Optimal. Leaf size=134 \[ \frac {e F_1\left (1+m;\frac {5}{4},\frac {5}{4};2+m;\frac {a+b \sin (c+d x)}{a-b},\frac {a+b \sin (c+d x)}{a+b}\right ) (a+b \sin (c+d x))^{1+m} \left (1-\frac {a+b \sin (c+d x)}{a-b}\right )^{5/4} \left (1-\frac {a+b \sin (c+d x)}{a+b}\right )^{5/4}}{b d (1+m) (e \cos (c+d x))^{5/2}} \]
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Rubi [A]
time = 0.07, 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 \left (1-\frac {a+b \sin (c+d x)}{a-b}\right )^{5/4} \left (1-\frac {a+b \sin (c+d x)}{a+b}\right )^{5/4} (a+b \sin (c+d x))^{m+1} F_1\left (m+1;\frac {5}{4},\frac {5}{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) (e \cos (c+d x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 143
Rule 2783
Rubi steps
\begin {align*} \int \frac {(a+b \sin (c+d x))^m}{(e \cos (c+d x))^{3/2}} \, dx &=\frac {\left (e \left (1-\frac {a+b \sin (c+d x)}{a-b}\right )^{5/4} \left (1-\frac {a+b \sin (c+d x)}{a+b}\right )^{5/4}\right ) \text {Subst}\left (\int \frac {(a+b x)^m}{\left (-\frac {b}{a-b}-\frac {b x}{a-b}\right )^{5/4} \left (\frac {b}{a+b}-\frac {b x}{a+b}\right )^{5/4}} \, dx,x,\sin (c+d x)\right )}{d (e \cos (c+d x))^{5/2}}\\ &=\frac {e F_1\left (1+m;\frac {5}{4},\frac {5}{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} \left (1-\frac {a+b \sin (c+d x)}{a-b}\right )^{5/4} \left (1-\frac {a+b \sin (c+d x)}{a+b}\right )^{5/4}}{b d (1+m) (e \cos (c+d x))^{5/2}}\\ \end {align*}
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Mathematica [F]
time = 1.67, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b \sin (c+d x))^m}{(e \cos (c+d x))^{3/2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \sin \left (d x +c \right )\right )^{m}}{\left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \sin {\left (c + d x \right )}\right )^{m}}{\left (e \cos {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \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 (a+b\,\sin \left (c+d\,x\right )\right )}^m}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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