Optimal. Leaf size=187 \[ -\frac {4 x^2 \sqrt {c-\frac {c}{a^2 x^2}}}{a^2 \sqrt {1-a^2 x^2}}+\frac {a x^5 \sqrt {c-\frac {c}{a^2 x^2}}}{4 \sqrt {1-a^2 x^2}}-\frac {x^4 \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-a^2 x^2}}+\frac {2 x^3 \sqrt {c-\frac {c}{a^2 x^2}}}{a \sqrt {1-a^2 x^2}}+\frac {4 x \sqrt {c-\frac {c}{a^2 x^2}} \log (a x+1)}{a^3 \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.27, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6160, 6150, 88} \[ \frac {a x^5 \sqrt {c-\frac {c}{a^2 x^2}}}{4 \sqrt {1-a^2 x^2}}-\frac {x^4 \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-a^2 x^2}}+\frac {2 x^3 \sqrt {c-\frac {c}{a^2 x^2}}}{a \sqrt {1-a^2 x^2}}-\frac {4 x^2 \sqrt {c-\frac {c}{a^2 x^2}}}{a^2 \sqrt {1-a^2 x^2}}+\frac {4 x \sqrt {c-\frac {c}{a^2 x^2}} \log (a x+1)}{a^3 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6150
Rule 6160
Rubi steps
\begin {align*} \int e^{-3 \tanh ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^3 \, dx &=\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int e^{-3 \tanh ^{-1}(a x)} x^2 \sqrt {1-a^2 x^2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \frac {x^2 (1-a x)^2}{1+a x} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (\sqrt {c-\frac {c}{a^2 x^2}} x\right ) \int \left (-\frac {4}{a^2}+\frac {4 x}{a}-3 x^2+a x^3+\frac {4}{a^2 (1+a x)}\right ) \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {4 \sqrt {c-\frac {c}{a^2 x^2}} x^2}{a^2 \sqrt {1-a^2 x^2}}+\frac {2 \sqrt {c-\frac {c}{a^2 x^2}} x^3}{a \sqrt {1-a^2 x^2}}-\frac {\sqrt {c-\frac {c}{a^2 x^2}} x^4}{\sqrt {1-a^2 x^2}}+\frac {a \sqrt {c-\frac {c}{a^2 x^2}} x^5}{4 \sqrt {1-a^2 x^2}}+\frac {4 \sqrt {c-\frac {c}{a^2 x^2}} x \log (1+a x)}{a^3 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 70, normalized size = 0.37 \[ \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {4 \log (a x+1)}{a^3}-\frac {4 x}{a^2}+\frac {a x^4}{4}+\frac {2 x^2}{a}-x^3\right )}{\sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 419, normalized size = 2.24 \[ \left [\frac {8 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {-c} \log \left (\frac {a^{6} c x^{6} + 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} - 4 \, a c x + {\left (a^{5} x^{5} + 4 \, a^{4} x^{4} + 6 \, a^{3} x^{3} + 4 \, a^{2} x^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 2 \, c}{a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1}\right ) - {\left (a^{5} x^{5} - 4 \, a^{4} x^{4} + 8 \, a^{3} x^{3} - 16 \, a^{2} x^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{4 \, {\left (a^{6} x^{2} - a^{4}\right )}}, -\frac {16 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {c} \arctan \left (\frac {{\left (a^{2} x^{2} + 2 \, a x + 2\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {c} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{3} c x^{3} + 2 \, a^{2} c x^{2} - a c x - 2 \, c}\right ) + {\left (a^{5} x^{5} - 4 \, a^{4} x^{4} + 8 \, a^{3} x^{3} - 16 \, a^{2} x^{2}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{4 \, {\left (a^{6} x^{2} - a^{4}\right )}}\right ] \]
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 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \sqrt {c - \frac {c}{a^{2} x^{2}}} x^{3}}{{\left (a x + 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 85, normalized size = 0.45 \[ -\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \sqrt {-a^{2} x^{2}+1}\, \left (x^{4} a^{4}-4 x^{3} a^{3}+8 a^{2} x^{2}-16 a x +16 \ln \left (a x +1\right )\right )}{4 \left (a^{2} x^{2}-1\right ) a^{3}} \]
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 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \sqrt {c - \frac {c}{a^{2} x^{2}}} x^{3}}{{\left (a x + 1\right )}^{3}}\,{d x} \]
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 \[ \int \frac {x^3\,\sqrt {c-\frac {c}{a^2\,x^2}}\,{\left (1-a^2\,x^2\right )}^{3/2}}{{\left (a\,x+1\right )}^3} \,d x \]
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 \[ \int \frac {x^{3} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \sqrt {- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )}}{\left (a x + 1\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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