Optimal. Leaf size=150 \[ -\frac {3 x^2 \sqrt {c-a^2 c x^2}}{2 \sqrt {1-a^2 x^2}}-\frac {4 x \sqrt {c-a^2 c x^2}}{a \sqrt {1-a^2 x^2}}-\frac {4 \sqrt {c-a^2 c x^2} \log (1-a x)}{a^2 \sqrt {1-a^2 x^2}}-\frac {a x^3 \sqrt {c-a^2 c x^2}}{3 \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.15, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6153, 6150, 77} \[ -\frac {a x^3 \sqrt {c-a^2 c x^2}}{3 \sqrt {1-a^2 x^2}}-\frac {3 x^2 \sqrt {c-a^2 c x^2}}{2 \sqrt {1-a^2 x^2}}-\frac {4 x \sqrt {c-a^2 c x^2}}{a \sqrt {1-a^2 x^2}}-\frac {4 \sqrt {c-a^2 c x^2} \log (1-a x)}{a^2 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 77
Rule 6150
Rule 6153
Rubi steps
\begin {align*} \int e^{3 \tanh ^{-1}(a x)} x \sqrt {c-a^2 c x^2} \, dx &=\frac {\sqrt {c-a^2 c x^2} \int e^{3 \tanh ^{-1}(a x)} x \sqrt {1-a^2 x^2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\sqrt {c-a^2 c x^2} \int \frac {x (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 x-a x^2-\frac {4}{a (-1+a x)}\right ) \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {4 x \sqrt {c-a^2 c x^2}}{a \sqrt {1-a^2 x^2}}-\frac {3 x^2 \sqrt {c-a^2 c x^2}}{2 \sqrt {1-a^2 x^2}}-\frac {a x^3 \sqrt {c-a^2 c x^2}}{3 \sqrt {1-a^2 x^2}}-\frac {4 \sqrt {c-a^2 c x^2} \log (1-a x)}{a^2 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 64, normalized size = 0.43 \[ \frac {\sqrt {c-a^2 c x^2} \left (-\frac {4 \log (1-a x)}{a^2}-\frac {a x^3}{3}-\frac {4 x}{a}-\frac {3 x^2}{2}\right )}{\sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 367, normalized size = 2.45 \[ \left [\frac {12 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {c} \log \left (\frac {a^{6} c x^{6} - 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} + 4 \, a c x + {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} \sqrt {c} - 2 \, c}{a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1}\right ) + {\left (2 \, a^{3} x^{3} + 9 \, a^{2} x^{2} + 24 \, a x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{6 \, {\left (a^{4} x^{2} - a^{2}\right )}}, -\frac {24 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} {\left (a^{2} x^{2} - 2 \, a x + 2\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c}}{a^{4} c x^{4} - 2 \, a^{3} c x^{3} - a^{2} c x^{2} + 2 \, a c x}\right ) - {\left (2 \, a^{3} x^{3} + 9 \, a^{2} x^{2} + 24 \, a x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{6 \, {\left (a^{4} x^{2} - a^{2}\right )}}\right ] \]
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 \[ \int \frac {\sqrt {-a^{2} c x^{2} + c} {\left (a x + 1\right )}^{3} x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 72, normalized size = 0.48 \[ \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 x^{3} a^{3}+9 a^{2} x^{2}+24 a x +24 \ln \left (a x -1\right )\right )}{6 \left (a^{2} x^{2}-1\right ) a^{2}} \]
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 = -0.01 \[ \int \frac {x\,\sqrt {c-a^2\,c\,x^2}\,{\left (a\,x+1\right )}^3}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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