Integrand size = 25, antiderivative size = 163 \[ \int \frac {\sqrt [3]{\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}} \, dx=\frac {3 \operatorname {AppellF1}\left (\frac {4}{3},1,\frac {1}{2},\frac {7}{3},-i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right ) \tan ^{\frac {4}{3}}(c+d x) \sqrt {1+\frac {b \tan (c+d x)}{a}}}{8 d \sqrt {a+b \tan (c+d x)}}+\frac {3 \operatorname {AppellF1}\left (\frac {4}{3},1,\frac {1}{2},\frac {7}{3},i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right ) \tan ^{\frac {4}{3}}(c+d x) \sqrt {1+\frac {b \tan (c+d x)}{a}}}{8 d \sqrt {a+b \tan (c+d x)}} \]
[Out]
Time = 0.30 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3656, 926, 129, 525, 524} \[ \int \frac {\sqrt [3]{\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}} \, dx=\frac {3 \tan ^{\frac {4}{3}}(c+d x) \sqrt {\frac {b \tan (c+d x)}{a}+1} \operatorname {AppellF1}\left (\frac {4}{3},1,\frac {1}{2},\frac {7}{3},-i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right )}{8 d \sqrt {a+b \tan (c+d x)}}+\frac {3 \tan ^{\frac {4}{3}}(c+d x) \sqrt {\frac {b \tan (c+d x)}{a}+1} \operatorname {AppellF1}\left (\frac {4}{3},1,\frac {1}{2},\frac {7}{3},i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right )}{8 d \sqrt {a+b \tan (c+d x)}} \]
[In]
[Out]
Rule 129
Rule 524
Rule 525
Rule 926
Rule 3656
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\sqrt [3]{x}}{\sqrt {a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (\frac {i \sqrt [3]{x}}{2 (i-x) \sqrt {a+b x}}+\frac {i \sqrt [3]{x}}{2 (i+x) \sqrt {a+b x}}\right ) \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {i \text {Subst}\left (\int \frac {\sqrt [3]{x}}{(i-x) \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac {i \text {Subst}\left (\int \frac {\sqrt [3]{x}}{(i+x) \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d} \\ & = \frac {(3 i) \text {Subst}\left (\int \frac {x^3}{\left (i-x^3\right ) \sqrt {a+b x^3}} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{2 d}+\frac {(3 i) \text {Subst}\left (\int \frac {x^3}{\left (i+x^3\right ) \sqrt {a+b x^3}} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{2 d} \\ & = \frac {\left (3 i \sqrt {1+\frac {b \tan (c+d x)}{a}}\right ) \text {Subst}\left (\int \frac {x^3}{\left (i-x^3\right ) \sqrt {1+\frac {b x^3}{a}}} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{2 d \sqrt {a+b \tan (c+d x)}}+\frac {\left (3 i \sqrt {1+\frac {b \tan (c+d x)}{a}}\right ) \text {Subst}\left (\int \frac {x^3}{\left (i+x^3\right ) \sqrt {1+\frac {b x^3}{a}}} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{2 d \sqrt {a+b \tan (c+d x)}} \\ & = \frac {3 \operatorname {AppellF1}\left (\frac {4}{3},1,\frac {1}{2},\frac {7}{3},-i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right ) \tan ^{\frac {4}{3}}(c+d x) \sqrt {1+\frac {b \tan (c+d x)}{a}}}{8 d \sqrt {a+b \tan (c+d x)}}+\frac {3 \operatorname {AppellF1}\left (\frac {4}{3},1,\frac {1}{2},\frac {7}{3},i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right ) \tan ^{\frac {4}{3}}(c+d x) \sqrt {1+\frac {b \tan (c+d x)}{a}}}{8 d \sqrt {a+b \tan (c+d x)}} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(6076\) vs. \(2(163)=326\).
Time = 58.56 (sec) , antiderivative size = 6076, normalized size of antiderivative = 37.28 \[ \int \frac {\sqrt [3]{\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}} \, dx=\text {Result too large to show} \]
[In]
[Out]
\[\int \frac {\tan ^{\frac {1}{3}}\left (d x +c \right )}{\sqrt {a +b \tan \left (d x +c \right )}}d x\]
[In]
[Out]
Timed out. \[ \int \frac {\sqrt [3]{\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {\sqrt [3]{\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}} \, dx=\int \frac {\sqrt [3]{\tan {\left (c + d x \right )}}}{\sqrt {a + b \tan {\left (c + d x \right )}}}\, dx \]
[In]
[Out]
\[ \int \frac {\sqrt [3]{\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}} \, dx=\int { \frac {\tan \left (d x + c\right )^{\frac {1}{3}}}{\sqrt {b \tan \left (d x + c\right ) + a}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\sqrt [3]{\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}} \, dx=\text {Timed out} \]
[In]
[Out]
Timed out. \[ \int \frac {\sqrt [3]{\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}} \, dx=\int \frac {{\mathrm {tan}\left (c+d\,x\right )}^{1/3}}{\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}} \,d x \]
[In]
[Out]