Integrand size = 19, antiderivative size = 32 \[ \int \frac {1}{(a+b x)^{5/3} \sqrt [3]{c+d x}} \, dx=-\frac {3 (c+d x)^{2/3}}{2 (b c-a d) (a+b x)^{2/3}} \]
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Time = 0.00 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {37} \[ \int \frac {1}{(a+b x)^{5/3} \sqrt [3]{c+d x}} \, dx=-\frac {3 (c+d x)^{2/3}}{2 (a+b x)^{2/3} (b c-a d)} \]
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Rule 37
Rubi steps \begin{align*} \text {integral}& = -\frac {3 (c+d x)^{2/3}}{2 (b c-a d) (a+b x)^{2/3}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(a+b x)^{5/3} \sqrt [3]{c+d x}} \, dx=-\frac {3 (c+d x)^{2/3}}{2 (b c-a d) (a+b x)^{2/3}} \]
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Time = 0.29 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84
method | result | size |
gosper | \(\frac {3 \left (d x +c \right )^{\frac {2}{3}}}{2 \left (b x +a \right )^{\frac {2}{3}} \left (a d -b c \right )}\) | \(27\) |
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none
Time = 0.23 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.31 \[ \int \frac {1}{(a+b x)^{5/3} \sqrt [3]{c+d x}} \, dx=-\frac {3 \, {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{2 \, {\left (a b c - a^{2} d + {\left (b^{2} c - a b d\right )} x\right )}} \]
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\[ \int \frac {1}{(a+b x)^{5/3} \sqrt [3]{c+d x}} \, dx=\int \frac {1}{\left (a + b x\right )^{\frac {5}{3}} \sqrt [3]{c + d x}}\, dx \]
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\[ \int \frac {1}{(a+b x)^{5/3} \sqrt [3]{c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{3}} {\left (d x + c\right )}^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {1}{(a+b x)^{5/3} \sqrt [3]{c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{3}} {\left (d x + c\right )}^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(a+b x)^{5/3} \sqrt [3]{c+d x}} \, dx=\int \frac {1}{{\left (a+b\,x\right )}^{5/3}\,{\left (c+d\,x\right )}^{1/3}} \,d x \]
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