Optimal. Leaf size=47 \[ -\frac {\sqrt {a+b x^3}}{3 x^3}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 \sqrt {a}} \]
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Rubi [A] time = 0.03, antiderivative size = 47, normalized size of antiderivative = 1.00, 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, 47, 63, 208} \[ -\frac {\sqrt {a+b x^3}}{3 x^3}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 \sqrt {a}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rule 266
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x^3}}{x^4} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x^2} \, dx,x,x^3\right )\\ &=-\frac {\sqrt {a+b x^3}}{3 x^3}+\frac {1}{6} b \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^3\right )\\ &=-\frac {\sqrt {a+b x^3}}{3 x^3}+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^3}\right )\\ &=-\frac {\sqrt {a+b x^3}}{3 x^3}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 \sqrt {a}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 59, normalized size = 1.26 \[ -\frac {b x^3 \sqrt {\frac {b x^3}{a}+1} \tanh ^{-1}\left (\sqrt {\frac {b x^3}{a}+1}\right )+a+b x^3}{3 x^3 \sqrt {a+b x^3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 108, normalized size = 2.30 \[ \left [\frac {\sqrt {a} b x^{3} \log \left (\frac {b x^{3} - 2 \, \sqrt {b x^{3} + a} \sqrt {a} + 2 \, a}{x^{3}}\right ) - 2 \, \sqrt {b x^{3} + a} a}{6 \, a x^{3}}, \frac {\sqrt {-a} b x^{3} \arctan \left (\frac {\sqrt {b x^{3} + a} \sqrt {-a}}{a}\right ) - \sqrt {b x^{3} + a} a}{3 \, a x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 46, normalized size = 0.98 \[ \frac {\frac {b^{2} \arctan \left (\frac {\sqrt {b x^{3} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {\sqrt {b x^{3} + a} b}{x^{3}}}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 36, normalized size = 0.77 \[ -\frac {b \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{3 \sqrt {a}}-\frac {\sqrt {b \,x^{3}+a}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.94, size = 53, normalized size = 1.13 \[ \frac {b \log \left (\frac {\sqrt {b x^{3} + a} - \sqrt {a}}{\sqrt {b x^{3} + a} + \sqrt {a}}\right )}{6 \, \sqrt {a}} - \frac {\sqrt {b x^{3} + a}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.26, size = 56, normalized size = 1.19 \[ \frac {b\,\ln \left (\frac {{\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )}^3\,\left (\sqrt {b\,x^3+a}+\sqrt {a}\right )}{x^6}\right )}{6\,\sqrt {a}}-\frac {\sqrt {b\,x^3+a}}{3\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.84, size = 49, normalized size = 1.04 \[ - \frac {\sqrt {b} \sqrt {\frac {a}{b x^{3}} + 1}}{3 x^{\frac {3}{2}}} - \frac {b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{3 \sqrt {a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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