Optimal. Leaf size=95 \[ \frac{b^2 \sqrt{a+b x^3}}{24 a^2 x^3}-\frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{24 a^{5/2}}-\frac{b \sqrt{a+b x^3}}{36 a x^6}-\frac{\sqrt{a+b x^3}}{9 x^9} \]
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Rubi [A] time = 0.0532572, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {266, 47, 51, 63, 208} \[ \frac{b^2 \sqrt{a+b x^3}}{24 a^2 x^3}-\frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{24 a^{5/2}}-\frac{b \sqrt{a+b x^3}}{36 a x^6}-\frac{\sqrt{a+b x^3}}{9 x^9} \]
Antiderivative was successfully verified.
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Rule 266
Rule 47
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x^3}}{x^{10}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^4} \, dx,x,x^3\right )\\ &=-\frac{\sqrt{a+b x^3}}{9 x^9}+\frac{1}{18} b \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{a+b x}} \, dx,x,x^3\right )\\ &=-\frac{\sqrt{a+b x^3}}{9 x^9}-\frac{b \sqrt{a+b x^3}}{36 a x^6}-\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,x^3\right )}{24 a}\\ &=-\frac{\sqrt{a+b x^3}}{9 x^9}-\frac{b \sqrt{a+b x^3}}{36 a x^6}+\frac{b^2 \sqrt{a+b x^3}}{24 a^2 x^3}+\frac{b^3 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^3\right )}{48 a^2}\\ &=-\frac{\sqrt{a+b x^3}}{9 x^9}-\frac{b \sqrt{a+b x^3}}{36 a x^6}+\frac{b^2 \sqrt{a+b x^3}}{24 a^2 x^3}+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^3}\right )}{24 a^2}\\ &=-\frac{\sqrt{a+b x^3}}{9 x^9}-\frac{b \sqrt{a+b x^3}}{36 a x^6}+\frac{b^2 \sqrt{a+b x^3}}{24 a^2 x^3}-\frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{24 a^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0088009, size = 39, normalized size = 0.41 \[ \frac{2 b^3 \left (a+b x^3\right )^{3/2} \, _2F_1\left (\frac{3}{2},4;\frac{5}{2};\frac{b x^3}{a}+1\right )}{9 a^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 76, normalized size = 0.8 \begin{align*} -{\frac{{b}^{3}}{24}{\it Artanh} \left ({\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{5}{2}}}}-{\frac{1}{9\,{x}^{9}}\sqrt{b{x}^{3}+a}}-{\frac{b}{36\,{x}^{6}a}\sqrt{b{x}^{3}+a}}+{\frac{{b}^{2}}{24\,{x}^{3}{a}^{2}}\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 = 1.49517, size = 370, normalized size = 3.89 \begin{align*} \left [\frac{3 \, \sqrt{a} b^{3} x^{9} \log \left (\frac{b x^{3} - 2 \, \sqrt{b x^{3} + a} \sqrt{a} + 2 \, a}{x^{3}}\right ) + 2 \,{\left (3 \, a b^{2} x^{6} - 2 \, a^{2} b x^{3} - 8 \, a^{3}\right )} \sqrt{b x^{3} + a}}{144 \, a^{3} x^{9}}, \frac{3 \, \sqrt{-a} b^{3} x^{9} \arctan \left (\frac{\sqrt{b x^{3} + a} \sqrt{-a}}{a}\right ) +{\left (3 \, a b^{2} x^{6} - 2 \, a^{2} b x^{3} - 8 \, a^{3}\right )} \sqrt{b x^{3} + a}}{72 \, a^{3} x^{9}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.30894, size = 129, normalized size = 1.36 \begin{align*} - \frac{a}{9 \sqrt{b} x^{\frac{21}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} - \frac{5 \sqrt{b}}{36 x^{\frac{15}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} + \frac{b^{\frac{3}{2}}}{72 a x^{\frac{9}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} + \frac{b^{\frac{5}{2}}}{24 a^{2} x^{\frac{3}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} - \frac{b^{3} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{24 a^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13218, size = 108, normalized size = 1.14 \begin{align*} \frac{1}{72} \, b^{3}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} + \frac{3 \,{\left (b x^{3} + a\right )}^{\frac{5}{2}} - 8 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} a - 3 \, \sqrt{b x^{3} + a} a^{2}}{a^{2} b^{3} x^{9}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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