Integrand size = 14, antiderivative size = 83 \[ \int \left (c \sin ^m(a+b x)\right )^{3/2} \, dx=\frac {2 c \cos (a+b x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (2+3 m),\frac {3 (2+m)}{4},\sin ^2(a+b x)\right ) \sin ^{1+m}(a+b x) \sqrt {c \sin ^m(a+b x)}}{b (2+3 m) \sqrt {\cos ^2(a+b x)}} \]
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Time = 0.03 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3287, 2722} \[ \int \left (c \sin ^m(a+b x)\right )^{3/2} \, dx=\frac {2 c \cos (a+b x) \sin ^{m+1}(a+b x) \sqrt {c \sin ^m(a+b x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (3 m+2),\frac {3 (m+2)}{4},\sin ^2(a+b x)\right )}{b (3 m+2) \sqrt {\cos ^2(a+b x)}} \]
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Rule 2722
Rule 3287
Rubi steps \begin{align*} \text {integral}& = \left (c \sin ^{-\frac {m}{2}}(a+b x) \sqrt {c \sin ^m(a+b x)}\right ) \int \sin ^{\frac {3 m}{2}}(a+b x) \, dx \\ & = \frac {2 c \cos (a+b x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (2+3 m),\frac {3 (2+m)}{4},\sin ^2(a+b x)\right ) \sin ^{1+m}(a+b x) \sqrt {c \sin ^m(a+b x)}}{b (2+3 m) \sqrt {\cos ^2(a+b x)}} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.87 \[ \int \left (c \sin ^m(a+b x)\right )^{3/2} \, dx=\frac {2 \sqrt {\cos ^2(a+b x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (2+3 m),\frac {3 (2+m)}{4},\sin ^2(a+b x)\right ) \left (c \sin ^m(a+b x)\right )^{3/2} \tan (a+b x)}{b (2+3 m)} \]
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\[\int {\left (c \left (\sin ^{m}\left (b x +a \right )\right )\right )}^{\frac {3}{2}}d x\]
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Exception generated. \[ \int \left (c \sin ^m(a+b x)\right )^{3/2} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \left (c \sin ^m(a+b x)\right )^{3/2} \, dx=\int \left (c \sin ^{m}{\left (a + b x \right )}\right )^{\frac {3}{2}}\, dx \]
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\[ \int \left (c \sin ^m(a+b x)\right )^{3/2} \, dx=\int { \left (c \sin \left (b x + a\right )^{m}\right )^{\frac {3}{2}} \,d x } \]
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\[ \int \left (c \sin ^m(a+b x)\right )^{3/2} \, dx=\int { \left (c \sin \left (b x + a\right )^{m}\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int \left (c \sin ^m(a+b x)\right )^{3/2} \, dx=\int {\left (c\,{\sin \left (a+b\,x\right )}^m\right )}^{3/2} \,d x \]
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