Optimal. Leaf size=154 \[ \frac{2 e^2 \left (7 a^2+2 b^2\right ) \sqrt{\sin (c+d x)} F\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{21 d \sqrt{e \sin (c+d x)}}-\frac{2 e \left (7 a^2+2 b^2\right ) \cos (c+d x) \sqrt{e \sin (c+d x)}}{21 d}+\frac{18 a b (e \sin (c+d x))^{5/2}}{35 d e}+\frac{2 b (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}{7 d e} \]
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Rubi [A] time = 0.168032, antiderivative size = 154, 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 = {2692, 2669, 2635, 2642, 2641} \[ \frac{2 e^2 \left (7 a^2+2 b^2\right ) \sqrt{\sin (c+d x)} F\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{21 d \sqrt{e \sin (c+d x)}}-\frac{2 e \left (7 a^2+2 b^2\right ) \cos (c+d x) \sqrt{e \sin (c+d x)}}{21 d}+\frac{18 a b (e \sin (c+d x))^{5/2}}{35 d e}+\frac{2 b (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}{7 d e} \]
Antiderivative was successfully verified.
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Rule 2692
Rule 2669
Rule 2635
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^2 (e \sin (c+d x))^{3/2} \, dx &=\frac{2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{5/2}}{7 d e}+\frac{2}{7} \int \left (\frac{7 a^2}{2}+b^2+\frac{9}{2} a b \cos (c+d x)\right ) (e \sin (c+d x))^{3/2} \, dx\\ &=\frac{18 a b (e \sin (c+d x))^{5/2}}{35 d e}+\frac{2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{5/2}}{7 d e}+\frac{1}{7} \left (7 a^2+2 b^2\right ) \int (e \sin (c+d x))^{3/2} \, dx\\ &=-\frac{2 \left (7 a^2+2 b^2\right ) e \cos (c+d x) \sqrt{e \sin (c+d x)}}{21 d}+\frac{18 a b (e \sin (c+d x))^{5/2}}{35 d e}+\frac{2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{5/2}}{7 d e}+\frac{1}{21} \left (\left (7 a^2+2 b^2\right ) e^2\right ) \int \frac{1}{\sqrt{e \sin (c+d x)}} \, dx\\ &=-\frac{2 \left (7 a^2+2 b^2\right ) e \cos (c+d x) \sqrt{e \sin (c+d x)}}{21 d}+\frac{18 a b (e \sin (c+d x))^{5/2}}{35 d e}+\frac{2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{5/2}}{7 d e}+\frac{\left (\left (7 a^2+2 b^2\right ) e^2 \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)}} \, dx}{21 \sqrt{e \sin (c+d x)}}\\ &=\frac{2 \left (7 a^2+2 b^2\right ) e^2 F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{21 d \sqrt{e \sin (c+d x)}}-\frac{2 \left (7 a^2+2 b^2\right ) e \cos (c+d x) \sqrt{e \sin (c+d x)}}{21 d}+\frac{18 a b (e \sin (c+d x))^{5/2}}{35 d e}+\frac{2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{5/2}}{7 d e}\\ \end{align*}
Mathematica [A] time = 0.811974, size = 117, normalized size = 0.76 \[ \frac{(e \sin (c+d x))^{3/2} \left (-\frac{1}{2} \csc (c+d x) \left (5 \left (28 a^2+5 b^2\right ) \cos (c+d x)+3 b (28 a \cos (2 (c+d x))-28 a+5 b \cos (3 (c+d x)))\right )-\frac{10 \left (7 a^2+2 b^2\right ) F\left (\left .\frac{1}{4} (-2 c-2 d x+\pi )\right |2\right )}{\sin ^{\frac{3}{2}}(c+d x)}\right )}{105 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.75, size = 229, normalized size = 1.5 \begin{align*} -{\frac{{e}^{2}}{105\,d\cos \left ( dx+c \right ) } \left ( 30\,{b}^{2}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}+35\,\sqrt{1-\sin \left ( dx+c \right ) }\sqrt{2+2\,\sin \left ( dx+c \right ) }\sqrt{\sin \left ( dx+c \right ) }{\it EllipticF} \left ( \sqrt{1-\sin \left ( dx+c \right ) },1/2\,\sqrt{2} \right ){a}^{2}+10\,\sqrt{1-\sin \left ( dx+c \right ) }\sqrt{2+2\,\sin \left ( dx+c \right ) }\sqrt{\sin \left ( dx+c \right ) }{\it EllipticF} \left ( \sqrt{1-\sin \left ( dx+c \right ) },1/2\,\sqrt{2} \right ){b}^{2}+84\,ab\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}+70\,{a}^{2}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}-10\,{b}^{2}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}-84\,ab\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) \right ){\frac{1}{\sqrt{e\sin \left ( dx+c \right ) }}}} \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 (b \cos \left (d x + c\right ) + a\right )}^{2} \left (e \sin \left (d x + c\right )\right )^{\frac{3}{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 (b^{2} e \cos \left (d x + c\right )^{2} + 2 \, a b e \cos \left (d x + c\right ) + a^{2} e\right )} \sqrt{e \sin \left (d x + c\right )} \sin \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(-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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