Optimal. Leaf size=303 \[ -\frac {1115 \sqrt {c-\frac {c}{a x}}}{64 a^4 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}+\frac {1115 \sqrt {c-\frac {c}{a x}} \tanh ^{-1}\left (\sqrt {\frac {1}{a x}+1}\right )}{64 a^4 \sqrt {1-\frac {1}{a x}}}-\frac {1115 x \sqrt {c-\frac {c}{a x}}}{192 a^3 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}+\frac {223 x^2 \sqrt {c-\frac {c}{a x}}}{96 a^2 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}+\frac {x^4 \sqrt {c-\frac {c}{a x}}}{4 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}-\frac {25 x^3 \sqrt {c-\frac {c}{a x}}}{24 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}} \]
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Rubi [A] time = 0.32, antiderivative size = 306, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6182, 6180, 89, 78, 51, 63, 208} \[ \frac {1115 x^2 \sqrt {\frac {1}{a x}+1} \sqrt {c-\frac {c}{a x}}}{96 a^2 \sqrt {1-\frac {1}{a x}}}-\frac {223 x^2 \sqrt {c-\frac {c}{a x}}}{24 a^2 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}-\frac {1115 x \sqrt {\frac {1}{a x}+1} \sqrt {c-\frac {c}{a x}}}{64 a^3 \sqrt {1-\frac {1}{a x}}}+\frac {1115 \sqrt {c-\frac {c}{a x}} \tanh ^{-1}\left (\sqrt {\frac {1}{a x}+1}\right )}{64 a^4 \sqrt {1-\frac {1}{a x}}}+\frac {x^4 \sqrt {c-\frac {c}{a x}}}{4 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}-\frac {25 x^3 \sqrt {c-\frac {c}{a x}}}{24 a \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 89
Rule 208
Rule 6180
Rule 6182
Rubi steps
\begin {align*} \int e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a x}} x^3 \, dx &=\frac {\sqrt {c-\frac {c}{a x}} \int e^{-3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a x}} x^3 \, dx}{\sqrt {1-\frac {1}{a x}}}\\ &=-\frac {\sqrt {c-\frac {c}{a x}} \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^2}{x^5 \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-\frac {1}{a x}}}\\ &=\frac {\sqrt {c-\frac {c}{a x}} x^4}{4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {\sqrt {c-\frac {c}{a x}} \operatorname {Subst}\left (\int \frac {-\frac {25}{2 a}+\frac {4 x}{a^2}}{x^4 \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{4 \sqrt {1-\frac {1}{a x}}}\\ &=-\frac {25 \sqrt {c-\frac {c}{a x}} x^3}{24 a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {c-\frac {c}{a x}} x^4}{4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {\left (223 \sqrt {c-\frac {c}{a x}}\right ) \operatorname {Subst}\left (\int \frac {1}{x^3 \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{48 a^2 \sqrt {1-\frac {1}{a x}}}\\ &=-\frac {223 \sqrt {c-\frac {c}{a x}} x^2}{24 a^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {25 \sqrt {c-\frac {c}{a x}} x^3}{24 a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {c-\frac {c}{a x}} x^4}{4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {\left (1115 \sqrt {c-\frac {c}{a x}}\right ) \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{48 a^2 \sqrt {1-\frac {1}{a x}}}\\ &=-\frac {223 \sqrt {c-\frac {c}{a x}} x^2}{24 a^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {1115 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^2}{96 a^2 \sqrt {1-\frac {1}{a x}}}-\frac {25 \sqrt {c-\frac {c}{a x}} x^3}{24 a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {c-\frac {c}{a x}} x^4}{4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {\left (1115 \sqrt {c-\frac {c}{a x}}\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{64 a^3 \sqrt {1-\frac {1}{a x}}}\\ &=-\frac {1115 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x}{64 a^3 \sqrt {1-\frac {1}{a x}}}-\frac {223 \sqrt {c-\frac {c}{a x}} x^2}{24 a^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {1115 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^2}{96 a^2 \sqrt {1-\frac {1}{a x}}}-\frac {25 \sqrt {c-\frac {c}{a x}} x^3}{24 a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {c-\frac {c}{a x}} x^4}{4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {\left (1115 \sqrt {c-\frac {c}{a x}}\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{128 a^4 \sqrt {1-\frac {1}{a x}}}\\ &=-\frac {1115 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x}{64 a^3 \sqrt {1-\frac {1}{a x}}}-\frac {223 \sqrt {c-\frac {c}{a x}} x^2}{24 a^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {1115 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^2}{96 a^2 \sqrt {1-\frac {1}{a x}}}-\frac {25 \sqrt {c-\frac {c}{a x}} x^3}{24 a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {c-\frac {c}{a x}} x^4}{4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}-\frac {\left (1115 \sqrt {c-\frac {c}{a x}}\right ) \operatorname {Subst}\left (\int \frac {1}{-a+a x^2} \, dx,x,\sqrt {1+\frac {1}{a x}}\right )}{64 a^3 \sqrt {1-\frac {1}{a x}}}\\ &=-\frac {1115 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x}{64 a^3 \sqrt {1-\frac {1}{a x}}}-\frac {223 \sqrt {c-\frac {c}{a x}} x^2}{24 a^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {1115 \sqrt {1+\frac {1}{a x}} \sqrt {c-\frac {c}{a x}} x^2}{96 a^2 \sqrt {1-\frac {1}{a x}}}-\frac {25 \sqrt {c-\frac {c}{a x}} x^3}{24 a \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {\sqrt {c-\frac {c}{a x}} x^4}{4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {1115 \sqrt {c-\frac {c}{a x}} \tanh ^{-1}\left (\sqrt {1+\frac {1}{a x}}\right )}{64 a^4 \sqrt {1-\frac {1}{a x}}}\\ \end {align*}
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Mathematica [A] time = 0.63, size = 167, normalized size = 0.55 \[ \frac {3345 \sqrt {c} \log \left (2 a^2 \sqrt {c} x^2 \sqrt {1-\frac {1}{a^2 x^2}} \sqrt {c-\frac {c}{a x}}+c \left (2 a^2 x^2-a x-1\right )\right )+\frac {2 a^2 x^2 \sqrt {1-\frac {1}{a^2 x^2}} \left (48 a^4 x^4-200 a^3 x^3+446 a^2 x^2-1115 a x-3345\right ) \sqrt {c-\frac {c}{a x}}}{a^2 x^2-1}-3345 \sqrt {c} \log (1-a x)}{384 a^4} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.73, size = 353, normalized size = 1.17 \[ \left [\frac {3345 \, {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x + 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (48 \, a^{5} x^{5} - 200 \, a^{4} x^{4} + 446 \, a^{3} x^{3} - 1115 \, a^{2} x^{2} - 3345 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{768 \, {\left (a^{5} x - a^{4}\right )}}, -\frac {3345 \, {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) - 2 \, {\left (48 \, a^{5} x^{5} - 200 \, a^{4} x^{4} + 446 \, a^{3} x^{3} - 1115 \, a^{2} x^{2} - 3345 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{384 \, {\left (a^{5} x - a^{4}\right )}}\right ] \]
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 = 0.07, size = 197, normalized size = 0.65 \[ \frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (96 a^{\frac {9}{2}} \sqrt {\left (a x +1\right ) x}\, x^{4}-400 a^{\frac {7}{2}} x^{3} \sqrt {\left (a x +1\right ) x}+892 a^{\frac {5}{2}} x^{2} \sqrt {\left (a x +1\right ) x}-2230 a^{\frac {3}{2}} x \sqrt {\left (a x +1\right ) x}+3345 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) x a -6690 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+3345 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right )\right )}{384 \left (a x -1\right )^{2} a^{\frac {7}{2}} \sqrt {\left (a x +1\right ) x}} \]
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 \[ \int \sqrt {c - \frac {c}{a x}} x^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}\,{d x} \]
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 \[ \int x^3\,\sqrt {c-\frac {c}{a\,x}}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2} \,d x \]
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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