Optimal. Leaf size=71 \[ \frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{12 a^{3/2}}-\frac{b \sqrt{a+b x^3}}{12 a x^3}-\frac{\sqrt{a+b x^3}}{6 x^6} \]
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Rubi [A] time = 0.0370353, antiderivative size = 71, 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, 51, 63, 208} \[ \frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{12 a^{3/2}}-\frac{b \sqrt{a+b x^3}}{12 a x^3}-\frac{\sqrt{a+b x^3}}{6 x^6} \]
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^7} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^3} \, dx,x,x^3\right )\\ &=-\frac{\sqrt{a+b x^3}}{6 x^6}+\frac{1}{12} b \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,x^3\right )\\ &=-\frac{\sqrt{a+b x^3}}{6 x^6}-\frac{b \sqrt{a+b x^3}}{12 a x^3}-\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^3\right )}{24 a}\\ &=-\frac{\sqrt{a+b x^3}}{6 x^6}-\frac{b \sqrt{a+b x^3}}{12 a x^3}-\frac{b \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^3}\right )}{12 a}\\ &=-\frac{\sqrt{a+b x^3}}{6 x^6}-\frac{b \sqrt{a+b x^3}}{12 a x^3}+\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{12 a^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0083865, size = 39, normalized size = 0.55 \[ -\frac{2 b^2 \left (a+b x^3\right )^{3/2} \, _2F_1\left (\frac{3}{2},3;\frac{5}{2};\frac{b x^3}{a}+1\right )}{9 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 56, normalized size = 0.8 \begin{align*}{\frac{{b}^{2}}{12}{\it Artanh} \left ({\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{3}{2}}}}-{\frac{1}{6\,{x}^{6}}\sqrt{b{x}^{3}+a}}-{\frac{b}{12\,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 = 1.48214, size = 316, normalized size = 4.45 \begin{align*} \left [\frac{\sqrt{a} b^{2} x^{6} \log \left (\frac{b x^{3} + 2 \, \sqrt{b x^{3} + a} \sqrt{a} + 2 \, a}{x^{3}}\right ) - 2 \,{\left (a b x^{3} + 2 \, a^{2}\right )} \sqrt{b x^{3} + a}}{24 \, a^{2} x^{6}}, -\frac{\sqrt{-a} b^{2} x^{6} \arctan \left (\frac{\sqrt{b x^{3} + a} \sqrt{-a}}{a}\right ) +{\left (a b x^{3} + 2 \, a^{2}\right )} \sqrt{b x^{3} + a}}{12 \, a^{2} x^{6}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.03743, size = 100, normalized size = 1.41 \begin{align*} - \frac{a}{6 \sqrt{b} x^{\frac{15}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} - \frac{\sqrt{b}}{4 x^{\frac{9}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} - \frac{b^{\frac{3}{2}}}{12 a x^{\frac{3}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} + \frac{b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{12 a^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26567, size = 84, normalized size = 1.18 \begin{align*} -\frac{1}{12} \, b^{2}{\left (\frac{\arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{{\left (b x^{3} + a\right )}^{\frac{3}{2}} + \sqrt{b x^{3} + a} a}{a b^{2} x^{6}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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