Optimal. Leaf size=121 \[ \frac{9 x (a d+9 b c)}{140 a^4 b \sqrt [3]{a+b x^3}}+\frac{3 x (a d+9 b c)}{140 a^3 b \left (a+b x^3\right )^{4/3}}+\frac{x (a d+9 b c)}{70 a^2 b \left (a+b x^3\right )^{7/3}}+\frac{x (b c-a d)}{10 a b \left (a+b x^3\right )^{10/3}} \]
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Rubi [A] time = 0.0352966, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {385, 192, 191} \[ \frac{9 x (a d+9 b c)}{140 a^4 b \sqrt [3]{a+b x^3}}+\frac{3 x (a d+9 b c)}{140 a^3 b \left (a+b x^3\right )^{4/3}}+\frac{x (a d+9 b c)}{70 a^2 b \left (a+b x^3\right )^{7/3}}+\frac{x (b c-a d)}{10 a b \left (a+b x^3\right )^{10/3}} \]
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 )^{13/3}} \, dx &=\frac{(b c-a d) x}{10 a b \left (a+b x^3\right )^{10/3}}+\frac{(9 b c+a d) \int \frac{1}{\left (a+b x^3\right )^{10/3}} \, dx}{10 a b}\\ &=\frac{(b c-a d) x}{10 a b \left (a+b x^3\right )^{10/3}}+\frac{(9 b c+a d) x}{70 a^2 b \left (a+b x^3\right )^{7/3}}+\frac{(3 (9 b c+a d)) \int \frac{1}{\left (a+b x^3\right )^{7/3}} \, dx}{35 a^2 b}\\ &=\frac{(b c-a d) x}{10 a b \left (a+b x^3\right )^{10/3}}+\frac{(9 b c+a d) x}{70 a^2 b \left (a+b x^3\right )^{7/3}}+\frac{3 (9 b c+a d) x}{140 a^3 b \left (a+b x^3\right )^{4/3}}+\frac{(9 (9 b c+a d)) \int \frac{1}{\left (a+b x^3\right )^{4/3}} \, dx}{140 a^3 b}\\ &=\frac{(b c-a d) x}{10 a b \left (a+b x^3\right )^{10/3}}+\frac{(9 b c+a d) x}{70 a^2 b \left (a+b x^3\right )^{7/3}}+\frac{3 (9 b c+a d) x}{140 a^3 b \left (a+b x^3\right )^{4/3}}+\frac{9 (9 b c+a d) x}{140 a^4 b \sqrt [3]{a+b x^3}}\\ \end{align*}
Mathematica [A] time = 0.0314036, size = 80, normalized size = 0.66 \[ \frac{x \left (15 a^2 b x^3 \left (21 c+2 d x^3\right )+35 a^3 \left (4 c+d x^3\right )+9 a b^2 x^6 \left (30 c+d x^3\right )+81 b^3 c x^9\right )}{140 a^4 \left (a+b x^3\right )^{10/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 81, normalized size = 0.7 \begin{align*}{\frac{x \left ( 9\,a{b}^{2}d{x}^{9}+81\,{b}^{3}c{x}^{9}+30\,{a}^{2}bd{x}^{6}+270\,a{b}^{2}c{x}^{6}+35\,{a}^{3}d{x}^{3}+315\,{a}^{2}bc{x}^{3}+140\,c{a}^{3} \right ) }{140\,{a}^{4}} \left ( b{x}^{3}+a \right ) ^{-{\frac{10}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.94365, size = 162, normalized size = 1.34 \begin{align*} \frac{{\left (14 \, b^{2} - \frac{40 \,{\left (b x^{3} + a\right )} b}{x^{3}} + \frac{35 \,{\left (b x^{3} + a\right )}^{2}}{x^{6}}\right )} d x^{10}}{140 \,{\left (b x^{3} + a\right )}^{\frac{10}{3}} a^{3}} - \frac{{\left (14 \, b^{3} - \frac{60 \,{\left (b x^{3} + a\right )} b^{2}}{x^{3}} + \frac{105 \,{\left (b x^{3} + a\right )}^{2} b}{x^{6}} - \frac{140 \,{\left (b x^{3} + a\right )}^{3}}{x^{9}}\right )} c x^{10}}{140 \,{\left (b x^{3} + a\right )}^{\frac{10}{3}} a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65692, size = 263, normalized size = 2.17 \begin{align*} \frac{{\left (9 \,{\left (9 \, b^{3} c + a b^{2} d\right )} x^{10} + 30 \,{\left (9 \, a b^{2} c + a^{2} b d\right )} x^{7} + 140 \, a^{3} c x + 35 \,{\left (9 \, a^{2} b c + a^{3} d\right )} x^{4}\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{140 \,{\left (a^{4} b^{4} x^{12} + 4 \, a^{5} b^{3} x^{9} + 6 \, a^{6} b^{2} x^{6} + 4 \, a^{7} b x^{3} + a^{8}\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{13}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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