Optimal. Leaf size=58 \[ \frac{(A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 a^{3/2}}-\frac{A \sqrt{a+b x^3}}{3 a x^3} \]
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Rubi [A] time = 0.0457008, antiderivative size = 58, 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, 78, 63, 208} \[ \frac{(A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 a^{3/2}}-\frac{A \sqrt{a+b x^3}}{3 a x^3} \]
Antiderivative was successfully verified.
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Rule 446
Rule 78
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{A+B x^3}{x^4 \sqrt{a+b x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{A+B x}{x^2 \sqrt{a+b x}} \, dx,x,x^3\right )\\ &=-\frac{A \sqrt{a+b x^3}}{3 a x^3}+\frac{\left (-\frac{A b}{2}+a B\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^3\right )}{3 a}\\ &=-\frac{A \sqrt{a+b x^3}}{3 a x^3}+\frac{\left (2 \left (-\frac{A b}{2}+a B\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^3}\right )}{3 a b}\\ &=-\frac{A \sqrt{a+b x^3}}{3 a x^3}+\frac{(A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 a^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0388455, size = 57, normalized size = 0.98 \[ \frac{1}{3} \left (\frac{(A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{A \sqrt{a+b x^3}}{a x^3}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 62, normalized size = 1.1 \begin{align*} A \left ( -{\frac{1}{3\,a{x}^{3}}\sqrt{b{x}^{3}+a}}+{\frac{b}{3}{\it Artanh} \left ({\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{3}{2}}}} \right ) -{\frac{2\,B}{3}{\it Artanh} \left ({\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){\frac{1}{\sqrt{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 = 1.78028, size = 302, normalized size = 5.21 \begin{align*} \left [-\frac{{\left (2 \, B a - A b\right )} \sqrt{a} x^{3} \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} A a}{6 \, a^{2} x^{3}}, \frac{{\left (2 \, B a - A b\right )} \sqrt{-a} x^{3} \arctan \left (\frac{\sqrt{b x^{3} + a} \sqrt{-a}}{a}\right ) - \sqrt{b x^{3} + a} A a}{3 \, a^{2} x^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 16.3022, size = 80, normalized size = 1.38 \begin{align*} - \frac{A \sqrt{b} \sqrt{\frac{a}{b x^{3}} + 1}}{3 a x^{\frac{3}{2}}} + \frac{A b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{3 a^{\frac{3}{2}}} - \frac{2 B \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{3 \sqrt{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15022, size = 84, normalized size = 1.45 \begin{align*} \frac{\frac{{\left (2 \, B a b - A b^{2}\right )} \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} - \frac{\sqrt{b x^{3} + a} A b}{a x^{3}}}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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