Optimal. Leaf size=268 \[ \frac {3 \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{9/2}}{a \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {3 \left (28 a-\frac {17}{x}\right ) \left (1+\frac {1}{a x}\right )^{3/2} \left (c-\frac {c}{a x}\right )^{9/2}}{35 a^2 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {9 \left (a-\frac {1}{x}\right )^2 \left (1+\frac {1}{a x}\right )^{3/2} \left (c-\frac {c}{a x}\right )^{9/2}}{7 a^3 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {\left (a-\frac {1}{x}\right )^3 \left (1+\frac {1}{a x}\right )^{3/2} \left (c-\frac {c}{a x}\right )^{9/2} x}{a^3 \left (1-\frac {1}{a x}\right )^{9/2}}-\frac {3 \left (c-\frac {c}{a x}\right )^{9/2} \tanh ^{-1}\left (\sqrt {1+\frac {1}{a x}}\right )}{a \left (1-\frac {1}{a x}\right )^{9/2}} \]
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Rubi [A]
time = 0.11, antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6317, 6314, 99,
158, 152, 52, 65, 214} \begin {gather*} \frac {9 \left (a-\frac {1}{x}\right )^2 \left (\frac {1}{a x}+1\right )^{3/2} \left (c-\frac {c}{a x}\right )^{9/2}}{7 a^3 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {x \left (a-\frac {1}{x}\right )^3 \left (\frac {1}{a x}+1\right )^{3/2} \left (c-\frac {c}{a x}\right )^{9/2}}{a^3 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {3 \left (28 a-\frac {17}{x}\right ) \left (\frac {1}{a x}+1\right )^{3/2} \left (c-\frac {c}{a x}\right )^{9/2}}{35 a^2 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {3 \sqrt {\frac {1}{a x}+1} \left (c-\frac {c}{a x}\right )^{9/2}}{a \left (1-\frac {1}{a x}\right )^{9/2}}-\frac {3 \left (c-\frac {c}{a x}\right )^{9/2} \tanh ^{-1}\left (\sqrt {\frac {1}{a x}+1}\right )}{a \left (1-\frac {1}{a x}\right )^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 99
Rule 152
Rule 158
Rule 214
Rule 6314
Rule 6317
Rubi steps
\begin {align*} \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx &=\frac {\left (c-\frac {c}{a x}\right )^{9/2} \int e^{3 \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^{9/2} \, dx}{\left (1-\frac {1}{a x}\right )^{9/2}}\\ &=-\frac {\left (c-\frac {c}{a x}\right )^{9/2} \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^3 \left (1+\frac {x}{a}\right )^{3/2}}{x^2} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{9/2}}\\ &=\frac {\left (a-\frac {1}{x}\right )^3 \left (1+\frac {1}{a x}\right )^{3/2} \left (c-\frac {c}{a x}\right )^{9/2} x}{a^3 \left (1-\frac {1}{a x}\right )^{9/2}}-\frac {\left (c-\frac {c}{a x}\right )^{9/2} \text {Subst}\left (\int \frac {\left (-\frac {3}{2 a}-\frac {9 x}{2 a^2}\right ) \left (1-\frac {x}{a}\right )^2 \sqrt {1+\frac {x}{a}}}{x} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{9/2}}\\ &=\frac {9 \left (a-\frac {1}{x}\right )^2 \left (1+\frac {1}{a x}\right )^{3/2} \left (c-\frac {c}{a x}\right )^{9/2}}{7 a^3 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {\left (a-\frac {1}{x}\right )^3 \left (1+\frac {1}{a x}\right )^{3/2} \left (c-\frac {c}{a x}\right )^{9/2} x}{a^3 \left (1-\frac {1}{a x}\right )^{9/2}}-\frac {\left (2 a \left (c-\frac {c}{a x}\right )^{9/2}\right ) \text {Subst}\left (\int \frac {\left (-\frac {21}{4 a^2}-\frac {51 x}{4 a^3}\right ) \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}}{x} \, dx,x,\frac {1}{x}\right )}{7 \left (1-\frac {1}{a x}\right )^{9/2}}\\ &=\frac {3 \left (28 a-\frac {17}{x}\right ) \left (1+\frac {1}{a x}\right )^{3/2} \left (c-\frac {c}{a x}\right )^{9/2}}{35 a^2 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {9 \left (a-\frac {1}{x}\right )^2 \left (1+\frac {1}{a x}\right )^{3/2} \left (c-\frac {c}{a x}\right )^{9/2}}{7 a^3 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {\left (a-\frac {1}{x}\right )^3 \left (1+\frac {1}{a x}\right )^{3/2} \left (c-\frac {c}{a x}\right )^{9/2} x}{a^3 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {\left (3 \left (c-\frac {c}{a x}\right )^{9/2}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{x} \, dx,x,\frac {1}{x}\right )}{2 a \left (1-\frac {1}{a x}\right )^{9/2}}\\ &=\frac {3 \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{9/2}}{a \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {3 \left (28 a-\frac {17}{x}\right ) \left (1+\frac {1}{a x}\right )^{3/2} \left (c-\frac {c}{a x}\right )^{9/2}}{35 a^2 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {9 \left (a-\frac {1}{x}\right )^2 \left (1+\frac {1}{a x}\right )^{3/2} \left (c-\frac {c}{a x}\right )^{9/2}}{7 a^3 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {\left (a-\frac {1}{x}\right )^3 \left (1+\frac {1}{a x}\right )^{3/2} \left (c-\frac {c}{a x}\right )^{9/2} x}{a^3 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {\left (3 \left (c-\frac {c}{a x}\right )^{9/2}\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a \left (1-\frac {1}{a x}\right )^{9/2}}\\ &=\frac {3 \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{9/2}}{a \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {3 \left (28 a-\frac {17}{x}\right ) \left (1+\frac {1}{a x}\right )^{3/2} \left (c-\frac {c}{a x}\right )^{9/2}}{35 a^2 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {9 \left (a-\frac {1}{x}\right )^2 \left (1+\frac {1}{a x}\right )^{3/2} \left (c-\frac {c}{a x}\right )^{9/2}}{7 a^3 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {\left (a-\frac {1}{x}\right )^3 \left (1+\frac {1}{a x}\right )^{3/2} \left (c-\frac {c}{a x}\right )^{9/2} x}{a^3 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {\left (3 \left (c-\frac {c}{a x}\right )^{9/2}\right ) \text {Subst}\left (\int \frac {1}{-a+a x^2} \, dx,x,\sqrt {1+\frac {1}{a x}}\right )}{\left (1-\frac {1}{a x}\right )^{9/2}}\\ &=\frac {3 \sqrt {1+\frac {1}{a x}} \left (c-\frac {c}{a x}\right )^{9/2}}{a \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {3 \left (28 a-\frac {17}{x}\right ) \left (1+\frac {1}{a x}\right )^{3/2} \left (c-\frac {c}{a x}\right )^{9/2}}{35 a^2 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {9 \left (a-\frac {1}{x}\right )^2 \left (1+\frac {1}{a x}\right )^{3/2} \left (c-\frac {c}{a x}\right )^{9/2}}{7 a^3 \left (1-\frac {1}{a x}\right )^{9/2}}+\frac {\left (a-\frac {1}{x}\right )^3 \left (1+\frac {1}{a x}\right )^{3/2} \left (c-\frac {c}{a x}\right )^{9/2} x}{a^3 \left (1-\frac {1}{a x}\right )^{9/2}}-\frac {3 \left (c-\frac {c}{a x}\right )^{9/2} \tanh ^{-1}\left (\sqrt {1+\frac {1}{a x}}\right )}{a \left (1-\frac {1}{a x}\right )^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 109, normalized size = 0.41 \begin {gather*} \frac {c^4 \sqrt {c-\frac {c}{a x}} \left (\sqrt {1+\frac {1}{a x}} \left (10-26 a x-12 a^2 x^2+164 a^3 x^3+35 a^4 x^4\right )-105 a^3 x^3 \tanh ^{-1}\left (\sqrt {1+\frac {1}{a x}}\right )\right )}{35 a^4 \sqrt {1-\frac {1}{a x}} x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 178, normalized size = 0.66
method | result | size |
default | \(\frac {\left (a x -1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, c^{4} \left (70 a^{\frac {9}{2}} \sqrt {x \left (a x +1\right )}\, x^{4}+328 a^{\frac {7}{2}} x^{3} \sqrt {x \left (a x +1\right )}-105 \ln \left (\frac {2 \sqrt {x \left (a x +1\right )}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) a^{4} x^{4}-24 a^{\frac {5}{2}} x^{2} \sqrt {x \left (a x +1\right )}-52 a^{\frac {3}{2}} x \sqrt {x \left (a x +1\right )}+20 \sqrt {x \left (a x +1\right )}\, \sqrt {a}\right )}{70 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) x^{3} a^{\frac {9}{2}} \sqrt {x \left (a x +1\right )}}\) | \(178\) |
risch | \(\frac {\left (35 a^{5} x^{5}+199 a^{4} x^{4}+152 a^{3} x^{3}-38 a^{2} x^{2}-16 a x +10\right ) c^{4} \sqrt {\frac {c \left (a x -1\right )}{a x}}}{35 x^{3} a^{4} \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}-\frac {3 \ln \left (\frac {\frac {1}{2} a c +c \,a^{2} x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+a c x}\right ) c^{4} \sqrt {\frac {c \left (a x -1\right )}{a x}}\, \sqrt {c a x \left (a x +1\right )}}{2 \sqrt {a^{2} c}\, \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\) | \(184\) |
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 = 0.38, size = 437, normalized size = 1.63 \begin {gather*} \left [\frac {105 \, {\left (a^{4} c^{4} x^{4} - a^{3} c^{4} x^{3}\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 (35 \, a^{5} c^{4} x^{5} + 199 \, a^{4} c^{4} x^{4} + 152 \, a^{3} c^{4} x^{3} - 38 \, a^{2} c^{4} x^{2} - 16 \, a c^{4} x + 10 \, c^{4}\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{140 \, {\left (a^{5} x^{4} - a^{4} x^{3}\right )}}, \frac {105 \, {\left (a^{4} c^{4} x^{4} - a^{3} c^{4} x^{3}\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 (35 \, a^{5} c^{4} x^{5} + 199 \, a^{4} c^{4} x^{4} + 152 \, a^{3} c^{4} x^{3} - 38 \, a^{2} c^{4} x^{2} - 16 \, a c^{4} x + 10 \, c^{4}\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{70 \, {\left (a^{5} x^{4} - a^{4} x^{3}\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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 = -0.00 \begin {gather*} \int \frac {{\left (c-\frac {c}{a\,x}\right )}^{9/2}}{{\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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