 a b^{14}}{2 x^{\frac{10}{3}}} - \frac{b^{15}}{3 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19536, size = 225, normalized size = 1. \begin{align*} -\frac{1307504 \, b^{15} x^{5} + 17651304 \, a b^{14} x^{\frac{14}{3}} + 112326480 \, a^{2} b^{13} x^{\frac{13}{3}} + 446185740 \, a^{3} b^{12} x^{4} + 1235591280 \, a^{4} b^{11} x^{\frac{11}{3}} + 2524136472 \, a^{5} b^{10} x^{\frac{10}{3}} + 3926434512 \, a^{6} b^{9} x^{3} + 4732755885 \, a^{7} b^{8} x^{\frac{8}{3}} + 4454358480 \, a^{8} b^{7} x^{\frac{7}{3}} + 3272028760 \, a^{9} b^{6} x^{2} + 1859890032 \, a^{10} b^{5} x^{\frac{5}{3}} + 803134332 \, a^{11} b^{4} x^{\frac{4}{3}} + 254963280 \, a^{12} b^{3} x + 56163240 \, a^{13} b^{2} x^{\frac{2}{3}} + 7674480 \, a^{14} b x^{\frac{1}{3}} + 490314 \, a^{15}}{3922512 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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