Optimal. Leaf size=62 \[ \frac {a \sqrt {-1+c x} \sqrt {1+c x}}{3 x^3}+\frac {\left (3 b+2 a c^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}{3 x} \]
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Rubi [A]
time = 0.04, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {465, 97}
\begin {gather*} \frac {\sqrt {c x-1} \sqrt {c x+1} \left (2 a c^2+3 b\right )}{3 x}+\frac {a \sqrt {c x-1} \sqrt {c x+1}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 97
Rule 465
Rubi steps
\begin {align*} \int \frac {a+b x^2}{x^4 \sqrt {-1+c x} \sqrt {1+c x}} \, dx &=\frac {a \sqrt {-1+c x} \sqrt {1+c x}}{3 x^3}+\frac {1}{3} \left (3 b+2 a c^2\right ) \int \frac {1}{x^2 \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {a \sqrt {-1+c x} \sqrt {1+c x}}{3 x^3}+\frac {\left (3 b+2 a c^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}{3 x}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 42, normalized size = 0.68 \begin {gather*} \frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+3 b x^2+2 a c^2 x^2\right )}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
2.
time = 0.28, size = 41, normalized size = 0.66
method | result | size |
gosper | \(\frac {\sqrt {c x +1}\, \sqrt {c x -1}\, \left (2 a \,c^{2} x^{2}+3 b \,x^{2}+a \right )}{3 x^{3}}\) | \(37\) |
risch | \(\frac {\sqrt {c x +1}\, \sqrt {c x -1}\, \left (2 a \,c^{2} x^{2}+3 b \,x^{2}+a \right )}{3 x^{3}}\) | \(37\) |
default | \(\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \mathrm {csgn}\left (c \right )^{2} \left (2 a \,c^{2} x^{2}+3 b \,x^{2}+a \right )}{3 x^{3}}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 54, normalized size = 0.87 \begin {gather*} \frac {2 \, \sqrt {c^{2} x^{2} - 1} a c^{2}}{3 \, x} + \frac {\sqrt {c^{2} x^{2} - 1} b}{x} + \frac {\sqrt {c^{2} x^{2} - 1} a}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.35, size = 52, normalized size = 0.84 \begin {gather*} \frac {{\left (2 \, a c^{3} + 3 \, b c\right )} x^{3} + {\left ({\left (2 \, a c^{2} + 3 \, b\right )} x^{2} + a\right )} \sqrt {c x + 1} \sqrt {c x - 1}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 26.80, size = 146, normalized size = 2.35 \begin {gather*} - \frac {a c^{3} {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {9}{4}, \frac {11}{4}, 1 & \frac {5}{2}, \frac {5}{2}, 3 \\2, \frac {9}{4}, \frac {5}{2}, \frac {11}{4}, 3 & 0 \end {matrix} \middle | {\frac {1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} - \frac {i a c^{3} {G_{6, 6}^{2, 6}\left (\begin {matrix} \frac {3}{2}, \frac {7}{4}, 2, \frac {9}{4}, \frac {5}{2}, 1 & \\\frac {7}{4}, \frac {9}{4} & \frac {3}{2}, 2, 2, 0 \end {matrix} \middle | {\frac {e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} - \frac {b c {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {5}{4}, \frac {7}{4}, 1 & \frac {3}{2}, \frac {3}{2}, 2 \\1, \frac {5}{4}, \frac {3}{2}, \frac {7}{4}, 2 & 0 \end {matrix} \middle | {\frac {1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} - \frac {i b c {G_{6, 6}^{2, 6}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4}, \frac {3}{2}, 1 & \\\frac {3}{4}, \frac {5}{4} & \frac {1}{2}, 1, 1, 0 \end {matrix} \middle | {\frac {e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 116 vs.
\(2 (50) = 100\).
time = 0.63, size = 116, normalized size = 1.87 \begin {gather*} \frac {8 \, {\left (3 \, b c^{2} {\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{8} + 24 \, a c^{4} {\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{4} + 24 \, b c^{2} {\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{4} + 32 \, a c^{4} + 48 \, b c^{2}\right )}}{3 \, {\left ({\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{4} + 4\right )}^{3} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.44, size = 53, normalized size = 0.85 \begin {gather*} \frac {\sqrt {c\,x-1}\,\left (\left (\frac {2\,a\,c^3}{3}+b\,c\right )\,x^3+\left (\frac {2\,a\,c^2}{3}+b\right )\,x^2+\frac {a\,c\,x}{3}+\frac {a}{3}\right )}{x^3\,\sqrt {c\,x+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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