Integrand size = 15, antiderivative size = 52 \[ \int \left (a x^3+b x^6\right )^{5/3} \, dx=-\frac {3 a \left (a x^3+b x^6\right )^{8/3}}{88 b^2 x^8}+\frac {\left (a x^3+b x^6\right )^{8/3}}{11 b x^5} \]
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Time = 0.03 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2027, 2039} \[ \int \left (a x^3+b x^6\right )^{5/3} \, dx=\frac {\left (a x^3+b x^6\right )^{8/3}}{11 b x^5}-\frac {3 a \left (a x^3+b x^6\right )^{8/3}}{88 b^2 x^8} \]
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Rule 2027
Rule 2039
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a x^3+b x^6\right )^{8/3}}{11 b x^5}-\frac {(3 a) \int \frac {\left (a x^3+b x^6\right )^{5/3}}{x^3} \, dx}{11 b} \\ & = -\frac {3 a \left (a x^3+b x^6\right )^{8/3}}{88 b^2 x^8}+\frac {\left (a x^3+b x^6\right )^{8/3}}{11 b x^5} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.67 \[ \int \left (a x^3+b x^6\right )^{5/3} \, dx=\frac {\left (x^3 \left (a+b x^3\right )\right )^{8/3} \left (-3 a+8 b x^3\right )}{88 b^2 x^8} \]
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Time = 0.77 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.75
method | result | size |
gosper | \(-\frac {\left (b \,x^{3}+a \right ) \left (-8 b \,x^{3}+3 a \right ) \left (b \,x^{6}+a \,x^{3}\right )^{\frac {5}{3}}}{88 b^{2} x^{5}}\) | \(39\) |
trager | \(-\frac {\left (-8 b^{3} x^{9}-13 b^{2} x^{6} a -2 a^{2} b \,x^{3}+3 a^{3}\right ) \left (b \,x^{6}+a \,x^{3}\right )^{\frac {2}{3}}}{88 b^{2} x^{2}}\) | \(54\) |
risch | \(-\frac {\left (x^{3} \left (b \,x^{3}+a \right )\right )^{\frac {2}{3}} \left (-8 b^{3} x^{9}-13 b^{2} x^{6} a -2 a^{2} b \,x^{3}+3 a^{3}\right )}{88 x^{2} b^{2}}\) | \(54\) |
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Time = 0.26 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.02 \[ \int \left (a x^3+b x^6\right )^{5/3} \, dx=\frac {{\left (8 \, b^{3} x^{9} + 13 \, a b^{2} x^{6} + 2 \, a^{2} b x^{3} - 3 \, a^{3}\right )} {\left (b x^{6} + a x^{3}\right )}^{\frac {2}{3}}}{88 \, b^{2} x^{2}} \]
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\[ \int \left (a x^3+b x^6\right )^{5/3} \, dx=\int \left (a x^{3} + b x^{6}\right )^{\frac {5}{3}}\, dx \]
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none
Time = 0.21 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.88 \[ \int \left (a x^3+b x^6\right )^{5/3} \, dx=\frac {{\left (8 \, b^{3} x^{9} + 13 \, a b^{2} x^{6} + 2 \, a^{2} b x^{3} - 3 \, a^{3}\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{88 \, b^{2}} \]
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\[ \int \left (a x^3+b x^6\right )^{5/3} \, dx=\int { {\left (b x^{6} + a x^{3}\right )}^{\frac {5}{3}} \,d x } \]
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Time = 8.33 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.77 \[ \int \left (a x^3+b x^6\right )^{5/3} \, dx=-\frac {{\left (b\,x^3+a\right )}^2\,{\left (b\,x^6+a\,x^3\right )}^{2/3}\,\left (3\,a-8\,b\,x^3\right )}{88\,b^2\,x^2} \]
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