Optimal. Leaf size=170 \[ \frac {2 a^3 e^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}-\frac {10 a^3 (e \cos (c+d x))^{7/2}}{21 d e}-\frac {10 \left (a^3 \sin (c+d x)+a^3\right ) (e \cos (c+d x))^{7/2}}{33 d e}+\frac {2 a^3 e \sin (c+d x) (e \cos (c+d x))^{3/2}}{3 d}-\frac {2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{7/2}}{11 d e} \]
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Rubi [A] time = 0.19, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2678, 2669, 2635, 2640, 2639} \[ \frac {2 a^3 e^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}-\frac {10 a^3 (e \cos (c+d x))^{7/2}}{21 d e}-\frac {10 \left (a^3 \sin (c+d x)+a^3\right ) (e \cos (c+d x))^{7/2}}{33 d e}+\frac {2 a^3 e \sin (c+d x) (e \cos (c+d x))^{3/2}}{3 d}-\frac {2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{7/2}}{11 d e} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2639
Rule 2640
Rule 2669
Rule 2678
Rubi steps
\begin {align*} \int (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^3 \, dx &=-\frac {2 a (e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^2}{11 d e}+\frac {1}{11} (15 a) \int (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^2 \, dx\\ &=-\frac {2 a (e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^2}{11 d e}-\frac {10 (e \cos (c+d x))^{7/2} \left (a^3+a^3 \sin (c+d x)\right )}{33 d e}+\frac {1}{3} \left (5 a^2\right ) \int (e \cos (c+d x))^{5/2} (a+a \sin (c+d x)) \, dx\\ &=-\frac {10 a^3 (e \cos (c+d x))^{7/2}}{21 d e}-\frac {2 a (e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^2}{11 d e}-\frac {10 (e \cos (c+d x))^{7/2} \left (a^3+a^3 \sin (c+d x)\right )}{33 d e}+\frac {1}{3} \left (5 a^3\right ) \int (e \cos (c+d x))^{5/2} \, dx\\ &=-\frac {10 a^3 (e \cos (c+d x))^{7/2}}{21 d e}+\frac {2 a^3 e (e \cos (c+d x))^{3/2} \sin (c+d x)}{3 d}-\frac {2 a (e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^2}{11 d e}-\frac {10 (e \cos (c+d x))^{7/2} \left (a^3+a^3 \sin (c+d x)\right )}{33 d e}+\left (a^3 e^2\right ) \int \sqrt {e \cos (c+d x)} \, dx\\ &=-\frac {10 a^3 (e \cos (c+d x))^{7/2}}{21 d e}+\frac {2 a^3 e (e \cos (c+d x))^{3/2} \sin (c+d x)}{3 d}-\frac {2 a (e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^2}{11 d e}-\frac {10 (e \cos (c+d x))^{7/2} \left (a^3+a^3 \sin (c+d x)\right )}{33 d e}+\frac {\left (a^3 e^2 \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{\sqrt {\cos (c+d x)}}\\ &=-\frac {10 a^3 (e \cos (c+d x))^{7/2}}{21 d e}+\frac {2 a^3 e^2 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d \sqrt {\cos (c+d x)}}+\frac {2 a^3 e (e \cos (c+d x))^{3/2} \sin (c+d x)}{3 d}-\frac {2 a (e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^2}{11 d e}-\frac {10 (e \cos (c+d x))^{7/2} \left (a^3+a^3 \sin (c+d x)\right )}{33 d e}\\ \end {align*}
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Mathematica [C] time = 0.11, size = 66, normalized size = 0.39 \[ -\frac {32\ 2^{3/4} a^3 (e \cos (c+d x))^{7/2} \, _2F_1\left (-\frac {15}{4},\frac {7}{4};\frac {11}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{7 d e (\sin (c+d x)+1)^{7/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (3 \, a^{3} e^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{3} e^{2} \cos \left (d x + c\right )^{2} + {\left (a^{3} e^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{3} e^{2} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}, x\right ) \]
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 \[ \int \left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.03, size = 264, normalized size = 1.55 \[ \frac {2 a^{3} e^{3} \left (1344 \left (\sin ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2464 \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-4032 \left (\sin ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4928 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2928 \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3080 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+864 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+616 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-1908 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+231 \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}+804 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-111 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{231 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d} \]
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 \[ \int \left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}^{3}\,{d x} \]
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 \[ \int {\left (e\,\cos \left (c+d\,x\right )\right )}^{5/2}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^3 \,d x \]
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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