Optimal. Leaf size=197 \[ \frac {2 a e^2 \left (3 a^2+2 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 d \sqrt {\cos (c+d x)}}-\frac {2 b \left (43 a^2+12 b^2\right ) (e \cos (c+d x))^{7/2}}{231 d e}+\frac {2 a e \left (3 a^2+2 b^2\right ) \sin (c+d x) (e \cos (c+d x))^{3/2}}{15 d}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}-\frac {10 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{33 d e} \]
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Rubi [A] time = 0.29, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2692, 2862, 2669, 2635, 2640, 2639} \[ \frac {2 a e^2 \left (3 a^2+2 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 d \sqrt {\cos (c+d x)}}-\frac {2 b \left (43 a^2+12 b^2\right ) (e \cos (c+d x))^{7/2}}{231 d e}+\frac {2 a e \left (3 a^2+2 b^2\right ) \sin (c+d x) (e \cos (c+d x))^{3/2}}{15 d}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}-\frac {10 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{33 d e} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2639
Rule 2640
Rule 2669
Rule 2692
Rule 2862
Rubi steps
\begin {align*} \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3 \, dx &=-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}+\frac {2}{11} \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x)) \left (\frac {11 a^2}{2}+2 b^2+\frac {15}{2} a b \sin (c+d x)\right ) \, dx\\ &=-\frac {10 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{33 d e}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}+\frac {4}{99} \int (e \cos (c+d x))^{5/2} \left (\frac {33}{4} a \left (3 a^2+2 b^2\right )+\frac {3}{4} b \left (43 a^2+12 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=-\frac {2 b \left (43 a^2+12 b^2\right ) (e \cos (c+d x))^{7/2}}{231 d e}-\frac {10 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{33 d e}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}+\frac {1}{3} \left (a \left (3 a^2+2 b^2\right )\right ) \int (e \cos (c+d x))^{5/2} \, dx\\ &=-\frac {2 b \left (43 a^2+12 b^2\right ) (e \cos (c+d x))^{7/2}}{231 d e}+\frac {2 a \left (3 a^2+2 b^2\right ) e (e \cos (c+d x))^{3/2} \sin (c+d x)}{15 d}-\frac {10 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{33 d e}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}+\frac {1}{5} \left (a \left (3 a^2+2 b^2\right ) e^2\right ) \int \sqrt {e \cos (c+d x)} \, dx\\ &=-\frac {2 b \left (43 a^2+12 b^2\right ) (e \cos (c+d x))^{7/2}}{231 d e}+\frac {2 a \left (3 a^2+2 b^2\right ) e (e \cos (c+d x))^{3/2} \sin (c+d x)}{15 d}-\frac {10 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{33 d e}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}+\frac {\left (a \left (3 a^2+2 b^2\right ) 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 \left (43 a^2+12 b^2\right ) (e \cos (c+d x))^{7/2}}{231 d e}+\frac {2 a \left (3 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 )}{5 d \sqrt {\cos (c+d x)}}+\frac {2 a \left (3 a^2+2 b^2\right ) e (e \cos (c+d x))^{3/2} \sin (c+d x)}{15 d}-\frac {10 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{33 d e}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}\\ \end {align*}
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Mathematica [A] time = 1.41, size = 150, normalized size = 0.76 \[ \frac {(e \cos (c+d x))^{5/2} \left (1848 \left (3 a^3+2 a b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\cos ^{\frac {3}{2}}(c+d x) \left (1848 a^3 \sin (c+d x)-60 \left (33 a^2 b+4 b^3\right ) \cos (2 (c+d x))-1980 a^2 b+462 a b^2 \sin (c+d x)-770 a b^2 \sin (3 (c+d x))+105 b^3 \cos (4 (c+d x))-345 b^3\right )\right )}{4620 d \cos ^{\frac {5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.01, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (3 \, a b^{2} e^{2} \cos \left (d x + c\right )^{4} - {\left (a^{3} + 3 \, a b^{2}\right )} e^{2} \cos \left (d x + c\right )^{2} + {\left (b^{3} e^{2} \cos \left (d x + c\right )^{4} - {\left (3 \, a^{2} b + b^{3}\right )} e^{2} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}, x\right ) \]
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 \[ \int \left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 3.09, size = 534, normalized size = 2.71 \[ \frac {2 e^{3} \left (6720 b^{3} \left (\sin ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12320 a \,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 )-20160 b^{3} \left (\sin ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7920 a^{2} b \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24640 a \,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 )+22560 b^{3} \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1848 a^{3} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+15840 a^{2} b \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-17248 a \,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 )-11520 b^{3} \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1848 a^{3} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-11880 a^{2} b \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4928 a \,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 )+2340 b^{3} \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+693 \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^{3}+462 \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 \,b^{2}+462 a^{3} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3960 a^{2} b \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-462 a \,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 )+60 b^{3} \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-495 a^{2} b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )-60 b^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1155 \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 \[ \int \left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{3}\,{d x} \]
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 \[ \int {\left (e\,\cos \left (c+d\,x\right )\right )}^{5/2}\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^3 \,d x \]
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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