Optimal. Leaf size=112 \[ -\frac{1}{4} x^3 \sqrt{c-a^2 c x^2}-\frac{2 x^2 \sqrt{c-a^2 c x^2}}{3 a}-\frac{(21 a x+32) \sqrt{c-a^2 c x^2}}{24 a^3}+\frac{7 \sqrt{c} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{8 a^3} \]
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Rubi [A] time = 0.287331, 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 = {6151, 1809, 833, 780, 217, 203} \[ -\frac{1}{4} x^3 \sqrt{c-a^2 c x^2}-\frac{2 x^2 \sqrt{c-a^2 c x^2}}{3 a}-\frac{(21 a x+32) \sqrt{c-a^2 c x^2}}{24 a^3}+\frac{7 \sqrt{c} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{8 a^3} \]
Antiderivative was successfully verified.
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Rule 6151
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 = 0.10731, 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 [B] time = 0.041, size = 186, normalized size = 1.7 \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{1}{{a}^{3}}\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }}+2\,{\frac{c}{{a}^{2}\sqrt{{a}^{2}c}}\arctan \left ({\sqrt{{a}^{2}c}x{\frac{1}{\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 = 3.23165, 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 [A] time = 1.17479, size = 113, normalized size = 1.01 \begin{align*} -\frac{1}{24} \, \sqrt{-a^{2} c x^{2} + c}{\left ({\left (2 \,{\left (3 \, x + \frac{8}{a}\right )} x + \frac{21}{a^{2}}\right )} x + \frac{32}{a^{3}}\right )} - \frac{7 \, c \log \left ({\left | -\sqrt{-a^{2} c} x + \sqrt{-a^{2} c x^{2} + c} \right |}\right )}{8 \, a^{2} \sqrt{-c}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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