Integrand size = 20, antiderivative size = 93 \[ \int \frac {a-b x^3}{\left (a+b x^3\right )^{16/3}} \, dx=\frac {2 x}{13 \left (a+b x^3\right )^{13/3}}+\frac {11 x}{130 a \left (a+b x^3\right )^{10/3}}+\frac {99 x}{910 a^2 \left (a+b x^3\right )^{7/3}}+\frac {297 x}{1820 a^3 \left (a+b x^3\right )^{4/3}}+\frac {891 x}{1820 a^4 \sqrt [3]{a+b x^3}} \]
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Time = 0.02 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {393, 198, 197} \[ \int \frac {a-b x^3}{\left (a+b x^3\right )^{16/3}} \, dx=\frac {891 x}{1820 a^4 \sqrt [3]{a+b x^3}}+\frac {297 x}{1820 a^3 \left (a+b x^3\right )^{4/3}}+\frac {99 x}{910 a^2 \left (a+b x^3\right )^{7/3}}+\frac {11 x}{130 a \left (a+b x^3\right )^{10/3}}+\frac {2 x}{13 \left (a+b x^3\right )^{13/3}} \]
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Rule 197
Rule 198
Rule 393
Rubi steps \begin{align*} \text {integral}& = \frac {2 x}{13 \left (a+b x^3\right )^{13/3}}+\frac {11}{13} \int \frac {1}{\left (a+b x^3\right )^{13/3}} \, dx \\ & = \frac {2 x}{13 \left (a+b x^3\right )^{13/3}}+\frac {11 x}{130 a \left (a+b x^3\right )^{10/3}}+\frac {99 \int \frac {1}{\left (a+b x^3\right )^{10/3}} \, dx}{130 a} \\ & = \frac {2 x}{13 \left (a+b x^3\right )^{13/3}}+\frac {11 x}{130 a \left (a+b x^3\right )^{10/3}}+\frac {99 x}{910 a^2 \left (a+b x^3\right )^{7/3}}+\frac {297 \int \frac {1}{\left (a+b x^3\right )^{7/3}} \, dx}{455 a^2} \\ & = \frac {2 x}{13 \left (a+b x^3\right )^{13/3}}+\frac {11 x}{130 a \left (a+b x^3\right )^{10/3}}+\frac {99 x}{910 a^2 \left (a+b x^3\right )^{7/3}}+\frac {297 x}{1820 a^3 \left (a+b x^3\right )^{4/3}}+\frac {891 \int \frac {1}{\left (a+b x^3\right )^{4/3}} \, dx}{1820 a^3} \\ & = \frac {2 x}{13 \left (a+b x^3\right )^{13/3}}+\frac {11 x}{130 a \left (a+b x^3\right )^{10/3}}+\frac {99 x}{910 a^2 \left (a+b x^3\right )^{7/3}}+\frac {297 x}{1820 a^3 \left (a+b x^3\right )^{4/3}}+\frac {891 x}{1820 a^4 \sqrt [3]{a+b x^3}} \\ \end{align*}
Time = 0.72 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.67 \[ \int \frac {a-b x^3}{\left (a+b x^3\right )^{16/3}} \, dx=\frac {x \left (1820 a^4+5005 a^3 b x^3+6435 a^2 b^2 x^6+3861 a b^3 x^9+891 b^4 x^{12}\right )}{1820 a^4 \left (a+b x^3\right )^{13/3}} \]
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Time = 3.91 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.63
method | result | size |
gosper | \(\frac {x \left (891 b^{4} x^{12}+3861 a \,b^{3} x^{9}+6435 a^{2} b^{2} x^{6}+5005 a^{3} b \,x^{3}+1820 a^{4}\right )}{1820 \left (b \,x^{3}+a \right )^{\frac {13}{3}} a^{4}}\) | \(59\) |
trager | \(\frac {x \left (891 b^{4} x^{12}+3861 a \,b^{3} x^{9}+6435 a^{2} b^{2} x^{6}+5005 a^{3} b \,x^{3}+1820 a^{4}\right )}{1820 \left (b \,x^{3}+a \right )^{\frac {13}{3}} a^{4}}\) | \(59\) |
pseudoelliptic | \(\frac {x \left (891 b^{4} x^{12}+3861 a \,b^{3} x^{9}+6435 a^{2} b^{2} x^{6}+5005 a^{3} b \,x^{3}+1820 a^{4}\right )}{1820 \left (b \,x^{3}+a \right )^{\frac {13}{3}} a^{4}}\) | \(59\) |
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none
Time = 0.30 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.22 \[ \int \frac {a-b x^3}{\left (a+b x^3\right )^{16/3}} \, dx=\frac {{\left (891 \, b^{4} x^{13} + 3861 \, a b^{3} x^{10} + 6435 \, a^{2} b^{2} x^{7} + 5005 \, a^{3} b x^{4} + 1820 \, a^{4} x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{1820 \, {\left (a^{4} b^{5} x^{15} + 5 \, a^{5} b^{4} x^{12} + 10 \, a^{6} b^{3} x^{9} + 10 \, a^{7} b^{2} x^{6} + 5 \, a^{8} b x^{3} + a^{9}\right )}} \]
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Timed out. \[ \int \frac {a-b x^3}{\left (a+b x^3\right )^{16/3}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (73) = 146\).
Time = 0.19 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.65 \[ \int \frac {a-b x^3}{\left (a+b x^3\right )^{16/3}} \, dx=\frac {{\left (140 \, b^{3} - \frac {546 \, {\left (b x^{3} + a\right )} b^{2}}{x^{3}} + \frac {780 \, {\left (b x^{3} + a\right )}^{2} b}{x^{6}} - \frac {455 \, {\left (b x^{3} + a\right )}^{3}}{x^{9}}\right )} b x^{13}}{1820 \, {\left (b x^{3} + a\right )}^{\frac {13}{3}} a^{4}} + \frac {{\left (35 \, b^{4} - \frac {182 \, {\left (b x^{3} + a\right )} b^{3}}{x^{3}} + \frac {390 \, {\left (b x^{3} + a\right )}^{2} b^{2}}{x^{6}} - \frac {455 \, {\left (b x^{3} + a\right )}^{3} b}{x^{9}} + \frac {455 \, {\left (b x^{3} + a\right )}^{4}}{x^{12}}\right )} x^{13}}{455 \, {\left (b x^{3} + a\right )}^{\frac {13}{3}} a^{4}} \]
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\[ \int \frac {a-b x^3}{\left (a+b x^3\right )^{16/3}} \, dx=\int { -\frac {b x^{3} - a}{{\left (b x^{3} + a\right )}^{\frac {16}{3}}} \,d x } \]
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Time = 5.50 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.78 \[ \int \frac {a-b x^3}{\left (a+b x^3\right )^{16/3}} \, dx=\frac {2\,x}{13\,{\left (b\,x^3+a\right )}^{13/3}}+\frac {891\,x}{1820\,a^4\,{\left (b\,x^3+a\right )}^{1/3}}+\frac {297\,x}{1820\,a^3\,{\left (b\,x^3+a\right )}^{4/3}}+\frac {99\,x}{910\,a^2\,{\left (b\,x^3+a\right )}^{7/3}}+\frac {11\,x}{130\,a\,{\left (b\,x^3+a\right )}^{10/3}} \]
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