Optimal. Leaf size=253 \[ \frac{81 x \left (a^2 d^2+6 a b c d+45 b^2 c^2\right )}{7280 a^6 b^2 \sqrt [3]{a+b x^3}}+\frac{27 x \left (a^2 d^2+6 a b c d+45 b^2 c^2\right )}{7280 a^5 b^2 \left (a+b x^3\right )^{4/3}}+\frac{9 x \left (a^2 d^2+6 a b c d+45 b^2 c^2\right )}{3640 a^4 b^2 \left (a+b x^3\right )^{7/3}}+\frac{x \left (a^2 d^2+6 a b c d+45 b^2 c^2\right )}{520 a^3 b^2 \left (a+b x^3\right )^{10/3}}+\frac{x (b c-a d) (4 a d+15 b c)}{208 a^2 b^2 \left (a+b x^3\right )^{13/3}}+\frac{x \left (c+d x^3\right ) (b c-a d)}{16 a b \left (a+b x^3\right )^{16/3}} \]
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Rubi [A] time = 0.207149, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {413, 385, 192, 191} \[ \frac{81 x \left (a^2 d^2+6 a b c d+45 b^2 c^2\right )}{7280 a^6 b^2 \sqrt [3]{a+b x^3}}+\frac{27 x \left (a^2 d^2+6 a b c d+45 b^2 c^2\right )}{7280 a^5 b^2 \left (a+b x^3\right )^{4/3}}+\frac{9 x \left (a^2 d^2+6 a b c d+45 b^2 c^2\right )}{3640 a^4 b^2 \left (a+b x^3\right )^{7/3}}+\frac{x \left (a^2 d^2+6 a b c d+45 b^2 c^2\right )}{520 a^3 b^2 \left (a+b x^3\right )^{10/3}}+\frac{x (b c-a d) (4 a d+15 b c)}{208 a^2 b^2 \left (a+b x^3\right )^{13/3}}+\frac{x \left (c+d x^3\right ) (b c-a d)}{16 a b \left (a+b x^3\right )^{16/3}} \]
Antiderivative was successfully verified.
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Rule 413
Rule 385
Rule 192
Rule 191
Rubi steps
\begin{align*} \int \frac{\left (c+d x^3\right )^2}{\left (a+b x^3\right )^{19/3}} \, dx &=\frac{(b c-a d) x \left (c+d x^3\right )}{16 a b \left (a+b x^3\right )^{16/3}}+\frac{\int \frac{c (15 b c+a d)+4 d (3 b c+a d) x^3}{\left (a+b x^3\right )^{16/3}} \, dx}{16 a b}\\ &=\frac{(b c-a d) (15 b c+4 a d) x}{208 a^2 b^2 \left (a+b x^3\right )^{13/3}}+\frac{(b c-a d) x \left (c+d x^3\right )}{16 a b \left (a+b x^3\right )^{16/3}}+\frac{\left (45 b^2 c^2+6 a b c d+a^2 d^2\right ) \int \frac{1}{\left (a+b x^3\right )^{13/3}} \, dx}{52 a^2 b^2}\\ &=\frac{(b c-a d) (15 b c+4 a d) x}{208 a^2 b^2 \left (a+b x^3\right )^{13/3}}+\frac{\left (45 b^2 c^2+6 a b c d+a^2 d^2\right ) x}{520 a^3 b^2 \left (a+b x^3\right )^{10/3}}+\frac{(b c-a d) x \left (c+d x^3\right )}{16 a b \left (a+b x^3\right )^{16/3}}+\frac{\left (9 \left (45 b^2 c^2+6 a b c d+a^2 d^2\right )\right ) \int \frac{1}{\left (a+b x^3\right )^{10/3}} \, dx}{520 a^3 b^2}\\ &=\frac{(b c-a d) (15 b c+4 a d) x}{208 a^2 b^2 \left (a+b x^3\right )^{13/3}}+\frac{\left (45 b^2 c^2+6 a b c d+a^2 d^2\right ) x}{520 a^3 b^2 \left (a+b x^3\right )^{10/3}}+\frac{9 \left (45 b^2 c^2+6 a b c d+a^2 d^2\right ) x}{3640 a^4 b^2 \left (a+b x^3\right )^{7/3}}+\frac{(b c-a d) x \left (c+d x^3\right )}{16 a b \left (a+b x^3\right )^{16/3}}+\frac{\left (27 \left (45 b^2 c^2+6 a b c d+a^2 d^2\right )\right ) \int \frac{1}{\left (a+b x^3\right )^{7/3}} \, dx}{1820 a^4 b^2}\\ &=\frac{(b c-a d) (15 b c+4 a d) x}{208 a^2 b^2 \left (a+b x^3\right )^{13/3}}+\frac{\left (45 b^2 c^2+6 a b c d+a^2 d^2\right ) x}{520 a^3 b^2 \left (a+b x^3\right )^{10/3}}+\frac{9 \left (45 b^2 c^2+6 a b c d+a^2 d^2\right ) x}{3640 a^4 b^2 \left (a+b x^3\right )^{7/3}}+\frac{27 \left (45 b^2 c^2+6 a b c d+a^2 d^2\right ) x}{7280 a^5 b^2 \left (a+b x^3\right )^{4/3}}+\frac{(b c-a d) x \left (c+d x^3\right )}{16 a b \left (a+b x^3\right )^{16/3}}+\frac{\left (81 \left (45 b^2 c^2+6 a b c d+a^2 d^2\right )\right ) \int \frac{1}{\left (a+b x^3\right )^{4/3}} \, dx}{7280 a^5 b^2}\\ &=\frac{(b c-a d) (15 b c+4 a d) x}{208 a^2 b^2 \left (a+b x^3\right )^{13/3}}+\frac{\left (45 b^2 c^2+6 a b c d+a^2 d^2\right ) x}{520 a^3 b^2 \left (a+b x^3\right )^{10/3}}+\frac{9 \left (45 b^2 c^2+6 a b c d+a^2 d^2\right ) x}{3640 a^4 b^2 \left (a+b x^3\right )^{7/3}}+\frac{27 \left (45 b^2 c^2+6 a b c d+a^2 d^2\right ) x}{7280 a^5 b^2 \left (a+b x^3\right )^{4/3}}+\frac{81 \left (45 b^2 c^2+6 a b c d+a^2 d^2\right ) x}{7280 a^6 b^2 \sqrt [3]{a+b x^3}}+\frac{(b c-a d) x \left (c+d x^3\right )}{16 a b \left (a+b x^3\right )^{16/3}}\\ \end{align*}
Mathematica [A] time = 5.08619, size = 169, normalized size = 0.67 \[ \frac{x \left (81 a^2 b^3 x^9 \left (520 c^2+32 c d x^3+d^2 x^6\right )+144 a^3 b^2 x^6 \left (325 c^2+39 c d x^3+3 d^2 x^6\right )+156 a^4 b x^3 \left (175 c^2+40 c d x^3+6 d^2 x^6\right )+520 a^5 \left (14 c^2+7 c d x^3+2 d^2 x^6\right )+486 a b^4 c x^{12} \left (40 c+d x^3\right )+3645 b^5 c^2 x^{15}\right )}{7280 a^6 \left (a+b x^3\right )^{16/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 197, normalized size = 0.8 \begin{align*}{\frac{x \left ( 81\,{a}^{2}{b}^{3}{d}^{2}{x}^{15}+486\,a{b}^{4}cd{x}^{15}+3645\,{b}^{5}{c}^{2}{x}^{15}+432\,{a}^{3}{b}^{2}{d}^{2}{x}^{12}+2592\,{a}^{2}{b}^{3}cd{x}^{12}+19440\,a{b}^{4}{c}^{2}{x}^{12}+936\,{a}^{4}b{d}^{2}{x}^{9}+5616\,{a}^{3}{b}^{2}cd{x}^{9}+42120\,{a}^{2}{b}^{3}{c}^{2}{x}^{9}+1040\,{a}^{5}{d}^{2}{x}^{6}+6240\,{a}^{4}bcd{x}^{6}+46800\,{a}^{3}{b}^{2}{c}^{2}{x}^{6}+3640\,{a}^{5}cd{x}^{3}+27300\,{a}^{4}b{c}^{2}{x}^{3}+7280\,{c}^{2}{a}^{5} \right ) }{7280\,{a}^{6}} \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 [A] time = 0.969908, size = 352, normalized size = 1.39 \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 )} d^{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 )} c d x^{16}}{3640 \,{\left (b x^{3} + a\right )}^{\frac{16}{3}} a^{5}} - \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 )} c^{2} x^{16}}{1456 \,{\left (b x^{3} + a\right )}^{\frac{16}{3}} a^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.06553, size = 544, normalized size = 2.15 \begin{align*} \frac{{\left (81 \,{\left (45 \, b^{5} c^{2} + 6 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} x^{16} + 432 \,{\left (45 \, a b^{4} c^{2} + 6 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} x^{13} + 936 \,{\left (45 \, a^{2} b^{3} c^{2} + 6 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} x^{10} + 7280 \, a^{5} c^{2} x + 1040 \,{\left (45 \, a^{3} b^{2} c^{2} + 6 \, a^{4} b c d + a^{5} d^{2}\right )} x^{7} + 1820 \,{\left (15 \, a^{4} b c^{2} + 2 \, a^{5} c d\right )} x^{4}\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{7280 \,{\left (a^{6} b^{6} x^{18} + 6 \, a^{7} b^{5} x^{15} + 15 \, a^{8} b^{4} x^{12} + 20 \, a^{9} b^{3} x^{9} + 15 \, a^{10} b^{2} x^{6} + 6 \, a^{11} b x^{3} + a^{12}\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 (d x^{3} + c\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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