Optimal. Leaf size=84 \[ -\frac{1}{3} x^2 \sqrt{c-a^2 c x^2}-\frac{(3 a x+5) \sqrt{c-a^2 c x^2}}{3 a^2}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{a^2} \]
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Rubi [A] time = 0.182244, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6151, 1809, 780, 217, 203} \[ -\frac{1}{3} x^2 \sqrt{c-a^2 c x^2}-\frac{(3 a x+5) \sqrt{c-a^2 c x^2}}{3 a^2}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{a^2} \]
Antiderivative was successfully verified.
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Rule 6151
Rule 1809
Rule 780
Rule 217
Rule 203
Rubi steps
\begin{align*} \int e^{2 \tanh ^{-1}(a x)} x \sqrt{c-a^2 c x^2} \, dx &=c \int \frac{x (1+a x)^2}{\sqrt{c-a^2 c x^2}} \, dx\\ &=-\frac{1}{3} x^2 \sqrt{c-a^2 c x^2}-\frac{\int \frac{x \left (-5 a^2 c-6 a^3 c x\right )}{\sqrt{c-a^2 c x^2}} \, dx}{3 a^2}\\ &=-\frac{1}{3} x^2 \sqrt{c-a^2 c x^2}-\frac{(5+3 a x) \sqrt{c-a^2 c x^2}}{3 a^2}+\frac{c \int \frac{1}{\sqrt{c-a^2 c x^2}} \, dx}{a}\\ &=-\frac{1}{3} x^2 \sqrt{c-a^2 c x^2}-\frac{(5+3 a x) \sqrt{c-a^2 c x^2}}{3 a^2}+\frac{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 )}{a}\\ &=-\frac{1}{3} x^2 \sqrt{c-a^2 c x^2}-\frac{(5+3 a x) \sqrt{c-a^2 c x^2}}{3 a^2}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c-a^2 c x^2}}\right )}{a^2}\\ \end{align*}
Mathematica [A] time = 0.0897938, size = 79, normalized size = 0.94 \[ -\frac{\left (a^2 x^2+3 a x+5\right ) \sqrt{c-a^2 c x^2}+3 \sqrt{c} \tan ^{-1}\left (\frac{a x \sqrt{c-a^2 c x^2}}{\sqrt{c} \left (a^2 x^2-1\right )}\right )}{3 a^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.038, size = 164, normalized size = 2. \begin{align*}{\frac{1}{3\,{a}^{2}c} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}}-{\frac{x}{a}\sqrt{-{a}^{2}c{x}^{2}+c}}-{\frac{c}{a}\arctan \left ({x\sqrt{{a}^{2}c}{\frac{1}{\sqrt{-{a}^{2}c{x}^{2}+c}}}} \right ){\frac{1}{\sqrt{{a}^{2}c}}}}-2\,{\frac{1}{{a}^{2}}\sqrt{-c{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,ac \left ( x-{a}^{-1} \right ) }}+2\,{\frac{c}{a\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 = 2.86243, size = 347, normalized size = 4.13 \begin{align*} \left [-\frac{2 \, \sqrt{-a^{2} c x^{2} + c}{\left (a^{2} x^{2} + 3 \, a x + 5\right )} - 3 \, \sqrt{-c} \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt{-a^{2} c x^{2} + c} a \sqrt{-c} x - c\right )}{6 \, a^{2}}, -\frac{\sqrt{-a^{2} c x^{2} + c}{\left (a^{2} x^{2} + 3 \, a x + 5\right )} + 3 \, \sqrt{c} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c} a \sqrt{c} x}{a^{2} c x^{2} - c}\right )}{3 \, a^{2}}\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 \sqrt{- a^{2} c x^{2} + c}}{a x - 1}\, dx - \int \frac{a x^{2} \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 = 1.17876, size = 99, normalized size = 1.18 \begin{align*} -\frac{1}{3} \, \sqrt{-a^{2} c x^{2} + c}{\left ({\left (x + \frac{3}{a}\right )} x + \frac{5}{a^{2}}\right )} - \frac{c \log \left ({\left | -\sqrt{-a^{2} c} x + \sqrt{-a^{2} c x^{2} + c} \right |}\right )}{a \sqrt{-c}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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