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 = 0.0226765, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, 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 1958
Rule 94
Rule 93
Rule 204
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*}
Mathematica [A] time = 0.0362176, 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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Maple [B] time = 0.02, size = 106, normalized size = 2.3 \begin{align*} -{\frac{1+7\,x}{x}\sqrt{{\frac{-1+5\,x}{1+7\,x}}} \left ( - \left ( 35\,{x}^{2}-2\,x-1 \right ) ^{{\frac{3}{2}}}+35\,\sqrt{35\,{x}^{2}-2\,x-1}{x}^{2}+6\,\arctan \left ({\frac{1+x}{\sqrt{35\,{x}^{2}-2\,x-1}}} \right ) x-2\,\sqrt{35\,{x}^{2}-2\,x-1}x \right ){\frac{1}{\sqrt{ \left ( -1+5\,x \right ) \left ( 1+7\,x \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.51165, size = 72, normalized size = 1.57 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.87202, size = 111, normalized size = 2.41 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{5 x - 1}{7 x + 1}}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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