Optimal. Leaf size=155 \[ -\frac {x^2 \sqrt {1-a^2 x^2}}{2 a^2 \sqrt {c-a^2 c x^2}}-\frac {x^3 \sqrt {1-a^2 x^2}}{3 a \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2} \log (1-a x)}{a^4 \sqrt {c-a^2 c x^2}}-\frac {x \sqrt {1-a^2 x^2}}{a^3 \sqrt {c-a^2 c x^2}} \]
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Rubi [A] time = 0.21, antiderivative size = 155, 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, 43} \[ -\frac {x^3 \sqrt {1-a^2 x^2}}{3 a \sqrt {c-a^2 c x^2}}-\frac {x^2 \sqrt {1-a^2 x^2}}{2 a^2 \sqrt {c-a^2 c x^2}}-\frac {x \sqrt {1-a^2 x^2}}{a^3 \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2} \log (1-a x)}{a^4 \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6150
Rule 6153
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x^3}{\sqrt {c-a^2 c x^2}} \, dx &=\frac {\sqrt {1-a^2 x^2} \int \frac {e^{\tanh ^{-1}(a x)} x^3}{\sqrt {1-a^2 x^2}} \, dx}{\sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int \frac {x^3}{1-a x} \, dx}{\sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int \left (-\frac {1}{a^3}-\frac {x}{a^2}-\frac {x^2}{a}-\frac {1}{a^3 (-1+a x)}\right ) \, dx}{\sqrt {c-a^2 c x^2}}\\ &=-\frac {x \sqrt {1-a^2 x^2}}{a^3 \sqrt {c-a^2 c x^2}}-\frac {x^2 \sqrt {1-a^2 x^2}}{2 a^2 \sqrt {c-a^2 c x^2}}-\frac {x^3 \sqrt {1-a^2 x^2}}{3 a \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2} \log (1-a x)}{a^4 \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 63, normalized size = 0.41 \[ -\frac {\sqrt {1-a^2 x^2} \left (a x \left (2 a^2 x^2+3 a x+6\right )+6 \log (1-a x)\right )}{6 a^4 \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 371, normalized size = 2.39 \[ \left [\frac {3 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {c} \log \left (\frac {a^{6} c x^{6} - 4 \, a^{5} c x^{5} + 5 \, a^{4} c x^{4} - 4 \, a^{2} c x^{2} + 4 \, a c x + {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} \sqrt {c} - 2 \, c}{a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1}\right ) + {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + 6 \, a x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{6 \, {\left (a^{6} c x^{2} - a^{4} c\right )}}, -\frac {6 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} {\left (a^{2} x^{2} - 2 \, a x + 2\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-c}}{a^{4} c x^{4} - 2 \, a^{3} c x^{3} - a^{2} c x^{2} + 2 \, a c x}\right ) - {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + 6 \, a x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{6 \, {\left (a^{6} c x^{2} - a^{4} c\right )}}\right ] \]
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 \[ \int \frac {{\left (a x + 1\right )} x^{3}}{\sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 75, normalized size = 0.48 \[ \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 x^{3} a^{3}+3 a^{2} x^{2}+6 a x +6 \ln \left (a x -1\right )\right )}{6 \left (a^{2} x^{2}-1\right ) c \,a^{4}} \]
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 \[ \frac {-\frac {1}{6} \, a {\left (\frac {2 \, {\left (a^{2} x^{3} + 3 \, x\right )}}{a^{4}} - \frac {3 \, \log \left (a x + 1\right )}{a^{5}} + \frac {3 \, \log \left (a x - 1\right )}{a^{5}}\right )}}{\sqrt {c}} + \frac {\log \left (x^{2} - \frac {1}{a^{2}}\right )}{2 \, a^{4} \sqrt {c}} + \frac {\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}}{2 \, a^{4} c} \]
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 \[ \int \frac {x^3\,\left (a\,x+1\right )}{\sqrt {c-a^2\,c\,x^2}\,\sqrt {1-a^2\,x^2}} \,d x \]
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 \[ \int \frac {x^{3} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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