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 = 0.04, antiderivative size = 58, normalized size of antiderivative = 1.00, 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 47
Rule 50
Rule 63
Rule 208
Rule 266
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*}
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Mathematica [C] time = 0.01, 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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fricas [A] time = 0.82, size = 121, normalized size = 2.09 \[ \left [\frac {3 \, \sqrt {a} b x^{3} \log \left (\frac {b x^{3} - 2 \, \sqrt {b x^{3} + a} \sqrt {a} + 2 \, a}{x^{3}}\right ) + 2 \, {\left (2 \, b x^{3} - a\right )} \sqrt {b x^{3} + a}}{6 \, x^{3}}, \frac {3 \, \sqrt {-a} b x^{3} \arctan \left (\frac {\sqrt {b x^{3} + a} \sqrt {-a}}{a}\right ) + {\left (2 \, b x^{3} - a\right )} \sqrt {b x^{3} + a}}{3 \, x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 63, normalized size = 1.09 \[ \frac {\frac {3 \, a b^{2} \arctan \left (\frac {\sqrt {b x^{3} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + 2 \, \sqrt {b x^{3} + a} b^{2} - \frac {\sqrt {b x^{3} + a} a b}{x^{3}}}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 49, normalized size = 0.84 \[ -\sqrt {a}\, b \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )+\frac {2 \sqrt {b \,x^{3}+a}\, b}{3}-\frac {\sqrt {b \,x^{3}+a}\, a}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.99, size = 66, normalized size = 1.14 \[ \frac {1}{2} \, \sqrt {a} b \log \left (\frac {\sqrt {b x^{3} + a} - \sqrt {a}}{\sqrt {b x^{3} + a} + \sqrt {a}}\right ) + \frac {2}{3} \, \sqrt {b x^{3} + a} b - \frac {\sqrt {b x^{3} + a} a}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.31, size = 69, normalized size = 1.19 \[ \frac {2\,b\,\sqrt {b\,x^3+a}}{3}-\frac {a\,\sqrt {b\,x^3+a}}{3\,x^3}+\frac {\sqrt {a}\,b\,\ln \left (\frac {{\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )}^3\,\left (\sqrt {b\,x^3+a}+\sqrt {a}\right )}{x^6}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 5.25, size = 100, normalized size = 1.72 \[ - \sqrt {a} b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )} - \frac {a^{2}}{3 \sqrt {b} x^{\frac {9}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} + \frac {a \sqrt {b}}{3 x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} + \frac {2 b^{\frac {3}{2}} x^{\frac {3}{2}}}{3 \sqrt {\frac {a}{b x^{3}} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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