Optimal. Leaf size=77 \[ -\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left (a+b x^2\right )}-\frac{b \sqrt{a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )} \]
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Rubi [A] time = 0.0230154, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {1112, 14} \[ -\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left (a+b x^2\right )}-\frac{b \sqrt{a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 1112
Rule 14
Rubi steps
\begin{align*} \int \frac{\sqrt{a^2+2 a b x^2+b^2 x^4}}{x^4} \, dx &=\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \int \frac{a b+b^2 x^2}{x^4} \, dx}{a b+b^2 x^2}\\ &=\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \int \left (\frac{a b}{x^4}+\frac{b^2}{x^2}\right ) \, dx}{a b+b^2 x^2}\\ &=-\frac{a \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left (a+b x^2\right )}-\frac{b \sqrt{a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )}\\ \end{align*}
Mathematica [A] time = 0.0069249, size = 37, normalized size = 0.48 \[ -\frac{\sqrt{\left (a+b x^2\right )^2} \left (a+3 b x^2\right )}{3 x^3 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 34, normalized size = 0.4 \begin{align*} -{\frac{3\,b{x}^{2}+a}{3\,{x}^{3} \left ( b{x}^{2}+a \right ) }\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01073, size = 18, normalized size = 0.23 \begin{align*} -\frac{3 \, b x^{2} + a}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42792, size = 32, normalized size = 0.42 \begin{align*} -\frac{3 \, b x^{2} + a}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.286232, size = 14, normalized size = 0.18 \begin{align*} - \frac{a + 3 b x^{2}}{3 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13392, size = 41, normalized size = 0.53 \begin{align*} -\frac{3 \, b x^{2} \mathrm{sgn}\left (b x^{2} + a\right ) + a \mathrm{sgn}\left (b x^{2} + a\right )}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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