Optimal. Leaf size=123 \[ \frac {46 a^2 x \sqrt {c-\frac {c}{a x}}}{3 \sqrt {1-a x} \sqrt {a x+1}}+\frac {20 a \sqrt {c-\frac {c}{a x}}}{3 \sqrt {1-a x} \sqrt {a x+1}}-\frac {2 \sqrt {c-\frac {c}{a x}}}{3 x \sqrt {1-a x} \sqrt {a x+1}} \]
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Rubi [A] time = 0.22, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {6134, 6129, 89, 78, 37} \[ \frac {46 a^2 x \sqrt {c-\frac {c}{a x}}}{3 \sqrt {1-a x} \sqrt {a x+1}}+\frac {20 a \sqrt {c-\frac {c}{a x}}}{3 \sqrt {1-a x} \sqrt {a x+1}}-\frac {2 \sqrt {c-\frac {c}{a x}}}{3 x \sqrt {1-a x} \sqrt {a x+1}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 78
Rule 89
Rule 6129
Rule 6134
Rubi steps
\begin {align*} \int \frac {e^{-3 \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 \frac {e^{-3 \tanh ^{-1}(a x)} \sqrt {1-a x}}{x^{5/2}} \, dx}{\sqrt {1-a x}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {(1-a x)^2}{x^{5/2} (1+a x)^{3/2}} \, dx}{\sqrt {1-a x}}\\ &=-\frac {2 \sqrt {c-\frac {c}{a x}}}{3 x \sqrt {1-a x} \sqrt {1+a x}}+\frac {\left (2 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {-5 a+\frac {3 a^2 x}{2}}{x^{3/2} (1+a x)^{3/2}} \, dx}{3 \sqrt {1-a x}}\\ &=\frac {20 a \sqrt {c-\frac {c}{a x}}}{3 \sqrt {1-a x} \sqrt {1+a x}}-\frac {2 \sqrt {c-\frac {c}{a x}}}{3 x \sqrt {1-a x} \sqrt {1+a x}}+\frac {\left (23 a^2 \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {1}{\sqrt {x} (1+a x)^{3/2}} \, dx}{3 \sqrt {1-a x}}\\ &=\frac {20 a \sqrt {c-\frac {c}{a x}}}{3 \sqrt {1-a x} \sqrt {1+a x}}-\frac {2 \sqrt {c-\frac {c}{a x}}}{3 x \sqrt {1-a x} \sqrt {1+a x}}+\frac {46 a^2 \sqrt {c-\frac {c}{a x}} x}{3 \sqrt {1-a x} \sqrt {1+a x}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 50, normalized size = 0.41 \[ \frac {2 \left (23 a^2 x^2+10 a x-1\right ) \sqrt {c-\frac {c}{a x}}}{3 x \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 58, normalized size = 0.47 \[ -\frac {2 \, {\left (23 \, a^{2} x^{2} + 10 \, a x - 1\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{3 \, {\left (a^{2} x^{3} - x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 61, normalized size = 0.50 \[ \frac {2 \left (23 a^{2} x^{2}+10 a x -1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3 \left (a x +1\right )^{2} x \left (a x -1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \sqrt {c - \frac {c}{a x}}}{{\left (a x + 1\right )}^{3} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.12, size = 80, normalized size = 0.65 \[ \frac {\sqrt {c-\frac {c}{a\,x}}\,\left (\frac {46\,x^2\,\sqrt {1-a^2\,x^2}}{3}-\frac {2\,\sqrt {1-a^2\,x^2}}{3\,a^2}+\frac {20\,x\,\sqrt {1-a^2\,x^2}}{3\,a}\right )}{\frac {x}{a^2}-x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right )} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{x^{2} \left (a x + 1\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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