Optimal. Leaf size=172 \[ -\frac{2 \sqrt{1-a^2 x^2} \sqrt{c-\frac{c}{a x}}}{7 x^3 (1-a x)}+\frac{208 a^3 \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{105 \sqrt{1-a x}}-\frac{104 a^2 \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{105 x \sqrt{1-a x}}+\frac{26 a \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{35 x^2 \sqrt{1-a x}} \]
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Rubi [A] time = 0.263483, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6134, 6128, 879, 848, 45, 37} \[ -\frac{2 \sqrt{1-a^2 x^2} \sqrt{c-\frac{c}{a x}}}{7 x^3 (1-a x)}+\frac{208 a^3 \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{105 \sqrt{1-a x}}-\frac{104 a^2 \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{105 x \sqrt{1-a x}}+\frac{26 a \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{35 x^2 \sqrt{1-a x}} \]
Antiderivative was successfully verified.
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Rule 6134
Rule 6128
Rule 879
Rule 848
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{e^{-\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^{-\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} \sqrt{1-a^2 x^2}} \, dx}{\sqrt{1-a x}}\\ &=-\frac{2 \sqrt{c-\frac{c}{a x}} \sqrt{1-a^2 x^2}}{7 x^3 (1-a x)}-\frac{\left (13 a \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{\sqrt{1-a x}}{x^{7/2} \sqrt{1-a^2 x^2}} \, dx}{7 \sqrt{1-a x}}\\ &=-\frac{2 \sqrt{c-\frac{c}{a x}} \sqrt{1-a^2 x^2}}{7 x^3 (1-a x)}-\frac{\left (13 a \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{1}{x^{7/2} \sqrt{1+a x}} \, dx}{7 \sqrt{1-a x}}\\ &=\frac{26 a \sqrt{c-\frac{c}{a x}} \sqrt{1+a x}}{35 x^2 \sqrt{1-a x}}-\frac{2 \sqrt{c-\frac{c}{a x}} \sqrt{1-a^2 x^2}}{7 x^3 (1-a x)}+\frac{\left (52 a^2 \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{1}{x^{5/2} \sqrt{1+a x}} \, dx}{35 \sqrt{1-a x}}\\ &=\frac{26 a \sqrt{c-\frac{c}{a x}} \sqrt{1+a x}}{35 x^2 \sqrt{1-a x}}-\frac{104 a^2 \sqrt{c-\frac{c}{a x}} \sqrt{1+a x}}{105 x \sqrt{1-a x}}-\frac{2 \sqrt{c-\frac{c}{a x}} \sqrt{1-a^2 x^2}}{7 x^3 (1-a x)}-\frac{\left (104 a^3 \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{1}{x^{3/2} \sqrt{1+a x}} \, dx}{105 \sqrt{1-a x}}\\ &=\frac{208 a^3 \sqrt{c-\frac{c}{a x}} \sqrt{1+a x}}{105 \sqrt{1-a x}}+\frac{26 a \sqrt{c-\frac{c}{a x}} \sqrt{1+a x}}{35 x^2 \sqrt{1-a x}}-\frac{104 a^2 \sqrt{c-\frac{c}{a x}} \sqrt{1+a x}}{105 x \sqrt{1-a x}}-\frac{2 \sqrt{c-\frac{c}{a x}} \sqrt{1-a^2 x^2}}{7 x^3 (1-a x)}\\ \end{align*}
Mathematica [A] time = 0.033091, size = 63, normalized size = 0.37 \[ \frac{2 \sqrt{a x+1} \left (104 a^3 x^3-52 a^2 x^2+39 a x-15\right ) \sqrt{c-\frac{c}{a x}}}{105 x^3 \sqrt{1-a x}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.084, size = 62, normalized size = 0.4 \begin{align*} -{\frac{208\,{x}^{3}{a}^{3}-104\,{a}^{2}{x}^{2}+78\,ax-30}{105\,{x}^{3} \left ( ax-1 \right ) }\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}}\sqrt{-{a}^{2}{x}^{2}+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{\sqrt{-a^{2} x^{2} + 1} \sqrt{c - \frac{c}{a x}}}{{\left (a x + 1\right )} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.5101, size = 143, normalized size = 0.83 \begin{align*} -\frac{2 \,{\left (104 \, a^{3} x^{3} - 52 \, a^{2} x^{2} + 39 \, a x - 15\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{105 \,{\left (a x^{4} - x^{3}\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 )} \sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}{x^{4} \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{\sqrt{-a^{2} x^{2} + 1} \sqrt{c - \frac{c}{a x}}}{{\left (a x + 1\right )} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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