Optimal. Leaf size=195 \[ \frac {184 (c-a c x)^{3/2}}{5 a^3 x^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}+\frac {2 x \left (a-\frac {1}{x}\right )^3 (c-a c x)^{3/2}}{5 a^3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}+\frac {16 (c-a c x)^{3/2}}{a^2 x \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}-\frac {8 (c-a c x)^{3/2}}{5 a \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}} \]
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Rubi [A] time = 0.19, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6176, 6181, 94, 89, 78, 37} \[ \frac {184 (c-a c x)^{3/2}}{5 a^3 x^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}+\frac {2 x \left (a-\frac {1}{x}\right )^3 (c-a c x)^{3/2}}{5 a^3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}+\frac {16 (c-a c x)^{3/2}}{a^2 x \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}-\frac {8 (c-a c x)^{3/2}}{5 a \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 78
Rule 89
Rule 94
Rule 6176
Rule 6181
Rubi steps
\begin {align*} \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{3/2} \, dx &=\frac {(c-a c x)^{3/2} \int e^{-3 \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^{3/2} x^{3/2} \, dx}{\left (1-\frac {1}{a x}\right )^{3/2} x^{3/2}}\\ &=-\frac {\left (\left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^3}{x^{7/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{3/2}}\\ &=\frac {2 \left (a-\frac {1}{x}\right )^3 x (c-a c x)^{3/2}}{5 a^3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}+\frac {\left (12 \left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^2}{x^{5/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{5 a \left (1-\frac {1}{a x}\right )^{3/2}}\\ &=-\frac {8 (c-a c x)^{3/2}}{5 a \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}+\frac {2 \left (a-\frac {1}{x}\right )^3 x (c-a c x)^{3/2}}{5 a^3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}+\frac {\left (8 \left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {-\frac {5}{a}+\frac {3 x}{2 a^2}}{x^{3/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{5 a \left (1-\frac {1}{a x}\right )^{3/2}}\\ &=-\frac {8 (c-a c x)^{3/2}}{5 a \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}+\frac {16 (c-a c x)^{3/2}}{a^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x}+\frac {2 \left (a-\frac {1}{x}\right )^3 x (c-a c x)^{3/2}}{5 a^3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}+\frac {\left (92 \left (\frac {1}{x}\right )^{3/2} (c-a c x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{5 a^3 \left (1-\frac {1}{a x}\right )^{3/2}}\\ &=-\frac {8 (c-a c x)^{3/2}}{5 a \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}+\frac {184 (c-a c x)^{3/2}}{5 a^3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x^2}+\frac {16 (c-a c x)^{3/2}}{a^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}} x}+\frac {2 \left (a-\frac {1}{x}\right )^3 x (c-a c x)^{3/2}}{5 a^3 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 57, normalized size = 0.29 \[ -\frac {2 c \left (a^3 x^3-7 a^2 x^2+43 a x+91\right ) \sqrt {c-a c x}}{5 a^2 x \sqrt {1-\frac {1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 63, normalized size = 0.32 \[ -\frac {2 \, {\left (a^{3} c x^{3} - 7 \, a^{2} c x^{2} + 43 \, a c x + 91 \, c\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{5 \, {\left (a^{2} x - a\right )}} \]
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 = 0.04, size = 63, normalized size = 0.32 \[ \frac {2 \left (a x +1\right ) \left (x^{3} a^{3}-7 a^{2} x^{2}+43 a x +91\right ) \left (-a c x +c \right )^{\frac {3}{2}} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{5 a \left (a x -1\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 93, normalized size = 0.48 \[ -\frac {2 \, {\left (a^{4} \sqrt {-c} c x^{4} - 6 \, a^{3} \sqrt {-c} c x^{3} + 36 \, a^{2} \sqrt {-c} c x^{2} + 134 \, a \sqrt {-c} c x + 91 \, \sqrt {-c} c\right )} {\left (a x - 1\right )}^{2}}{5 \, {\left (a^{3} x^{2} - 2 \, a^{2} x + a\right )} {\left (a x + 1\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.36, size = 81, normalized size = 0.42 \[ -\frac {2\,c\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (a^2\,x^2-6\,a\,x+37\right )}{5\,a}-\frac {256\,c\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{5\,a\,\left (a\,x-1\right )} \]
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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