Optimal. Leaf size=74 \[ \frac{b x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )} \]
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Rubi [A] time = 0.020172, antiderivative size = 74, 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} \[ \frac{b x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}-\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )} \]
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^3} \, dx &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \int \frac{a b+b^2 x^3}{x^3} \, dx}{a b+b^2 x^3}\\ &=\frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \int \left (b^2+\frac{a b}{x^3}\right ) \, dx}{a b+b^2 x^3}\\ &=-\frac{a \sqrt{a^2+2 a b x^3+b^2 x^6}}{2 x^2 \left (a+b x^3\right )}+\frac{b x \sqrt{a^2+2 a b x^3+b^2 x^6}}{a+b x^3}\\ \end{align*}
Mathematica [A] time = 0.0075755, size = 37, normalized size = 0.5 \[ -\frac{\left (a-2 b x^3\right ) \sqrt{\left (a+b x^3\right )^2}}{2 x^2 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 34, normalized size = 0.5 \begin{align*} -{\frac{-2\,b{x}^{3}+a}{2\,{x}^{2} \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 = 1.01841, size = 20, normalized size = 0.27 \begin{align*} \frac{2 \, b x^{3} - a}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73779, size = 31, normalized size = 0.42 \begin{align*} \frac{2 \, b x^{3} - a}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.273665, size = 8, normalized size = 0.11 \begin{align*} - \frac{a}{2 x^{2}} + b x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10113, size = 35, normalized size = 0.47 \begin{align*} b x \mathrm{sgn}\left (b x^{3} + a\right ) - \frac{a \mathrm{sgn}\left (b x^{3} + a\right )}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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