Optimal. Leaf size=237 \[ -\frac{104 a^3 \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{21 \sqrt{1-a x}}-\frac{32 a^2 \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{21 x \sqrt{1-a x}}+\frac{4 \sqrt{2} a^{7/2} \sqrt{x} \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{\sqrt{1-a x}}-\frac{6 a \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{7 x^2 \sqrt{1-a x}}-\frac{2 \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{7 x^3 \sqrt{1-a x}} \]
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Rubi [A] time = 0.277296, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {6134, 6129, 98, 152, 12, 93, 206} \[ -\frac{104 a^3 \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{21 \sqrt{1-a x}}-\frac{32 a^2 \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{21 x \sqrt{1-a x}}+\frac{4 \sqrt{2} a^{7/2} \sqrt{x} \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )}{\sqrt{1-a x}}-\frac{6 a \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{7 x^2 \sqrt{1-a x}}-\frac{2 \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{7 x^3 \sqrt{1-a x}} \]
Antiderivative was successfully verified.
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Rule 6134
Rule 6129
Rule 98
Rule 152
Rule 12
Rule 93
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{3 \tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^4} \, dx &=\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{e^{3 \tanh ^{-1}(a x)} \sqrt{1-a x}}{x^{9/2}} \, dx}{\sqrt{1-a x}}\\ &=\frac{\left (\sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{(1+a x)^{3/2}}{x^{9/2} (1-a x)} \, dx}{\sqrt{1-a x}}\\ &=-\frac{2 \sqrt{c-\frac{c}{a x}} \sqrt{1+a x}}{7 x^3 \sqrt{1-a x}}-\frac{\left (2 \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{-\frac{15 a}{2}-\frac{13 a^2 x}{2}}{x^{7/2} (1-a x) \sqrt{1+a x}} \, dx}{7 \sqrt{1-a x}}\\ &=-\frac{2 \sqrt{c-\frac{c}{a x}} \sqrt{1+a x}}{7 x^3 \sqrt{1-a x}}-\frac{6 a \sqrt{c-\frac{c}{a x}} \sqrt{1+a x}}{7 x^2 \sqrt{1-a x}}+\frac{\left (4 \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{20 a^2+15 a^3 x}{x^{5/2} (1-a x) \sqrt{1+a x}} \, dx}{35 \sqrt{1-a x}}\\ &=-\frac{2 \sqrt{c-\frac{c}{a x}} \sqrt{1+a x}}{7 x^3 \sqrt{1-a x}}-\frac{6 a \sqrt{c-\frac{c}{a x}} \sqrt{1+a x}}{7 x^2 \sqrt{1-a x}}-\frac{32 a^2 \sqrt{c-\frac{c}{a x}} \sqrt{1+a x}}{21 x \sqrt{1-a x}}-\frac{\left (8 \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{-\frac{65 a^3}{2}-20 a^4 x}{x^{3/2} (1-a x) \sqrt{1+a x}} \, dx}{105 \sqrt{1-a x}}\\ &=-\frac{104 a^3 \sqrt{c-\frac{c}{a x}} \sqrt{1+a x}}{21 \sqrt{1-a x}}-\frac{2 \sqrt{c-\frac{c}{a x}} \sqrt{1+a x}}{7 x^3 \sqrt{1-a x}}-\frac{6 a \sqrt{c-\frac{c}{a x}} \sqrt{1+a x}}{7 x^2 \sqrt{1-a x}}-\frac{32 a^2 \sqrt{c-\frac{c}{a x}} \sqrt{1+a x}}{21 x \sqrt{1-a x}}+\frac{\left (16 \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{105 a^4}{4 \sqrt{x} (1-a x) \sqrt{1+a x}} \, dx}{105 \sqrt{1-a x}}\\ &=-\frac{104 a^3 \sqrt{c-\frac{c}{a x}} \sqrt{1+a x}}{21 \sqrt{1-a x}}-\frac{2 \sqrt{c-\frac{c}{a x}} \sqrt{1+a x}}{7 x^3 \sqrt{1-a x}}-\frac{6 a \sqrt{c-\frac{c}{a x}} \sqrt{1+a x}}{7 x^2 \sqrt{1-a x}}-\frac{32 a^2 \sqrt{c-\frac{c}{a x}} \sqrt{1+a x}}{21 x \sqrt{1-a x}}+\frac{\left (4 a^4 \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{1}{\sqrt{x} (1-a x) \sqrt{1+a x}} \, dx}{\sqrt{1-a x}}\\ &=-\frac{104 a^3 \sqrt{c-\frac{c}{a x}} \sqrt{1+a x}}{21 \sqrt{1-a x}}-\frac{2 \sqrt{c-\frac{c}{a x}} \sqrt{1+a x}}{7 x^3 \sqrt{1-a x}}-\frac{6 a \sqrt{c-\frac{c}{a x}} \sqrt{1+a x}}{7 x^2 \sqrt{1-a x}}-\frac{32 a^2 \sqrt{c-\frac{c}{a x}} \sqrt{1+a x}}{21 x \sqrt{1-a x}}+\frac{\left (8 a^4 \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 )}{\sqrt{1-a x}}\\ &=-\frac{104 a^3 \sqrt{c-\frac{c}{a x}} \sqrt{1+a x}}{21 \sqrt{1-a x}}-\frac{2 \sqrt{c-\frac{c}{a x}} \sqrt{1+a x}}{7 x^3 \sqrt{1-a x}}-\frac{6 a \sqrt{c-\frac{c}{a x}} \sqrt{1+a x}}{7 x^2 \sqrt{1-a x}}-\frac{32 a^2 \sqrt{c-\frac{c}{a x}} \sqrt{1+a x}}{21 x \sqrt{1-a x}}+\frac{4 \sqrt{2} a^{7/2} \sqrt{c-\frac{c}{a x}} \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{1+a x}}\right )}{\sqrt{1-a x}}\\ \end{align*}
Mathematica [A] time = 0.0709776, size = 109, normalized size = 0.46 \[ \frac{2 \sqrt{c-\frac{c}{a x}} \left (42 \sqrt{2} a^{7/2} x^{7/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{x}}{\sqrt{a x+1}}\right )-\sqrt{a x+1} \left (52 a^3 x^3+16 a^2 x^2+9 a x+3\right )\right )}{21 x^3 \sqrt{1-a x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.145, size = 209, normalized size = 0.9 \begin{align*}{\frac{\sqrt{2}}{21\,{x}^{3} \left ( ax-1 \right ) }\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}}\sqrt{-{a}^{2}{x}^{2}+1} \left ( 52\,{a}^{3}\sqrt{2}\sqrt{-{a}^{-1}}{x}^{3}\sqrt{- \left ( ax+1 \right ) x}+42\,{a}^{3}\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 ){x}^{4}+16\,{a}^{2}\sqrt{2}\sqrt{-{a}^{-1}}{x}^{2}\sqrt{- \left ( ax+1 \right ) x}+9\,x\sqrt{- \left ( ax+1 \right ) x}a\sqrt{2}\sqrt{-{a}^{-1}}+3\,\sqrt{- \left ( ax+1 \right ) x}\sqrt{2}\sqrt{-{a}^{-1}} \right ){\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}}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.51777, size = 760, normalized size = 3.21 \begin{align*} \left [\frac{21 \, \sqrt{2}{\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt{-c} \log \left (-\frac{17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x + 4 \, \sqrt{2}{\left (3 \, 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^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 2 \,{\left (52 \, a^{3} x^{3} + 16 \, a^{2} x^{2} + 9 \, a x + 3\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{21 \,{\left (a x^{4} - x^{3}\right )}}, -\frac{2 \,{\left (21 \, \sqrt{2}{\left (a^{4} x^{4} - a^{3} x^{3}\right )} \sqrt{c} \arctan \left (\frac{2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) -{\left (52 \, a^{3} x^{3} + 16 \, a^{2} x^{2} + 9 \, a x + 3\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}\right )}}{21 \,{\left (a x^{4} - x^{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{\sqrt{- c \left (-1 + \frac{1}{a x}\right )} \left (a x + 1\right )^{3}}{x^{4} \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}}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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