Integrand size = 19, antiderivative size = 74 \[ \int (a+b x)^{7/6} \sqrt [6]{c+d x} \, dx=\frac {6 (a+b x)^{13/6} \sqrt [6]{c+d x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {13}{6},\frac {19}{6},-\frac {d (a+b x)}{b c-a d}\right )}{13 b \sqrt [6]{\frac {b (c+d x)}{b c-a d}}} \]
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Time = 0.02 (sec) , antiderivative size = 74, 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} \[ \int (a+b x)^{7/6} \sqrt [6]{c+d x} \, dx=\frac {6 (a+b x)^{13/6} \sqrt [6]{c+d x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {13}{6},\frac {19}{6},-\frac {d (a+b x)}{b c-a d}\right )}{13 b \sqrt [6]{\frac {b (c+d x)}{b c-a d}}} \]
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Rule 71
Rule 72
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [6]{c+d x} \int (a+b x)^{7/6} \sqrt [6]{\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \, dx}{\sqrt [6]{\frac {b (c+d x)}{b c-a d}}} \\ & = \frac {6 (a+b x)^{13/6} \sqrt [6]{c+d x} \, _2F_1\left (-\frac {1}{6},\frac {13}{6};\frac {19}{6};-\frac {d (a+b x)}{b c-a d}\right )}{13 b \sqrt [6]{\frac {b (c+d x)}{b c-a d}}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.99 \[ \int (a+b x)^{7/6} \sqrt [6]{c+d x} \, dx=\frac {6 (a+b x)^{13/6} \sqrt [6]{c+d x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {13}{6},\frac {19}{6},\frac {d (a+b x)}{-b c+a d}\right )}{13 b \sqrt [6]{\frac {b (c+d x)}{b c-a d}}} \]
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\[\int \left (b x +a \right )^{\frac {7}{6}} \left (d x +c \right )^{\frac {1}{6}}d x\]
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\[ \int (a+b x)^{7/6} \sqrt [6]{c+d x} \, dx=\int { {\left (b x + a\right )}^{\frac {7}{6}} {\left (d x + c\right )}^{\frac {1}{6}} \,d x } \]
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\[ \int (a+b x)^{7/6} \sqrt [6]{c+d x} \, dx=\int \left (a + b x\right )^{\frac {7}{6}} \sqrt [6]{c + d x}\, dx \]
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\[ \int (a+b x)^{7/6} \sqrt [6]{c+d x} \, dx=\int { {\left (b x + a\right )}^{\frac {7}{6}} {\left (d x + c\right )}^{\frac {1}{6}} \,d x } \]
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\[ \int (a+b x)^{7/6} \sqrt [6]{c+d x} \, dx=\int { {\left (b x + a\right )}^{\frac {7}{6}} {\left (d x + c\right )}^{\frac {1}{6}} \,d x } \]
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Timed out. \[ \int (a+b x)^{7/6} \sqrt [6]{c+d x} \, dx=\int {\left (a+b\,x\right )}^{7/6}\,{\left (c+d\,x\right )}^{1/6} \,d x \]
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