Integrand size = 19, antiderivative size = 66 \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{17/6}} \, dx=\frac {6 (a+b x)^{5/6}}{11 (b c-a d) (c+d x)^{11/6}}+\frac {36 b (a+b x)^{5/6}}{55 (b c-a d)^2 (c+d x)^{5/6}} \]
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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} \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{17/6}} \, dx=\frac {36 b (a+b x)^{5/6}}{55 (c+d x)^{5/6} (b c-a d)^2}+\frac {6 (a+b x)^{5/6}}{11 (c+d x)^{11/6} (b c-a d)} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = \frac {6 (a+b x)^{5/6}}{11 (b c-a d) (c+d x)^{11/6}}+\frac {(6 b) \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{11/6}} \, dx}{11 (b c-a d)} \\ & = \frac {6 (a+b x)^{5/6}}{11 (b c-a d) (c+d x)^{11/6}}+\frac {36 b (a+b x)^{5/6}}{55 (b c-a d)^2 (c+d x)^{5/6}} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.70 \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{17/6}} \, dx=\frac {6 (a+b x)^{5/6} (11 b c-5 a d+6 b d x)}{55 (b c-a d)^2 (c+d x)^{11/6}} \]
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Time = 0.77 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.82
method | result | size |
gosper | \(-\frac {6 \left (b x +a \right )^{\frac {5}{6}} \left (-6 b d x +5 a d -11 b c \right )}{55 \left (d x +c \right )^{\frac {11}{6}} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(54\) |
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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 \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{17/6}} \, dx=\frac {6 \, {\left (6 \, b d x + 11 \, b c - 5 \, a d\right )} {\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {1}{6}}}{55 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2} + {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2} + 2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} x\right )}} \]
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Timed out. \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{17/6}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{17/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {17}{6}}} \,d x } \]
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\[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{17/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {17}{6}}} \,d x } \]
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Time = 1.05 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.92 \[ \int \frac {1}{\sqrt [6]{a+b x} (c+d x)^{17/6}} \, dx=\frac {{\left (c+d\,x\right )}^{1/6}\,\left (\frac {x\,\left (66\,c\,b^2+6\,a\,d\,b\right )}{55\,d^2\,{\left (a\,d-b\,c\right )}^2}-\frac {30\,a^2\,d-66\,a\,b\,c}{55\,d^2\,{\left (a\,d-b\,c\right )}^2}+\frac {36\,b^2\,x^2}{55\,d\,{\left (a\,d-b\,c\right )}^2}\right )}{x^2\,{\left (a+b\,x\right )}^{1/6}+\frac {c^2\,{\left (a+b\,x\right )}^{1/6}}{d^2}+\frac {2\,c\,x\,{\left (a+b\,x\right )}^{1/6}}{d}} \]
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