Integrand size = 19, antiderivative size = 66 \[ \int \frac {1}{(a+b x)^{8/3} \sqrt [3]{c+d x}} \, dx=-\frac {3 (c+d x)^{2/3}}{5 (b c-a d) (a+b x)^{5/3}}+\frac {9 d (c+d x)^{2/3}}{10 (b c-a d)^2 (a+b x)^{2/3}} \]
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Time = 0.01 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {47, 37} \[ \int \frac {1}{(a+b x)^{8/3} \sqrt [3]{c+d x}} \, dx=\frac {9 d (c+d x)^{2/3}}{10 (a+b x)^{2/3} (b c-a d)^2}-\frac {3 (c+d x)^{2/3}}{5 (a+b x)^{5/3} (b c-a d)} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = -\frac {3 (c+d x)^{2/3}}{5 (b c-a d) (a+b x)^{5/3}}-\frac {(3 d) \int \frac {1}{(a+b x)^{5/3} \sqrt [3]{c+d x}} \, dx}{5 (b c-a d)} \\ & = -\frac {3 (c+d x)^{2/3}}{5 (b c-a d) (a+b x)^{5/3}}+\frac {9 d (c+d x)^{2/3}}{10 (b c-a d)^2 (a+b x)^{2/3}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.70 \[ \int \frac {1}{(a+b x)^{8/3} \sqrt [3]{c+d x}} \, dx=-\frac {3 (c+d x)^{2/3} (2 b c-5 a d-3 b d x)}{10 (b c-a d)^2 (a+b x)^{5/3}} \]
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Time = 0.29 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.82
method | result | size |
gosper | \(\frac {3 \left (d x +c \right )^{\frac {2}{3}} \left (3 b d x +5 a d -2 b c \right )}{10 \left (b x +a \right )^{\frac {5}{3}} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(54\) |
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Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (54) = 108\).
Time = 0.23 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.79 \[ \int \frac {1}{(a+b x)^{8/3} \sqrt [3]{c+d x}} \, dx=\frac {3 \, {\left (3 \, b d x - 2 \, b c + 5 \, a d\right )} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{10 \, {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} + {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2} + 2 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x\right )}} \]
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\[ \int \frac {1}{(a+b x)^{8/3} \sqrt [3]{c+d x}} \, dx=\int \frac {1}{\left (a + b x\right )^{\frac {8}{3}} \sqrt [3]{c + d x}}\, dx \]
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\[ \int \frac {1}{(a+b x)^{8/3} \sqrt [3]{c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {8}{3}} {\left (d x + c\right )}^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {1}{(a+b x)^{8/3} \sqrt [3]{c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {8}{3}} {\left (d x + c\right )}^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(a+b x)^{8/3} \sqrt [3]{c+d x}} \, dx=\int \frac {1}{{\left (a+b\,x\right )}^{8/3}\,{\left (c+d\,x\right )}^{1/3}} \,d x \]
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