Optimal. Leaf size=179 \[ -\frac{x \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{8 a^2 \sqrt{1-a x}}+\frac{\sqrt{x} \sqrt{c-\frac{c}{a x}} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{8 a^{5/2} \sqrt{1-a x}}+\frac{x^3 \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{3 \sqrt{1-a x}}+\frac{x^2 \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{12 a \sqrt{1-a x}} \]
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Rubi [A] time = 0.250841, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {6134, 6128, 848, 50, 54, 215} \[ -\frac{x \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{8 a^2 \sqrt{1-a x}}+\frac{\sqrt{x} \sqrt{c-\frac{c}{a x}} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{8 a^{5/2} \sqrt{1-a x}}+\frac{x^3 \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{3 \sqrt{1-a x}}+\frac{x^2 \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{12 a \sqrt{1-a x}} \]
Antiderivative was successfully verified.
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Rule 6134
Rule 6128
Rule 848
Rule 50
Rule 54
Rule 215
Rubi steps
\begin{align*} \int e^{\tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x^2 \, dx &=\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int e^{\tanh ^{-1}(a x)} x^{3/2} \sqrt{1-a x} \, dx}{\sqrt{1-a x}}\\ &=\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{x^{3/2} \sqrt{1-a^2 x^2}}{\sqrt{1-a x}} \, dx}{\sqrt{1-a x}}\\ &=\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int x^{3/2} \sqrt{1+a x} \, dx}{\sqrt{1-a x}}\\ &=\frac{\sqrt{c-\frac{c}{a x}} x^3 \sqrt{1+a x}}{3 \sqrt{1-a x}}+\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{x^{3/2}}{\sqrt{1+a x}} \, dx}{6 \sqrt{1-a x}}\\ &=\frac{\sqrt{c-\frac{c}{a x}} x^2 \sqrt{1+a x}}{12 a \sqrt{1-a x}}+\frac{\sqrt{c-\frac{c}{a x}} x^3 \sqrt{1+a x}}{3 \sqrt{1-a x}}-\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{\sqrt{x}}{\sqrt{1+a x}} \, dx}{8 a \sqrt{1-a x}}\\ &=-\frac{\sqrt{c-\frac{c}{a x}} x \sqrt{1+a x}}{8 a^2 \sqrt{1-a x}}+\frac{\sqrt{c-\frac{c}{a x}} x^2 \sqrt{1+a x}}{12 a \sqrt{1-a x}}+\frac{\sqrt{c-\frac{c}{a x}} x^3 \sqrt{1+a x}}{3 \sqrt{1-a x}}+\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{1}{\sqrt{x} \sqrt{1+a x}} \, dx}{16 a^2 \sqrt{1-a x}}\\ &=-\frac{\sqrt{c-\frac{c}{a x}} x \sqrt{1+a x}}{8 a^2 \sqrt{1-a x}}+\frac{\sqrt{c-\frac{c}{a x}} x^2 \sqrt{1+a x}}{12 a \sqrt{1-a x}}+\frac{\sqrt{c-\frac{c}{a x}} x^3 \sqrt{1+a x}}{3 \sqrt{1-a x}}+\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+a x^2}} \, dx,x,\sqrt{x}\right )}{8 a^2 \sqrt{1-a x}}\\ &=-\frac{\sqrt{c-\frac{c}{a x}} x \sqrt{1+a x}}{8 a^2 \sqrt{1-a x}}+\frac{\sqrt{c-\frac{c}{a x}} x^2 \sqrt{1+a x}}{12 a \sqrt{1-a x}}+\frac{\sqrt{c-\frac{c}{a x}} x^3 \sqrt{1+a x}}{3 \sqrt{1-a x}}+\frac{\sqrt{c-\frac{c}{a x}} \sqrt{x} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{8 a^{5/2} \sqrt{1-a x}}\\ \end{align*}
Mathematica [A] time = 0.0625353, size = 88, normalized size = 0.49 \[ \frac{\sqrt{x} \sqrt{c-\frac{c}{a x}} \left (\sqrt{a} \sqrt{x} \sqrt{a x+1} \left (8 a^2 x^2+2 a x-3\right )+3 \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )\right )}{24 a^{5/2} \sqrt{1-a x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.133, size = 125, normalized size = 0.7 \begin{align*} -{\frac{x}{48\,ax-48}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}}\sqrt{-{a}^{2}{x}^{2}+1} \left ( 16\,{a}^{5/2}{x}^{2}\sqrt{- \left ( ax+1 \right ) x}+4\,{a}^{3/2}x\sqrt{- \left ( ax+1 \right ) x}-6\,\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}-3\,\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ) \right ){a}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{- \left ( ax+1 \right ) x}}}} \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 )} \sqrt{c - \frac{c}{a x}} x^{2}}{\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 = 2.02362, size = 617, normalized size = 3.45 \begin{align*} \left [\frac{3 \,{\left (a x - 1\right )} \sqrt{-c} \log \left (-\frac{8 \, a^{3} c x^{3} - 7 \, a c x + 4 \,{\left (2 \, a^{2} x^{2} + a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right ) - 4 \,{\left (8 \, a^{3} x^{3} + 2 \, a^{2} x^{2} - 3 \, a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{96 \,{\left (a^{4} x - a^{3}\right )}}, -\frac{3 \,{\left (a x - 1\right )} \sqrt{c} \arctan \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \,{\left (8 \, a^{3} x^{3} + 2 \, a^{2} x^{2} - 3 \, a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{48 \,{\left (a^{4} x - a^{3}\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^{2} \sqrt{- c \left (-1 + \frac{1}{a x}\right )} \left (a x + 1\right )}{\sqrt{- \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 )} \sqrt{c - \frac{c}{a x}} x^{2}}{\sqrt{-a^{2} x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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