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