Integrand size = 29, antiderivative size = 46 \[ \int \frac {a+b x^2}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx=\frac {b \sqrt {-1+c x} \sqrt {1+c x}}{c^2}+a \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {471, 94, 211} \[ \int \frac {a+b x^2}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx=a \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c^2} \]
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Rule 94
Rule 211
Rule 471
Rubi steps \begin{align*} \text {integral}& = \frac {b \sqrt {-1+c x} \sqrt {1+c x}}{c^2}+a \int \frac {1}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = \frac {b \sqrt {-1+c x} \sqrt {1+c x}}{c^2}+(a c) \text {Subst}\left (\int \frac {1}{c+c x^2} \, dx,x,\sqrt {-1+c x} \sqrt {1+c x}\right ) \\ & = \frac {b \sqrt {-1+c x} \sqrt {1+c x}}{c^2}+a \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.98 \[ \int \frac {a+b x^2}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx=\frac {b \sqrt {-1+c x} \sqrt {1+c x}}{c^2}+2 a \arctan \left (\sqrt {\frac {-1+c x}{1+c x}}\right ) \]
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Time = 4.01 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.35
method | result | size |
default | \(\frac {\left (-\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) a \,c^{2}+\sqrt {c^{2} x^{2}-1}\, b \right ) \sqrt {c x -1}\, \sqrt {c x +1}}{\sqrt {c^{2} x^{2}-1}\, c^{2}}\) | \(62\) |
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Time = 0.26 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.04 \[ \int \frac {a+b x^2}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx=\frac {2 \, a c^{2} \arctan \left (-c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + \sqrt {c x + 1} \sqrt {c x - 1} b}{c^{2}} \]
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Result contains complex when optimal does not.
Time = 15.95 (sec) , antiderivative size = 162, normalized size of antiderivative = 3.52 \[ \int \frac {a+b x^2}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx=- \frac {a {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {3}{4}, \frac {5}{4}, 1 & 1, 1, \frac {3}{2} \\\frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4}, \frac {3}{2} & 0 \end {matrix} \middle | {\frac {1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} + \frac {i a {G_{6, 6}^{2, 6}\left (\begin {matrix} 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, 1 & \\\frac {1}{4}, \frac {3}{4} & 0, \frac {1}{2}, \frac {1}{2}, 0 \end {matrix} \middle | {\frac {e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} + \frac {b {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{4} & 0, 0, \frac {1}{2}, 1 \\- \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, 0 & \end {matrix} \middle | {\frac {1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} c^{2}} + \frac {i b {G_{6, 6}^{2, 6}\left (\begin {matrix} -1, - \frac {3}{4}, - \frac {1}{2}, - \frac {1}{4}, 0, 1 & \\- \frac {3}{4}, - \frac {1}{4} & -1, - \frac {1}{2}, - \frac {1}{2}, 0 \end {matrix} \middle | {\frac {e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} c^{2}} \]
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Time = 0.36 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.63 \[ \int \frac {a+b x^2}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx=-a \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\sqrt {c^{2} x^{2} - 1} b}{c^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.98 \[ \int \frac {a+b x^2}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx=-2 \, a \arctan \left (\frac {1}{2} \, {\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{2}\right ) + \frac {\sqrt {c x + 1} \sqrt {c x - 1} b}{c^{2}} \]
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Time = 7.66 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.67 \[ \int \frac {a+b x^2}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx=\frac {b\,\sqrt {c\,x-1}\,\sqrt {c\,x+1}}{c^2}-a\,\left (\ln \left (\frac {{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {c\,x+1}-1\right )}^2}+1\right )-\ln \left (\frac {\sqrt {c\,x-1}-\mathrm {i}}{\sqrt {c\,x+1}-1}\right )\right )\,1{}\mathrm {i} \]
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