Optimal. Leaf size=223 \[ \frac {3 a^{7/2} x^{9/2} \left (c-\frac {c}{a x}\right )^{9/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{(1-a x)^{9/2}}-\frac {3 a^3 x^4 \sqrt {a x+1} \left (c-\frac {c}{a x}\right )^{9/2}}{(1-a x)^{9/2}}+\frac {3 a^2 x^3 (6-17 a x) (a x+1)^{3/2} \left (c-\frac {c}{a x}\right )^{9/2}}{35 (1-a x)^{9/2}}+\frac {6 a x^2 (a x+1)^{3/2} \left (c-\frac {c}{a x}\right )^{9/2}}{35 (1-a x)^{5/2}}-\frac {2 x (a x+1)^{3/2} \left (c-\frac {c}{a x}\right )^{9/2}}{7 (1-a x)^{3/2}} \]
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Rubi [A] time = 0.18, antiderivative size = 223, 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 = {6134, 6129, 97, 150, 143, 47, 54, 215} \[ \frac {3 a^2 x^3 (6-17 a x) (a x+1)^{3/2} \left (c-\frac {c}{a x}\right )^{9/2}}{35 (1-a x)^{9/2}}-\frac {3 a^3 x^4 \sqrt {a x+1} \left (c-\frac {c}{a x}\right )^{9/2}}{(1-a x)^{9/2}}+\frac {3 a^{7/2} x^{9/2} \left (c-\frac {c}{a x}\right )^{9/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{(1-a x)^{9/2}}+\frac {6 a x^2 (a x+1)^{3/2} \left (c-\frac {c}{a x}\right )^{9/2}}{35 (1-a x)^{5/2}}-\frac {2 x (a x+1)^{3/2} \left (c-\frac {c}{a x}\right )^{9/2}}{7 (1-a x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 54
Rule 97
Rule 143
Rule 150
Rule 215
Rule 6129
Rule 6134
Rubi steps
\begin {align*} \int e^{3 \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{9/2} \, dx &=\frac {\left (\left (c-\frac {c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac {e^{3 \tanh ^{-1}(a x)} (1-a x)^{9/2}}{x^{9/2}} \, dx}{(1-a x)^{9/2}}\\ &=\frac {\left (\left (c-\frac {c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac {(1-a x)^3 (1+a x)^{3/2}}{x^{9/2}} \, dx}{(1-a x)^{9/2}}\\ &=-\frac {2 \left (c-\frac {c}{a x}\right )^{9/2} x (1+a x)^{3/2}}{7 (1-a x)^{3/2}}+\frac {\left (2 \left (c-\frac {c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac {(1-a x)^2 \sqrt {1+a x} \left (-\frac {3 a}{2}-\frac {9 a^2 x}{2}\right )}{x^{7/2}} \, dx}{7 (1-a x)^{9/2}}\\ &=\frac {6 a \left (c-\frac {c}{a x}\right )^{9/2} x^2 (1+a x)^{3/2}}{35 (1-a x)^{5/2}}-\frac {2 \left (c-\frac {c}{a x}\right )^{9/2} x (1+a x)^{3/2}}{7 (1-a x)^{3/2}}+\frac {\left (4 \left (c-\frac {c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac {(1-a x) \sqrt {1+a x} \left (-\frac {27 a^2}{4}+\frac {51 a^3 x}{4}\right )}{x^{5/2}} \, dx}{35 (1-a x)^{9/2}}\\ &=\frac {3 a^2 \left (c-\frac {c}{a x}\right )^{9/2} x^3 (6-17 a x) (1+a x)^{3/2}}{35 (1-a x)^{9/2}}+\frac {6 a \left (c-\frac {c}{a x}\right )^{9/2} x^2 (1+a x)^{3/2}}{35 (1-a x)^{5/2}}-\frac {2 \left (c-\frac {c}{a x}\right )^{9/2} x (1+a x)^{3/2}}{7 (1-a x)^{3/2}}+\frac {\left (3 a^3 \left (c-\frac {c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac {\sqrt {1+a x}}{x^{3/2}} \, dx}{2 (1-a x)^{9/2}}\\ &=-\frac {3 a^3 \left (c-\frac {c}{a x}\right )^{9/2} x^4 \sqrt {1+a x}}{(1-a x)^{9/2}}+\frac {3 a^2 \left (c-\frac {c}{a x}\right )^{9/2} x^3 (6-17 a x) (1+a x)^{3/2}}{35 (1-a x)^{9/2}}+\frac {6 a \left (c-\frac {c}{a x}\right )^{9/2} x^2 (1+a x)^{3/2}}{35 (1-a x)^{5/2}}-\frac {2 \left (c-\frac {c}{a x}\right )^{9/2} x (1+a x)^{3/2}}{7 (1-a x)^{3/2}}+\frac {\left (3 a^4 \left (c-\frac {c}{a x}\right )^{9/2} x^{9/2}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+a x}} \, dx}{2 (1-a x)^{9/2}}\\ &=-\frac {3 a^3 \left (c-\frac {c}{a x}\right )^{9/2} x^4 \sqrt {1+a x}}{(1-a x)^{9/2}}+\frac {3 a^2 \left (c-\frac {c}{a x}\right )^{9/2} x^3 (6-17 a x) (1+a x)^{3/2}}{35 (1-a x)^{9/2}}+\frac {6 a \left (c-\frac {c}{a x}\right )^{9/2} x^2 (1+a x)^{3/2}}{35 (1-a x)^{5/2}}-\frac {2 \left (c-\frac {c}{a x}\right )^{9/2} x (1+a x)^{3/2}}{7 (1-a x)^{3/2}}+\frac {\left (3 a^4 \left (c-\frac {c}{a x}\right )^{9/2} x^{9/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+a x^2}} \, dx,x,\sqrt {x}\right )}{(1-a x)^{9/2}}\\ &=-\frac {3 a^3 \left (c-\frac {c}{a x}\right )^{9/2} x^4 \sqrt {1+a x}}{(1-a x)^{9/2}}+\frac {3 a^2 \left (c-\frac {c}{a x}\right )^{9/2} x^3 (6-17 a x) (1+a x)^{3/2}}{35 (1-a x)^{9/2}}+\frac {6 a \left (c-\frac {c}{a x}\right )^{9/2} x^2 (1+a x)^{3/2}}{35 (1-a x)^{5/2}}-\frac {2 \left (c-\frac {c}{a x}\right )^{9/2} x (1+a x)^{3/2}}{7 (1-a x)^{3/2}}+\frac {3 a^{7/2} \left (c-\frac {c}{a x}\right )^{9/2} x^{9/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{(1-a x)^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 85, normalized size = 0.38 \[ -\frac {c^4 \sqrt {c-\frac {c}{a x}} \left (35 a^2 x^2 \, _2F_1\left (-\frac {3}{2},-\frac {3}{2};-\frac {1}{2};-a x\right )+\left (35 a^2 x^2-46 a x+10\right ) (a x+1)^{5/2}\right )}{35 a^4 x^3 \sqrt {1-a x}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.60, size = 386, normalized size = 1.73 \[ \left [\frac {105 \, {\left (a^{4} c^{4} x^{4} - a^{3} c^{4} x^{3}\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (35 \, a^{4} c^{4} x^{4} + 164 \, a^{3} c^{4} x^{3} - 12 \, a^{2} c^{4} x^{2} - 26 \, a c^{4} x + 10 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{140 \, {\left (a^{5} x^{4} - a^{4} x^{3}\right )}}, -\frac {105 \, {\left (a^{4} c^{4} x^{4} - a^{3} c^{4} x^{3}\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \, {\left (35 \, a^{4} c^{4} x^{4} + 164 \, a^{3} c^{4} x^{3} - 12 \, a^{2} c^{4} x^{2} - 26 \, a c^{4} x + 10 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{70 \, {\left (a^{5} x^{4} - a^{4} x^{3}\right )}}\right ] \]
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 \[ \int \frac {{\left (a x + 1\right )}^{3} {\left (c - \frac {c}{a x}\right )}^{\frac {9}{2}}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 172, normalized size = 0.77 \[ \frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c^{4} \sqrt {-a^{2} x^{2}+1}\, \left (70 a^{\frac {9}{2}} \sqrt {-\left (a x +1\right ) x}\, x^{4}+105 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right ) x^{4} a^{4}+328 a^{\frac {7}{2}} x^{3} \sqrt {-\left (a x +1\right ) x}-24 a^{\frac {5}{2}} x^{2} \sqrt {-\left (a x +1\right ) x}-52 a^{\frac {3}{2}} x \sqrt {-\left (a x +1\right ) x}+20 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}\right )}{70 x^{3} a^{\frac {9}{2}} \left (a x -1\right ) \sqrt {-\left (a x +1\right ) x}} \]
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 \[ \int \frac {{\left (a x + 1\right )}^{3} {\left (c - \frac {c}{a x}\right )}^{\frac {9}{2}}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c-\frac {c}{a\,x}\right )}^{9/2}\,{\left (a\,x+1\right )}^3}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{\frac {9}{2}} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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