Optimal. Leaf size=60 \[ \frac {(a x+1)^2}{3 a^2 \left (c-a^2 c x^2\right )^{3/2}}-\frac {2 (a x+1)}{3 a^2 c \sqrt {c-a^2 c x^2}} \]
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Rubi [A] time = 0.12, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6151, 789, 637} \[ \frac {(a x+1)^2}{3 a^2 \left (c-a^2 c x^2\right )^{3/2}}-\frac {2 (a x+1)}{3 a^2 c \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 637
Rule 789
Rule 6151
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)} x}{\left (c-a^2 c x^2\right )^{3/2}} \, dx &=c \int \frac {x (1+a x)^2}{\left (c-a^2 c x^2\right )^{5/2}} \, dx\\ &=\frac {(1+a x)^2}{3 a^2 \left (c-a^2 c x^2\right )^{3/2}}-\frac {2 \int \frac {1+a x}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{3 a}\\ &=\frac {(1+a x)^2}{3 a^2 \left (c-a^2 c x^2\right )^{3/2}}-\frac {2 (1+a x)}{3 a^2 c \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 38, normalized size = 0.63 \[ \frac {(2 a x-1) \sqrt {c-a^2 c x^2}}{3 a^2 c^2 (a x-1)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 50, normalized size = 0.83 \[ \frac {\sqrt {-a^{2} c x^{2} + c} {\left (2 \, a x - 1\right )}}{3 \, {\left (a^{4} c^{2} x^{2} - 2 \, a^{3} c^{2} x + a^{2} c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.31, size = 117, normalized size = 1.95 \[ -\frac {{\left (a c + 3 \, \sqrt {-a^{2} c} \sqrt {c}\right )} \mathrm {sgn}\relax (x)}{3 \, {\left (a^{3} c^{\frac {5}{2}} - \sqrt {-a^{2} c} a^{2} c^{2}\right )}} - \frac {2 \, {\left (a \sqrt {c} + 3 \, \sqrt {-a^{2} c + \frac {c}{x^{2}}} - \frac {3 \, \sqrt {c}}{x}\right )}}{3 \, {\left (a \sqrt {c} + \sqrt {-a^{2} c + \frac {c}{x^{2}}} - \frac {\sqrt {c}}{x}\right )}^{3} \sqrt {c} \mathrm {sgn}\relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 32, normalized size = 0.53 \[ \frac {\left (2 a x -1\right ) \left (a x +1\right )^{2}}{3 \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.48, size = 218, normalized size = 3.63 \[ \frac {1}{3} \, a {\left (\frac {a}{\sqrt {-a^{2} c x^{2} + c} a^{5} c x + \sqrt {-a^{2} c x^{2} + c} a^{4} c} - \frac {a}{\sqrt {-a^{2} c x^{2} + c} a^{5} c x - \sqrt {-a^{2} c x^{2} + c} a^{4} c} - \frac {1}{\sqrt {-a^{2} c x^{2} + c} a^{4} c x + \sqrt {-a^{2} c x^{2} + c} a^{3} c} - \frac {1}{\sqrt {-a^{2} c x^{2} + c} a^{4} c x - \sqrt {-a^{2} c x^{2} + c} a^{3} c} - \frac {2 \, x}{\sqrt {-a^{2} c x^{2} + c} a^{2} c} - \frac {3}{\sqrt {-a^{2} c x^{2} + c} a^{3} c}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.02, size = 34, normalized size = 0.57 \[ \frac {\sqrt {c-a^2\,c\,x^2}\,\left (2\,a\,x-1\right )}{3\,a^2\,c^2\,{\left (a\,x-1\right )}^2} \]
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 \[ - \int \frac {x}{- a^{3} c x^{3} \sqrt {- a^{2} c x^{2} + c} + a^{2} c x^{2} \sqrt {- a^{2} c x^{2} + c} + a c x \sqrt {- a^{2} c x^{2} + c} - c \sqrt {- a^{2} c x^{2} + c}}\, dx - \int \frac {a x^{2}}{- a^{3} c x^{3} \sqrt {- a^{2} c x^{2} + c} + a^{2} c x^{2} \sqrt {- a^{2} c x^{2} + c} + a c x \sqrt {- a^{2} c x^{2} + c} - c \sqrt {- a^{2} c x^{2} + c}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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