Optimal. Leaf size=117 \[ \frac{81 x}{182 a^4 \sqrt [3]{a+b x^3}}+\frac{27 x}{182 a^3 \left (a+b x^3\right )^{4/3}}+\frac{9 x}{91 a^2 \left (a+b x^3\right )^{7/3}}+\frac{x}{13 a \left (a+b x^3\right )^{10/3}}+\frac{11 x}{104 \left (a+b x^3\right )^{13/3}}+\frac{x \left (a-b x^3\right )}{8 \left (a+b x^3\right )^{16/3}} \]
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Rubi [A] time = 0.0442149, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {413, 385, 192, 191} \[ \frac{81 x}{182 a^4 \sqrt [3]{a+b x^3}}+\frac{27 x}{182 a^3 \left (a+b x^3\right )^{4/3}}+\frac{9 x}{91 a^2 \left (a+b x^3\right )^{7/3}}+\frac{x}{13 a \left (a+b x^3\right )^{10/3}}+\frac{11 x}{104 \left (a+b x^3\right )^{13/3}}+\frac{x \left (a-b x^3\right )}{8 \left (a+b x^3\right )^{16/3}} \]
Antiderivative was successfully verified.
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Rule 413
Rule 385
Rule 192
Rule 191
Rubi steps
\begin{align*} \int \frac{\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{19/3}} \, dx &=\frac{x \left (a-b x^3\right )}{8 \left (a+b x^3\right )^{16/3}}+\frac{\int \frac{14 a^2 b-8 a b^2 x^3}{\left (a+b x^3\right )^{16/3}} \, dx}{16 a b}\\ &=\frac{x \left (a-b x^3\right )}{8 \left (a+b x^3\right )^{16/3}}+\frac{11 x}{104 \left (a+b x^3\right )^{13/3}}+\frac{10}{13} \int \frac{1}{\left (a+b x^3\right )^{13/3}} \, dx\\ &=\frac{x \left (a-b x^3\right )}{8 \left (a+b x^3\right )^{16/3}}+\frac{11 x}{104 \left (a+b x^3\right )^{13/3}}+\frac{x}{13 a \left (a+b x^3\right )^{10/3}}+\frac{9 \int \frac{1}{\left (a+b x^3\right )^{10/3}} \, dx}{13 a}\\ &=\frac{x \left (a-b x^3\right )}{8 \left (a+b x^3\right )^{16/3}}+\frac{11 x}{104 \left (a+b x^3\right )^{13/3}}+\frac{x}{13 a \left (a+b x^3\right )^{10/3}}+\frac{9 x}{91 a^2 \left (a+b x^3\right )^{7/3}}+\frac{54 \int \frac{1}{\left (a+b x^3\right )^{7/3}} \, dx}{91 a^2}\\ &=\frac{x \left (a-b x^3\right )}{8 \left (a+b x^3\right )^{16/3}}+\frac{11 x}{104 \left (a+b x^3\right )^{13/3}}+\frac{x}{13 a \left (a+b x^3\right )^{10/3}}+\frac{9 x}{91 a^2 \left (a+b x^3\right )^{7/3}}+\frac{27 x}{182 a^3 \left (a+b x^3\right )^{4/3}}+\frac{81 \int \frac{1}{\left (a+b x^3\right )^{4/3}} \, dx}{182 a^3}\\ &=\frac{x \left (a-b x^3\right )}{8 \left (a+b x^3\right )^{16/3}}+\frac{11 x}{104 \left (a+b x^3\right )^{13/3}}+\frac{x}{13 a \left (a+b x^3\right )^{10/3}}+\frac{9 x}{91 a^2 \left (a+b x^3\right )^{7/3}}+\frac{27 x}{182 a^3 \left (a+b x^3\right )^{4/3}}+\frac{81 x}{182 a^4 \sqrt [3]{a+b x^3}}\\ \end{align*}
Mathematica [A] time = 0.0373948, size = 73, normalized size = 0.62 \[ \frac{x \left (1872 a^2 b^3 x^9+2080 a^3 b^2 x^6+1183 a^4 b x^3+364 a^5+864 a b^4 x^{12}+162 b^5 x^{15}\right )}{364 a^4 \left (a+b x^3\right )^{16/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 70, normalized size = 0.6 \begin{align*}{\frac{x \left ( 162\,{b}^{5}{x}^{15}+864\,{b}^{4}{x}^{12}a+1872\,{b}^{3}{x}^{9}{a}^{2}+2080\,{a}^{3}{b}^{2}{x}^{6}+1183\,b{x}^{3}{a}^{4}+364\,{a}^{5} \right ) }{364\,{a}^{4}} \left ( b{x}^{3}+a \right ) ^{-{\frac{16}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.980965, size = 347, normalized size = 2.97 \begin{align*} -\frac{{\left (455 \, b^{3} - \frac{1680 \,{\left (b x^{3} + a\right )} b^{2}}{x^{3}} + \frac{2184 \,{\left (b x^{3} + a\right )}^{2} b}{x^{6}} - \frac{1040 \,{\left (b x^{3} + a\right )}^{3}}{x^{9}}\right )} b^{2} x^{16}}{7280 \,{\left (b x^{3} + a\right )}^{\frac{16}{3}} a^{4}} - \frac{{\left (455 \, b^{4} - \frac{2240 \,{\left (b x^{3} + a\right )} b^{3}}{x^{3}} + \frac{4368 \,{\left (b x^{3} + a\right )}^{2} b^{2}}{x^{6}} - \frac{4160 \,{\left (b x^{3} + a\right )}^{3} b}{x^{9}} + \frac{1820 \,{\left (b x^{3} + a\right )}^{4}}{x^{12}}\right )} b x^{16}}{3640 \,{\left (b x^{3} + a\right )}^{\frac{16}{3}} a^{4}} - \frac{{\left (91 \, b^{5} - \frac{560 \,{\left (b x^{3} + a\right )} b^{4}}{x^{3}} + \frac{1456 \,{\left (b x^{3} + a\right )}^{2} b^{3}}{x^{6}} - \frac{2080 \,{\left (b x^{3} + a\right )}^{3} b^{2}}{x^{9}} + \frac{1820 \,{\left (b x^{3} + a\right )}^{4} b}{x^{12}} - \frac{1456 \,{\left (b x^{3} + a\right )}^{5}}{x^{15}}\right )} x^{16}}{1456 \,{\left (b x^{3} + a\right )}^{\frac{16}{3}} a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.18981, size = 311, normalized size = 2.66 \begin{align*} \frac{{\left (162 \, b^{5} x^{16} + 864 \, a b^{4} x^{13} + 1872 \, a^{2} b^{3} x^{10} + 2080 \, a^{3} b^{2} x^{7} + 1183 \, a^{4} b x^{4} + 364 \, a^{5} x\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{364 \,{\left (a^{4} b^{6} x^{18} + 6 \, a^{5} b^{5} x^{15} + 15 \, a^{6} b^{4} x^{12} + 20 \, a^{7} b^{3} x^{9} + 15 \, a^{8} b^{2} x^{6} + 6 \, a^{9} b x^{3} + a^{10}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{3} - a\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac{19}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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