Optimal. Leaf size=134 \[ \frac {e F_1\left (1+m;-\frac {1}{4},-\frac {1}{4};2+m;\frac {a+b \sin (c+d x)}{a-b},\frac {a+b \sin (c+d x)}{a+b}\right ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^{1+m}}{b d (1+m) \sqrt [4]{1-\frac {a+b \sin (c+d x)}{a-b}} \sqrt [4]{1-\frac {a+b \sin (c+d x)}{a+b}}} \]
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Rubi [A]
time = 0.08, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2783, 143}
\begin {gather*} \frac {e \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^{m+1} F_1\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 [4]{1-\frac {a+b \sin (c+d x)}{a-b}} \sqrt [4]{1-\frac {a+b \sin (c+d x)}{a+b}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 143
Rule 2783
Rubi steps
\begin {align*} \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^m \, dx &=\frac {\left (e \sqrt {e \cos (c+d x)}\right ) \text {Subst}\left (\int (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 [4]{1-\frac {a+b \sin (c+d x)}{a-b}} \sqrt [4]{1-\frac {a+b \sin (c+d x)}{a+b}}}\\ &=\frac {e F_1\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 ) \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^{1+m}}{b d (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}}}\\ \end {align*}
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Mathematica [F]
time = 5.65, size = 0, normalized size = 0.00 \begin {gather*} \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^m \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}} \left (a +b \sin \left (d x +c \right )\right )^{m}\, dx\]
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 {\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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