Optimal. Leaf size=46 \[ -\frac {\sqrt {5 x-1} \sqrt {7 x+1}}{x}-12 \tan ^{-1}\left (\frac {\sqrt {7 x+1}}{\sqrt {5 x-1}}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {1958, 94, 93, 204} \[ -\frac {\sqrt {5 x-1} \sqrt {7 x+1}}{x}-12 \tan ^{-1}\left (\frac {\sqrt {7 x+1}}{\sqrt {5 x-1}}\right ) \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 204
Rule 1958
Rubi steps
\begin {align*} \int \frac {\sqrt {\frac {-1+5 x}{1+7 x}}}{x^2} \, dx &=\int \frac {\sqrt {-1+5 x}}{x^2 \sqrt {1+7 x}} \, dx\\ &=-\frac {\sqrt {-1+5 x} \sqrt {1+7 x}}{x}+6 \int \frac {1}{x \sqrt {-1+5 x} \sqrt {1+7 x}} \, dx\\ &=-\frac {\sqrt {-1+5 x} \sqrt {1+7 x}}{x}+12 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\frac {\sqrt {1+7 x}}{\sqrt {-1+5 x}}\right )\\ &=-\frac {\sqrt {-1+5 x} \sqrt {1+7 x}}{x}-12 \tan ^{-1}\left (\frac {\sqrt {1+7 x}}{\sqrt {-1+5 x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 79, normalized size = 1.72 \[ \frac {\sqrt {\frac {5 x-1}{7 x+1}} \left (12 x \sqrt {7 x+1} \tan ^{-1}\left (\frac {\sqrt {5 x-1}}{\sqrt {7 x+1}}\right )-\sqrt {5 x-1} (7 x+1)\right )}{x \sqrt {5 x-1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 46, normalized size = 1.00 \[ \frac {12 \, x \arctan \left (\sqrt {\frac {5 \, x - 1}{7 \, x + 1}}\right ) - {\left (7 \, x + 1\right )} \sqrt {\frac {5 \, x - 1}{7 \, x + 1}}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.40, size = 114, normalized size = 2.48 \[ {\left (\sqrt {35} - 12 \, \arctan \left (\frac {1}{7} \, \sqrt {35}\right )\right )} \mathrm {sgn}\left (7 \, x + 1\right ) + 12 \, \arctan \left (-\sqrt {35} x + \sqrt {35 \, x^{2} - 2 \, x - 1}\right ) \mathrm {sgn}\left (7 \, x + 1\right ) - \frac {2 \, {\left ({\left (\sqrt {35} x - \sqrt {35 \, x^{2} - 2 \, x - 1}\right )} \mathrm {sgn}\left (7 \, x + 1\right ) + \sqrt {35} \mathrm {sgn}\left (7 \, x + 1\right )\right )}}{{\left (\sqrt {35} x - \sqrt {35 \, x^{2} - 2 \, x - 1}\right )}^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 103, normalized size = 2.24 \[ \frac {\sqrt {\frac {5 x -1}{7 x +1}}\, \left (7 x +1\right ) \left (-35 \sqrt {35 x^{2}-2 x -1}\, x^{2}-6 x \arctan \left (\frac {x +1}{\sqrt {35 x^{2}-2 x -1}}\right )+2 \sqrt {35 x^{2}-2 x -1}\, x +\left (35 x^{2}-2 x -1\right )^{\frac {3}{2}}\right )}{\sqrt {\left (5 x -1\right ) \left (7 x +1\right )}\, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.43, size = 53, normalized size = 1.15 \[ -\frac {12 \, \sqrt {\frac {5 \, x - 1}{7 \, x + 1}}}{\frac {5 \, x - 1}{7 \, x + 1} + 1} + 12 \, \arctan \left (\sqrt {\frac {5 \, x - 1}{7 \, x + 1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.24, size = 74, normalized size = 1.61 \[ 12\,\mathrm {atan}\left (\frac {\sqrt {5}\,\sqrt {7}\,\sqrt {35}\,\sqrt {\frac {5\,x-1}{7\,x+1}}}{35}\right )-\frac {12\,\sqrt {5}\,\sqrt {7}\,\sqrt {35}\,\sqrt {\frac {5\,x-1}{7\,x+1}}}{25\,\left (\frac {7\,x-\frac {7}{5}}{7\,x+1}+\frac {7}{5}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {5 x - 1}{7 x + 1}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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