Optimal. Leaf size=72 \[ -\frac{b^2 \left (a+b \sqrt [3]{x}\right )^{16}}{816 a^3 x^{16/3}}+\frac{b \left (a+b \sqrt [3]{x}\right )^{16}}{51 a^2 x^{17/3}}-\frac{\left (a+b \sqrt [3]{x}\right )^{16}}{6 a x^6} \]
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Rubi [A] time = 0.0232657, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {266, 45, 37} \[ -\frac{b^2 \left (a+b \sqrt [3]{x}\right )^{16}}{816 a^3 x^{16/3}}+\frac{b \left (a+b \sqrt [3]{x}\right )^{16}}{51 a^2 x^{17/3}}-\frac{\left (a+b \sqrt [3]{x}\right )^{16}}{6 a x^6} \]
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^7} \, dx &=3 \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{19}} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{\left (a+b \sqrt [3]{x}\right )^{16}}{6 a x^6}-\frac{b \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{18}} \, dx,x,\sqrt [3]{x}\right )}{3 a}\\ &=-\frac{\left (a+b \sqrt [3]{x}\right )^{16}}{6 a x^6}+\frac{b \left (a+b \sqrt [3]{x}\right )^{16}}{51 a^2 x^{17/3}}+\frac{b^2 \operatorname{Subst}\left (\int \frac{(a+b x)^{15}}{x^{17}} \, dx,x,\sqrt [3]{x}\right )}{51 a^2}\\ &=-\frac{\left (a+b \sqrt [3]{x}\right )^{16}}{6 a x^6}+\frac{b \left (a+b \sqrt [3]{x}\right )^{16}}{51 a^2 x^{17/3}}-\frac{b^2 \left (a+b \sqrt [3]{x}\right )^{16}}{816 a^3 x^{16/3}}\\ \end{align*}
Mathematica [A] time = 0.0105986, size = 45, normalized size = 0.62 \[ -\frac{\left (a+b \sqrt [3]{x}\right )^{16} \left (136 a^2-16 a b \sqrt [3]{x}+b^2 x^{2/3}\right )}{816 a^3 x^6} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.01, size = 168, normalized size = 2.3 \begin{align*} -{\frac{{b}^{15}}{x}}-63\,{\frac{{a}^{2}{b}^{13}}{{x}^{5/3}}}-{\frac{5005\,{a}^{9}{b}^{6}}{4\,{x}^{4}}}-91\,{\frac{{a}^{12}{b}^{3}}{{x}^{5}}}-1755\,{\frac{{a}^{8}{b}^{7}}{{x}^{11/3}}}-{\frac{45\,{a}^{14}b}{17}{x}^{-{\frac{17}{3}}}}-{\frac{585\,{a}^{11}{b}^{4}}{2}{x}^{-{\frac{14}{3}}}}-{\frac{45\,a{b}^{14}}{4}{x}^{-{\frac{4}{3}}}}-{\frac{{a}^{15}}{6\,{x}^{6}}}-{\frac{3861\,{a}^{7}{b}^{8}}{2}{x}^{-{\frac{10}{3}}}}-{\frac{9009\,{a}^{5}{b}^{10}}{8}{x}^{-{\frac{8}{3}}}}-693\,{\frac{{a}^{10}{b}^{5}}{{x}^{13/3}}}-{\frac{455\,{a}^{3}{b}^{12}}{2\,{x}^{2}}}-585\,{\frac{{a}^{4}{b}^{11}}{{x}^{7/3}}}-{\frac{5005\,{a}^{6}{b}^{9}}{3\,{x}^{3}}}-{\frac{315\,{a}^{13}{b}^{2}}{16}{x}^{-{\frac{16}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.976921, size = 225, normalized size = 3.12 \begin{align*} -\frac{816 \, b^{15} x^{5} + 9180 \, a b^{14} x^{\frac{14}{3}} + 51408 \, a^{2} b^{13} x^{\frac{13}{3}} + 185640 \, a^{3} b^{12} x^{4} + 477360 \, a^{4} b^{11} x^{\frac{11}{3}} + 918918 \, a^{5} b^{10} x^{\frac{10}{3}} + 1361360 \, a^{6} b^{9} x^{3} + 1575288 \, a^{7} b^{8} x^{\frac{8}{3}} + 1432080 \, a^{8} b^{7} x^{\frac{7}{3}} + 1021020 \, a^{9} b^{6} x^{2} + 565488 \, a^{10} b^{5} x^{\frac{5}{3}} + 238680 \, a^{11} b^{4} x^{\frac{4}{3}} + 74256 \, a^{12} b^{3} x + 16065 \, a^{13} b^{2} x^{\frac{2}{3}} + 2160 \, a^{14} b x^{\frac{1}{3}} + 136 \, a^{15}}{816 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.49467, size = 441, normalized size = 6.12 \begin{align*} -\frac{816 \, b^{15} x^{5} + 185640 \, a^{3} b^{12} x^{4} + 1361360 \, a^{6} b^{9} x^{3} + 1021020 \, a^{9} b^{6} x^{2} + 74256 \, a^{12} b^{3} x + 136 \, a^{15} + 459 \,{\left (20 \, a b^{14} x^{4} + 1040 \, a^{4} b^{11} x^{3} + 3432 \, a^{7} b^{8} x^{2} + 1232 \, a^{10} b^{5} x + 35 \, a^{13} b^{2}\right )} x^{\frac{2}{3}} + 54 \,{\left (952 \, a^{2} b^{13} x^{4} + 17017 \, a^{5} b^{10} x^{3} + 26520 \, a^{8} b^{7} x^{2} + 4420 \, a^{11} b^{4} x + 40 \, a^{14} b\right )} x^{\frac{1}{3}}}{816 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 13.4109, size = 209, normalized size = 2.9 \begin{align*} - \frac{a^{15}}{6 x^{6}} - \frac{45 a^{14} b}{17 x^{\frac{17}{3}}} - \frac{315 a^{13} b^{2}}{16 x^{\frac{16}{3}}} - \frac{91 a^{12} b^{3}}{x^{5}} - \frac{585 a^{11} b^{4}}{2 x^{\frac{14}{3}}} - \frac{693 a^{10} b^{5}}{x^{\frac{13}{3}}} - \frac{5005 a^{9} b^{6}}{4 x^{4}} - \frac{1755 a^{8} b^{7}}{x^{\frac{11}{3}}} - \frac{3861 a^{7} b^{8}}{2 x^{\frac{10}{3}}} - \frac{5005 a^{6} b^{9}}{3 x^{3}} - \frac{9009 a^{5} b^{10}}{8 x^{\frac{8}{3}}} - \frac{585 a^{4} b^{11}}{x^{\frac{7}{3}}} - \frac{455 a^{3} b^{12}}{2 x^{2}} - \frac{63 a^{2} b^{13}}{x^{\frac{5}{3}}} - \frac{45 a b^{14}}{4 x^{\frac{4}{3}}} - \frac{b^{15}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.23157, size = 225, normalized size = 3.12 \begin{align*} -\frac{816 \, b^{15} x^{5} + 9180 \, a b^{14} x^{\frac{14}{3}} + 51408 \, a^{2} b^{13} x^{\frac{13}{3}} + 185640 \, a^{3} b^{12} x^{4} + 477360 \, a^{4} b^{11} x^{\frac{11}{3}} + 918918 \, a^{5} b^{10} x^{\frac{10}{3}} + 1361360 \, a^{6} b^{9} x^{3} + 1575288 \, a^{7} b^{8} x^{\frac{8}{3}} + 1432080 \, a^{8} b^{7} x^{\frac{7}{3}} + 1021020 \, a^{9} b^{6} x^{2} + 565488 \, a^{10} b^{5} x^{\frac{5}{3}} + 238680 \, a^{11} b^{4} x^{\frac{4}{3}} + 74256 \, a^{12} b^{3} x + 16065 \, a^{13} b^{2} x^{\frac{2}{3}} + 2160 \, a^{14} b x^{\frac{1}{3}} + 136 \, a^{15}}{816 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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