Optimal. Leaf size=58 \[ -\frac{\left (a+b x^3\right )^{3/2}}{3 x^3}+b \sqrt{a+b x^3}-\sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right ) \]
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Rubi [A] time = 0.0359271, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {266, 47, 50, 63, 208} \[ -\frac{\left (a+b x^3\right )^{3/2}}{3 x^3}+b \sqrt{a+b x^3}-\sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
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Rule 266
Rule 47
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (a+b x^3\right )^{3/2}}{x^4} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x^2} \, dx,x,x^3\right )\\ &=-\frac{\left (a+b x^3\right )^{3/2}}{3 x^3}+\frac{1}{2} b \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,x^3\right )\\ &=b \sqrt{a+b x^3}-\frac{\left (a+b x^3\right )^{3/2}}{3 x^3}+\frac{1}{2} (a b) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^3\right )\\ &=b \sqrt{a+b x^3}-\frac{\left (a+b x^3\right )^{3/2}}{3 x^3}+a \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^3}\right )\\ &=b \sqrt{a+b x^3}-\frac{\left (a+b x^3\right )^{3/2}}{3 x^3}-\sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [C] time = 0.0091124, size = 37, normalized size = 0.64 \[ \frac{2 b \left (a+b x^3\right )^{5/2} \, _2F_1\left (2,\frac{5}{2};\frac{7}{2};\frac{b x^3}{a}+1\right )}{15 a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 49, normalized size = 0.8 \begin{align*} -{\frac{a}{3\,{x}^{3}}\sqrt{b{x}^{3}+a}}+{\frac{2\,b}{3}\sqrt{b{x}^{3}+a}}-b{\it Artanh} \left ({\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ) \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 = 1.41634, size = 285, normalized size = 4.91 \begin{align*} \left [\frac{3 \, \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 \,{\left (2 \, b x^{3} - a\right )} \sqrt{b x^{3} + a}}{6 \, x^{3}}, \frac{3 \, \sqrt{-a} b x^{3} \arctan \left (\frac{\sqrt{b x^{3} + a} \sqrt{-a}}{a}\right ) +{\left (2 \, b x^{3} - a\right )} \sqrt{b x^{3} + a}}{3 \, x^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.8285, size = 100, normalized size = 1.72 \begin{align*} - \sqrt{a} b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )} - \frac{a^{2}}{3 \sqrt{b} x^{\frac{9}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} + \frac{a \sqrt{b}}{3 x^{\frac{3}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} + \frac{2 b^{\frac{3}{2}} x^{\frac{3}{2}}}{3 \sqrt{\frac{a}{b x^{3}} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10525, size = 77, normalized size = 1.33 \begin{align*} \frac{1}{3} \,{\left (\frac{3 \, a \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + 2 \, \sqrt{b x^{3} + a} - \frac{\sqrt{b x^{3} + a} a}{b x^{3}}\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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