Optimal. Leaf size=134 \[ \frac {e \left (1-\frac {a+b \sin (c+d x)}{a-b}\right )^{7/4} \left (1-\frac {a+b \sin (c+d x)}{a+b}\right )^{7/4} (a+b \sin (c+d x))^{m+1} F_1\left (m+1;\frac {7}{4},\frac {7}{4};m+2;\frac {a+b \sin (c+d x)}{a-b},\frac {a+b \sin (c+d x)}{a+b}\right )}{b d (m+1) (e \cos (c+d x))^{7/2}} \]
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Rubi [A] time = 0.10, 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 = {2704, 138} \[ \frac {e \left (1-\frac {a+b \sin (c+d x)}{a-b}\right )^{7/4} \left (1-\frac {a+b \sin (c+d x)}{a+b}\right )^{7/4} (a+b \sin (c+d x))^{m+1} F_1\left (m+1;\frac {7}{4},\frac {7}{4};m+2;\frac {a+b \sin (c+d x)}{a-b},\frac {a+b \sin (c+d x)}{a+b}\right )}{b d (m+1) (e \cos (c+d x))^{7/2}} \]
Antiderivative was successfully verified.
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Rule 138
Rule 2704
Rubi steps
\begin {align*} \int \frac {(a+b \sin (c+d x))^m}{(e \cos (c+d x))^{5/2}} \, dx &=\frac {\left (e \left (1-\frac {a+b \sin (c+d x)}{a-b}\right )^{7/4} \left (1-\frac {a+b \sin (c+d x)}{a+b}\right )^{7/4}\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^m}{\left (-\frac {b}{a-b}-\frac {b x}{a-b}\right )^{7/4} \left (\frac {b}{a+b}-\frac {b x}{a+b}\right )^{7/4}} \, dx,x,\sin (c+d x)\right )}{d (e \cos (c+d x))^{7/2}}\\ &=\frac {e F_1\left (1+m;\frac {7}{4},\frac {7}{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 )^{7/4} \left (1-\frac {a+b \sin (c+d x)}{a+b}\right )^{7/4}}{b d (1+m) (e \cos (c+d x))^{7/2}}\\ \end {align*}
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Mathematica [F] time = 2.10, size = 0, normalized size = 0.00 \[ \int \frac {(a+b \sin (c+d x))^m}{(e \cos (c+d x))^{5/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.91, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e \cos \left (d x + c\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{m}}{e^{3} \cos \left (d x + c\right )^{3}}, x\right ) \]
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 \[ \int \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{m}}{\left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.18, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \sin \left (d x +c \right )\right )^{m}}{\left (e \cos \left (d x +c \right )\right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{m}}{\left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\sin \left (c+d\,x\right )\right )}^m}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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