Integrand size = 19, antiderivative size = 30 \[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{7/6}} \, dx=\frac {6 \sqrt [6]{a+b x}}{(b c-a d) \sqrt [6]{c+d x}} \]
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Time = 0.00 (sec) , antiderivative size = 30, 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} \[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{7/6}} \, dx=\frac {6 \sqrt [6]{a+b x}}{\sqrt [6]{c+d x} (b c-a d)} \]
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Rule 37
Rubi steps \begin{align*} \text {integral}& = \frac {6 \sqrt [6]{a+b x}}{(b c-a d) \sqrt [6]{c+d x}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{7/6}} \, dx=\frac {6 \sqrt [6]{a+b x}}{(b c-a d) \sqrt [6]{c+d x}} \]
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Time = 0.40 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90
method | result | size |
gosper | \(-\frac {6 \left (b x +a \right )^{\frac {1}{6}}}{\left (d x +c \right )^{\frac {1}{6}} \left (a d -b c \right )}\) | \(27\) |
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none
Time = 0.22 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40 \[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{7/6}} \, dx=\frac {6 \, {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {5}{6}}}{b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x} \]
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\[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{7/6}} \, dx=\int \frac {1}{\left (a + b x\right )^{\frac {5}{6}} \left (c + d x\right )^{\frac {7}{6}}}\, dx \]
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\[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{7/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {7}{6}}} \,d x } \]
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\[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{7/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {7}{6}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{7/6}} \, dx=\int \frac {1}{{\left (a+b\,x\right )}^{5/6}\,{\left (c+d\,x\right )}^{7/6}} \,d x \]
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