Optimal. Leaf size=145 \[ \frac {32}{3} a x \sqrt {a+a \cos (x)}+\frac {16}{9} a x \cos ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)}+\frac {4}{3} a x^2 \cos \left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)} \sin \left (\frac {x}{2}\right )-\frac {224}{9} a \sqrt {a+a \cos (x)} \tan \left (\frac {x}{2}\right )+\frac {8}{3} a x^2 \sqrt {a+a \cos (x)} \tan \left (\frac {x}{2}\right )+\frac {32}{27} a \sqrt {a+a \cos (x)} \sin ^2\left (\frac {x}{2}\right ) \tan \left (\frac {x}{2}\right ) \]
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Rubi [A]
time = 0.10, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3400, 3392,
3377, 2717, 2713} \begin {gather*} \frac {4}{3} a x^2 \sin \left (\frac {x}{2}\right ) \cos \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a}+\frac {8}{3} a x^2 \tan \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a}+\frac {16}{9} a x \cos ^2\left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a}+\frac {32}{3} a x \sqrt {a \cos (x)+a}-\frac {224}{9} a \tan \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a}+\frac {32}{27} a \sin ^2\left (\frac {x}{2}\right ) \tan \left (\frac {x}{2}\right ) \sqrt {a \cos (x)+a} \end {gather*}
Antiderivative was successfully verified.
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Rule 2713
Rule 2717
Rule 3377
Rule 3392
Rule 3400
Rubi steps
\begin {align*} \int x^2 (a+a \cos (x))^{3/2} \, dx &=\left (2 a \sqrt {a+a \cos (x)} \sec \left (\frac {x}{2}\right )\right ) \int x^2 \cos ^3\left (\frac {x}{2}\right ) \, dx\\ &=\frac {16}{9} a x \cos ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)}+\frac {4}{3} a x^2 \cos \left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)} \sin \left (\frac {x}{2}\right )+\frac {1}{3} \left (4 a \sqrt {a+a \cos (x)} \sec \left (\frac {x}{2}\right )\right ) \int x^2 \cos \left (\frac {x}{2}\right ) \, dx-\frac {1}{9} \left (16 a \sqrt {a+a \cos (x)} \sec \left (\frac {x}{2}\right )\right ) \int \cos ^3\left (\frac {x}{2}\right ) \, dx\\ &=\frac {16}{9} a x \cos ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)}+\frac {4}{3} a x^2 \cos \left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)} \sin \left (\frac {x}{2}\right )+\frac {8}{3} a x^2 \sqrt {a+a \cos (x)} \tan \left (\frac {x}{2}\right )+\frac {1}{9} \left (32 a \sqrt {a+a \cos (x)} \sec \left (\frac {x}{2}\right )\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin \left (\frac {x}{2}\right )\right )-\frac {1}{3} \left (16 a \sqrt {a+a \cos (x)} \sec \left (\frac {x}{2}\right )\right ) \int x \sin \left (\frac {x}{2}\right ) \, dx\\ &=\frac {32}{3} a x \sqrt {a+a \cos (x)}+\frac {16}{9} a x \cos ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)}+\frac {4}{3} a x^2 \cos \left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)} \sin \left (\frac {x}{2}\right )-\frac {32}{9} a \sqrt {a+a \cos (x)} \tan \left (\frac {x}{2}\right )+\frac {8}{3} a x^2 \sqrt {a+a \cos (x)} \tan \left (\frac {x}{2}\right )+\frac {32}{27} a \sqrt {a+a \cos (x)} \sin ^2\left (\frac {x}{2}\right ) \tan \left (\frac {x}{2}\right )-\frac {1}{3} \left (32 a \sqrt {a+a \cos (x)} \sec \left (\frac {x}{2}\right )\right ) \int \cos \left (\frac {x}{2}\right ) \, dx\\ &=\frac {32}{3} a x \sqrt {a+a \cos (x)}+\frac {16}{9} a x \cos ^2\left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)}+\frac {4}{3} a x^2 \cos \left (\frac {x}{2}\right ) \sqrt {a+a \cos (x)} \sin \left (\frac {x}{2}\right )-\frac {224}{9} a \sqrt {a+a \cos (x)} \tan \left (\frac {x}{2}\right )+\frac {8}{3} a x^2 \sqrt {a+a \cos (x)} \tan \left (\frac {x}{2}\right )+\frac {32}{27} a \sqrt {a+a \cos (x)} \sin ^2\left (\frac {x}{2}\right ) \tan \left (\frac {x}{2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 54, normalized size = 0.37 \begin {gather*} \frac {2}{27} a \sqrt {a (1+\cos (x))} \left (156 x+\left (-328+45 x^2\right ) \tan \left (\frac {x}{2}\right )+\cos (x) \left (12 x+\left (-8+9 x^2\right ) \tan \left (\frac {x}{2}\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int x^{2} \left (a +a \cos \left (x \right )\right )^{\frac {3}{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 72, normalized size = 0.50 \begin {gather*} \frac {1}{27} \, {\left (81 \, \sqrt {2} a x^{2} \sin \left (\frac {1}{2} \, x\right ) + 12 \, \sqrt {2} a x \cos \left (\frac {3}{2} \, x\right ) + 324 \, \sqrt {2} a x \cos \left (\frac {1}{2} \, x\right ) - 648 \, \sqrt {2} a \sin \left (\frac {1}{2} \, x\right ) + {\left (9 \, \sqrt {2} a x^{2} - 8 \, \sqrt {2} a\right )} \sin \left (\frac {3}{2} \, x\right )\right )} \sqrt {a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \left (a \left (\cos {\left (x \right )} + 1\right )\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 85, normalized size = 0.59 \begin {gather*} \frac {1}{27} \, \sqrt {2} {\left (12 \, a x \cos \left (\frac {3}{2} \, x\right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) + 324 \, a x \cos \left (\frac {1}{2} \, x\right ) \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) + {\left (9 \, a x^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) - 8 \, a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right )\right )} \sin \left (\frac {3}{2} \, x\right ) + 81 \, {\left (a x^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) - 8 \, a \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right )\right )} \sin \left (\frac {1}{2} \, x\right )\right )} \sqrt {a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^2\,{\left (a+a\,\cos \left (x\right )\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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