Optimal. Leaf size=95 \[ \frac {6 a e^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 d \sqrt {\cos (c+d x)}}+\frac {2 a e \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 d}-\frac {2 b (e \cos (c+d x))^{7/2}}{7 d e} \]
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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} \[ \frac {6 a e^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 d \sqrt {\cos (c+d x)}}+\frac {2 a e \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 d}-\frac {2 b (e \cos (c+d x))^{7/2}}{7 d e} \]
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+b \sin (c+d x)) \, dx &=-\frac {2 b (e \cos (c+d x))^{7/2}}{7 d e}+a \int (e \cos (c+d x))^{5/2} \, dx\\ &=-\frac {2 b (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 b (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 b (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 [A] time = 0.49, size = 79, normalized size = 0.83 \[ \frac {(e \cos (c+d x))^{5/2} \left (\cos ^{\frac {3}{2}}(c+d x) (14 a \sin (c+d x)-5 b \cos (2 (c+d x))-5 b)+42 a E\left (\left .\frac {1}{2} (c+d x)\right |2\right )\right )}{35 d \cos ^{\frac {5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b e^{2} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) + a e^{2} \cos \left (d x + c\right )^{2}\right )} \sqrt {e \cos \left (d x + c\right )}, x\right ) \]
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 \[ \int \left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.41, size = 222, normalized size = 2.34 \[ \frac {2 e^{3} \left (-80 b \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+56 a \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+160 b \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-56 a \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-120 b \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+21 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a +14 a \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+40 b \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d} \]
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 \[ \int \left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}\,{d x} \]
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 \[ \int {\left (e\,\cos \left (c+d\,x\right )\right )}^{5/2}\,\left (a+b\,\sin \left (c+d\,x\right )\right ) \,d x \]
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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