Optimal. Leaf size=213 \[ \frac{2672 a^3 \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{105 \sqrt{1-a x}}-\frac{1336 a^2 \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{105 x \sqrt{1-a x}}+\frac{334 a^2 \sqrt{c-\frac{c}{a x}}}{35 x \sqrt{1-a x} \sqrt{a x+1}}+\frac{44 a \sqrt{c-\frac{c}{a x}}}{35 x^2 \sqrt{1-a x} \sqrt{a x+1}}-\frac{2 \sqrt{c-\frac{c}{a x}}}{7 x^3 \sqrt{1-a x} \sqrt{a x+1}} \]
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Rubi [A] time = 0.244291, antiderivative size = 213, 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, 6129, 89, 78, 45, 37} \[ \frac{2672 a^3 \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{105 \sqrt{1-a x}}-\frac{1336 a^2 \sqrt{a x+1} \sqrt{c-\frac{c}{a x}}}{105 x \sqrt{1-a x}}+\frac{334 a^2 \sqrt{c-\frac{c}{a x}}}{35 x \sqrt{1-a x} \sqrt{a x+1}}+\frac{44 a \sqrt{c-\frac{c}{a x}}}{35 x^2 \sqrt{1-a x} \sqrt{a x+1}}-\frac{2 \sqrt{c-\frac{c}{a x}}}{7 x^3 \sqrt{1-a x} \sqrt{a x+1}} \]
Antiderivative was successfully verified.
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Rule 6134
Rule 6129
Rule 89
Rule 78
Rule 45
Rule 37
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)^2}{x^{9/2} (1+a x)^{3/2}} \, dx}{\sqrt{1-a x}}\\ &=-\frac{2 \sqrt{c-\frac{c}{a x}}}{7 x^3 \sqrt{1-a x} \sqrt{1+a x}}+\frac{\left (2 \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{-11 a+\frac{7 a^2 x}{2}}{x^{7/2} (1+a x)^{3/2}} \, dx}{7 \sqrt{1-a x}}\\ &=-\frac{2 \sqrt{c-\frac{c}{a x}}}{7 x^3 \sqrt{1-a x} \sqrt{1+a x}}+\frac{44 a \sqrt{c-\frac{c}{a x}}}{35 x^2 \sqrt{1-a x} \sqrt{1+a x}}+\frac{\left (167 a^2 \sqrt{c-\frac{c}{a x}} \sqrt{x}\right ) \int \frac{1}{x^{5/2} (1+a x)^{3/2}} \, dx}{35 \sqrt{1-a x}}\\ &=-\frac{2 \sqrt{c-\frac{c}{a x}}}{7 x^3 \sqrt{1-a x} \sqrt{1+a x}}+\frac{44 a \sqrt{c-\frac{c}{a x}}}{35 x^2 \sqrt{1-a x} \sqrt{1+a x}}+\frac{334 a^2 \sqrt{c-\frac{c}{a x}}}{35 x \sqrt{1-a x} \sqrt{1+a x}}+\frac{\left (668 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{2 \sqrt{c-\frac{c}{a x}}}{7 x^3 \sqrt{1-a x} \sqrt{1+a x}}+\frac{44 a \sqrt{c-\frac{c}{a x}}}{35 x^2 \sqrt{1-a x} \sqrt{1+a x}}+\frac{334 a^2 \sqrt{c-\frac{c}{a x}}}{35 x \sqrt{1-a x} \sqrt{1+a x}}-\frac{1336 a^2 \sqrt{c-\frac{c}{a x}} \sqrt{1+a x}}{105 x \sqrt{1-a x}}-\frac{\left (1336 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{2 \sqrt{c-\frac{c}{a x}}}{7 x^3 \sqrt{1-a x} \sqrt{1+a x}}+\frac{44 a \sqrt{c-\frac{c}{a x}}}{35 x^2 \sqrt{1-a x} \sqrt{1+a x}}+\frac{334 a^2 \sqrt{c-\frac{c}{a x}}}{35 x \sqrt{1-a x} \sqrt{1+a x}}+\frac{2672 a^3 \sqrt{c-\frac{c}{a x}} \sqrt{1+a x}}{105 \sqrt{1-a x}}-\frac{1336 a^2 \sqrt{c-\frac{c}{a x}} \sqrt{1+a x}}{105 x \sqrt{1-a x}}\\ \end{align*}
Mathematica [A] time = 0.0331991, size = 66, normalized size = 0.31 \[ \frac{2 \left (1336 a^4 x^4+668 a^3 x^3-167 a^2 x^2+66 a x-15\right ) \sqrt{c-\frac{c}{a x}}}{105 x^3 \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.082, size = 77, normalized size = 0.4 \begin{align*}{\frac{2672\,{x}^{4}{a}^{4}+1336\,{x}^{3}{a}^{3}-334\,{a}^{2}{x}^{2}+132\,ax-30}{105\, \left ( ax+1 \right ) ^{2}{x}^{3} \left ( ax-1 \right ) ^{2}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}} \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^{2} x^{2} + 1\right )}^{\frac{3}{2}} \sqrt{c - \frac{c}{a x}}}{{\left (a x + 1\right )}^{3} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.10802, size = 167, normalized size = 0.78 \begin{align*} -\frac{2 \,{\left (1336 \, a^{4} x^{4} + 668 \, a^{3} x^{3} - 167 \, a^{2} x^{2} + 66 \, a x - 15\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a c x - c}{a x}}}{105 \,{\left (a^{2} x^{5} - x^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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^{2} x^{2} + 1\right )}^{\frac{3}{2}} \sqrt{c - \frac{c}{a x}}}{{\left (a x + 1\right )}^{3} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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