Optimal. Leaf size=102 \[ -\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{x (c-a c x)^{3/2}}-\frac {a c \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}+a \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right ) \]
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Rubi [A] time = 0.11, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {879, 865, 875, 208} \[ -\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{x (c-a c x)^{3/2}}-\frac {a c \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}+a \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right ) \]
Antiderivative was successfully verified.
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Rule 208
Rule 865
Rule 875
Rule 879
Rubi steps
\begin {align*} \int \frac {\sqrt {c-a c x} \sqrt {1-a^2 x^2}}{x^2} \, dx &=-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{x (c-a c x)^{3/2}}-\frac {1}{2} (a c) \int \frac {\sqrt {1-a^2 x^2}}{x \sqrt {c-a c x}} \, dx\\ &=-\frac {a c \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{x (c-a c x)^{3/2}}-\frac {1}{2} a \int \frac {\sqrt {c-a c x}}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {a c \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{x (c-a c x)^{3/2}}-\left (a^3 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{-a^2 c+a^2 c^2 x^2} \, dx,x,\frac {\sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right )\\ &=-\frac {a c \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}-\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{x (c-a c x)^{3/2}}+a \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.10, size = 93, normalized size = 0.91 \[ \frac {\sqrt {1-a^2 x^2} \left (a \sqrt {c} x \tanh ^{-1}\left (\sqrt {c} \sqrt {\frac {a x+1}{c}}\right )-c (2 a x+1) \sqrt {\frac {a x+1}{c}}\right )}{x \sqrt {\frac {a x+1}{c}} \sqrt {c-a c x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 217, normalized size = 2.13 \[ \left [\frac {{\left (a^{2} x^{2} - a x\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + a c x - 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{a x^{2} - x}\right ) + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} {\left (2 \, a x + 1\right )}}{2 \, {\left (a x^{2} - x\right )}}, \frac {{\left (a^{2} x^{2} - a x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} {\left (2 \, a x + 1\right )}}{a x^{2} - x}\right ] \]
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 = 0.05, size = 95, normalized size = 0.93 \[ \frac {\left (-a c x \arctanh \left (\frac {\sqrt {\left (a x +1\right ) c}}{\sqrt {c}}\right )+2 \sqrt {\left (a x +1\right ) c}\, a \sqrt {c}\, x +\sqrt {\left (a x +1\right ) c}\, \sqrt {c}\right ) \sqrt {-\left (a x -1\right ) c}\, \sqrt {-a^{2} x^{2}+1}}{\left (a x -1\right ) \sqrt {\left (a x +1\right ) c}\, \sqrt {c}\, x} \]
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 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {1-a^2\,x^2}\,\sqrt {c-a\,c\,x}}{x^2} \,d x \]
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 \[ \int \frac {\sqrt {- c \left (a x - 1\right )} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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