Integrand size = 19, antiderivative size = 79 \[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{11/6}} \, dx=\frac {6 \sqrt [6]{a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {11}{6},\frac {7}{6},-\frac {d (a+b x)}{b c-a d}\right )}{(b c-a d) (c+d x)^{5/6}} \]
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Time = 0.02 (sec) , antiderivative size = 79, 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 = {72, 71} \[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{11/6}} \, dx=\frac {6 \sqrt [6]{a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {11}{6},\frac {7}{6},-\frac {d (a+b x)}{b c-a d}\right )}{(c+d x)^{5/6} (b c-a d)} \]
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Rule 71
Rule 72
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6}\right ) \int \frac {1}{(a+b x)^{5/6} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{11/6}} \, dx}{(b c-a d) (c+d x)^{5/6}} \\ & = \frac {6 \sqrt [6]{a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6} \, _2F_1\left (\frac {1}{6},\frac {11}{6};\frac {7}{6};-\frac {d (a+b x)}{b c-a d}\right )}{(b c-a d) (c+d x)^{5/6}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{11/6}} \, dx=\frac {6 \sqrt [6]{a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{11/6} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {11}{6},\frac {7}{6},\frac {d (a+b x)}{-b c+a d}\right )}{b (c+d x)^{11/6}} \]
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\[\int \frac {1}{\left (b x +a \right )^{\frac {5}{6}} \left (d x +c \right )^{\frac {11}{6}}}d x\]
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\[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{11/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {11}{6}}} \,d x } \]
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\[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{11/6}} \, dx=\int \frac {1}{\left (a + b x\right )^{\frac {5}{6}} \left (c + d x\right )^{\frac {11}{6}}}\, dx \]
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\[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{11/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {11}{6}}} \,d x } \]
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\[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{11/6}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{6}} {\left (d x + c\right )}^{\frac {11}{6}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(a+b x)^{5/6} (c+d x)^{11/6}} \, dx=\int \frac {1}{{\left (a+b\,x\right )}^{5/6}\,{\left (c+d\,x\right )}^{11/6}} \,d x \]
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