Optimal. Leaf size=133 \[ \frac {9 \sqrt {a} x^{3/2} \left (c-\frac {c}{a x}\right )^{3/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{(1-a x)^{3/2}}-\frac {a x^2 (23-a x) \left (c-\frac {c}{a x}\right )^{3/2}}{(1-a x)^{3/2} \sqrt {a x+1}}-\frac {2 x \sqrt {1-a x} \left (c-\frac {c}{a x}\right )^{3/2}}{\sqrt {a x+1}} \]
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Rubi [A] time = 0.18, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6134, 6129, 98, 143, 54, 215} \[ -\frac {a x^2 (23-a x) \left (c-\frac {c}{a x}\right )^{3/2}}{(1-a x)^{3/2} \sqrt {a x+1}}+\frac {9 \sqrt {a} x^{3/2} \left (c-\frac {c}{a x}\right )^{3/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{(1-a x)^{3/2}}-\frac {2 x \sqrt {1-a x} \left (c-\frac {c}{a x}\right )^{3/2}}{\sqrt {a x+1}} \]
Antiderivative was successfully verified.
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Rule 54
Rule 98
Rule 143
Rule 215
Rule 6129
Rule 6134
Rubi steps
\begin {align*} \int e^{-3 \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^{3/2} \, dx &=\frac {\left (\left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}\right ) \int \frac {e^{-3 \tanh ^{-1}(a x)} (1-a x)^{3/2}}{x^{3/2}} \, dx}{(1-a x)^{3/2}}\\ &=\frac {\left (\left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}\right ) \int \frac {(1-a x)^3}{x^{3/2} (1+a x)^{3/2}} \, dx}{(1-a x)^{3/2}}\\ &=-\frac {2 \left (c-\frac {c}{a x}\right )^{3/2} x \sqrt {1-a x}}{\sqrt {1+a x}}-\frac {\left (2 \left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}\right ) \int \frac {(1-a x) \left (\frac {7 a}{2}+\frac {a^2 x}{2}\right )}{\sqrt {x} (1+a x)^{3/2}} \, dx}{(1-a x)^{3/2}}\\ &=-\frac {2 \left (c-\frac {c}{a x}\right )^{3/2} x \sqrt {1-a x}}{\sqrt {1+a x}}-\frac {a \left (c-\frac {c}{a x}\right )^{3/2} x^2 (23-a x)}{(1-a x)^{3/2} \sqrt {1+a x}}+\frac {\left (9 a \left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+a x}} \, dx}{2 (1-a x)^{3/2}}\\ &=-\frac {2 \left (c-\frac {c}{a x}\right )^{3/2} x \sqrt {1-a x}}{\sqrt {1+a x}}-\frac {a \left (c-\frac {c}{a x}\right )^{3/2} x^2 (23-a x)}{(1-a x)^{3/2} \sqrt {1+a x}}+\frac {\left (9 a \left (c-\frac {c}{a x}\right )^{3/2} x^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+a x^2}} \, dx,x,\sqrt {x}\right )}{(1-a x)^{3/2}}\\ &=-\frac {2 \left (c-\frac {c}{a x}\right )^{3/2} x \sqrt {1-a x}}{\sqrt {1+a x}}-\frac {a \left (c-\frac {c}{a x}\right )^{3/2} x^2 (23-a x)}{(1-a x)^{3/2} \sqrt {1+a x}}+\frac {9 \sqrt {a} \left (c-\frac {c}{a x}\right )^{3/2} x^{3/2} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{(1-a x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 80, normalized size = 0.60 \[ \frac {c \sqrt {c-\frac {c}{a x}} \left (a^2 x^2+19 a x-9 \sqrt {a} \sqrt {x} \sqrt {a x+1} \sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )+2\right )}{a \sqrt {1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.56, size = 298, normalized size = 2.24 \[ \left [\frac {9 \, {\left (a^{2} c x^{2} - c\right )} \sqrt {-c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{2} x^{2} + a x\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) - 4 \, {\left (a^{2} c x^{2} + 19 \, a c x + 2 \, c\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{4 \, {\left (a^{3} x^{2} - a\right )}}, \frac {9 \, {\left (a^{2} c x^{2} - c\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} a \sqrt {c} x \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \, {\left (a^{2} c x^{2} + 19 \, a c x + 2 \, c\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a^{3} x^{2} - a\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 164, normalized size = 1.23 \[ -\frac {\sqrt {\frac {c \left (a x -1\right )}{a x}}\, c \left (2 a^{\frac {5}{2}} x^{2} \sqrt {-\left (a x +1\right ) x}+9 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right ) x^{2} a^{2}+38 a^{\frac {3}{2}} x \sqrt {-\left (a x +1\right ) x}+9 \arctan \left (\frac {2 a x +1}{2 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}}\right ) x a +4 \sqrt {a}\, \sqrt {-\left (a x +1\right ) x}\right ) \sqrt {-a^{2} x^{2}+1}}{2 a^{\frac {3}{2}} \left (a x +1\right ) \sqrt {-\left (a x +1\right ) x}\, \left (a x -1\right )} \]
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 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (c - \frac {c}{a x}\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3}}\,{d x} \]
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 \[ \int \frac {{\left (c-\frac {c}{a\,x}\right )}^{3/2}\,{\left (1-a^2\,x^2\right )}^{3/2}}{{\left (a\,x+1\right )}^3} \,d x \]
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 \[ \int \frac {\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{\frac {3}{2}} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{\left (a x + 1\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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