Optimal. Leaf size=50 \[ \frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 a^{3/2}}-\frac{\sqrt{a+b x^3}}{3 a x^3} \]
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Rubi [A] time = 0.0264228, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 51, 63, 208} \[ \frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 a^{3/2}}-\frac{\sqrt{a+b x^3}}{3 a x^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x^4 \sqrt{a+b x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,x^3\right )\\ &=-\frac{\sqrt{a+b x^3}}{3 a x^3}-\frac{b \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^3\right )}{6 a}\\ &=-\frac{\sqrt{a+b x^3}}{3 a x^3}-\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^3}\right )}{3 a}\\ &=-\frac{\sqrt{a+b x^3}}{3 a x^3}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 a^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.048069, size = 64, normalized size = 1.28 \[ \frac{2 b \sqrt{a+b x^3} \left (\frac{\tanh ^{-1}\left (\sqrt{\frac{b x^3}{a}+1}\right )}{2 \sqrt{\frac{b x^3}{a}+1}}-\frac{a}{2 b x^3}\right )}{3 a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 39, normalized size = 0.8 \begin{align*}{\frac{b}{3}{\it Artanh} \left ({\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{3}{2}}}}-{\frac{1}{3\,a{x}^{3}}\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 = 1.56232, size = 265, normalized size = 5.3 \begin{align*} \left [\frac{\sqrt{a} b 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}{6 \, a^{2} x^{3}}, -\frac{\sqrt{-a} b x^{3} \arctan \left (\frac{\sqrt{b x^{3} + a} \sqrt{-a}}{a}\right ) + \sqrt{b x^{3} + 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 = 2.48261, size = 49, normalized size = 0.98 \begin{align*} - \frac{\sqrt{b} \sqrt{\frac{a}{b x^{3}} + 1}}{3 a x^{\frac{3}{2}}} + \frac{b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{3 a^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1062, size = 65, normalized size = 1.3 \begin{align*} -\frac{1}{3} \, b{\left (\frac{\arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{\sqrt{b x^{3} + a}}{a b x^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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