Optimal. Leaf size=98 \[ \frac {423 x}{910 a^3 \sqrt [3]{a+b x^3}}+\frac {141 x}{910 a^2 \left (a+b x^3\right )^{4/3}}+\frac {47 x}{455 a \left (a+b x^3\right )^{7/3}}+\frac {8 x}{65 \left (a+b x^3\right )^{10/3}}+\frac {2 x \left (a-b x^3\right )}{13 \left (a+b x^3\right )^{13/3}} \]
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Rubi [A] time = 0.04, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {413, 385, 192, 191} \[ \frac {423 x}{910 a^3 \sqrt [3]{a+b x^3}}+\frac {141 x}{910 a^2 \left (a+b x^3\right )^{4/3}}+\frac {47 x}{455 a \left (a+b x^3\right )^{7/3}}+\frac {8 x}{65 \left (a+b x^3\right )^{10/3}}+\frac {2 x \left (a-b x^3\right )}{13 \left (a+b x^3\right )^{13/3}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 385
Rule 413
Rubi steps
\begin {align*} \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{16/3}} \, dx &=\frac {2 x \left (a-b x^3\right )}{13 \left (a+b x^3\right )^{13/3}}+\frac {\int \frac {11 a^2 b-5 a b^2 x^3}{\left (a+b x^3\right )^{13/3}} \, dx}{13 a b}\\ &=\frac {2 x \left (a-b x^3\right )}{13 \left (a+b x^3\right )^{13/3}}+\frac {8 x}{65 \left (a+b x^3\right )^{10/3}}+\frac {47}{65} \int \frac {1}{\left (a+b x^3\right )^{10/3}} \, dx\\ &=\frac {2 x \left (a-b x^3\right )}{13 \left (a+b x^3\right )^{13/3}}+\frac {8 x}{65 \left (a+b x^3\right )^{10/3}}+\frac {47 x}{455 a \left (a+b x^3\right )^{7/3}}+\frac {282 \int \frac {1}{\left (a+b x^3\right )^{7/3}} \, dx}{455 a}\\ &=\frac {2 x \left (a-b x^3\right )}{13 \left (a+b x^3\right )^{13/3}}+\frac {8 x}{65 \left (a+b x^3\right )^{10/3}}+\frac {47 x}{455 a \left (a+b x^3\right )^{7/3}}+\frac {141 x}{910 a^2 \left (a+b x^3\right )^{4/3}}+\frac {423 \int \frac {1}{\left (a+b x^3\right )^{4/3}} \, dx}{910 a^2}\\ &=\frac {2 x \left (a-b x^3\right )}{13 \left (a+b x^3\right )^{13/3}}+\frac {8 x}{65 \left (a+b x^3\right )^{10/3}}+\frac {47 x}{455 a \left (a+b x^3\right )^{7/3}}+\frac {141 x}{910 a^2 \left (a+b x^3\right )^{4/3}}+\frac {423 x}{910 a^3 \sqrt [3]{a+b x^3}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 62, normalized size = 0.63 \[ \frac {x \left (910 a^4+2275 a^3 b x^3+3055 a^2 b^2 x^6+1833 a b^3 x^9+423 b^4 x^{12}\right )}{910 a^3 \left (a+b x^3\right )^{13/3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 113, normalized size = 1.15 \[ \frac {{\left (423 \, b^{4} x^{13} + 1833 \, a b^{3} x^{10} + 3055 \, a^{2} b^{2} x^{7} + 2275 \, a^{3} b x^{4} + 910 \, a^{4} x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{910 \, {\left (a^{3} b^{5} x^{15} + 5 \, a^{4} b^{4} x^{12} + 10 \, a^{5} b^{3} x^{9} + 10 \, a^{6} b^{2} x^{6} + 5 \, a^{7} b x^{3} + a^{8}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{3} - a\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac {16}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 59, normalized size = 0.60 \[ \frac {\left (423 b^{4} x^{12}+1833 b^{3} x^{9} a +3055 b^{2} x^{6} a^{2}+2275 b \,x^{3} a^{3}+910 a^{4}\right ) x}{910 \left (b \,x^{3}+a \right )^{\frac {13}{3}} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.57, size = 206, normalized size = 2.10 \[ \frac {{\left (35 \, b^{2} - \frac {91 \, {\left (b x^{3} + a\right )} b}{x^{3}} + \frac {65 \, {\left (b x^{3} + a\right )}^{2}}{x^{6}}\right )} b^{2} x^{13}}{455 \, {\left (b x^{3} + a\right )}^{\frac {13}{3}} a^{3}} + \frac {{\left (140 \, b^{3} - \frac {546 \, {\left (b x^{3} + a\right )} b^{2}}{x^{3}} + \frac {780 \, {\left (b x^{3} + a\right )}^{2} b}{x^{6}} - \frac {455 \, {\left (b x^{3} + a\right )}^{3}}{x^{9}}\right )} b x^{13}}{910 \, {\left (b x^{3} + a\right )}^{\frac {13}{3}} a^{3}} + \frac {{\left (35 \, b^{4} - \frac {182 \, {\left (b x^{3} + a\right )} b^{3}}{x^{3}} + \frac {390 \, {\left (b x^{3} + a\right )}^{2} b^{2}}{x^{6}} - \frac {455 \, {\left (b x^{3} + a\right )}^{3} b}{x^{9}} + \frac {455 \, {\left (b x^{3} + a\right )}^{4}}{x^{12}}\right )} x^{13}}{455 \, {\left (b x^{3} + a\right )}^{\frac {13}{3}} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.44, size = 71, normalized size = 0.72 \[ \frac {423\,x}{910\,a^3\,{\left (b\,x^3+a\right )}^{1/3}}-\frac {2\,x}{65\,{\left (b\,x^3+a\right )}^{10/3}}+\frac {141\,x}{910\,a^2\,{\left (b\,x^3+a\right )}^{4/3}}+\frac {47\,x}{455\,a\,{\left (b\,x^3+a\right )}^{7/3}}+\frac {4\,a\,x}{13\,{\left (b\,x^3+a\right )}^{13/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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