Optimal. Leaf size=60 \[ \frac {1}{2} \left (a c^2+2 b\right ) \tan ^{-1}\left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {a \sqrt {c x-1} \sqrt {c x+1}}{2 x^2} \]
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Rubi [A] time = 0.06, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {454, 92, 205} \[ \frac {1}{2} \left (a c^2+2 b\right ) \tan ^{-1}\left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {a \sqrt {c x-1} \sqrt {c x+1}}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 92
Rule 205
Rule 454
Rubi steps
\begin {align*} \int \frac {a+b x^2}{x^3 \sqrt {-1+c x} \sqrt {1+c x}} \, dx &=\frac {a \sqrt {-1+c x} \sqrt {1+c x}}{2 x^2}+\frac {1}{2} \left (2 b+a c^2\right ) \int \frac {1}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {a \sqrt {-1+c x} \sqrt {1+c x}}{2 x^2}+\frac {1}{2} \left (c \left (2 b+a c^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{c+c x^2} \, dx,x,\sqrt {-1+c x} \sqrt {1+c x}\right )\\ &=\frac {a \sqrt {-1+c x} \sqrt {1+c x}}{2 x^2}+\frac {1}{2} \left (2 b+a c^2\right ) \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )\\ \end {align*}
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Mathematica [A] time = 0.07, size = 77, normalized size = 1.28 \[ \frac {x^2 \sqrt {c^2 x^2-1} \left (a c^2+2 b\right ) \tan ^{-1}\left (\sqrt {c^2 x^2-1}\right )+a \left (c^2 x^2-1\right )}{2 x^2 \sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.07, size = 57, normalized size = 0.95 \[ \frac {2 \, {\left (a c^{2} + 2 \, b\right )} x^{2} \arctan \left (-c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + \sqrt {c x + 1} \sqrt {c x - 1} a}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 114, normalized size = 1.90 \[ -\frac {{\left (a c^{3} + 2 \, b c\right )} \arctan \left (\frac {1}{2} \, {\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{2}\right ) + \frac {2 \, {\left (a c^{3} {\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{6} - 4 \, a c^{3} {\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{2}\right )}}{{\left ({\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{4} + 4\right )}^{2}}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 84, normalized size = 1.40 \[ -\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (a \,c^{2} x^{2} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )+2 b \,x^{2} \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )-\sqrt {c^{2} x^{2}-1}\, a \right )}{2 \sqrt {c^{2} x^{2}-1}\, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.23, size = 45, normalized size = 0.75 \[ -\frac {1}{2} \, a c^{2} \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) - b \arcsin \left (\frac {1}{c {\left | x \right |}}\right ) + \frac {\sqrt {c^{2} x^{2} - 1} a}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.67, size = 297, normalized size = 4.95 \[ \frac {\frac {a\,c^2\,1{}\mathrm {i}}{32}+\frac {a\,c^2\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}{16\,{\left (\sqrt {c\,x+1}-1\right )}^2}-\frac {a\,c^2\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^4\,15{}\mathrm {i}}{32\,{\left (\sqrt {c\,x+1}-1\right )}^4}}{\frac {{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {c\,x+1}-1\right )}^2}+\frac {2\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {c\,x+1}-1\right )}^4}+\frac {{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {c\,x+1}-1\right )}^6}}-b\,\left (\ln \left (\frac {{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {c\,x+1}-1\right )}^2}+1\right )-\ln \left (\frac {\sqrt {c\,x-1}-\mathrm {i}}{\sqrt {c\,x+1}-1}\right )\right )\,1{}\mathrm {i}-\frac {a\,c^2\,\ln \left (\frac {{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {c\,x+1}-1\right )}^2}+1\right )\,1{}\mathrm {i}}{2}+\frac {a\,c^2\,\ln \left (\frac {\sqrt {c\,x-1}-\mathrm {i}}{\sqrt {c\,x+1}-1}\right )\,1{}\mathrm {i}}{2}+\frac {a\,c^2\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}{32\,{\left (\sqrt {c\,x+1}-1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 63.62, size = 141, normalized size = 2.35 \[ - \frac {a c^{2} {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {7}{4}, \frac {9}{4}, 1 & 2, 2, \frac {5}{2} \\\frac {3}{2}, \frac {7}{4}, 2, \frac {9}{4}, \frac {5}{2} & 0 \end {matrix} \middle | {\frac {1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} + \frac {i a c^{2} {G_{6, 6}^{2, 6}\left (\begin {matrix} 1, \frac {5}{4}, \frac {3}{2}, \frac {7}{4}, 2, 1 & \\\frac {5}{4}, \frac {7}{4} & 1, \frac {3}{2}, \frac {3}{2}, 0 \end {matrix} \middle | {\frac {e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} - \frac {b {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {3}{4}, \frac {5}{4}, 1 & 1, 1, \frac {3}{2} \\\frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4}, \frac {3}{2} & 0 \end {matrix} \middle | {\frac {1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} + \frac {i b {G_{6, 6}^{2, 6}\left (\begin {matrix} 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, 1 & \\\frac {1}{4}, \frac {3}{4} & 0, \frac {1}{2}, \frac {1}{2}, 0 \end {matrix} \middle | {\frac {e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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