Optimal. Leaf size=112 \[ -\frac{1}{4} x^3 \sqrt{c-a^2 c x^2}+\frac{2 x^2 \sqrt{c-a^2 c x^2}}{3 a}+\frac{(32-21 a x) \sqrt{c-a^2 c x^2}}{24 a^3}+\frac{7 \sqrt{c} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{8 a^3} \]
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Rubi [A] time = 0.28425, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6152, 1809, 833, 780, 217, 203} \[ -\frac{1}{4} x^3 \sqrt{c-a^2 c x^2}+\frac{2 x^2 \sqrt{c-a^2 c x^2}}{3 a}+\frac{(32-21 a x) \sqrt{c-a^2 c x^2}}{24 a^3}+\frac{7 \sqrt{c} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{8 a^3} \]
Antiderivative was successfully verified.
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Rule 6152
Rule 1809
Rule 833
Rule 780
Rule 217
Rule 203
Rubi steps
\begin{align*} \int e^{-2 \tanh ^{-1}(a x)} x^2 \sqrt{c-a^2 c x^2} \, dx &=c \int \frac{x^2 (1-a x)^2}{\sqrt{c-a^2 c x^2}} \, dx\\ &=-\frac{1}{4} x^3 \sqrt{c-a^2 c x^2}-\frac{\int \frac{x^2 \left (-7 a^2 c+8 a^3 c x\right )}{\sqrt{c-a^2 c x^2}} \, dx}{4 a^2}\\ &=\frac{2 x^2 \sqrt{c-a^2 c x^2}}{3 a}-\frac{1}{4} x^3 \sqrt{c-a^2 c x^2}+\frac{\int \frac{x \left (-16 a^3 c^2+21 a^4 c^2 x\right )}{\sqrt{c-a^2 c x^2}} \, dx}{12 a^4 c}\\ &=\frac{2 x^2 \sqrt{c-a^2 c x^2}}{3 a}-\frac{1}{4} x^3 \sqrt{c-a^2 c x^2}+\frac{(32-21 a x) \sqrt{c-a^2 c x^2}}{24 a^3}+\frac{(7 c) \int \frac{1}{\sqrt{c-a^2 c x^2}} \, dx}{8 a^2}\\ &=\frac{2 x^2 \sqrt{c-a^2 c x^2}}{3 a}-\frac{1}{4} x^3 \sqrt{c-a^2 c x^2}+\frac{(32-21 a x) \sqrt{c-a^2 c x^2}}{24 a^3}+\frac{(7 c) \operatorname{Subst}\left (\int \frac{1}{1+a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c-a^2 c x^2}}\right )}{8 a^2}\\ &=\frac{2 x^2 \sqrt{c-a^2 c x^2}}{3 a}-\frac{1}{4} x^3 \sqrt{c-a^2 c x^2}+\frac{(32-21 a x) \sqrt{c-a^2 c x^2}}{24 a^3}+\frac{7 \sqrt{c} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{8 a^3}\\ \end{align*}
Mathematica [A] time = 0.0949311, size = 88, normalized size = 0.79 \[ \frac{\left (-6 a^3 x^3+16 a^2 x^2-21 a x+32\right ) \sqrt{c-a^2 c x^2}-21 \sqrt{c} \tan ^{-1}\left (\frac{a x \sqrt{c-a^2 c x^2}}{\sqrt{c} \left (a^2 x^2-1\right )}\right )}{24 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 178, normalized size = 1.6 \begin{align*}{\frac{x}{4\,{a}^{2}c} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{9\,x}{8\,{a}^{2}}\sqrt{-{a}^{2}c{x}^{2}+c}}-{\frac{9\,c}{8\,{a}^{2}}\arctan \left ({x\sqrt{{a}^{2}c}{\frac{1}{\sqrt{-{a}^{2}c{x}^{2}+c}}}} \right ){\frac{1}{\sqrt{{a}^{2}c}}}}-{\frac{2}{3\,{a}^{3}c} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}+2\,{\frac{\sqrt{-c{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,ac \left ( x+{a}^{-1} \right ) }}{{a}^{3}}}+2\,{\frac{c}{{a}^{2}\sqrt{{a}^{2}c}}\arctan \left ({\frac{\sqrt{{a}^{2}c}x}{\sqrt{-c{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,ac \left ( x+{a}^{-1} \right ) }}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.36431, size = 398, normalized size = 3.55 \begin{align*} \left [-\frac{2 \,{\left (6 \, a^{3} x^{3} - 16 \, a^{2} x^{2} + 21 \, a x - 32\right )} \sqrt{-a^{2} c x^{2} + c} - 21 \, \sqrt{-c} \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right )}{48 \, a^{3}}, -\frac{{\left (6 \, a^{3} x^{3} - 16 \, a^{2} x^{2} + 21 \, a x - 32\right )} \sqrt{-a^{2} c x^{2} + c} + 21 \, \sqrt{c} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right )}{24 \, a^{3}}\right ] \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 - \frac{x^{2} \sqrt{- a^{2} c x^{2} + c}}{a x + 1}\, dx - \int \frac{a x^{3} \sqrt{- a^{2} c x^{2} + c}}{a x + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.26439, size = 305, normalized size = 2.72 \begin{align*} -\frac{{\left (336 \, a^{5} c^{\frac{3}{2}} \arctan \left (\frac{\sqrt{-c + \frac{2 \, c}{a x + 1}}}{\sqrt{c}}\right ) \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right ) + \frac{{\left (75 \, a^{5}{\left (c - \frac{2 \, c}{a x + 1}\right )}^{3} c^{2} \sqrt{-c + \frac{2 \, c}{a x + 1}} \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right ) - 83 \, a^{5}{\left (c - \frac{2 \, c}{a x + 1}\right )}^{2} c^{3} \sqrt{-c + \frac{2 \, c}{a x + 1}} \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right ) - 21 \, a^{5} c^{5} \sqrt{-c + \frac{2 \, c}{a x + 1}} \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right ) - 77 \, a^{5} c^{4}{\left (-c + \frac{2 \, c}{a x + 1}\right )}^{\frac{3}{2}} \mathrm{sgn}\left (\frac{1}{a x + 1}\right ) \mathrm{sgn}\left (a\right )\right )}{\left (a x + 1\right )}^{4}}{c^{4}}\right )}{\left | a \right |}}{192 \, a^{9} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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