Optimal. Leaf size=95 \[ \frac {6 a e^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 d \sqrt {\cos (c+d x)}}-\frac {2 a (e \cos (c+d x))^{7/2}}{7 d e}+\frac {2 a e \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 d} \]
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Rubi [A] time = 0.07, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2669, 2635, 2640, 2639} \[ \frac {6 a e^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 d \sqrt {\cos (c+d x)}}-\frac {2 a (e \cos (c+d x))^{7/2}}{7 d e}+\frac {2 a e \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 d} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2639
Rule 2640
Rule 2669
Rubi steps
\begin {align*} \int (e \cos (c+d x))^{5/2} (a+a \sin (c+d x)) \, dx &=-\frac {2 a (e \cos (c+d x))^{7/2}}{7 d e}+a \int (e \cos (c+d x))^{5/2} \, dx\\ &=-\frac {2 a (e \cos (c+d x))^{7/2}}{7 d e}+\frac {2 a e (e \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {1}{5} \left (3 a e^2\right ) \int \sqrt {e \cos (c+d x)} \, dx\\ &=-\frac {2 a (e \cos (c+d x))^{7/2}}{7 d e}+\frac {2 a e (e \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {\left (3 a e^2 \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 \sqrt {\cos (c+d x)}}\\ &=-\frac {2 a (e \cos (c+d x))^{7/2}}{7 d e}+\frac {6 a e^2 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)}}+\frac {2 a e (e \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}\\ \end {align*}
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Mathematica [C] time = 2.84, size = 264, normalized size = 2.78 \[ \frac {a e^3 \csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right ) \left (168 (\cos (d x)-i \sin (d x)) \sqrt {i \sin (2 (c+d x))+\cos (2 (c+d x))+1} \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right )+56 (\cos (d x)+i \sin (d x)) \sqrt {i \sin (2 (c+d x))+\cos (2 (c+d x))+1} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right )+20 \sin (c+2 d x)-20 \sin (3 c+2 d x)+5 \sin (3 c+4 d x)-5 \sin (5 c+4 d x)-182 \cos (2 c+d x)+14 \cos (2 c+3 d x)-14 \cos (4 c+3 d x)-30 \sin (c)-154 \cos (d x)\right )}{560 d \sqrt {e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a e^{2} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) + a e^{2} \cos \left (d x + c\right )^{2}\right )} \sqrt {e \cos \left (d x + c\right )}, x\right ) \]
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 \[ \int \left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.64, size = 214, normalized size = 2.25 \[ \frac {2 a \,e^{3} \left (-80 \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+56 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+160 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-56 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-120 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+21 \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}+14 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+40 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d} \]
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 \[ \int \left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}\,{d x} \]
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 \[ \int {\left (e\,\cos \left (c+d\,x\right )\right )}^{5/2}\,\left (a+a\,\sin \left (c+d\,x\right )\right ) \,d x \]
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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