Optimal. Leaf size=151 \[ \frac{81 x (a d+12 b c)}{1820 a^5 b \sqrt [3]{a+b x^3}}+\frac{27 x (a d+12 b c)}{1820 a^4 b \left (a+b x^3\right )^{4/3}}+\frac{9 x (a d+12 b c)}{910 a^3 b \left (a+b x^3\right )^{7/3}}+\frac{x (a d+12 b c)}{130 a^2 b \left (a+b x^3\right )^{10/3}}+\frac{x (b c-a d)}{13 a b \left (a+b x^3\right )^{13/3}} \]
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Rubi [A] time = 0.0469943, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {385, 192, 191} \[ \frac{81 x (a d+12 b c)}{1820 a^5 b \sqrt [3]{a+b x^3}}+\frac{27 x (a d+12 b c)}{1820 a^4 b \left (a+b x^3\right )^{4/3}}+\frac{9 x (a d+12 b c)}{910 a^3 b \left (a+b x^3\right )^{7/3}}+\frac{x (a d+12 b c)}{130 a^2 b \left (a+b x^3\right )^{10/3}}+\frac{x (b c-a d)}{13 a b \left (a+b x^3\right )^{13/3}} \]
Antiderivative was successfully verified.
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Rule 385
Rule 192
Rule 191
Rubi steps
\begin{align*} \int \frac{c+d x^3}{\left (a+b x^3\right )^{16/3}} \, dx &=\frac{(b c-a d) x}{13 a b \left (a+b x^3\right )^{13/3}}+\frac{(12 b c+a d) \int \frac{1}{\left (a+b x^3\right )^{13/3}} \, dx}{13 a b}\\ &=\frac{(b c-a d) x}{13 a b \left (a+b x^3\right )^{13/3}}+\frac{(12 b c+a d) x}{130 a^2 b \left (a+b x^3\right )^{10/3}}+\frac{(9 (12 b c+a d)) \int \frac{1}{\left (a+b x^3\right )^{10/3}} \, dx}{130 a^2 b}\\ &=\frac{(b c-a d) x}{13 a b \left (a+b x^3\right )^{13/3}}+\frac{(12 b c+a d) x}{130 a^2 b \left (a+b x^3\right )^{10/3}}+\frac{9 (12 b c+a d) x}{910 a^3 b \left (a+b x^3\right )^{7/3}}+\frac{(27 (12 b c+a d)) \int \frac{1}{\left (a+b x^3\right )^{7/3}} \, dx}{455 a^3 b}\\ &=\frac{(b c-a d) x}{13 a b \left (a+b x^3\right )^{13/3}}+\frac{(12 b c+a d) x}{130 a^2 b \left (a+b x^3\right )^{10/3}}+\frac{9 (12 b c+a d) x}{910 a^3 b \left (a+b x^3\right )^{7/3}}+\frac{27 (12 b c+a d) x}{1820 a^4 b \left (a+b x^3\right )^{4/3}}+\frac{(81 (12 b c+a d)) \int \frac{1}{\left (a+b x^3\right )^{4/3}} \, dx}{1820 a^4 b}\\ &=\frac{(b c-a d) x}{13 a b \left (a+b x^3\right )^{13/3}}+\frac{(12 b c+a d) x}{130 a^2 b \left (a+b x^3\right )^{10/3}}+\frac{9 (12 b c+a d) x}{910 a^3 b \left (a+b x^3\right )^{7/3}}+\frac{27 (12 b c+a d) x}{1820 a^4 b \left (a+b x^3\right )^{4/3}}+\frac{81 (12 b c+a d) x}{1820 a^5 b \sqrt [3]{a+b x^3}}\\ \end{align*}
Mathematica [A] time = 0.0375034, size = 100, normalized size = 0.66 \[ \frac{x \left (351 a^2 b^2 x^6 \left (20 c+d x^3\right )+195 a^3 b x^3 \left (28 c+3 d x^3\right )+455 a^4 \left (4 c+d x^3\right )+81 a b^3 x^9 \left (52 c+d x^3\right )+972 b^4 c x^{12}\right )}{1820 a^5 \left (a+b x^3\right )^{13/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 105, normalized size = 0.7 \begin{align*}{\frac{x \left ( 81\,a{b}^{3}d{x}^{12}+972\,{b}^{4}c{x}^{12}+351\,{a}^{2}{b}^{2}d{x}^{9}+4212\,a{b}^{3}c{x}^{9}+585\,{a}^{3}bd{x}^{6}+7020\,{a}^{2}{b}^{2}c{x}^{6}+455\,{a}^{4}d{x}^{3}+5460\,{a}^{3}bc{x}^{3}+1820\,c{a}^{4} \right ) }{1820\,{a}^{5}} \left ( b{x}^{3}+a \right ) ^{-{\frac{13}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.973927, size = 208, normalized size = 1.38 \begin{align*} -\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 )} d x^{13}}{1820 \,{\left (b x^{3} + a\right )}^{\frac{13}{3}} a^{4}} + \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 )} c x^{13}}{455 \,{\left (b x^{3} + a\right )}^{\frac{13}{3}} a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78946, size = 352, normalized size = 2.33 \begin{align*} \frac{{\left (81 \,{\left (12 \, b^{4} c + a b^{3} d\right )} x^{13} + 351 \,{\left (12 \, a b^{3} c + a^{2} b^{2} d\right )} x^{10} + 585 \,{\left (12 \, a^{2} b^{2} c + a^{3} b d\right )} x^{7} + 1820 \, a^{4} c x + 455 \,{\left (12 \, a^{3} b c + a^{4} d\right )} x^{4}\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{1820 \,{\left (a^{5} b^{5} x^{15} + 5 \, a^{6} b^{4} x^{12} + 10 \, a^{7} b^{3} x^{9} + 10 \, a^{8} b^{2} x^{6} + 5 \, 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{d x^{3} + c}{{\left (b x^{3} + a\right )}^{\frac{16}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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