Optimal. Leaf size=112 \[ -\frac {9 a c^2 (1-a x)^{3/2}}{\sqrt {a x+1} (c-a c x)^{3/2}}-\frac {c^2 (1-a x)^{3/2}}{x \sqrt {a x+1} (c-a c x)^{3/2}}+\frac {7 a c^2 (1-a x)^{3/2} \tanh ^{-1}\left (\sqrt {a x+1}\right )}{(c-a c x)^{3/2}} \]
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Rubi [A] time = 0.12, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6130, 23, 89, 78, 63, 208} \[ -\frac {9 a c^2 (1-a x)^{3/2}}{\sqrt {a x+1} (c-a c x)^{3/2}}-\frac {c^2 (1-a x)^{3/2}}{x \sqrt {a x+1} (c-a c x)^{3/2}}+\frac {7 a c^2 (1-a x)^{3/2} \tanh ^{-1}\left (\sqrt {a x+1}\right )}{(c-a c x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 23
Rule 63
Rule 78
Rule 89
Rule 208
Rule 6130
Rubi steps
\begin {align*} \int \frac {e^{-3 \tanh ^{-1}(a x)} \sqrt {c-a c x}}{x^2} \, dx &=\int \frac {(1-a x)^{3/2} \sqrt {c-a c x}}{x^2 (1+a x)^{3/2}} \, dx\\ &=\frac {(1-a x)^{3/2} \int \frac {(c-a c x)^2}{x^2 (1+a x)^{3/2}} \, dx}{(c-a c x)^{3/2}}\\ &=-\frac {c^2 (1-a x)^{3/2}}{x \sqrt {1+a x} (c-a c x)^{3/2}}+\frac {(1-a x)^{3/2} \int \frac {-\frac {7 a c^2}{2}+a^2 c^2 x}{x (1+a x)^{3/2}} \, dx}{(c-a c x)^{3/2}}\\ &=-\frac {9 a c^2 (1-a x)^{3/2}}{\sqrt {1+a x} (c-a c x)^{3/2}}-\frac {c^2 (1-a x)^{3/2}}{x \sqrt {1+a x} (c-a c x)^{3/2}}-\frac {\left (7 a c^2 (1-a x)^{3/2}\right ) \int \frac {1}{x \sqrt {1+a x}} \, dx}{2 (c-a c x)^{3/2}}\\ &=-\frac {9 a c^2 (1-a x)^{3/2}}{\sqrt {1+a x} (c-a c x)^{3/2}}-\frac {c^2 (1-a x)^{3/2}}{x \sqrt {1+a x} (c-a c x)^{3/2}}-\frac {\left (7 c^2 (1-a x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{a}+\frac {x^2}{a}} \, dx,x,\sqrt {1+a x}\right )}{(c-a c x)^{3/2}}\\ &=-\frac {9 a c^2 (1-a x)^{3/2}}{\sqrt {1+a x} (c-a c x)^{3/2}}-\frac {c^2 (1-a x)^{3/2}}{x \sqrt {1+a x} (c-a c x)^{3/2}}+\frac {7 a c^2 (1-a x)^{3/2} \tanh ^{-1}\left (\sqrt {1+a x}\right )}{(c-a c x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 64, normalized size = 0.57 \[ \frac {c \sqrt {1-a x} \left (-9 a x+7 a x \sqrt {a x+1} \tanh ^{-1}\left (\sqrt {a x+1}\right )-1\right )}{x \sqrt {a x+1} \sqrt {c-a c x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 223, normalized size = 1.99 \[ \left [\frac {7 \, {\left (a^{3} x^{3} - a x\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + a c x - 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{a x^{2} - x}\right ) + 2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} {\left (9 \, a x + 1\right )}}{2 \, {\left (a^{2} x^{3} - x\right )}}, \frac {7 \, {\left (a^{3} x^{3} - a x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} {\left (9 \, a x + 1\right )}}{a^{2} x^{3} - x}\right ] \]
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 = 0.05, size = 82, normalized size = 0.73 \[ \frac {\left (-7 \arctanh \left (\frac {\sqrt {c \left (a x +1\right )}}{\sqrt {c}}\right ) x a \sqrt {c \left (a x +1\right )}+9 x a \sqrt {c}+\sqrt {c}\right ) \sqrt {-c \left (a x -1\right )}\, \sqrt {-a^{2} x^{2}+1}}{\left (a x -1\right ) \sqrt {c}\, \left (a x +1\right ) x} \]
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 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \sqrt {-a c x + c}}{{\left (a x + 1\right )}^{3} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (1-a^2\,x^2\right )}^{3/2}\,\sqrt {c-a\,c\,x}}{x^2\,{\left (a\,x+1\right )}^3} \,d x \]
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 \[ \int \frac {\sqrt {- c \left (a x - 1\right )} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{x^{2} \left (a x + 1\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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