Optimal. Leaf size=84 \[ \frac {10 a \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{3 \sqrt {1-a x}}-\frac {2 \sqrt {1-a^2 x^2} \sqrt {c-\frac {c}{a x}}}{3 x (1-a x)} \]
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Rubi [A] time = 0.26, antiderivative size = 84, 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, 6128, 879, 848, 37} \[ \frac {10 a \sqrt {a x+1} \sqrt {c-\frac {c}{a x}}}{3 \sqrt {1-a x}}-\frac {2 \sqrt {1-a^2 x^2} \sqrt {c-\frac {c}{a x}}}{3 x (1-a x)} \]
Antiderivative was successfully verified.
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Rule 37
Rule 848
Rule 879
Rule 6128
Rule 6134
Rubi steps
\begin {align*} \int \frac {e^{-\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^{-\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)^{3/2}}{x^{5/2} \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}}{3 x (1-a x)}-\frac {\left (5 a \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {\sqrt {1-a x}}{x^{3/2} \sqrt {1-a^2 x^2}} \, dx}{3 \sqrt {1-a x}}\\ &=-\frac {2 \sqrt {c-\frac {c}{a x}} \sqrt {1-a^2 x^2}}{3 x (1-a x)}-\frac {\left (5 a \sqrt {c-\frac {c}{a x}} \sqrt {x}\right ) \int \frac {1}{x^{3/2} \sqrt {1+a x}} \, dx}{3 \sqrt {1-a x}}\\ &=\frac {10 a \sqrt {c-\frac {c}{a x}} \sqrt {1+a x}}{3 \sqrt {1-a x}}-\frac {2 \sqrt {c-\frac {c}{a x}} \sqrt {1-a^2 x^2}}{3 x (1-a x)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 47, normalized size = 0.56 \[ \frac {2 \sqrt {a x+1} (5 a x-1) \sqrt {c-\frac {c}{a x}}}{3 x \sqrt {1-a x}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.46, size = 48, normalized size = 0.57 \[ -\frac {2 \, \sqrt {-a^{2} x^{2} + 1} {\left (5 \, a x - 1\right )} \sqrt {\frac {a c x - c}{a x}}}{3 \, {\left (a x^{2} - x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {c - \frac {c}{a x}}}{{\left (a x + 1\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 46, normalized size = 0.55 \[ -\frac {2 \left (5 a x -1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {-a^{2} x^{2}+1}}{3 \left (a x -1\right ) x} \]
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 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {c - \frac {c}{a x}}}{{\left (a x + 1\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.01, size = 60, normalized size = 0.71 \[ \frac {\sqrt {c-\frac {c}{a\,x}}\,\left (\frac {10\,x\,\sqrt {1-a^2\,x^2}}{3}-\frac {2\,\sqrt {1-a^2\,x^2}}{3\,a}\right )}{\frac {x}{a}-x^2} \]
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 \[ \int \frac {\sqrt {- c \left (-1 + \frac {1}{a x}\right )} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{x^{2} \left (a x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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