Integrand size = 19, antiderivative size = 72 \[ \int \frac {(c+d x)^{5/6}}{(a+b x)^{7/6}} \, dx=-\frac {6 (c+d x)^{5/6} \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},-\frac {1}{6},\frac {5}{6},-\frac {d (a+b x)}{b c-a d}\right )}{b \sqrt [6]{a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6}} \]
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Time = 0.02 (sec) , antiderivative size = 72, 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} \[ \int \frac {(c+d x)^{5/6}}{(a+b x)^{7/6}} \, dx=-\frac {6 (c+d x)^{5/6} \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},-\frac {1}{6},\frac {5}{6},-\frac {d (a+b x)}{b c-a d}\right )}{b \sqrt [6]{a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6}} \]
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Rule 71
Rule 72
Rubi steps \begin{align*} \text {integral}& = \frac {(c+d x)^{5/6} \int \frac {\left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{5/6}}{(a+b x)^{7/6}} \, dx}{\left (\frac {b (c+d x)}{b c-a d}\right )^{5/6}} \\ & = -\frac {6 (c+d x)^{5/6} \, _2F_1\left (-\frac {5}{6},-\frac {1}{6};\frac {5}{6};-\frac {d (a+b x)}{b c-a d}\right )}{b \sqrt [6]{a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.99 \[ \int \frac {(c+d x)^{5/6}}{(a+b x)^{7/6}} \, dx=-\frac {6 (c+d x)^{5/6} \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},-\frac {1}{6},\frac {5}{6},\frac {d (a+b x)}{-b c+a d}\right )}{b \sqrt [6]{a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{5/6}} \]
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\[\int \frac {\left (d x +c \right )^{\frac {5}{6}}}{\left (b x +a \right )^{\frac {7}{6}}}d x\]
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\[ \int \frac {(c+d x)^{5/6}}{(a+b x)^{7/6}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{6}}}{{\left (b x + a\right )}^{\frac {7}{6}}} \,d x } \]
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\[ \int \frac {(c+d x)^{5/6}}{(a+b x)^{7/6}} \, dx=\int \frac {\left (c + d x\right )^{\frac {5}{6}}}{\left (a + b x\right )^{\frac {7}{6}}}\, dx \]
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\[ \int \frac {(c+d x)^{5/6}}{(a+b x)^{7/6}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{6}}}{{\left (b x + a\right )}^{\frac {7}{6}}} \,d x } \]
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\[ \int \frac {(c+d x)^{5/6}}{(a+b x)^{7/6}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{6}}}{{\left (b x + a\right )}^{\frac {7}{6}}} \,d x } \]
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Timed out. \[ \int \frac {(c+d x)^{5/6}}{(a+b x)^{7/6}} \, dx=\int \frac {{\left (c+d\,x\right )}^{5/6}}{{\left (a+b\,x\right )}^{7/6}} \,d x \]
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