Integrand size = 25, antiderivative size = 134 \[ \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^m \, dx=\frac {e \operatorname {AppellF1}\left (1+m,\frac {1}{4},\frac {1}{4},2+m,\frac {a+b \sin (c+d x)}{a-b},\frac {a+b \sin (c+d x)}{a+b}\right ) (a+b \sin (c+d x))^{1+m} \sqrt [4]{1-\frac {a+b \sin (c+d x)}{a-b}} \sqrt [4]{1-\frac {a+b \sin (c+d x)}{a+b}}}{b d (1+m) \sqrt {e \cos (c+d x)}} \]
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Time = 0.07 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2783, 143} \[ \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^m \, dx=\frac {e \sqrt [4]{1-\frac {a+b \sin (c+d x)}{a-b}} \sqrt [4]{1-\frac {a+b \sin (c+d x)}{a+b}} (a+b \sin (c+d x))^{m+1} \operatorname {AppellF1}\left (m+1,\frac {1}{4},\frac {1}{4},m+2,\frac {a+b \sin (c+d x)}{a-b},\frac {a+b \sin (c+d x)}{a+b}\right )}{b d (m+1) \sqrt {e \cos (c+d x)}} \]
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Rule 143
Rule 2783
Rubi steps \begin{align*} \text {integral}& = \frac {\left (e \sqrt [4]{1-\frac {a+b \sin (c+d x)}{a-b}} \sqrt [4]{1-\frac {a+b \sin (c+d x)}{a+b}}\right ) \text {Subst}\left (\int \frac {(a+b x)^m}{\sqrt [4]{-\frac {b}{a-b}-\frac {b x}{a-b}} \sqrt [4]{\frac {b}{a+b}-\frac {b x}{a+b}}} \, dx,x,\sin (c+d x)\right )}{d \sqrt {e \cos (c+d x)}} \\ & = \frac {e \operatorname {AppellF1}\left (1+m,\frac {1}{4},\frac {1}{4},2+m,\frac {a+b \sin (c+d x)}{a-b},\frac {a+b \sin (c+d x)}{a+b}\right ) (a+b \sin (c+d x))^{1+m} \sqrt [4]{1-\frac {a+b \sin (c+d x)}{a-b}} \sqrt [4]{1-\frac {a+b \sin (c+d x)}{a+b}}}{b d (1+m) \sqrt {e \cos (c+d x)}} \\ \end{align*}
\[ \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^m \, dx=\int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^m \, dx \]
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\[\int \sqrt {e \cos \left (d x +c \right )}\, \left (a +b \sin \left (d x +c \right )\right )^{m}d x\]
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\[ \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^m \, dx=\int { \sqrt {e \cos \left (d x + c\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{m} \,d x } \]
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\[ \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^m \, dx=\int \sqrt {e \cos {\left (c + d x \right )}} \left (a + b \sin {\left (c + d x \right )}\right )^{m}\, dx \]
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\[ \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^m \, dx=\int { \sqrt {e \cos \left (d x + c\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{m} \,d x } \]
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\[ \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^m \, dx=\int { \sqrt {e \cos \left (d x + c\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{m} \,d x } \]
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Timed out. \[ \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^m \, dx=\int \sqrt {e\,\cos \left (c+d\,x\right )}\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^m \,d x \]
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