Optimal. Leaf size=77 \[ -\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )}-\frac{b \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )} \]
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Rubi [A] time = 0.0209954, 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 = {1355, 14} \[ -\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )}-\frac{b \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Rule 1355
Rule 14
Rubi steps
\begin{align*} \int \frac{\sqrt{a^2+2 a b x^3+b^2 x^6}}{x^5} \, dx &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \int \frac{a b+b^2 x^3}{x^5} \, dx}{a b+b^2 x^3}\\ &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \int \left (\frac{a b}{x^5}+\frac{b^2}{x^2}\right ) \, dx}{a b+b^2 x^3}\\ &=-\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6}}{4 x^4 \left (a+b x^3\right )}-\frac{b \sqrt{a^2+2 a b x^3+b^2 x^6}}{x \left (a+b x^3\right )}\\ \end{align*}
Mathematica [A] time = 0.008404, size = 37, normalized size = 0.48 \[ -\frac{\sqrt{\left (a+b x^3\right )^2} \left (a+4 b x^3\right )}{4 x^4 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 34, normalized size = 0.4 \begin{align*} -{\frac{4\,b{x}^{3}+a}{4\,{x}^{4} \left ( b{x}^{3}+a \right ) }\sqrt{ \left ( b{x}^{3}+a \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02588, size = 18, normalized size = 0.23 \begin{align*} -\frac{4 \, b x^{3} + a}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80482, size = 32, normalized size = 0.42 \begin{align*} -\frac{4 \, b x^{3} + a}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.308046, size = 14, normalized size = 0.18 \begin{align*} - \frac{a + 4 b x^{3}}{4 x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13396, size = 41, normalized size = 0.53 \begin{align*} -\frac{4 \, b x^{3} \mathrm{sgn}\left (b x^{3} + a\right ) + a \mathrm{sgn}\left (b x^{3} + a\right )}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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