Optimal. Leaf size=138 \[ -\frac{64 a^3 (7 A+5 B) \cos (e+f x)}{105 f \sqrt{a \sin (e+f x)+a}}-\frac{16 a^2 (7 A+5 B) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{105 f}-\frac{2 a (7 A+5 B) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{35 f}-\frac{2 B \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{7 f} \]
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Rubi [A] time = 0.112403, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2751, 2647, 2646} \[ -\frac{64 a^3 (7 A+5 B) \cos (e+f x)}{105 f \sqrt{a \sin (e+f x)+a}}-\frac{16 a^2 (7 A+5 B) \cos (e+f x) \sqrt{a \sin (e+f x)+a}}{105 f}-\frac{2 a (7 A+5 B) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{35 f}-\frac{2 B \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{7 f} \]
Antiderivative was successfully verified.
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Rule 2751
Rule 2647
Rule 2646
Rubi steps
\begin{align*} \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) \, dx &=-\frac{2 B \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{7 f}+\frac{1}{7} (7 A+5 B) \int (a+a \sin (e+f x))^{5/2} \, dx\\ &=-\frac{2 a (7 A+5 B) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{35 f}-\frac{2 B \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{7 f}+\frac{1}{35} (8 a (7 A+5 B)) \int (a+a \sin (e+f x))^{3/2} \, dx\\ &=-\frac{16 a^2 (7 A+5 B) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{105 f}-\frac{2 a (7 A+5 B) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{35 f}-\frac{2 B \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{7 f}+\frac{1}{105} \left (32 a^2 (7 A+5 B)\right ) \int \sqrt{a+a \sin (e+f x)} \, dx\\ &=-\frac{64 a^3 (7 A+5 B) \cos (e+f x)}{105 f \sqrt{a+a \sin (e+f x)}}-\frac{16 a^2 (7 A+5 B) \cos (e+f x) \sqrt{a+a \sin (e+f x)}}{105 f}-\frac{2 a (7 A+5 B) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{35 f}-\frac{2 B \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{7 f}\\ \end{align*}
Mathematica [A] time = 1.5225, size = 119, normalized size = 0.86 \[ -\frac{a^2 \sqrt{a (\sin (e+f x)+1)} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) ((392 A+505 B) \sin (e+f x)-6 (7 A+20 B) \cos (2 (e+f x))+1246 A-15 B \sin (3 (e+f x))+1040 B)}{210 f \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.897, size = 99, normalized size = 0.7 \begin{align*}{\frac{ \left ( 2+2\,\sin \left ( fx+e \right ) \right ){a}^{3} \left ( -1+\sin \left ( fx+e \right ) \right ) \left ( -15\,B \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) + \left ( 98\,A+130\,B \right ) \sin \left ( fx+e \right ) + \left ( -21\,A-60\,B \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}+322\,A+290\,B \right ) }{105\,f\cos \left ( fx+e \right ) }{\frac{1}{\sqrt{a+a\sin \left ( fx+e \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 \sin \left (f x + e\right ) + A\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61816, size = 483, normalized size = 3.5 \begin{align*} \frac{2 \,{\left (15 \, B a^{2} \cos \left (f x + e\right )^{4} + 3 \,{\left (7 \, A + 20 \, B\right )} a^{2} \cos \left (f x + e\right )^{3} -{\left (77 \, A + 85 \, B\right )} a^{2} \cos \left (f x + e\right )^{2} - 2 \,{\left (161 \, A + 145 \, B\right )} a^{2} \cos \left (f x + e\right ) - 32 \,{\left (7 \, A + 5 \, B\right )} a^{2} +{\left (15 \, B a^{2} \cos \left (f x + e\right )^{3} - 3 \,{\left (7 \, A + 15 \, B\right )} a^{2} \cos \left (f x + e\right )^{2} - 2 \,{\left (49 \, A + 65 \, B\right )} a^{2} \cos \left (f x + e\right ) + 32 \,{\left (7 \, A + 5 \, B\right )} a^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt{a \sin \left (f x + e\right ) + a}}{105 \,{\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\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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