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 = 0.04, antiderivative size = 65, normalized size of antiderivative = 1.00, 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 74
Rule 460
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*}
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Mathematica [A] time = 0.05, size = 52, normalized size = 0.80 \[ \frac {\left (c^2 x^2-1\right ) \left (3 a c^2+b \left (c^2 x^2+2\right )\right )}{3 c^4 \sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.04, size = 37, normalized size = 0.57 \[ \frac {{\left (b c^{2} x^{2} + 3 \, a c^{2} + 2 \, b\right )} \sqrt {c x + 1} \sqrt {c x - 1}}{3 \, c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 59, normalized size = 0.91 \[ \frac {\sqrt {c x + 1} \sqrt {c x - 1} {\left ({\left (c x + 1\right )} {\left (\frac {{\left (c x + 1\right )} b}{c^{3}} - \frac {2 \, b}{c^{3}}\right )} + \frac {3 \, {\left (a c^{11} + b c^{9}\right )}}{c^{12}}\right )}}{3 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 38, normalized size = 0.58 \[ \frac {\sqrt {c x +1}\, \sqrt {c x -1}\, \left (b \,c^{2} x^{2}+3 a \,c^{2}+2 b \right )}{3 c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 54, normalized size = 0.83 \[ \frac {\sqrt {c^{2} x^{2} - 1} b x^{2}}{3 \, c^{2}} + \frac {\sqrt {c^{2} x^{2} - 1} a}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1} b}{3 \, c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.36, size = 66, normalized size = 1.02 \[ \frac {\sqrt {c\,x-1}\,\left (\frac {3\,a\,c^2+2\,b}{3\,c^4}+\frac {b\,x^3}{3\,c}+\frac {b\,x^2}{3\,c^2}+\frac {x\,\left (3\,a\,c^3+2\,b\,c\right )}{3\,c^4}\right )}{\sqrt {c\,x+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 41.91, size = 202, normalized size = 3.11 \[ \frac {a {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{4} & 0, 0, \frac {1}{2}, 1 \\- \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, 0 & \end {matrix} \middle | {\frac {1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} c^{2}} + \frac {i a {G_{6, 6}^{2, 6}\left (\begin {matrix} -1, - \frac {3}{4}, - \frac {1}{2}, - \frac {1}{4}, 0, 1 & \\- \frac {3}{4}, - \frac {1}{4} & -1, - \frac {1}{2}, - \frac {1}{2}, 0 \end {matrix} \middle | {\frac {e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} c^{2}} + \frac {b {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {5}{4}, - \frac {3}{4} & -1, -1, - \frac {1}{2}, 1 \\- \frac {3}{2}, - \frac {5}{4}, -1, - \frac {3}{4}, - \frac {1}{2}, 0 & \end {matrix} \middle | {\frac {1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} c^{4}} + \frac {i b {G_{6, 6}^{2, 6}\left (\begin {matrix} -2, - \frac {7}{4}, - \frac {3}{2}, - \frac {5}{4}, -1, 1 & \\- \frac {7}{4}, - \frac {5}{4} & -2, - \frac {3}{2}, - \frac {3}{2}, 0 \end {matrix} \middle | {\frac {e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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