Optimal. Leaf size=193 \[ -\frac{10 e^3 \left (11 a^2+2 b^2\right ) \cos (c+d x) \sqrt{e \sin (c+d x)}}{231 d}+\frac{10 e^4 \left (11 a^2+2 b^2\right ) \sqrt{\sin (c+d x)} F\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{231 d \sqrt{e \sin (c+d x)}}-\frac{2 e \left (11 a^2+2 b^2\right ) \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac{26 a b (e \sin (c+d x))^{9/2}}{99 d e}+\frac{2 b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))}{11 d e} \]
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Rubi [A] time = 0.189719, antiderivative size = 193, 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 = {2692, 2669, 2635, 2642, 2641} \[ -\frac{10 e^3 \left (11 a^2+2 b^2\right ) \cos (c+d x) \sqrt{e \sin (c+d x)}}{231 d}+\frac{10 e^4 \left (11 a^2+2 b^2\right ) \sqrt{\sin (c+d x)} F\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{231 d \sqrt{e \sin (c+d x)}}-\frac{2 e \left (11 a^2+2 b^2\right ) \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac{26 a b (e \sin (c+d x))^{9/2}}{99 d e}+\frac{2 b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))}{11 d e} \]
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))^{7/2} \, dx &=\frac{2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{11 d e}+\frac{2}{11} \int \left (\frac{11 a^2}{2}+b^2+\frac{13}{2} a b \cos (c+d x)\right ) (e \sin (c+d x))^{7/2} \, dx\\ &=\frac{26 a b (e \sin (c+d x))^{9/2}}{99 d e}+\frac{2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{11 d e}+\frac{1}{11} \left (11 a^2+2 b^2\right ) \int (e \sin (c+d x))^{7/2} \, dx\\ &=-\frac{2 \left (11 a^2+2 b^2\right ) e \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac{26 a b (e \sin (c+d x))^{9/2}}{99 d e}+\frac{2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{11 d e}+\frac{1}{77} \left (5 \left (11 a^2+2 b^2\right ) e^2\right ) \int (e \sin (c+d x))^{3/2} \, dx\\ &=-\frac{10 \left (11 a^2+2 b^2\right ) e^3 \cos (c+d x) \sqrt{e \sin (c+d x)}}{231 d}-\frac{2 \left (11 a^2+2 b^2\right ) e \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac{26 a b (e \sin (c+d x))^{9/2}}{99 d e}+\frac{2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{11 d e}+\frac{1}{231} \left (5 \left (11 a^2+2 b^2\right ) e^4\right ) \int \frac{1}{\sqrt{e \sin (c+d x)}} \, dx\\ &=-\frac{10 \left (11 a^2+2 b^2\right ) e^3 \cos (c+d x) \sqrt{e \sin (c+d x)}}{231 d}-\frac{2 \left (11 a^2+2 b^2\right ) e \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac{26 a b (e \sin (c+d x))^{9/2}}{99 d e}+\frac{2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{11 d e}+\frac{\left (5 \left (11 a^2+2 b^2\right ) e^4 \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)}} \, dx}{231 \sqrt{e \sin (c+d x)}}\\ &=\frac{10 \left (11 a^2+2 b^2\right ) e^4 F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{231 d \sqrt{e \sin (c+d x)}}-\frac{10 \left (11 a^2+2 b^2\right ) e^3 \cos (c+d x) \sqrt{e \sin (c+d x)}}{231 d}-\frac{2 \left (11 a^2+2 b^2\right ) e \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac{26 a b (e \sin (c+d x))^{9/2}}{99 d e}+\frac{2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{11 d e}\\ \end{align*}
Mathematica [A] time = 1.76726, size = 157, normalized size = 0.81 \[ \frac{(e \sin (c+d x))^{7/2} \left (\frac{1}{6} \csc ^3(c+d x) \left (-6 \left (506 a^2+71 b^2\right ) \cos (c+d x)+396 a^2 \cos (3 (c+d x))-1232 a b \cos (2 (c+d x))+308 a b \cos (4 (c+d x))+924 a b-117 b^2 \cos (3 (c+d x))+63 b^2 \cos (5 (c+d x))\right )-\frac{40 \left (11 a^2+2 b^2\right ) F\left (\left .\frac{1}{4} (-2 c-2 d x+\pi )\right |2\right )}{\sin ^{\frac{7}{2}}(c+d x)}\right )}{924 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.855, size = 228, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ({\frac{4\,ab}{9\,e} \left ( e\sin \left ( dx+c \right ) \right ) ^{{\frac{9}{2}}}}-{\frac{{e}^{4}}{231\,\cos \left ( dx+c \right ) } \left ( -42\,{b}^{2}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{6}+ \left ( -66\,{a}^{2}+72\,{b}^{2} \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}\sin \left ( dx+c \right ) + \left ( 176\,{a}^{2}-10\,{b}^{2} \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) +55\,\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} \right ){\frac{1}{\sqrt{e\sin \left ( dx+c \right ) }}}} \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{7}{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^{3} \cos \left (d x + c\right )^{4} + 2 \, a b e^{3} \cos \left (d x + c\right )^{3} - 2 \, a b e^{3} \cos \left (d x + c\right ) +{\left (a^{2} - b^{2}\right )} e^{3} \cos \left (d x + c\right )^{2} - a^{2} e^{3}\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] 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{7}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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