Optimal. Leaf size=119 \[ -\frac {9 \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{5/2}}+\frac {9}{a c^2 \sqrt {c-\frac {c}{a x}}}-\frac {x}{\left (c-\frac {c}{a x}\right )^{5/2}}+\frac {3}{a c \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {9}{5 a \left (c-\frac {c}{a x}\right )^{5/2}} \]
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Rubi [A] time = 0.16, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6133, 25, 514, 375, 78, 51, 63, 208} \[ \frac {9}{a c^2 \sqrt {c-\frac {c}{a x}}}-\frac {9 \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{5/2}}-\frac {x}{\left (c-\frac {c}{a x}\right )^{5/2}}+\frac {3}{a c \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {9}{5 a \left (c-\frac {c}{a x}\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 25
Rule 51
Rule 63
Rule 78
Rule 208
Rule 375
Rule 514
Rule 6133
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{5/2}} \, dx &=\int \frac {1+a x}{\left (c-\frac {c}{a x}\right )^{5/2} (1-a x)} \, dx\\ &=-\frac {c \int \frac {1+a x}{\left (c-\frac {c}{a x}\right )^{7/2} x} \, dx}{a}\\ &=-\frac {c \int \frac {a+\frac {1}{x}}{\left (c-\frac {c}{a x}\right )^{7/2}} \, dx}{a}\\ &=\frac {c \operatorname {Subst}\left (\int \frac {a+x}{x^2 \left (c-\frac {c x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=-\frac {x}{\left (c-\frac {c}{a x}\right )^{5/2}}+\frac {(9 c) \operatorname {Subst}\left (\int \frac {1}{x \left (c-\frac {c x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{2 a}\\ &=\frac {9}{5 a \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {x}{\left (c-\frac {c}{a x}\right )^{5/2}}+\frac {9 \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 {9}{5 a \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {3}{a c \left (c-\frac {c}{a x}\right )^{3/2}}-\frac {x}{\left (c-\frac {c}{a x}\right )^{5/2}}+\frac {9 \operatorname {Subst}\left (\int \frac {1}{x \left (c-\frac {c x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{2 a c}\\ &=\frac {9}{5 a \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {3}{a c \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {9}{a c^2 \sqrt {c-\frac {c}{a x}}}-\frac {x}{\left (c-\frac {c}{a x}\right )^{5/2}}+\frac {9 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c-\frac {c x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a c^2}\\ &=\frac {9}{5 a \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {3}{a c \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {9}{a c^2 \sqrt {c-\frac {c}{a x}}}-\frac {x}{\left (c-\frac {c}{a x}\right )^{5/2}}-\frac {9 \operatorname {Subst}\left (\int \frac {1}{a-\frac {a x^2}{c}} \, dx,x,\sqrt {c-\frac {c}{a x}}\right )}{c^3}\\ &=\frac {9}{5 a \left (c-\frac {c}{a x}\right )^{5/2}}+\frac {3}{a c \left (c-\frac {c}{a x}\right )^{3/2}}+\frac {9}{a c^2 \sqrt {c-\frac {c}{a x}}}-\frac {x}{\left (c-\frac {c}{a x}\right )^{5/2}}-\frac {9 \tanh ^{-1}\left (\frac {\sqrt {c-\frac {c}{a x}}}{\sqrt {c}}\right )}{a c^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 59, normalized size = 0.50 \[ \frac {9 \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};1-\frac {1}{a x}\right )}{5 a \left (c-\frac {c}{a x}\right )^{5/2}}-\frac {x}{\left (c-\frac {c}{a x}\right )^{5/2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.44, size = 294, normalized size = 2.47 \[ \left [\frac {45 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {c} \log \left (-2 \, a c x + 2 \, a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}} + c\right ) - 2 \, {\left (5 \, a^{4} x^{4} - 69 \, a^{3} x^{3} + 105 \, a^{2} x^{2} - 45 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{10 \, {\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}}, \frac {45 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {\frac {a c x - c}{a x}}}{c}\right ) - {\left (5 \, a^{4} x^{4} - 69 \, a^{3} x^{3} + 105 \, a^{2} x^{2} - 45 \, a x\right )} \sqrt {\frac {a c x - c}{a x}}}{5 \, {\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 165, normalized size = 1.39 \[ \frac {a {\left (\frac {2 \, {\left (2 \, c^{2} + \frac {5 \, {\left (a c x - c\right )} c}{a x} + \frac {20 \, {\left (a c x - c\right )}^{2}}{a^{2} x^{2}}\right )} x^{2}}{{\left (a c x - c\right )}^{2} c \sqrt {\frac {a c x - c}{a x}}} + \frac {45 \, \arctan \left (\frac {\sqrt {\frac {a c x - c}{a x}}}{\sqrt {-c}}\right )}{a^{2} \sqrt {-c} c} - \frac {5 \, \sqrt {\frac {a c x - c}{a x}}}{a^{2} {\left (c - \frac {a c x - c}{a x}\right )} c}\right )}}{5 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 328, normalized size = 2.76 \[ -\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (90 a^{\frac {9}{2}} \sqrt {\left (a x -1\right ) x}\, x^{4}+45 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) x^{4} a^{4}-80 a^{\frac {7}{2}} \left (\left (a x -1\right ) x \right )^{\frac {3}{2}} x^{2}-360 a^{\frac {7}{2}} \sqrt {\left (a x -1\right ) x}\, x^{3}-180 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) x^{3} a^{3}+132 a^{\frac {5}{2}} \left (\left (a x -1\right ) x \right )^{\frac {3}{2}} x +540 a^{\frac {5}{2}} \sqrt {\left (a x -1\right ) x}\, x^{2}+270 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) x^{2} a^{2}-60 a^{\frac {3}{2}} \left (\left (a x -1\right ) x \right )^{\frac {3}{2}}-360 a^{\frac {3}{2}} \sqrt {\left (a x -1\right ) x}\, x -180 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right ) x a +90 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+45 \ln \left (\frac {2 \sqrt {\left (a x -1\right ) x}\, \sqrt {a}+2 a x -1}{2 \sqrt {a}}\right )\right )}{10 \sqrt {\left (a x -1\right ) x}\, c^{3} \left (a x -1\right )^{4} \sqrt {a}} \]
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 \[ -\int \frac {{\left (a x + 1\right )}^{2}}{{\left (a^{2} x^{2} - 1\right )} {\left (c - \frac {c}{a x}\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int -\frac {{\left (a\,x+1\right )}^2}{{\left (c-\frac {c}{a\,x}\right )}^{5/2}\,\left (a^2\,x^2-1\right )} \,d x \]
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 \[ - \int \frac {a x}{a c^{2} x \sqrt {c - \frac {c}{a x}} - 3 c^{2} \sqrt {c - \frac {c}{a x}} + \frac {3 c^{2} \sqrt {c - \frac {c}{a x}}}{a x} - \frac {c^{2} \sqrt {c - \frac {c}{a x}}}{a^{2} x^{2}}}\, dx - \int \frac {1}{a c^{2} x \sqrt {c - \frac {c}{a x}} - 3 c^{2} \sqrt {c - \frac {c}{a x}} + \frac {3 c^{2} \sqrt {c - \frac {c}{a x}}}{a x} - \frac {c^{2} \sqrt {c - \frac {c}{a x}}}{a^{2} x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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