Optimal. Leaf size=84 \[ -\frac {2^{\frac {9}{4}+m} g \sqrt {g \cos (e+f x)} \, _2F_1\left (\frac {1}{4},-\frac {1}{4}-m;\frac {5}{4};\frac {1}{2} (1-\sin (e+f x))\right ) (1+\sin (e+f x))^{-\frac {1}{4}-m} (a+a \sin (e+f x))^m}{c f} \]
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Rubi [A]
time = 0.18, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2919, 2768, 72,
71} \begin {gather*} -\frac {g 2^{m+\frac {9}{4}} \sqrt {g \cos (e+f x)} (\sin (e+f x)+1)^{-m-\frac {1}{4}} (a \sin (e+f x)+a)^m \, _2F_1\left (\frac {1}{4},-m-\frac {1}{4};\frac {5}{4};\frac {1}{2} (1-\sin (e+f x))\right )}{c f} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 72
Rule 2768
Rule 2919
Rubi steps
\begin {align*} \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m}{c-c \sin (e+f x)} \, dx &=\frac {g^2 \int \frac {(a+a \sin (e+f x))^{1+m}}{\sqrt {g \cos (e+f x)}} \, dx}{a c}\\ &=\frac {\left (a g \sqrt {g \cos (e+f x)}\right ) \text {Subst}\left (\int \frac {(a+a x)^{\frac {1}{4}+m}}{(a-a x)^{3/4}} \, dx,x,\sin (e+f x)\right )}{c f \sqrt [4]{a-a \sin (e+f x)} \sqrt [4]{a+a \sin (e+f x)}}\\ &=\frac {\left (2^{\frac {1}{4}+m} a g \sqrt {g \cos (e+f x)} (a+a \sin (e+f x))^m \left (\frac {a+a \sin (e+f x)}{a}\right )^{-\frac {1}{4}-m}\right ) \text {Subst}\left (\int \frac {\left (\frac {1}{2}+\frac {x}{2}\right )^{\frac {1}{4}+m}}{(a-a x)^{3/4}} \, dx,x,\sin (e+f x)\right )}{c f \sqrt [4]{a-a \sin (e+f x)}}\\ &=-\frac {2^{\frac {9}{4}+m} g \sqrt {g \cos (e+f x)} \, _2F_1\left (\frac {1}{4},-\frac {1}{4}-m;\frac {5}{4};\frac {1}{2} (1-\sin (e+f x))\right ) (1+\sin (e+f x))^{-\frac {1}{4}-m} (a+a \sin (e+f x))^m}{c f}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 84, normalized size = 1.00 \begin {gather*} -\frac {2^{\frac {9}{4}+m} g \sqrt {g \cos (e+f x)} \, _2F_1\left (\frac {1}{4},-\frac {1}{4}-m;\frac {5}{4};\frac {1}{2} (1-\sin (e+f x))\right ) (1+\sin (e+f x))^{-\frac {1}{4}-m} (a (1+\sin (e+f x)))^m}{c f} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.12, size = 0, normalized size = 0.00 \[\int \frac {\left (g \cos \left (f x +e \right )\right )^{\frac {3}{2}} \left (a +a \sin \left (f x +e \right )\right )^{m}}{c -c \sin \left (f x +e \right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m}{c-c\,\sin \left (e+f\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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