Optimal. Leaf size=48 \[ \frac{2 B \sqrt{a+b x^3}}{3 b}-\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 \sqrt{a}} \]
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Rubi [A] time = 0.0316857, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {446, 80, 63, 208} \[ \frac{2 B \sqrt{a+b x^3}}{3 b}-\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 \sqrt{a}} \]
Antiderivative was successfully verified.
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Rule 446
Rule 80
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{A+B x^3}{x \sqrt{a+b x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{A+B x}{x \sqrt{a+b x}} \, dx,x,x^3\right )\\ &=\frac{2 B \sqrt{a+b x^3}}{3 b}+\frac{1}{3} A \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^3\right )\\ &=\frac{2 B \sqrt{a+b x^3}}{3 b}+\frac{(2 A) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^3}\right )}{3 b}\\ &=\frac{2 B \sqrt{a+b x^3}}{3 b}-\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 \sqrt{a}}\\ \end{align*}
Mathematica [A] time = 0.025787, size = 48, normalized size = 1. \[ \frac{2 B \sqrt{a+b x^3}}{3 b}-\frac{2 A \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 \sqrt{a}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 37, normalized size = 0.8 \begin{align*} -{\frac{2\,A}{3}{\it Artanh} \left ({\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){\frac{1}{\sqrt{a}}}}+{\frac{2\,B}{3\,b}\sqrt{b{x}^{3}+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8653, size = 252, normalized size = 5.25 \begin{align*} \left [\frac{A \sqrt{a} b \log \left (\frac{b x^{3} - 2 \, \sqrt{b x^{3} + a} \sqrt{a} + 2 \, a}{x^{3}}\right ) + 2 \, \sqrt{b x^{3} + a} B a}{3 \, a b}, \frac{2 \,{\left (A \sqrt{-a} b \arctan \left (\frac{\sqrt{b x^{3} + a} \sqrt{-a}}{a}\right ) + \sqrt{b x^{3} + a} B a\right )}}{3 \, a b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.97342, size = 65, normalized size = 1.35 \begin{align*} \frac{2 A \operatorname{atan}{\left (\frac{1}{\sqrt{- \frac{1}{a}} \sqrt{a + b x^{3}}} \right )}}{3 a \sqrt{- \frac{1}{a}}} - \frac{B \left (\begin{cases} - \frac{x^{3}}{\sqrt{a}} & \text{for}\: b = 0 \\- \frac{2 \sqrt{a + b x^{3}}}{b} & \text{otherwise} \end{cases}\right )}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14264, size = 54, normalized size = 1.12 \begin{align*} \frac{2 \, A \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{3 \, \sqrt{-a}} + \frac{2 \, \sqrt{b x^{3} + a} B}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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