Integrand size = 33, antiderivative size = 142 \[ \int \sec ^{\frac {3}{2}}(c+d x) (b \sec (c+d x))^n \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {2 C \sec ^{\frac {5}{2}}(c+d x) (b \sec (c+d x))^n \sin (c+d x)}{d (5+2 n)}+\frac {2 (C (3+2 n)+A (5+2 n)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (-1-2 n),\frac {1}{4} (3-2 n),\cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} (b \sec (c+d x))^n \sin (c+d x)}{d (1+2 n) (5+2 n) \sqrt {\sin ^2(c+d x)}} \]
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Time = 0.20 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {20, 4131, 3857, 2722} \[ \int \sec ^{\frac {3}{2}}(c+d x) (b \sec (c+d x))^n \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {2 (A (2 n+5)+C (2 n+3)) \sin (c+d x) \sqrt {\sec (c+d x)} (b \sec (c+d x))^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (-2 n-1),\frac {1}{4} (3-2 n),\cos ^2(c+d x)\right )}{d (2 n+1) (2 n+5) \sqrt {\sin ^2(c+d x)}}+\frac {2 C \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (b \sec (c+d x))^n}{d (2 n+5)} \]
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Rule 20
Rule 2722
Rule 3857
Rule 4131
Rubi steps \begin{align*} \text {integral}& = \left (\sec ^{-n}(c+d x) (b \sec (c+d x))^n\right ) \int \sec ^{\frac {3}{2}+n}(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx \\ & = \frac {2 C \sec ^{\frac {5}{2}}(c+d x) (b \sec (c+d x))^n \sin (c+d x)}{d (5+2 n)}+\frac {\left (\left (C \left (\frac {3}{2}+n\right )+A \left (\frac {5}{2}+n\right )\right ) \sec ^{-n}(c+d x) (b \sec (c+d x))^n\right ) \int \sec ^{\frac {3}{2}+n}(c+d x) \, dx}{\frac {5}{2}+n} \\ & = \frac {2 C \sec ^{\frac {5}{2}}(c+d x) (b \sec (c+d x))^n \sin (c+d x)}{d (5+2 n)}+\frac {\left (\left (C \left (\frac {3}{2}+n\right )+A \left (\frac {5}{2}+n\right )\right ) \cos ^{\frac {1}{2}+n}(c+d x) \sqrt {\sec (c+d x)} (b \sec (c+d x))^n\right ) \int \cos ^{-\frac {3}{2}-n}(c+d x) \, dx}{\frac {5}{2}+n} \\ & = \frac {2 C \sec ^{\frac {5}{2}}(c+d x) (b \sec (c+d x))^n \sin (c+d x)}{d (5+2 n)}+\frac {2 (C (3+2 n)+A (5+2 n)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (-1-2 n),\frac {1}{4} (3-2 n),\cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} (b \sec (c+d x))^n \sin (c+d x)}{d (1+2 n) (5+2 n) \sqrt {\sin ^2(c+d x)}} \\ \end{align*}
Time = 0.49 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00 \[ \int \sec ^{\frac {3}{2}}(c+d x) (b \sec (c+d x))^n \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \csc (c+d x) \sqrt {\sec (c+d x)} (b \sec (c+d x))^n \left (A (7+2 n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (3+2 n),\frac {1}{4} (7+2 n),\sec ^2(c+d x)\right )+C (3+2 n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (7+2 n),\frac {1}{4} (11+2 n),\sec ^2(c+d x)\right ) \sec ^2(c+d x)\right ) \sqrt {-\tan ^2(c+d x)}}{d (3+2 n) (7+2 n)} \]
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\[\int \sec \left (d x +c \right )^{\frac {3}{2}} \left (b \sec \left (d x +c \right )\right )^{n} \left (A +C \sec \left (d x +c \right )^{2}\right )d x\]
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\[ \int \sec ^{\frac {3}{2}}(c+d x) (b \sec (c+d x))^n \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int \sec ^{\frac {3}{2}}(c+d x) (b \sec (c+d x))^n \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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\[ \int \sec ^{\frac {3}{2}}(c+d x) (b \sec (c+d x))^n \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{\frac {3}{2}} \,d x } \]
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\[ \int \sec ^{\frac {3}{2}}(c+d x) (b \sec (c+d x))^n \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} \left (b \sec \left (d x + c\right )\right )^{n} \sec \left (d x + c\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int \sec ^{\frac {3}{2}}(c+d x) (b \sec (c+d x))^n \left (A+C \sec ^2(c+d x)\right ) \, dx=\int \left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^n\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \]
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