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 = 0.189465, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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 2678
Rule 2669
Rule 2635
Rule 2640
Rule 2639
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*}
Mathematica [C] time = 0.113621, 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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Maple [A] time = 0.494, size = 264, normalized size = 1.6 \begin{align*}{\frac{2\,{a}^{3}{e}^{3}}{231\,d} \left ( 1344\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{13}-2464\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{10}\cos \left ( 1/2\,dx+c/2 \right ) -4032\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{11}+4928\,\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}+2928\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{9}-3080\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}\cos \left ( 1/2\,dx+c/2 \right ) +864\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{7}+616\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}\cos \left ( 1/2\,dx+c/2 \right ) -1908\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}+231\,\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) +804\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}-111\,\sin \left ( 1/2\,dx+c/2 \right ) \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}e+e}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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