Optimal. Leaf size=231 \[ -\frac {2672 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{105 a^3 \sqrt {1-\frac {1}{a x}}}+\frac {1336 x \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{105 a^2 \sqrt {1-\frac {1}{a x}}}-\frac {334 x \sqrt {c-a c x}}{35 a^2 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}+\frac {2 x^3 \sqrt {c-a c x}}{7 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}-\frac {44 x^2 \sqrt {c-a c x}}{35 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}} \]
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Rubi [A] time = 0.26, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6176, 6181, 89, 78, 45, 37} \[ \frac {1336 x \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{105 a^2 \sqrt {1-\frac {1}{a x}}}-\frac {334 x \sqrt {c-a c x}}{35 a^2 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}-\frac {2672 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{105 a^3 \sqrt {1-\frac {1}{a x}}}+\frac {2 x^3 \sqrt {c-a c x}}{7 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}-\frac {44 x^2 \sqrt {c-a c x}}{35 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 78
Rule 89
Rule 6176
Rule 6181
Rubi steps
\begin {align*} \int e^{-3 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx &=\frac {\sqrt {c-a c x} \int e^{-3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a x}} x^{5/2} \, dx}{\sqrt {1-\frac {1}{a x}} \sqrt {x}}\\ &=-\frac {\left (\sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^2}{x^{9/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}}\\ &=\frac {2 x^3 \sqrt {c-a c x}}{7 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {\left (2 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \operatorname {Subst}\left (\int \frac {-\frac {11}{a}+\frac {7 x}{2 a^2}}{x^{7/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{7 \sqrt {1-\frac {1}{a x}}}\\ &=-\frac {44 x^2 \sqrt {c-a c x}}{35 a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {2 x^3 \sqrt {c-a c x}}{7 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {\left (167 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \operatorname {Subst}\left (\int \frac {1}{x^{5/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{35 a^2 \sqrt {1-\frac {1}{a x}}}\\ &=-\frac {334 x \sqrt {c-a c x}}{35 a^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {44 x^2 \sqrt {c-a c x}}{35 a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {2 x^3 \sqrt {c-a c x}}{7 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {\left (668 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \operatorname {Subst}\left (\int \frac {1}{x^{5/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{35 a^2 \sqrt {1-\frac {1}{a x}}}\\ &=-\frac {334 x \sqrt {c-a c x}}{35 a^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {1336 \sqrt {1+\frac {1}{a x}} x \sqrt {c-a c x}}{105 a^2 \sqrt {1-\frac {1}{a x}}}-\frac {44 x^2 \sqrt {c-a c x}}{35 a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {2 x^3 \sqrt {c-a c x}}{7 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {\left (1336 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \operatorname {Subst}\left (\int \frac {1}{x^{3/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{105 a^3 \sqrt {1-\frac {1}{a x}}}\\ &=-\frac {2672 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{105 a^3 \sqrt {1-\frac {1}{a x}}}-\frac {334 x \sqrt {c-a c x}}{35 a^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {1336 \sqrt {1+\frac {1}{a x}} x \sqrt {c-a c x}}{105 a^2 \sqrt {1-\frac {1}{a x}}}-\frac {44 x^2 \sqrt {c-a c x}}{35 a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {2 x^3 \sqrt {c-a c x}}{7 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 65, normalized size = 0.28 \[ \frac {2 \left (15 a^4 x^4-66 a^3 x^3+167 a^2 x^2-668 a x-1336\right ) \sqrt {c-a c x}}{105 a^4 x \sqrt {1-\frac {1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 69, normalized size = 0.30 \[ \frac {2 \, {\left (15 \, a^{4} x^{4} - 66 \, a^{3} x^{3} + 167 \, a^{2} x^{2} - 668 \, a x - 1336\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{105 \, {\left (a^{4} x - a^{3}\right )}} \]
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 = 72, normalized size = 0.31 \[ \frac {2 \left (a x +1\right ) \left (15 x^{4} a^{4}-66 x^{3} a^{3}+167 a^{2} x^{2}-668 a x -1336\right ) \sqrt {-a c x +c}\, \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{105 a^{3} \left (a x -1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 104, normalized size = 0.45 \[ \frac {2 \, {\left (15 \, a^{5} \sqrt {-c} x^{5} - 51 \, a^{4} \sqrt {-c} x^{4} + 101 \, a^{3} \sqrt {-c} x^{3} - 501 \, a^{2} \sqrt {-c} x^{2} - 2004 \, a \sqrt {-c} x - 1336 \, \sqrt {-c}\right )} {\left (a x - 1\right )}^{2}}{105 \, {\left (a^{5} x^{2} - 2 \, a^{4} x + a^{3}\right )} {\left (a x + 1\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.34, size = 88, normalized size = 0.38 \[ \frac {2\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (15\,a^3\,x^3-51\,a^2\,x^2+116\,a\,x-552\right )}{105\,a^3}-\frac {3776\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{105\,a^3\,\left (a\,x-1\right )} \]
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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