Optimal. Leaf size=149 \[ \frac {2 e^2 \left (9 a^2+2 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{15 d \sqrt {\cos (c+d x)}}+\frac {2 e \left (9 a^2+2 b^2\right ) \sin (c+d x) (e \cos (c+d x))^{3/2}}{45 d}-\frac {22 a b (e \cos (c+d x))^{7/2}}{63 d e}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{9 d e} \]
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Rubi [A] time = 0.17, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2692, 2669, 2635, 2640, 2639} \[ \frac {2 e^2 \left (9 a^2+2 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{15 d \sqrt {\cos (c+d x)}}+\frac {2 e \left (9 a^2+2 b^2\right ) \sin (c+d x) (e \cos (c+d x))^{3/2}}{45 d}-\frac {22 a b (e \cos (c+d x))^{7/2}}{63 d e}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{9 d e} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2639
Rule 2640
Rule 2669
Rule 2692
Rubi steps
\begin {align*} \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2 \, dx &=-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{9 d e}+\frac {2}{9} \int (e \cos (c+d x))^{5/2} \left (\frac {9 a^2}{2}+b^2+\frac {11}{2} a b \sin (c+d x)\right ) \, dx\\ &=-\frac {22 a b (e \cos (c+d x))^{7/2}}{63 d e}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{9 d e}+\frac {1}{9} \left (9 a^2+2 b^2\right ) \int (e \cos (c+d x))^{5/2} \, dx\\ &=-\frac {22 a b (e \cos (c+d x))^{7/2}}{63 d e}+\frac {2 \left (9 a^2+2 b^2\right ) e (e \cos (c+d x))^{3/2} \sin (c+d x)}{45 d}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{9 d e}+\frac {1}{15} \left (\left (9 a^2+2 b^2\right ) e^2\right ) \int \sqrt {e \cos (c+d x)} \, dx\\ &=-\frac {22 a b (e \cos (c+d x))^{7/2}}{63 d e}+\frac {2 \left (9 a^2+2 b^2\right ) e (e \cos (c+d x))^{3/2} \sin (c+d x)}{45 d}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{9 d e}+\frac {\left (\left (9 a^2+2 b^2\right ) e^2 \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{15 \sqrt {\cos (c+d x)}}\\ &=-\frac {22 a b (e \cos (c+d x))^{7/2}}{63 d e}+\frac {2 \left (9 a^2+2 b^2\right ) e^2 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d \sqrt {\cos (c+d x)}}+\frac {2 \left (9 a^2+2 b^2\right ) e (e \cos (c+d x))^{3/2} \sin (c+d x)}{45 d}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{9 d e}\\ \end {align*}
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Mathematica [A] time = 0.92, size = 113, normalized size = 0.76 \[ \frac {(e \cos (c+d x))^{5/2} \left (84 \left (9 a^2+2 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\cos ^{\frac {3}{2}}(c+d x) \left (21 \left (12 a^2+b^2\right ) \sin (c+d x)-5 b (36 a+7 b \sin (3 (c+d x)))-180 a b \cos (2 (c+d x))\right )\right )}{630 d \cos ^{\frac {5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.98, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (b^{2} e^{2} \cos \left (d x + c\right )^{4} - 2 \, a b e^{2} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - {\left (a^{2} + b^{2}\right )} e^{2} \cos \left (d x + c\right )^{2}\right )} \sqrt {e \cos \left (d x + c\right )}, x\right ) \]
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 \[ \int \left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.51, size = 408, normalized size = 2.74 \[ \frac {2 e^{3} \left (-1120 b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1440 a b \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2240 b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+504 a^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2880 a b \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1568 b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-504 a^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2160 a b \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+448 b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+189 \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^{2}+42 \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 ) b^{2}+126 a^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+720 a b \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-42 b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-90 a b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{315 \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 \[ \int \left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{2}\,{d x} \]
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 \[ \int {\left (e\,\cos \left (c+d\,x\right )\right )}^{5/2}\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^2 \,d x \]
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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