Optimal. Leaf size=271 \[ \frac{16 a x \sin ^2\left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}}{9 f^2}+\frac{32 a x \sqrt{a \sin (e+f x)+a}}{3 f^2}+\frac{224 a \cot \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}}{9 f^3}-\frac{32 a \cos ^2\left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \cot \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}}{27 f^3}-\frac{4 a x^2 \sin \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \cos \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}}{3 f}-\frac{8 a x^2 \cot \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}}{3 f} \]
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Rubi [A] time = 0.179307, antiderivative size = 271, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {3319, 3311, 3296, 2638, 2633} \[ \frac{16 a x \sin ^2\left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}}{9 f^2}+\frac{32 a x \sqrt{a \sin (e+f x)+a}}{3 f^2}+\frac{224 a \cot \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}}{9 f^3}-\frac{32 a \cos ^2\left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \cot \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}}{27 f^3}-\frac{4 a x^2 \sin \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \cos \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}}{3 f}-\frac{8 a x^2 \cot \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right ) \sqrt{a \sin (e+f x)+a}}{3 f} \]
Antiderivative was successfully verified.
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Rule 3319
Rule 3311
Rule 3296
Rule 2638
Rule 2633
Rubi steps
\begin{align*} \int x^2 (a+a \sin (e+f x))^{3/2} \, dx &=\left (2 a \csc \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}\right ) \int x^2 \sin ^3\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \, dx\\ &=-\frac{4 a x^2 \cos \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{3 f}+\frac{16 a x \sin ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{9 f^2}+\frac{1}{3} \left (4 a \csc \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}\right ) \int x^2 \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \, dx-\frac{\left (16 a \csc \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}\right ) \int \sin ^3\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \, dx}{9 f^2}\\ &=-\frac{8 a x^2 \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{3 f}-\frac{4 a x^2 \cos \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{3 f}+\frac{16 a x \sin ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{9 f^2}+\frac{\left (32 a \csc \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )\right )}{9 f^3}+\frac{\left (16 a \csc \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}\right ) \int x \cos \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \, dx}{3 f}\\ &=\frac{32 a x \sqrt{a+a \sin (e+f x)}}{3 f^2}+\frac{32 a \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{9 f^3}-\frac{8 a x^2 \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{3 f}-\frac{32 a \cos ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{27 f^3}-\frac{4 a x^2 \cos \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{3 f}+\frac{16 a x \sin ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{9 f^2}-\frac{\left (32 a \csc \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}\right ) \int \cos \left (\frac{e}{2}-\frac{\pi }{4}+\frac{f x}{2}\right ) \, dx}{3 f^2}\\ &=\frac{32 a x \sqrt{a+a \sin (e+f x)}}{3 f^2}+\frac{224 a \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{9 f^3}-\frac{8 a x^2 \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{3 f}-\frac{32 a \cos ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{27 f^3}-\frac{4 a x^2 \cos \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{3 f}+\frac{16 a x \sin ^2\left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \sqrt{a+a \sin (e+f x)}}{9 f^2}\\ \end{align*}
Mathematica [A] time = 0.822052, size = 191, normalized size = 0.7 \[ \frac{2 a \sqrt{a (\sin (e+f x)+1)} \left (-\frac{4 \left (\sin \left (\frac{e}{2}\right ) \left (-9 f^2 x^2-39 f x+80\right )+\cos \left (\frac{e}{2}\right ) \left (9 f^2 x^2-39 f x-80\right )\right )}{\sin \left (\frac{e}{2}\right )+\cos \left (\frac{e}{2}\right )}-\cos (f x) \left (\cos (e) \left (9 f^2 x^2-8\right )-12 f x \sin (e)\right )+\sin (f x) \left (\sin (e) \left (9 f^2 x^2-8\right )+12 f x \cos (e)\right )+\frac{8 \left (9 f^2 x^2-80\right ) \sin \left (\frac{f x}{2}\right )}{\left (\sin \left (\frac{e}{2}\right )+\cos \left (\frac{e}{2}\right )\right ) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )}\right )}{27 f^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.039, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( a+a\sin \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}\, dx \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 (a \sin \left (f x + e\right ) + a\right )}^{\frac{3}{2}} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \left (a \left (\sin{\left (e + f x \right )} + 1\right )\right )^{\frac{3}{2}}\, dx \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 (a \sin \left (f x + e\right ) + a\right )}^{\frac{3}{2}} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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