Optimal. Leaf size=226 \[ -\frac {2 x^2 \sqrt {c-a^2 c x^2}}{a^2 \sqrt {1-a^2 x^2}}-\frac {a x^5 \sqrt {c-a^2 c x^2}}{5 \sqrt {1-a^2 x^2}}-\frac {3 x^4 \sqrt {c-a^2 c x^2}}{4 \sqrt {1-a^2 x^2}}-\frac {4 x^3 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-a^2 x^2}}-\frac {4 \sqrt {c-a^2 c x^2} \log (1-a x)}{a^4 \sqrt {1-a^2 x^2}}-\frac {4 x \sqrt {c-a^2 c x^2}}{a^3 \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.22, antiderivative size = 226, 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} \[ -\frac {a x^5 \sqrt {c-a^2 c x^2}}{5 \sqrt {1-a^2 x^2}}-\frac {3 x^4 \sqrt {c-a^2 c x^2}}{4 \sqrt {1-a^2 x^2}}-\frac {4 x^3 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-a^2 x^2}}-\frac {2 x^2 \sqrt {c-a^2 c x^2}}{a^2 \sqrt {1-a^2 x^2}}-\frac {4 x \sqrt {c-a^2 c x^2}}{a^3 \sqrt {1-a^2 x^2}}-\frac {4 \sqrt {c-a^2 c x^2} \log (1-a x)}{a^4 \sqrt {1-a^2 x^2}} \]
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 = 0.05, size = 79, normalized size = 0.35 \[ -\frac {\sqrt {c-a^2 c x^2} \left (a x \left (12 a^4 x^4+45 a^3 x^3+80 a^2 x^2+120 a x+240\right )+240 \log (1-a x)\right )}{60 a^4 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, 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 \[ \int \frac {\sqrt {-a^{2} c x^{2} + c} {\left (a x + 1\right )}^{3} x^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (12 x^{5} a^{5}+45 x^{4} a^{4}+80 x^{3} a^{3}+120 a^{2} x^{2}+240 a x +240 \ln \left (a x -1\right )\right )}{60 \left (a^{2} x^{2}-1\right ) a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \frac {x^3\,\sqrt {c-a^2\,c\,x^2}\,{\left (a\,x+1\right )}^3}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{3} \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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