Optimal. Leaf size=156 \[ \frac {c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a}+\frac {c^5 x \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{\left (c-\frac {c}{a x}\right )^{5/2}}-\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 a \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{a \sqrt {c-\frac {c}{a x}}} \]
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Rubi [A] time = 0.27, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6177, 879, 865, 875, 208} \[ \frac {c^5 x \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}{\left (c-\frac {c}{a x}\right )^{5/2}}-\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 a \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{a \sqrt {c-\frac {c}{a x}}}+\frac {c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 208
Rule 865
Rule 875
Rule 879
Rule 6177
Rubi steps
\begin {align*} \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{5/2} \, dx &=-\left (c^3 \operatorname {Subst}\left (\int \frac {\left (1-\frac {x^2}{a^2}\right )^{3/2}}{x^2 \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )\right )\\ &=\frac {c^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x}{\left (c-\frac {c}{a x}\right )^{5/2}}-\frac {c^4 \operatorname {Subst}\left (\int \frac {\left (1-\frac {x^2}{a^2}\right )^{3/2}}{x \left (c-\frac {c x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=-\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 a \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {c^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x}{\left (c-\frac {c}{a x}\right )^{5/2}}-\frac {c^3 \operatorname {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a^2}}}{x \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=-\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 a \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{a \sqrt {c-\frac {c}{a x}}}+\frac {c^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x}{\left (c-\frac {c}{a x}\right )^{5/2}}-\frac {c^2 \operatorname {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}}}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=-\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 a \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{a \sqrt {c-\frac {c}{a x}}}+\frac {c^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x}{\left (c-\frac {c}{a x}\right )^{5/2}}-\frac {c^4 \operatorname {Subst}\left (\int \frac {1}{-\frac {c}{a^2}+\frac {c^2 x^2}{a^2}} \, dx,x,\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a^3}\\ &=-\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 a \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {c^3 \sqrt {1-\frac {1}{a^2 x^2}}}{a \sqrt {c-\frac {c}{a x}}}+\frac {c^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x}{\left (c-\frac {c}{a x}\right )^{5/2}}+\frac {c^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 89, normalized size = 0.57 \[ \frac {c^2 \sqrt {c-\frac {c}{a x}} \left (\sqrt {\frac {1}{a x}+1} \left (3 a^2 x^2+2 a x+2\right )+3 a x \tanh ^{-1}\left (\sqrt {\frac {1}{a x}+1}\right )\right )}{3 a^2 x \sqrt {1-\frac {1}{a x}}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.53, size = 381, normalized size = 2.44 \[ \left [\frac {3 \, {\left (a^{2} c^{2} x^{2} - a c^{2} x\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (3 \, a^{3} c^{2} x^{3} + 5 \, a^{2} c^{2} x^{2} + 4 \, a c^{2} x + 2 \, c^{2}\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{12 \, {\left (a^{3} x^{2} - a^{2} x\right )}}, -\frac {3 \, {\left (a^{2} c^{2} x^{2} - a c^{2} x\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \, {\left (3 \, a^{3} c^{2} x^{3} + 5 \, a^{2} c^{2} x^{2} + 4 \, a c^{2} x + 2 \, c^{2}\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{6 \, {\left (a^{3} x^{2} - a^{2} x\right )}}\right ] \]
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 \[ \int \frac {{\left (c - \frac {c}{a x}\right )}^{\frac {5}{2}}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 144, normalized size = 0.92 \[ \frac {\left (a x -1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, c^{2} \left (6 a^{\frac {5}{2}} x^{2} \sqrt {\left (a x +1\right ) x}+3 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) x^{2} a^{2}+4 a^{\frac {3}{2}} x \sqrt {\left (a x +1\right ) x}+4 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}\right )}{6 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) x \,a^{\frac {5}{2}} \sqrt {\left (a x +1\right ) x}} \]
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 \[ \int \frac {{\left (c - \frac {c}{a x}\right )}^{\frac {5}{2}}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (c-\frac {c}{a\,x}\right )}^{5/2}}{{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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