Optimal. Leaf size=187 \[ -\frac {3 a \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x}{4 \left (a-\frac {1}{x}\right ) (c-a c x)^{3/2}}-\frac {a^2 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2} x}{2 \left (a-\frac {1}{x}\right )^2 (c-a c x)^{3/2}}-\frac {3 \sqrt {a} \left (1-\frac {1}{a x}\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )}{4 \sqrt {2} \left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}} \]
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Rubi [A]
time = 0.14, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6311, 6316, 96,
95, 212} \begin {gather*} -\frac {a^2 x \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{3/2}}{2 \left (a-\frac {1}{x}\right )^2 (c-a c x)^{3/2}}-\frac {3 a x \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}{4 \left (a-\frac {1}{x}\right ) (c-a c x)^{3/2}}-\frac {3 \sqrt {a} \left (1-\frac {1}{a x}\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )}{4 \sqrt {2} \left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 95
Rule 96
Rule 212
Rule 6311
Rule 6316
Rubi steps
\begin {align*} \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx &=\frac {\left (\left (1-\frac {1}{a x}\right )^{3/2} x^{3/2}\right ) \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (1-\frac {1}{a x}\right )^{3/2} x^{3/2}} \, dx}{(c-a c x)^{3/2}}\\ &=-\frac {\left (1-\frac {1}{a x}\right )^{3/2} \text {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{3/2}}{\sqrt {x} \left (1-\frac {x}{a}\right )^3} \, dx,x,\frac {1}{x}\right )}{\left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}}\\ &=-\frac {a^2 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2} x}{2 \left (a-\frac {1}{x}\right )^2 (c-a c x)^{3/2}}-\frac {\left (3 \left (1-\frac {1}{a x}\right )^{3/2}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{\sqrt {x} \left (1-\frac {x}{a}\right )^2} \, dx,x,\frac {1}{x}\right )}{4 \left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}}\\ &=-\frac {3 a \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x}{4 \left (a-\frac {1}{x}\right ) (c-a c x)^{3/2}}-\frac {a^2 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2} x}{2 \left (a-\frac {1}{x}\right )^2 (c-a c x)^{3/2}}-\frac {\left (3 \left (1-\frac {1}{a x}\right )^{3/2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{8 \left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}}\\ &=-\frac {3 a \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x}{4 \left (a-\frac {1}{x}\right ) (c-a c x)^{3/2}}-\frac {a^2 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2} x}{2 \left (a-\frac {1}{x}\right )^2 (c-a c x)^{3/2}}-\frac {\left (3 \left (1-\frac {1}{a x}\right )^{3/2}\right ) \text {Subst}\left (\int \frac {1}{1-\frac {2 x^2}{a}} \, dx,x,\frac {\sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{a x}}}\right )}{4 \left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}}\\ &=-\frac {3 a \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x}{4 \left (a-\frac {1}{x}\right ) (c-a c x)^{3/2}}-\frac {a^2 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{3/2} x}{2 \left (a-\frac {1}{x}\right )^2 (c-a c x)^{3/2}}-\frac {3 \sqrt {a} \left (1-\frac {1}{a x}\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )}{4 \sqrt {2} \left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 125, normalized size = 0.67 \begin {gather*} \frac {\sqrt {1-\frac {1}{a x}} x \left (2 \sqrt {a} \sqrt {1+\frac {1}{a x}} (-1+5 a x)+3 \sqrt {2} \sqrt {\frac {1}{x}} (-1+a x)^2 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )\right )}{8 \sqrt {a} c (-1+a x)^2 \sqrt {c-a c x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 174, normalized size = 0.93
method | result | size |
default | \(-\frac {\sqrt {-c \left (a x -1\right )}\, \left (3 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) a^{2} c \,x^{2}-6 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) a c x +10 a x \sqrt {-c \left (a x +1\right )}\, \sqrt {c}+3 \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c -2 \sqrt {-c \left (a x +1\right )}\, \sqrt {c}\right )}{8 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x -1\right ) \left (a x +1\right ) c^{\frac {5}{2}} \sqrt {-c \left (a x +1\right )}\, a}\) | \(174\) |
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 = 0.36, size = 341, normalized size = 1.82 \begin {gather*} \left [-\frac {3 \, \sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + 4 \, {\left (5 \, a^{2} x^{2} + 4 \, a x - 1\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{16 \, {\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right )}}, -\frac {3 \, \sqrt {2} {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) + 2 \, {\left (5 \, a^{2} x^{2} + 4 \, a x - 1\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{8 \, {\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 78, normalized size = 0.42 \begin {gather*} \frac {\frac {3 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x - c}}{2 \, \sqrt {c}}\right )}{\sqrt {c}} - \frac {2 \, {\left (5 \, {\left (-a c x - c\right )}^{\frac {3}{2}} + 6 \, \sqrt {-a c x - c} c\right )}}{{\left (a c x - c\right )}^{2}}}{8 \, a {\left | c \right |}} \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 {1}{{\left (c-a\,c\,x\right )}^{3/2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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