Optimal. Leaf size=55 \[ \frac {\sqrt {a x^3+b} \left (2 a x^3+b\right )}{3 x^3}-\frac {1}{3} a \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {a x^3+b}}{\sqrt {b}}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 63, normalized size of antiderivative = 1.15, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {446, 78, 50, 63, 208} \begin {gather*} \frac {\left (a x^3+b\right )^{3/2}}{3 x^3}+\frac {1}{3} a \sqrt {a x^3+b}-\frac {1}{3} a \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {a x^3+b}}{\sqrt {b}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 78
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (-b+a x^3\right ) \sqrt {b+a x^3}}{x^4} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {(-b+a x) \sqrt {b+a x}}{x^2} \, dx,x,x^3\right )\\ &=\frac {\left (b+a x^3\right )^{3/2}}{3 x^3}+\frac {1}{6} a \operatorname {Subst}\left (\int \frac {\sqrt {b+a x}}{x} \, dx,x,x^3\right )\\ &=\frac {1}{3} a \sqrt {b+a x^3}+\frac {\left (b+a x^3\right )^{3/2}}{3 x^3}+\frac {1}{6} (a b) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {b+a x}} \, dx,x,x^3\right )\\ &=\frac {1}{3} a \sqrt {b+a x^3}+\frac {\left (b+a x^3\right )^{3/2}}{3 x^3}+\frac {1}{3} b \operatorname {Subst}\left (\int \frac {1}{-\frac {b}{a}+\frac {x^2}{a}} \, dx,x,\sqrt {b+a x^3}\right )\\ &=\frac {1}{3} a \sqrt {b+a x^3}+\frac {\left (b+a x^3\right )^{3/2}}{3 x^3}-\frac {1}{3} a \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b+a x^3}}{\sqrt {b}}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 54, normalized size = 0.98 \begin {gather*} \frac {1}{3} \left (\frac {\sqrt {a x^3+b} \left (2 a x^3+b\right )}{x^3}-a \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {a x^3+b}}{\sqrt {b}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.07, size = 55, normalized size = 1.00 \begin {gather*} \frac {\sqrt {b+a x^3} \left (b+2 a x^3\right )}{3 x^3}-\frac {1}{3} a \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b+a x^3}}{\sqrt {b}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 115, normalized size = 2.09 \begin {gather*} \left [\frac {a \sqrt {b} x^{3} \log \left (\frac {a x^{3} - 2 \, \sqrt {a x^{3} + b} \sqrt {b} + 2 \, b}{x^{3}}\right ) + 2 \, {\left (2 \, a x^{3} + b\right )} \sqrt {a x^{3} + b}}{6 \, x^{3}}, \frac {a \sqrt {-b} x^{3} \arctan \left (\frac {\sqrt {a x^{3} + b} \sqrt {-b}}{b}\right ) + {\left (2 \, a x^{3} + b\right )} \sqrt {a x^{3} + b}}{3 \, x^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 61, normalized size = 1.11 \begin {gather*} \frac {\frac {a^{2} b \arctan \left (\frac {\sqrt {a x^{3} + b}}{\sqrt {-b}}\right )}{\sqrt {-b}} + 2 \, \sqrt {a x^{3} + b} a^{2} + \frac {\sqrt {a x^{3} + b} a b}{x^{3}}}{3 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 44, normalized size = 0.80
method | result | size |
risch | \(\frac {\sqrt {a \,x^{3}+b}\, \left (2 a \,x^{3}+b \right )}{3 x^{3}}-\frac {a \sqrt {b}\, \arctanh \left (\frac {\sqrt {a \,x^{3}+b}}{\sqrt {b}}\right )}{3}\) | \(44\) |
elliptic | \(\frac {b \sqrt {a \,x^{3}+b}}{3 x^{3}}+\frac {2 a \sqrt {a \,x^{3}+b}}{3}-\frac {a \sqrt {b}\, \arctanh \left (\frac {\sqrt {a \,x^{3}+b}}{\sqrt {b}}\right )}{3}\) | \(49\) |
default | \(-b \left (-\frac {\sqrt {a \,x^{3}+b}}{3 x^{3}}-\frac {a \arctanh \left (\frac {\sqrt {a \,x^{3}+b}}{\sqrt {b}}\right )}{3 \sqrt {b}}\right )+a \left (\frac {2 \sqrt {a \,x^{3}+b}}{3}-\frac {2 \sqrt {b}\, \arctanh \left (\frac {\sqrt {a \,x^{3}+b}}{\sqrt {b}}\right )}{3}\right )\) | \(73\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 107, normalized size = 1.95 \begin {gather*} \frac {1}{3} \, {\left (\sqrt {b} \log \left (\frac {\sqrt {a x^{3} + b} - \sqrt {b}}{\sqrt {a x^{3} + b} + \sqrt {b}}\right ) + 2 \, \sqrt {a x^{3} + b}\right )} a - \frac {1}{6} \, {\left (\frac {a \log \left (\frac {\sqrt {a x^{3} + b} - \sqrt {b}}{\sqrt {a x^{3} + b} + \sqrt {b}}\right )}{\sqrt {b}} - \frac {2 \, \sqrt {a x^{3} + b}}{x^{3}}\right )} b \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.81, size = 69, normalized size = 1.25 \begin {gather*} \frac {2\,a\,\sqrt {a\,x^3+b}}{3}+\frac {b\,\sqrt {a\,x^3+b}}{3\,x^3}+\frac {a\,\sqrt {b}\,\ln \left (\frac {{\left (\sqrt {a\,x^3+b}-\sqrt {b}\right )}^3\,\left (\sqrt {a\,x^3+b}+\sqrt {b}\right )}{x^6}\right )}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 25.80, size = 102, normalized size = 1.85 \begin {gather*} \frac {2 a^{\frac {3}{2}} x^{\frac {3}{2}}}{3 \sqrt {1 + \frac {b}{a x^{3}}}} + \frac {\sqrt {a} b \sqrt {1 + \frac {b}{a x^{3}}}}{3 x^{\frac {3}{2}}} + \frac {2 \sqrt {a} b}{3 x^{\frac {3}{2}} \sqrt {1 + \frac {b}{a x^{3}}}} - \frac {a \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} x^{\frac {3}{2}}} \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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