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 = 0.04, antiderivative size = 117, normalized size of antiderivative = 1.00, 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 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 )^{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*}
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Mathematica [A] time = 0.04, size = 73, normalized size = 0.62 \[ \frac {x \left (364 a^5+1183 a^4 b x^3+2080 a^3 b^2 x^6+1872 a^2 b^3 x^9+864 a b^4 x^{12}+162 b^5 x^{15}\right )}{364 a^4 \left (a+b x^3\right )^{16/3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.99, size = 135, normalized size = 1.15 \[ \frac {{\left (162 \, b^{5} x^{16} + 864 \, a b^{4} x^{13} + 1872 \, a^{2} b^{3} x^{10} + 2080 \, a^{3} b^{2} x^{7} + 1183 \, a^{4} b x^{4} + 364 \, a^{5} x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{364 \, {\left (a^{4} b^{6} x^{18} + 6 \, a^{5} b^{5} x^{15} + 15 \, a^{6} b^{4} x^{12} + 20 \, a^{7} b^{3} x^{9} + 15 \, a^{8} b^{2} x^{6} + 6 \, a^{9} b x^{3} + a^{10}\right )}} \]
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 \[ \int \frac {{\left (b x^{3} - a\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac {19}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 70, normalized size = 0.60 \[ \frac {\left (162 b^{5} x^{15}+864 a \,b^{4} x^{12}+1872 a^{2} b^{3} x^{9}+2080 a^{3} b^{2} x^{6}+1183 a^{4} b \,x^{3}+364 a^{5}\right ) x}{364 \left (b \,x^{3}+a \right )^{\frac {16}{3}} a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.51, size = 257, normalized size = 2.20 \[ -\frac {{\left (455 \, b^{3} - \frac {1680 \, {\left (b x^{3} + a\right )} b^{2}}{x^{3}} + \frac {2184 \, {\left (b x^{3} + a\right )}^{2} b}{x^{6}} - \frac {1040 \, {\left (b x^{3} + a\right )}^{3}}{x^{9}}\right )} b^{2} x^{16}}{7280 \, {\left (b x^{3} + a\right )}^{\frac {16}{3}} a^{4}} - \frac {{\left (455 \, b^{4} - \frac {2240 \, {\left (b x^{3} + a\right )} b^{3}}{x^{3}} + \frac {4368 \, {\left (b x^{3} + a\right )}^{2} b^{2}}{x^{6}} - \frac {4160 \, {\left (b x^{3} + a\right )}^{3} b}{x^{9}} + \frac {1820 \, {\left (b x^{3} + a\right )}^{4}}{x^{12}}\right )} b x^{16}}{3640 \, {\left (b x^{3} + a\right )}^{\frac {16}{3}} a^{4}} - \frac {{\left (91 \, b^{5} - \frac {560 \, {\left (b x^{3} + a\right )} b^{4}}{x^{3}} + \frac {1456 \, {\left (b x^{3} + a\right )}^{2} b^{3}}{x^{6}} - \frac {2080 \, {\left (b x^{3} + a\right )}^{3} b^{2}}{x^{9}} + \frac {1820 \, {\left (b x^{3} + a\right )}^{4} b}{x^{12}} - \frac {1456 \, {\left (b x^{3} + a\right )}^{5}}{x^{15}}\right )} x^{16}}{1456 \, {\left (b x^{3} + a\right )}^{\frac {16}{3}} a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.43, size = 86, normalized size = 0.74 \[ \frac {81\,x}{182\,a^4\,{\left (b\,x^3+a\right )}^{1/3}}-\frac {x}{52\,{\left (b\,x^3+a\right )}^{13/3}}+\frac {27\,x}{182\,a^3\,{\left (b\,x^3+a\right )}^{4/3}}+\frac {9\,x}{91\,a^2\,{\left (b\,x^3+a\right )}^{7/3}}+\frac {x}{13\,a\,{\left (b\,x^3+a\right )}^{10/3}}+\frac {a\,x}{4\,{\left (b\,x^3+a\right )}^{16/3}} \]
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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