Optimal. Leaf size=194 \[ -\frac{x^4 \sqrt{1-a^2 x^2}}{4 a \sqrt{c-a^2 c x^2}}-\frac{x^3 \sqrt{1-a^2 x^2}}{3 a^2 \sqrt{c-a^2 c x^2}}-\frac{x^2 \sqrt{1-a^2 x^2}}{2 a^3 \sqrt{c-a^2 c x^2}}-\frac{x \sqrt{1-a^2 x^2}}{a^4 \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2} \log (1-a x)}{a^5 \sqrt{c-a^2 c x^2}} \]
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Rubi [A] time = 0.212941, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {6153, 6150, 43} \[ -\frac{x^4 \sqrt{1-a^2 x^2}}{4 a \sqrt{c-a^2 c x^2}}-\frac{x^3 \sqrt{1-a^2 x^2}}{3 a^2 \sqrt{c-a^2 c x^2}}-\frac{x^2 \sqrt{1-a^2 x^2}}{2 a^3 \sqrt{c-a^2 c x^2}}-\frac{x \sqrt{1-a^2 x^2}}{a^4 \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2} \log (1-a x)}{a^5 \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 6153
Rule 6150
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)} x^4}{\sqrt{c-a^2 c x^2}} \, dx &=\frac{\sqrt{1-a^2 x^2} \int \frac{e^{\tanh ^{-1}(a x)} x^4}{\sqrt{1-a^2 x^2}} \, dx}{\sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2} \int \frac{x^4}{1-a x} \, dx}{\sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2} \int \left (-\frac{1}{a^4}-\frac{x}{a^3}-\frac{x^2}{a^2}-\frac{x^3}{a}-\frac{1}{a^4 (-1+a x)}\right ) \, dx}{\sqrt{c-a^2 c x^2}}\\ &=-\frac{x \sqrt{1-a^2 x^2}}{a^4 \sqrt{c-a^2 c x^2}}-\frac{x^2 \sqrt{1-a^2 x^2}}{2 a^3 \sqrt{c-a^2 c x^2}}-\frac{x^3 \sqrt{1-a^2 x^2}}{3 a^2 \sqrt{c-a^2 c x^2}}-\frac{x^4 \sqrt{1-a^2 x^2}}{4 a \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2} \log (1-a x)}{a^5 \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0451362, size = 71, normalized size = 0.37 \[ -\frac{\sqrt{1-a^2 x^2} \left (a x \left (3 a^3 x^3+4 a^2 x^2+6 a x+12\right )+12 \log (1-a x)\right )}{12 a^5 \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.085, size = 83, normalized size = 0.4 \begin{align*}{\frac{3\,{x}^{4}{a}^{4}+4\,{x}^{3}{a}^{3}+6\,{a}^{2}{x}^{2}+12\,ax+12\,\ln \left ( ax-1 \right ) }{ \left ( 12\,{a}^{2}{x}^{2}-12 \right ) c{a}^{5}}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) }} \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 \frac{{\left (a x + 1\right )} x^{4}}{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.46467, size = 828, normalized size = 4.27 \begin{align*} \left [\frac{6 \,{\left (a^{2} x^{2} - 1\right )} \sqrt{c} \log \left (\frac{a^{6} c x^{6} - 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} + 4 \, a c x +{\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x\right )} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1} \sqrt{c} - 2 \, c}{a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1}\right ) +{\left (3 \, a^{4} x^{4} + 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} + 12 \, a x\right )} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{12 \,{\left (a^{7} c x^{2} - a^{5} c\right )}}, -\frac{12 \,{\left (a^{2} x^{2} - 1\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-a^{2} c x^{2} + c}{\left (a^{2} x^{2} - 2 \, a x + 2\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c}}{a^{4} c x^{4} - 2 \, a^{3} c x^{3} - a^{2} c x^{2} + 2 \, a c x}\right ) -{\left (3 \, a^{4} x^{4} + 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} + 12 \, a x\right )} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{12 \,{\left (a^{7} c x^{2} - a^{5} c\right )}}\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^{4} \left (a x + 1\right )}{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )} \sqrt{- c \left (a x - 1\right ) \left (a x + 1\right )}}\, 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 \frac{{\left (a x + 1\right )} x^{4}}{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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