Optimal. Leaf size=112 \[ -\frac {9 a c^2 (1-a x)^{3/2}}{\sqrt {1+a x} (c-a c x)^{3/2}}-\frac {c^2 (1-a x)^{3/2}}{x \sqrt {1+a x} (c-a c x)^{3/2}}+\frac {7 a c^2 (1-a x)^{3/2} \tanh ^{-1}\left (\sqrt {1+a x}\right )}{(c-a c x)^{3/2}} \]
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Rubi [A]
time = 0.08, 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 = {6265, 23, 91,
79, 65, 214} \begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 23
Rule 65
Rule 79
Rule 91
Rule 214
Rule 6265
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 ) \text {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 = 0.02, size = 64, normalized size = 0.57 \begin {gather*} \frac {c \sqrt {1-a x} \left (-1-9 a x+7 a x \sqrt {1+a x} \tanh ^{-1}\left (\sqrt {1+a x}\right )\right )}{x \sqrt {1+a x} \sqrt {c-a c x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.10, size = 82, normalized size = 0.73
method | result | size |
default | \(\frac {\left (-7 \arctanh \left (\frac {\sqrt {\left (a x +1\right ) c}}{\sqrt {c}}\right ) a x \sqrt {\left (a x +1\right ) c}+9 a x \sqrt {c}+\sqrt {c}\right ) \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}{\left (a x -1\right ) \left (a x +1\right ) \sqrt {c}\, x}\) | \(82\) |
risch | \(\frac {\left (a x +1\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right ) c}{x \sqrt {\left (a x +1\right ) c}\, \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}+\frac {a \left (-\frac {14 \arctanh \left (\frac {\sqrt {c x a +c}}{\sqrt {c}}\right )}{\sqrt {c}}+\frac {16}{\sqrt {c x a +c}}\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right ) c}{2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}\) | \(152\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 223, normalized size = 1.99 \begin {gather*} \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 ] \end {gather*}
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 \begin {gather*} \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 \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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