Optimal. Leaf size=147 \[ \frac {45 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{8 a^3}-\frac {4 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a^3}-\frac {19 x \sqrt {c-\frac {c}{a x}}}{8 a^2}-\frac {1}{3} x^3 \sqrt {c-\frac {c}{a x}}+\frac {13 x^2 \sqrt {c-\frac {c}{a x}}}{12 a} \]
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Rubi [A] time = 0.31, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6133, 25, 514, 446, 98, 151, 156, 63, 208} \[ -\frac {19 x \sqrt {c-\frac {c}{a x}}}{8 a^2}+\frac {45 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{8 a^3}-\frac {4 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a^3}-\frac {1}{3} x^3 \sqrt {c-\frac {c}{a x}}+\frac {13 x^2 \sqrt {c-\frac {c}{a x}}}{12 a} \]
Antiderivative was successfully verified.
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Rule 25
Rule 63
Rule 98
Rule 151
Rule 156
Rule 208
Rule 446
Rule 514
Rule 6133
Rubi steps
\begin {align*} \int e^{-2 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^2 \, dx &=\int \frac {\sqrt {c-\frac {c}{a x}} x^2 (1-a x)}{1+a x} \, dx\\ &=-\frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{3/2} x^3}{1+a x} \, dx}{c}\\ &=-\frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{3/2} x^2}{a+\frac {1}{x}} \, dx}{c}\\ &=\frac {a \operatorname {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{3/2}}{x^4 (a+x)} \, dx,x,\frac {1}{x}\right )}{c}\\ &=-\frac {1}{3} \sqrt {c-\frac {c}{a x}} x^3-\frac {\operatorname {Subst}\left (\int \frac {\frac {13 c^2}{2}-\frac {11 c^2 x}{2 a}}{x^3 (a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{3 c}\\ &=\frac {13 \sqrt {c-\frac {c}{a x}} x^2}{12 a}-\frac {1}{3} \sqrt {c-\frac {c}{a x}} x^3+\frac {\operatorname {Subst}\left (\int \frac {\frac {57 c^3}{4}-\frac {39 c^3 x}{4 a}}{x^2 (a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{6 a c^2}\\ &=-\frac {19 \sqrt {c-\frac {c}{a x}} x}{8 a^2}+\frac {13 \sqrt {c-\frac {c}{a x}} x^2}{12 a}-\frac {1}{3} \sqrt {c-\frac {c}{a x}} x^3-\frac {\operatorname {Subst}\left (\int \frac {\frac {135 c^4}{8}-\frac {57 c^4 x}{8 a}}{x (a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{6 a^2 c^3}\\ &=-\frac {19 \sqrt {c-\frac {c}{a x}} x}{8 a^2}+\frac {13 \sqrt {c-\frac {c}{a x}} x^2}{12 a}-\frac {1}{3} \sqrt {c-\frac {c}{a x}} x^3-\frac {(45 c) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{16 a^3}+\frac {(4 c) \operatorname {Subst}\left (\int \frac {1}{(a+x) \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{a^3}\\ &=-\frac {19 \sqrt {c-\frac {c}{a x}} x}{8 a^2}+\frac {13 \sqrt {c-\frac {c}{a x}} x^2}{12 a}-\frac {1}{3} \sqrt {c-\frac {c}{a x}} x^3+\frac {45 \operatorname {Subst}\left (\int \frac {1}{a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )}{8 a^2}-\frac {8 \operatorname {Subst}\left (\int \frac {1}{2 a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )}{a^2}\\ &=-\frac {19 \sqrt {c-\frac {c}{a x}} x}{8 a^2}+\frac {13 \sqrt {c-\frac {c}{a x}} x^2}{12 a}-\frac {1}{3} \sqrt {c-\frac {c}{a x}} x^3+\frac {45 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{8 a^3}-\frac {4 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{a^3}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 108, normalized size = 0.73 \[ \frac {a x \left (-8 a^2 x^2+26 a x-57\right ) \sqrt {c-\frac {c}{a x}}+135 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )-96 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {2} \sqrt {c}}\right )}{24 a^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 256, normalized size = 1.74 \[ \left [\frac {96 \, \sqrt {2} \sqrt {c} \log \left (\frac {2 \, \sqrt {2} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} - 3 \, a c x + c}{a x + 1}\right ) - 2 \, {\left (8 \, a^{3} x^{3} - 26 \, a^{2} x^{2} + 57 \, a x\right )} \sqrt {\frac {a c x - c}{a x}} + 135 \, \sqrt {c} \log \left (-2 \, a c x - 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right )}{48 \, a^{3}}, \frac {96 \, \sqrt {2} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{2 \, c}\right ) - {\left (8 \, a^{3} x^{3} - 26 \, a^{2} x^{2} + 57 \, a x\right )} \sqrt {\frac {a c x - c}{a x}} - 135 \, \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right )}{24 \, a^{3}}\right ] \]
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 = 0.04, size = 237, normalized size = 1.61 \[ -\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (16 \left (a \,x^{2}-x \right )^{\frac {3}{2}} a^{\frac {7}{2}} \sqrt {\frac {1}{a}}-36 \sqrt {a \,x^{2}-x}\, a^{\frac {7}{2}} \sqrt {\frac {1}{a}}\, x +18 \sqrt {a \,x^{2}-x}\, a^{\frac {5}{2}} \sqrt {\frac {1}{a}}+96 a^{\frac {5}{2}} \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}-96 a^{\frac {3}{2}} \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {\frac {1}{a}}\, \sqrt {\left (a x -1\right ) x}\, a -3 a x +1}{a x +1}\right )-144 a^{2} \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}+9 \ln \left (\frac {2 \sqrt {a \,x^{2}-x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) \sqrt {\frac {1}{a}}\, a^{2}\right )}{48 \sqrt {\left (a x -1\right ) x}\, a^{\frac {9}{2}} \sqrt {\frac {1}{a}}} \]
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 \[ -\int \frac {{\left (a^{2} x^{2} - 1\right )} \sqrt {c - \frac {c}{a x}} x^{2}}{{\left (a x + 1\right )}^{2}}\,{d x} \]
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 \[ -\int \frac {x^2\,\sqrt {c-\frac {c}{a\,x}}\,\left (a^2\,x^2-1\right )}{{\left (a\,x+1\right )}^2} \,d x \]
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 \[ - \int \left (- \frac {x^{2} \sqrt {c - \frac {c}{a x}}}{a x + 1}\right )\, dx - \int \frac {a x^{3} \sqrt {c - \frac {c}{a x}}}{a x + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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