Optimal. Leaf size=83 \[ \frac{2 c \cos (a+b x) \sin ^{m+1}(a+b x) \sqrt{c \sin ^m(a+b x)} \, _2F_1\left (\frac{1}{2},\frac{1}{4} (3 m+2);\frac{3 (m+2)}{4};\sin ^2(a+b x)\right )}{b (3 m+2) \sqrt{\cos ^2(a+b x)}} \]
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Rubi [A] time = 0.0379316, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3208, 2643} \[ \frac{2 c \cos (a+b x) \sin ^{m+1}(a+b x) \sqrt{c \sin ^m(a+b x)} \, _2F_1\left (\frac{1}{2},\frac{1}{4} (3 m+2);\frac{3 (m+2)}{4};\sin ^2(a+b x)\right )}{b (3 m+2) \sqrt{\cos ^2(a+b x)}} \]
Antiderivative was successfully verified.
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Rule 3208
Rule 2643
Rubi steps
\begin{align*} \int \left (c \sin ^m(a+b x)\right )^{3/2} \, dx &=\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) \, _2F_1\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*}
Mathematica [A] time = 0.118059, size = 72, normalized size = 0.87 \[ \frac{2 \sqrt{\cos ^2(a+b x)} \tan (a+b x) \left (c \sin ^m(a+b x)\right )^{3/2} \, _2F_1\left (\frac{1}{2},\frac{1}{4} (3 m+2);\frac{3 (m+2)}{4};\sin ^2(a+b x)\right )}{b (3 m+2)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.19, size = 0, normalized size = 0. \begin{align*} \int \left ( c \left ( \sin \left ( bx+a \right ) \right ) ^{m} \right ) ^{{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c \sin \left (b x + a\right )^{m}\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c \sin \left (b x + a\right )^{m}\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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