Optimal. Leaf size=164 \[ \frac {c x^2 \sqrt {1-\frac {1}{a^2 x^2}}}{12 a \sqrt {c-\frac {c}{a x}}}-\frac {c x \sqrt {1-\frac {1}{a^2 x^2}}}{8 a^2 \sqrt {c-\frac {c}{a x}}}+\frac {c x^3 \sqrt {1-\frac {1}{a^2 x^2}}}{3 \sqrt {c-\frac {c}{a x}}}+\frac {\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{8 a^3} \]
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Rubi [A] time = 0.34, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6178, 863, 873, 875, 208} \[ \frac {c x^3 \sqrt {1-\frac {1}{a^2 x^2}}}{3 \sqrt {c-\frac {c}{a x}}}+\frac {c x^2 \sqrt {1-\frac {1}{a^2 x^2}}}{12 a \sqrt {c-\frac {c}{a x}}}-\frac {c x \sqrt {1-\frac {1}{a^2 x^2}}}{8 a^2 \sqrt {c-\frac {c}{a x}}}+\frac {\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{8 a^3} \]
Antiderivative was successfully verified.
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Rule 208
Rule 863
Rule 873
Rule 875
Rule 6178
Rubi steps
\begin {align*} \int e^{\coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx &=-\left (c \operatorname {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a^2}}}{x^4 \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )\right )\\ &=\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^3}{3 \sqrt {c-\frac {c}{a x}}}-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}}}{x^3 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{6 a}\\ &=\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^2}{12 a \sqrt {c-\frac {c}{a x}}}+\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^3}{3 \sqrt {c-\frac {c}{a x}}}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{8 a^2}\\ &=-\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x}{8 a^2 \sqrt {c-\frac {c}{a x}}}+\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^2}{12 a \sqrt {c-\frac {c}{a x}}}+\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^3}{3 \sqrt {c-\frac {c}{a x}}}-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}}}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{16 a^3}\\ &=-\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x}{8 a^2 \sqrt {c-\frac {c}{a x}}}+\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^2}{12 a \sqrt {c-\frac {c}{a x}}}+\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^3}{3 \sqrt {c-\frac {c}{a x}}}-\frac {c^2 \operatorname {Subst}\left (\int \frac {1}{-\frac {c}{a^2}+\frac {c^2 x^2}{a^2}} \, dx,x,\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{8 a^5}\\ &=-\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x}{8 a^2 \sqrt {c-\frac {c}{a x}}}+\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^2}{12 a \sqrt {c-\frac {c}{a x}}}+\frac {c \sqrt {1-\frac {1}{a^2 x^2}} x^3}{3 \sqrt {c-\frac {c}{a x}}}+\frac {\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{8 a^3}\\ \end {align*}
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Mathematica [A] time = 0.51, size = 147, normalized size = 0.90 \[ \frac {\frac {2 a^2 x^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (8 a^2 x^2+2 a x-3\right ) \sqrt {c-\frac {c}{a x}}}{a x-1}+3 \sqrt {c} \log \left (2 a^2 \sqrt {c} x^2 \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}}+c \left (2 a^2 x^2-a x-1\right )\right )-3 \sqrt {c} \log (1-a x)}{48 a^3} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.65, size = 337, normalized size = 2.05 \[ \left [\frac {3 \, {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (8 \, a^{4} x^{4} + 10 \, a^{3} x^{3} - a^{2} x^{2} - 3 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{96 \, {\left (a^{4} x - a^{3}\right )}}, -\frac {3 \, {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \, {\left (8 \, a^{4} x^{4} + 10 \, a^{3} x^{3} - a^{2} x^{2} - 3 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{48 \, {\left (a^{4} x - a^{3}\right )}}\right ] \]
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 \[ \int \frac {\sqrt {c - \frac {c}{a x}} x^{2}}{\sqrt {\frac {a x - 1}{a x + 1}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 121, normalized size = 0.74 \[ \frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (16 a^{\frac {5}{2}} x^{2} \sqrt {\left (a x +1\right ) x}+4 a^{\frac {3}{2}} x \sqrt {\left (a x +1\right ) x}-6 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+3 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right )\right )}{48 \sqrt {\frac {a x -1}{a x +1}}\, a^{\frac {5}{2}} \sqrt {\left (a x +1\right ) x}} \]
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 \[ \int \frac {\sqrt {c - \frac {c}{a x}} x^{2}}{\sqrt {\frac {a x - 1}{a x + 1}}}\,{d x} \]
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 \[ \int \frac {x^2\,\sqrt {c-\frac {c}{a\,x}}}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]
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 \[ \int \frac {x^{2} \sqrt {- c \left (-1 + \frac {1}{a x}\right )}}{\sqrt {\frac {a x - 1}{a x + 1}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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