Optimal. Leaf size=204 \[ -\frac{23 \sqrt{x} \sqrt{c-\frac{c}{a x}} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{4 a^{3/2} \sqrt{1-a x}}+\frac{4 \sqrt{2} \sqrt{x} \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{a^{3/2} \sqrt{1-a x}}-\frac{x (a x+1)^{3/2} \sqrt{c-\frac{c}{a x}}}{2 a \sqrt{1-a x}}-\frac{7 x \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{4 a \sqrt{1-a x}} \]
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Rubi [A] time = 0.227905, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {6134, 6129, 101, 154, 157, 54, 215, 93, 206} \[ -\frac{23 \sqrt{x} \sqrt{c-\frac{c}{a x}} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{4 a^{3/2} \sqrt{1-a x}}+\frac{4 \sqrt{2} \sqrt{x} \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{a^{3/2} \sqrt{1-a x}}-\frac{x (a x+1)^{3/2} \sqrt{c-\frac{c}{a x}}}{2 a \sqrt{1-a x}}-\frac{7 x \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{4 a \sqrt{1-a x}} \]
Antiderivative was successfully verified.
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Rule 6134
Rule 6129
Rule 101
Rule 154
Rule 157
Rule 54
Rule 215
Rule 93
Rule 206
Rubi steps
\begin{align*} \int e^{3 \tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x \, dx &=\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int e^{3 \tanh ^{-1}(a x)} \sqrt{x} \sqrt{1-a x} \, dx}{\sqrt{1-a x}}\\ &=\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{\sqrt{x} (1+a x)^{3/2}}{1-a x} \, dx}{\sqrt{1-a x}}\\ &=-\frac{\sqrt{c-\frac{c}{a x}} x (1+a x)^{3/2}}{2 a \sqrt{1-a x}}+\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{\sqrt{1+a x} \left (\frac{1}{2}+\frac{7 a x}{2}\right )}{\sqrt{x} (1-a x)} \, dx}{2 a \sqrt{1-a x}}\\ &=-\frac{7 \sqrt{c-\frac{c}{a x}} x \sqrt{1+a x}}{4 a \sqrt{1-a x}}-\frac{\sqrt{c-\frac{c}{a x}} x (1+a x)^{3/2}}{2 a \sqrt{1-a x}}-\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{-\frac{9 a}{4}-\frac{23 a^2 x}{4}}{\sqrt{x} (1-a x) \sqrt{1+a x}} \, dx}{2 a^2 \sqrt{1-a x}}\\ &=-\frac{7 \sqrt{c-\frac{c}{a x}} x \sqrt{1+a x}}{4 a \sqrt{1-a x}}-\frac{\sqrt{c-\frac{c}{a x}} x (1+a x)^{3/2}}{2 a \sqrt{1-a x}}-\frac{\left (23 \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{1}{\sqrt{x} \sqrt{1+a x}} \, dx}{8 a \sqrt{1-a x}}+\frac{\left (4 \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{1}{\sqrt{x} (1-a x) \sqrt{1+a x}} \, dx}{a \sqrt{1-a x}}\\ &=-\frac{7 \sqrt{c-\frac{c}{a x}} x \sqrt{1+a x}}{4 a \sqrt{1-a x}}-\frac{\sqrt{c-\frac{c}{a x}} x (1+a x)^{3/2}}{2 a \sqrt{1-a x}}-\frac{\left (23 \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 )}{4 a \sqrt{1-a x}}+\frac{\left (8 \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \operatorname{Subst}\left (\int \frac{1}{1-2 a x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{1+a x}}\right )}{a \sqrt{1-a x}}\\ &=-\frac{7 \sqrt{c-\frac{c}{a x}} x \sqrt{1+a x}}{4 a \sqrt{1-a x}}-\frac{\sqrt{c-\frac{c}{a x}} x (1+a x)^{3/2}}{2 a \sqrt{1-a x}}-\frac{23 \sqrt{c-\frac{c}{a x}} \sqrt{x} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{4 a^{3/2} \sqrt{1-a x}}+\frac{4 \sqrt{2} \sqrt{c-\frac{c}{a x}} \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{1+a x}}\right )}{a^{3/2} \sqrt{1-a x}}\\ \end{align*}
Mathematica [A] time = 0.072024, size = 114, normalized size = 0.56 \[ -\frac{\sqrt{c-\frac{c}{a x}} \left (\sqrt{a} x \sqrt{a x+1} (2 a x+9)+23 \sqrt{x} \sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )-16 \sqrt{2} \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )\right )}{4 a^{3/2} \sqrt{1-a x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.138, size = 191, normalized size = 0.9 \begin{align*}{\frac{x\sqrt{2}}{16\,ax-16}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}}\sqrt{-{a}^{2}{x}^{2}+1} \left ( 4\,\sqrt{- \left ( ax+1 \right ) x}{a}^{5/2}\sqrt{2}\sqrt{-{a}^{-1}}x+18\,\sqrt{- \left ( ax+1 \right ) x}{a}^{3/2}\sqrt{2}\sqrt{-{a}^{-1}}-23\,\arctan \left ( 1/2\,{\frac{2\,ax+1}{\sqrt{a}\sqrt{- \left ( ax+1 \right ) x}}} \right ) a\sqrt{2}\sqrt{-{a}^{-1}}+32\,\ln \left ({\frac{1}{ax-1} \left ( 2\,\sqrt{2}\sqrt{-{a}^{-1}}\sqrt{- \left ( ax+1 \right ) x}a-3\,ax-1 \right ) } \right ) \sqrt{a} \right ){a}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{- \left ( ax+1 \right ) x}}}{\frac{1}{\sqrt{-{a}^{-1}}}}} \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 )}^{3} \sqrt{c - \frac{c}{a x}} x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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{- c \left (-1 + \frac{1}{a x}\right )} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}\, 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 )}^{3} \sqrt{c - \frac{c}{a x}} x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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