Integrand size = 41, antiderivative size = 154 \[ \int \sec ^2(c+d x) (b \sec (c+d x))^{2/3} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 (11 A+8 C) \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},\frac {1}{2},\frac {1}{6},\cos ^2(c+d x)\right ) (b \sec (c+d x))^{5/3} \sin (c+d x)}{55 b d \sqrt {\sin ^2(c+d x)}}+\frac {3 B \operatorname {Hypergeometric2F1}\left (-\frac {4}{3},\frac {1}{2},-\frac {1}{3},\cos ^2(c+d x)\right ) (b \sec (c+d x))^{8/3} \sin (c+d x)}{8 b^2 d \sqrt {\sin ^2(c+d x)}}+\frac {3 C (b \sec (c+d x))^{8/3} \tan (c+d x)}{11 b^2 d} \]
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Time = 0.26 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {16, 4132, 3857, 2722, 4131} \[ \int \sec ^2(c+d x) (b \sec (c+d x))^{2/3} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 (11 A+8 C) \sin (c+d x) (b \sec (c+d x))^{5/3} \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},\frac {1}{2},\frac {1}{6},\cos ^2(c+d x)\right )}{55 b d \sqrt {\sin ^2(c+d x)}}+\frac {3 B \sin (c+d x) (b \sec (c+d x))^{8/3} \operatorname {Hypergeometric2F1}\left (-\frac {4}{3},\frac {1}{2},-\frac {1}{3},\cos ^2(c+d x)\right )}{8 b^2 d \sqrt {\sin ^2(c+d x)}}+\frac {3 C \tan (c+d x) (b \sec (c+d x))^{8/3}}{11 b^2 d} \]
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Rule 16
Rule 2722
Rule 3857
Rule 4131
Rule 4132
Rubi steps \begin{align*} \text {integral}& = \frac {\int (b \sec (c+d x))^{8/3} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx}{b^2} \\ & = \frac {\int (b \sec (c+d x))^{8/3} \left (A+C \sec ^2(c+d x)\right ) \, dx}{b^2}+\frac {B \int (b \sec (c+d x))^{11/3} \, dx}{b^3} \\ & = \frac {3 C (b \sec (c+d x))^{8/3} \tan (c+d x)}{11 b^2 d}+\frac {(11 A+8 C) \int (b \sec (c+d x))^{8/3} \, dx}{11 b^2}+\frac {\left (B \left (\frac {\cos (c+d x)}{b}\right )^{2/3} (b \sec (c+d x))^{2/3}\right ) \int \frac {1}{\left (\frac {\cos (c+d x)}{b}\right )^{11/3}} \, dx}{b^3} \\ & = \frac {3 C (b \sec (c+d x))^{8/3} \tan (c+d x)}{11 b^2 d}+\frac {3 B \operatorname {Hypergeometric2F1}\left (-\frac {4}{3},\frac {1}{2},-\frac {1}{3},\cos ^2(c+d x)\right ) \sec (c+d x) (b \sec (c+d x))^{2/3} \tan (c+d x)}{8 d \sqrt {\sin ^2(c+d x)}}+\frac {\left ((11 A+8 C) \left (\frac {\cos (c+d x)}{b}\right )^{2/3} (b \sec (c+d x))^{2/3}\right ) \int \frac {1}{\left (\frac {\cos (c+d x)}{b}\right )^{8/3}} \, dx}{11 b^2} \\ & = \frac {3 (11 A+8 C) \operatorname {Hypergeometric2F1}\left (-\frac {5}{6},\frac {1}{2},\frac {1}{6},\cos ^2(c+d x)\right ) (b \sec (c+d x))^{5/3} \sin (c+d x)}{55 b d \sqrt {\sin ^2(c+d x)}}+\frac {3 C (b \sec (c+d x))^{8/3} \tan (c+d x)}{11 b^2 d}+\frac {3 B \operatorname {Hypergeometric2F1}\left (-\frac {4}{3},\frac {1}{2},-\frac {1}{3},\cos ^2(c+d x)\right ) \sec (c+d x) (b \sec (c+d x))^{2/3} \tan (c+d x)}{8 d \sqrt {\sin ^2(c+d x)}} \\ \end{align*}
Time = 0.54 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.78 \[ \int \sec ^2(c+d x) (b \sec (c+d x))^{2/3} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \csc (c+d x) (b \sec (c+d x))^{5/3} \left (77 A \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4}{3},\frac {7}{3},\sec ^2(c+d x)\right )+56 B \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {11}{6},\frac {17}{6},\sec ^2(c+d x)\right ) \sec (c+d x)+44 C \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{3},\frac {10}{3},\sec ^2(c+d x)\right ) \sec ^2(c+d x)\right ) \sqrt {-\tan ^2(c+d x)}}{616 b d} \]
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\[\int \sec \left (d x +c \right )^{2} \left (b \sec \left (d x +c \right )\right )^{\frac {2}{3}} \left (A +B \sec \left (d x +c \right )+C \sec \left (d x +c \right )^{2}\right )d x\]
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\[ \int \sec ^2(c+d x) (b \sec (c+d x))^{2/3} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac {2}{3}} \sec \left (d x + c\right )^{2} \,d x } \]
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\[ \int \sec ^2(c+d x) (b \sec (c+d x))^{2/3} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \left (b \sec {\left (c + d x \right )}\right )^{\frac {2}{3}} \left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{2}{\left (c + d x \right )}\, dx \]
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\[ \int \sec ^2(c+d x) (b \sec (c+d x))^{2/3} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac {2}{3}} \sec \left (d x + c\right )^{2} \,d x } \]
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\[ \int \sec ^2(c+d x) (b \sec (c+d x))^{2/3} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \left (b \sec \left (d x + c\right )\right )^{\frac {2}{3}} \sec \left (d x + c\right )^{2} \,d x } \]
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Timed out. \[ \int \sec ^2(c+d x) (b \sec (c+d x))^{2/3} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \frac {{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{2/3}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\cos \left (c+d\,x\right )}^2} \,d x \]
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