Optimal. Leaf size=65 \[ \frac{\sqrt{c x-1} \sqrt{c x+1} \left (3 a c^2+2 b\right )}{3 c^4}+\frac{b x^2 \sqrt{c x-1} \sqrt{c x+1}}{3 c^2} \]
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Rubi [A] time = 0.0402388, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {460, 74} \[ \frac{\sqrt{c x-1} \sqrt{c x+1} \left (3 a c^2+2 b\right )}{3 c^4}+\frac{b x^2 \sqrt{c x-1} \sqrt{c x+1}}{3 c^2} \]
Antiderivative was successfully verified.
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Rule 460
Rule 74
Rubi steps
\begin{align*} \int \frac{x \left (a+b x^2\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx &=\frac{b x^2 \sqrt{-1+c x} \sqrt{1+c x}}{3 c^2}-\frac{1}{3} \left (-3 a-\frac{2 b}{c^2}\right ) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=\frac{\left (2 b+3 a c^2\right ) \sqrt{-1+c x} \sqrt{1+c x}}{3 c^4}+\frac{b x^2 \sqrt{-1+c x} \sqrt{1+c x}}{3 c^2}\\ \end{align*}
Mathematica [A] time = 0.0273294, size = 52, normalized size = 0.8 \[ \frac{\left (c^2 x^2-1\right ) \left (3 a c^2+b \left (c^2 x^2+2\right )\right )}{3 c^4 \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 38, normalized size = 0.6 \begin{align*}{\frac{b{x}^{2}{c}^{2}+3\,a{c}^{2}+2\,b}{3\,{c}^{4}}\sqrt{cx-1}\sqrt{cx+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.986657, size = 73, normalized size = 1.12 \begin{align*} \frac{\sqrt{c^{2} x^{2} - 1} b x^{2}}{3 \, c^{2}} + \frac{\sqrt{c^{2} x^{2} - 1} a}{c^{2}} + \frac{2 \, \sqrt{c^{2} x^{2} - 1} b}{3 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47574, size = 88, normalized size = 1.35 \begin{align*} \frac{{\left (b c^{2} x^{2} + 3 \, a c^{2} + 2 \, b\right )} \sqrt{c x + 1} \sqrt{c x - 1}}{3 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 27.9257, size = 202, normalized size = 3.11 \begin{align*} \frac{a{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{4} & 0, 0, \frac{1}{2}, 1 \\- \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{2}} + \frac{i a{G_{6, 6}^{2, 6}\left (\begin{matrix} -1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 1 & \\- \frac{3}{4}, - \frac{1}{4} & -1, - \frac{1}{2}, - \frac{1}{2}, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{2}} + \frac{b{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{5}{4}, - \frac{3}{4} & -1, -1, - \frac{1}{2}, 1 \\- \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{4}} + \frac{i b{G_{6, 6}^{2, 6}\left (\begin{matrix} -2, - \frac{7}{4}, - \frac{3}{2}, - \frac{5}{4}, -1, 1 & \\- \frac{7}{4}, - \frac{5}{4} & -2, - \frac{3}{2}, - \frac{3}{2}, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1637, size = 74, normalized size = 1.14 \begin{align*} \frac{{\left (3 \, a c^{11} + 3 \, b c^{9} +{\left ({\left (c x + 1\right )} b c^{9} - 2 \, b c^{9}\right )}{\left (c x + 1\right )}\right )} \sqrt{c x + 1} \sqrt{c x - 1}}{1920 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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