Optimal. Leaf size=95 \[ \frac {7 \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{3/2}}+\frac {x}{\left (c-\frac {c}{a x}\right )^{3/2}}-\frac {7}{a c \sqrt {c-\frac {c}{a x}}}-\frac {7}{3 a \left (c-\frac {c}{a x}\right )^{3/2}} \]
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Rubi [A] time = 0.20, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6167, 6133, 25, 514, 375, 78, 51, 63, 208} \[ \frac {7 \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{3/2}}+\frac {x}{\left (c-\frac {c}{a x}\right )^{3/2}}-\frac {7}{a c \sqrt {c-\frac {c}{a x}}}-\frac {7}{3 a \left (c-\frac {c}{a x}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 25
Rule 51
Rule 63
Rule 78
Rule 208
Rule 375
Rule 514
Rule 6133
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx &=-\int \frac {e^{2 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx\\ &=-\int \frac {1+a x}{\left (c-\frac {c}{a x}\right )^{3/2} (1-a x)} \, dx\\ &=\frac {c \int \frac {1+a x}{\left (c-\frac {c}{a x}\right )^{5/2} x} \, dx}{a}\\ &=\frac {c \int \frac {a+\frac {1}{x}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx}{a}\\ &=-\frac {c \operatorname {Subst}\left (\int \frac {a+x}{x^2 \left (c-\frac {c x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=\frac {x}{\left (c-\frac {c}{a x}\right )^{3/2}}-\frac {(7 c) \operatorname {Subst}\left (\int \frac {1}{x \left (c-\frac {c x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=-\frac {7}{3 a \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {x}{\left (c-\frac {c}{a x}\right )^{3/2}}-\frac {7 \operatorname {Subst}\left (\int \frac {1}{x \left (c-\frac {c x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=-\frac {7}{3 a \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {7}{a c \sqrt {c-\frac {c}{a x}}}+\frac {x}{\left (c-\frac {c}{a x}\right )^{3/2}}-\frac {7 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a c}\\ &=-\frac {7}{3 a \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {7}{a c \sqrt {c-\frac {c}{a x}}}+\frac {x}{\left (c-\frac {c}{a x}\right )^{3/2}}+\frac {7 \operatorname {Subst}\left (\int \frac {1}{a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )}{c^2}\\ &=-\frac {7}{3 a \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {7}{a c \sqrt {c-\frac {c}{a x}}}+\frac {x}{\left (c-\frac {c}{a x}\right )^{3/2}}+\frac {7 \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 55, normalized size = 0.58 \[ \frac {x \left (3 a x-7 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};1-\frac {1}{a x}\right )\right )}{3 c (a x-1) \sqrt {c-\frac {c}{a x}}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.54, size = 238, normalized size = 2.51 \[ \left [\frac {21 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {c} \log \left (-2 \, a c x - 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) + 2 \, {\left (3 \, a^{3} x^{3} - 28 \, a^{2} x^{2} + 21 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{6 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}}, -\frac {21 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right ) - {\left (3 \, a^{3} x^{3} - 28 \, a^{2} x^{2} + 21 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{3 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 136, normalized size = 1.43 \[ -\frac {a {\left (\frac {2 \, {\left (2 \, c + \frac {9 \, {\left (a c x - c\right )}}{a x}\right )} x}{{\left (a c x - c\right )} a \sqrt {\frac {a c x - c}{a x}}} + \frac {21 \, \arctan \left (\frac {\sqrt {\frac {a c x - c}{a x}}}{\sqrt {-c}}\right )}{a^{2} \sqrt {-c}} - \frac {3 \, \sqrt {\frac {a c x - c}{a x}}}{a^{2} {\left (c - \frac {a c x - c}{a x}\right )}}\right )}}{3 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 260, normalized size = 2.74 \[ \frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (42 a^{\frac {7}{2}} \sqrt {\left (a x -1\right ) x}\, x^{3}+21 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) x^{3} a^{3}-36 a^{\frac {5}{2}} \left (\left (a x -1\right ) x \right )^{\frac {3}{2}} x -126 a^{\frac {5}{2}} \sqrt {\left (a x -1\right ) x}\, x^{2}-63 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) x^{2} a^{2}+28 a^{\frac {3}{2}} \left (\left (a x -1\right ) x \right )^{\frac {3}{2}}+126 a^{\frac {3}{2}} \sqrt {\left (a x -1\right ) x}\, x +63 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) x a -42 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}-21 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right )\right )}{6 \sqrt {\left (a x -1\right ) x}\, c^{2} \sqrt {a}\, \left (a x -1\right )^{3}} \]
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 \[ \int \frac {a x + 1}{{\left (a x - 1\right )} {\left (c - \frac {c}{a x}\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {a\,x+1}{{\left (c-\frac {c}{a\,x}\right )}^{3/2}\,\left (a\,x-1\right )} \,d x \]
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 \[ \int \frac {a x + 1}{\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{\frac {3}{2}} \left (a x - 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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