Integrand size = 19, antiderivative size = 72 \[ \int \frac {1}{(a+b x)^{3/2} \sqrt [5]{c+d x}} \, dx=-\frac {2 \sqrt [5]{\frac {b (c+d x)}{b c-a d}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{5},\frac {1}{2},-\frac {d (a+b x)}{b c-a d}\right )}{b \sqrt {a+b x} \sqrt [5]{c+d x}} \]
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Time = 0.01 (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} \[ \int \frac {1}{(a+b x)^{3/2} \sqrt [5]{c+d x}} \, dx=-\frac {2 \sqrt [5]{\frac {b (c+d x)}{b c-a d}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{5},\frac {1}{2},-\frac {d (a+b x)}{b c-a d}\right )}{b \sqrt {a+b x} \sqrt [5]{c+d x}} \]
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Rule 71
Rule 72
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [5]{\frac {b (c+d x)}{b c-a d}} \int \frac {1}{(a+b x)^{3/2} \sqrt [5]{\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}}} \, dx}{\sqrt [5]{c+d x}} \\ & = -\frac {2 \sqrt [5]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (-\frac {1}{2},\frac {1}{5};\frac {1}{2};-\frac {d (a+b x)}{b c-a d}\right )}{b \sqrt {a+b x} \sqrt [5]{c+d x}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.99 \[ \int \frac {1}{(a+b x)^{3/2} \sqrt [5]{c+d x}} \, dx=-\frac {2 \sqrt [5]{\frac {b (c+d x)}{b c-a d}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{5},\frac {1}{2},\frac {d (a+b x)}{-b c+a d}\right )}{b \sqrt {a+b x} \sqrt [5]{c+d x}} \]
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\[\int \frac {1}{\left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {1}{5}}}d x\]
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\[ \int \frac {1}{(a+b x)^{3/2} \sqrt [5]{c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {1}{5}}} \,d x } \]
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\[ \int \frac {1}{(a+b x)^{3/2} \sqrt [5]{c+d x}} \, dx=\int \frac {1}{\left (a + b x\right )^{\frac {3}{2}} \sqrt [5]{c + d x}}\, dx \]
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\[ \int \frac {1}{(a+b x)^{3/2} \sqrt [5]{c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {1}{5}}} \,d x } \]
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\[ \int \frac {1}{(a+b x)^{3/2} \sqrt [5]{c+d x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{\frac {1}{5}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(a+b x)^{3/2} \sqrt [5]{c+d x}} \, dx=\int \frac {1}{{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{1/5}} \,d x \]
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