Optimal. Leaf size=137 \[ \frac{6 a^2 e^2 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{7 d \sqrt{e \cos (c+d x)}}-\frac{18 a^2 (e \cos (c+d x))^{5/2}}{35 d e}-\frac{2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{5/2}}{7 d e}+\frac{6 a^2 e \sin (c+d x) \sqrt{e \cos (c+d x)}}{7 d} \]
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Rubi [A] time = 0.124766, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2678, 2669, 2635, 2642, 2641} \[ \frac{6 a^2 e^2 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{7 d \sqrt{e \cos (c+d x)}}-\frac{18 a^2 (e \cos (c+d x))^{5/2}}{35 d e}-\frac{2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{5/2}}{7 d e}+\frac{6 a^2 e \sin (c+d x) \sqrt{e \cos (c+d x)}}{7 d} \]
Antiderivative was successfully verified.
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Rule 2678
Rule 2669
Rule 2635
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2 \, dx &=-\frac{2 (e \cos (c+d x))^{5/2} \left (a^2+a^2 \sin (c+d x)\right )}{7 d e}+\frac{1}{7} (9 a) \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x)) \, dx\\ &=-\frac{18 a^2 (e \cos (c+d x))^{5/2}}{35 d e}-\frac{2 (e \cos (c+d x))^{5/2} \left (a^2+a^2 \sin (c+d x)\right )}{7 d e}+\frac{1}{7} \left (9 a^2\right ) \int (e \cos (c+d x))^{3/2} \, dx\\ &=-\frac{18 a^2 (e \cos (c+d x))^{5/2}}{35 d e}+\frac{6 a^2 e \sqrt{e \cos (c+d x)} \sin (c+d x)}{7 d}-\frac{2 (e \cos (c+d x))^{5/2} \left (a^2+a^2 \sin (c+d x)\right )}{7 d e}+\frac{1}{7} \left (3 a^2 e^2\right ) \int \frac{1}{\sqrt{e \cos (c+d x)}} \, dx\\ &=-\frac{18 a^2 (e \cos (c+d x))^{5/2}}{35 d e}+\frac{6 a^2 e \sqrt{e \cos (c+d x)} \sin (c+d x)}{7 d}-\frac{2 (e \cos (c+d x))^{5/2} \left (a^2+a^2 \sin (c+d x)\right )}{7 d e}+\frac{\left (3 a^2 e^2 \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{7 \sqrt{e \cos (c+d x)}}\\ &=-\frac{18 a^2 (e \cos (c+d x))^{5/2}}{35 d e}+\frac{6 a^2 e^2 \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{7 d \sqrt{e \cos (c+d x)}}+\frac{6 a^2 e \sqrt{e \cos (c+d x)} \sin (c+d x)}{7 d}-\frac{2 (e \cos (c+d x))^{5/2} \left (a^2+a^2 \sin (c+d x)\right )}{7 d e}\\ \end{align*}
Mathematica [C] time = 0.0744977, size = 66, normalized size = 0.48 \[ -\frac{16 \sqrt [4]{2} a^2 (e \cos (c+d x))^{5/2} \, _2F_1\left (-\frac{9}{4},\frac{5}{4};\frac{9}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{5 d e (\sin (c+d x)+1)^{5/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.46, size = 203, normalized size = 1.5 \begin{align*} -{\frac{2\,{a}^{2}{e}^{2}}{35\,d} \left ( -80\,\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}+120\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}\cos \left ( 1/2\,dx+c/2 \right ) -112\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{7}+168\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}+15\,\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -20\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) -84\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}+14\,\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{3}{2}}{\left (a \sin \left (d x + c\right ) + a\right )}^{2}\,{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 (a^{2} e \cos \left (d x + c\right )^{3} - 2 \, a^{2} e \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 \, a^{2} e \cos \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{3}{2}}{\left (a \sin \left (d x + c\right ) + a\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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