Optimal. Leaf size=225 \[ -\frac {2 x^2 \sqrt {c-a^2 c x^2}}{a^2 \sqrt {1-a^2 x^2}}+\frac {a x^5 \sqrt {c-a^2 c x^2}}{5 \sqrt {1-a^2 x^2}}-\frac {3 x^4 \sqrt {c-a^2 c x^2}}{4 \sqrt {1-a^2 x^2}}+\frac {4 x^3 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-a^2 x^2}}-\frac {4 \sqrt {c-a^2 c x^2} \log (a x+1)}{a^4 \sqrt {1-a^2 x^2}}+\frac {4 x \sqrt {c-a^2 c x^2}}{a^3 \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.22, antiderivative size = 225, 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 = {6153, 6150, 88} \[ \frac {a x^5 \sqrt {c-a^2 c x^2}}{5 \sqrt {1-a^2 x^2}}-\frac {3 x^4 \sqrt {c-a^2 c x^2}}{4 \sqrt {1-a^2 x^2}}+\frac {4 x^3 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-a^2 x^2}}-\frac {2 x^2 \sqrt {c-a^2 c x^2}}{a^2 \sqrt {1-a^2 x^2}}+\frac {4 x \sqrt {c-a^2 c x^2}}{a^3 \sqrt {1-a^2 x^2}}-\frac {4 \sqrt {c-a^2 c x^2} \log (a x+1)}{a^4 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6150
Rule 6153
Rubi steps
\begin {align*} \int e^{-3 \tanh ^{-1}(a x)} x^3 \sqrt {c-a^2 c x^2} \, dx &=\frac {\sqrt {c-a^2 c x^2} \int e^{-3 \tanh ^{-1}(a x)} x^3 \sqrt {1-a^2 x^2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\sqrt {c-a^2 c x^2} \int \frac {x^3 (1-a x)^2}{1+a x} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\sqrt {c-a^2 c x^2} \int \left (\frac {4}{a^3}-\frac {4 x}{a^2}+\frac {4 x^2}{a}-3 x^3+a x^4-\frac {4}{a^3 (1+a x)}\right ) \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {4 x \sqrt {c-a^2 c x^2}}{a^3 \sqrt {1-a^2 x^2}}-\frac {2 x^2 \sqrt {c-a^2 c x^2}}{a^2 \sqrt {1-a^2 x^2}}+\frac {4 x^3 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-a^2 x^2}}-\frac {3 x^4 \sqrt {c-a^2 c x^2}}{4 \sqrt {1-a^2 x^2}}+\frac {a x^5 \sqrt {c-a^2 c x^2}}{5 \sqrt {1-a^2 x^2}}-\frac {4 \sqrt {c-a^2 c x^2} \log (1+a x)}{a^4 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 81, normalized size = 0.36 \[ \frac {\sqrt {c-a^2 c x^2} \left (-\frac {4 \log (a x+1)}{a^4}+\frac {4 x}{a^3}-\frac {2 x^2}{a^2}+\frac {a x^5}{5}+\frac {4 x^3}{3 a}-\frac {3 x^4}{4}\right )}{\sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.12, size = 399, normalized size = 1.77 \[ \left [\frac {120 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {c} \log \left (\frac {a^{6} c x^{6} + 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} - 4 \, a c x + {\left (a^{4} x^{4} + 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} + 4 \, a x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} \sqrt {c} - 2 \, c}{a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1}\right ) - {\left (12 \, a^{5} x^{5} - 45 \, a^{4} x^{4} + 80 \, a^{3} x^{3} - 120 \, a^{2} x^{2} + 240 \, a x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{60 \, {\left (a^{6} x^{2} - a^{4}\right )}}, -\frac {240 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} {\left (a^{2} x^{2} + 2 \, a x + 2\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c}}{a^{4} c x^{4} + 2 \, a^{3} c x^{3} - a^{2} c x^{2} - 2 \, a c x}\right ) + {\left (12 \, a^{5} x^{5} - 45 \, a^{4} x^{4} + 80 \, a^{3} x^{3} - 120 \, a^{2} x^{2} + 240 \, a x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{60 \, {\left (a^{6} x^{2} - a^{4}\right )}}\right ] \]
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 \[ \int \frac {\sqrt {-a^{2} c x^{2} + c} {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{3}}{{\left (a x + 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 88, normalized size = 0.39 \[ \frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \left (-12 x^{5} a^{5}+45 x^{4} a^{4}-80 x^{3} a^{3}+120 a^{2} x^{2}-240 a x +240 \ln \left (a x +1\right )\right )}{60 \left (a^{2} x^{2}-1\right ) a^{4}} \]
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 \[ \int \frac {\sqrt {-a^{2} c x^{2} + c} {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{3}}{{\left (a x + 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3\,\sqrt {c-a^2\,c\,x^2}\,{\left (1-a^2\,x^2\right )}^{3/2}}{{\left (a\,x+1\right )}^3} \,d x \]
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 \[ \int \frac {x^{3} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}{\left (a x + 1\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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