Integrand size = 25, antiderivative size = 25 \[ \int \sec ^{\frac {2}{3}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx=\text {Int}\left (\sec ^{\frac {2}{3}}(c+d x) (a+b \sec (c+d x))^{3/2},x\right ) \]
[Out]
Not integrable
Time = 0.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \sec ^{\frac {2}{3}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx=\int \sec ^{\frac {2}{3}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \int \sec ^{\frac {2}{3}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx \\ \end{align*}
Not integrable
Time = 135.55 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \sec ^{\frac {2}{3}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx=\int \sec ^{\frac {2}{3}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx \]
[In]
[Out]
Not integrable
Time = 0.48 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84
\[\int \sec \left (d x +c \right )^{\frac {2}{3}} \left (a +b \sec \left (d x +c \right )\right )^{\frac {3}{2}}d x\]
[In]
[Out]
Not integrable
Time = 1.05 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \sec ^{\frac {2}{3}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{\frac {2}{3}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \sec ^{\frac {2}{3}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx=\text {Timed out} \]
[In]
[Out]
Not integrable
Time = 0.74 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \sec ^{\frac {2}{3}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{\frac {2}{3}} \,d x } \]
[In]
[Out]
Not integrable
Time = 2.31 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \sec ^{\frac {2}{3}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{\frac {2}{3}} \,d x } \]
[In]
[Out]
Not integrable
Time = 14.75 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \sec ^{\frac {2}{3}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx=\int {\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{2/3} \,d x \]
[In]
[Out]