Optimal. Leaf size=142 \[ \frac {\left (b-3 a^3 x^3\right ) \sqrt [3]{a^3 x^3+b}}{4 x^4}-\frac {1}{3} a^4 \log \left (\sqrt [3]{a^3 x^3+b}-a x\right )-\frac {a^4 \tan ^{-1}\left (\frac {\sqrt {3} a x}{2 \sqrt [3]{a^3 x^3+b}+a x}\right )}{\sqrt {3}}+\frac {1}{6} a^4 \log \left (a x \sqrt [3]{a^3 x^3+b}+\left (a^3 x^3+b\right )^{2/3}+a^2 x^2\right ) \]
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Rubi [A] time = 0.10, antiderivative size = 151, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 9, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {451, 277, 331, 292, 31, 634, 617, 204, 628} \begin {gather*} -\frac {a^3 \sqrt [3]{a^3 x^3+b}}{x}+\frac {\left (a^3 x^3+b\right )^{4/3}}{4 x^4}-\frac {1}{3} a^4 \log \left (1-\frac {a x}{\sqrt [3]{a^3 x^3+b}}\right )-\frac {a^4 \tan ^{-1}\left (\frac {\frac {2 a x}{\sqrt [3]{a^3 x^3+b}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{6} a^4 \log \left (\frac {a x}{\sqrt [3]{a^3 x^3+b}}+\frac {a^2 x^2}{\left (a^3 x^3+b\right )^{2/3}}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 277
Rule 292
Rule 331
Rule 451
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {\left (-b+a^3 x^3\right ) \sqrt [3]{b+a^3 x^3}}{x^5} \, dx &=\frac {\left (b+a^3 x^3\right )^{4/3}}{4 x^4}+a^3 \int \frac {\sqrt [3]{b+a^3 x^3}}{x^2} \, dx\\ &=-\frac {a^3 \sqrt [3]{b+a^3 x^3}}{x}+\frac {\left (b+a^3 x^3\right )^{4/3}}{4 x^4}+a^6 \int \frac {x}{\left (b+a^3 x^3\right )^{2/3}} \, dx\\ &=-\frac {a^3 \sqrt [3]{b+a^3 x^3}}{x}+\frac {\left (b+a^3 x^3\right )^{4/3}}{4 x^4}+a^6 \operatorname {Subst}\left (\int \frac {x}{1-a^3 x^3} \, dx,x,\frac {x}{\sqrt [3]{b+a^3 x^3}}\right )\\ &=-\frac {a^3 \sqrt [3]{b+a^3 x^3}}{x}+\frac {\left (b+a^3 x^3\right )^{4/3}}{4 x^4}+\frac {1}{3} a^5 \operatorname {Subst}\left (\int \frac {1}{1-a x} \, dx,x,\frac {x}{\sqrt [3]{b+a^3 x^3}}\right )-\frac {1}{3} a^5 \operatorname {Subst}\left (\int \frac {1-a x}{1+a x+a^2 x^2} \, dx,x,\frac {x}{\sqrt [3]{b+a^3 x^3}}\right )\\ &=-\frac {a^3 \sqrt [3]{b+a^3 x^3}}{x}+\frac {\left (b+a^3 x^3\right )^{4/3}}{4 x^4}-\frac {1}{3} a^4 \log \left (1-\frac {a x}{\sqrt [3]{b+a^3 x^3}}\right )+\frac {1}{6} a^4 \operatorname {Subst}\left (\int \frac {a+2 a^2 x}{1+a x+a^2 x^2} \, dx,x,\frac {x}{\sqrt [3]{b+a^3 x^3}}\right )-\frac {1}{2} a^5 \operatorname {Subst}\left (\int \frac {1}{1+a x+a^2 x^2} \, dx,x,\frac {x}{\sqrt [3]{b+a^3 x^3}}\right )\\ &=-\frac {a^3 \sqrt [3]{b+a^3 x^3}}{x}+\frac {\left (b+a^3 x^3\right )^{4/3}}{4 x^4}-\frac {1}{3} a^4 \log \left (1-\frac {a x}{\sqrt [3]{b+a^3 x^3}}\right )+\frac {1}{6} a^4 \log \left (1+\frac {a^2 x^2}{\left (b+a^3 x^3\right )^{2/3}}+\frac {a x}{\sqrt [3]{b+a^3 x^3}}\right )+a^4 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 a x}{\sqrt [3]{b+a^3 x^3}}\right )\\ &=-\frac {a^3 \sqrt [3]{b+a^3 x^3}}{x}+\frac {\left (b+a^3 x^3\right )^{4/3}}{4 x^4}-\frac {a^4 \tan ^{-1}\left (\frac {1+\frac {2 a x}{\sqrt [3]{b+a^3 x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{3} a^4 \log \left (1-\frac {a x}{\sqrt [3]{b+a^3 x^3}}\right )+\frac {1}{6} a^4 \log \left (1+\frac {a^2 x^2}{\left (b+a^3 x^3\right )^{2/3}}+\frac {a x}{\sqrt [3]{b+a^3 x^3}}\right )\\ \end {align*}
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Mathematica [C] time = 0.06, size = 74, normalized size = 0.52 \begin {gather*} \frac {\sqrt [3]{a^3 x^3+b} \left (-\frac {4 a^3 x^3 \, _2F_1\left (-\frac {1}{3},-\frac {1}{3};\frac {2}{3};-\frac {a^3 x^3}{b}\right )}{\sqrt [3]{\frac {a^3 x^3}{b}+1}}+a^3 x^3+b\right )}{4 x^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.26, size = 142, normalized size = 1.00 \begin {gather*} \frac {\left (b-3 a^3 x^3\right ) \sqrt [3]{b+a^3 x^3}}{4 x^4}-\frac {a^4 \tan ^{-1}\left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{b+a^3 x^3}}\right )}{\sqrt {3}}-\frac {1}{3} a^4 \log \left (-a x+\sqrt [3]{b+a^3 x^3}\right )+\frac {1}{6} a^4 \log \left (a^2 x^2+a x \sqrt [3]{b+a^3 x^3}+\left (b+a^3 x^3\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a^{3} x^{3} + b\right )}^{\frac {1}{3}} {\left (a^{3} x^{3} - b\right )}}{x^{5}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (a^{3} x^{3}-b \right ) \left (a^{3} x^{3}+b \right )^{\frac {1}{3}}}{x^{5}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 133, normalized size = 0.94 \begin {gather*} \frac {1}{6} \, {\left (2 \, \sqrt {3} a \arctan \left (\frac {\sqrt {3} {\left (a + \frac {2 \, {\left (a^{3} x^{3} + b\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, a}\right ) + a \log \left (a^{2} + \frac {{\left (a^{3} x^{3} + b\right )}^{\frac {1}{3}} a}{x} + \frac {{\left (a^{3} x^{3} + b\right )}^{\frac {2}{3}}}{x^{2}}\right ) - 2 \, a \log \left (-a + \frac {{\left (a^{3} x^{3} + b\right )}^{\frac {1}{3}}}{x}\right ) - \frac {6 \, {\left (a^{3} x^{3} + b\right )}^{\frac {1}{3}}}{x}\right )} a^{3} + \frac {{\left (a^{3} x^{3} + b\right )}^{\frac {4}{3}}}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.63, size = 66, normalized size = 0.46 \begin {gather*} \frac {{\left (a^3\,x^3+b\right )}^{4/3}}{4\,x^4}-\frac {a^3\,{\left (a^3\,x^3+b\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{3},-\frac {1}{3};\ \frac {2}{3};\ -\frac {a^3\,x^3}{b}\right )}{x\,{\left (\frac {a^3\,x^3}{b}+1\right )}^{1/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.63, size = 114, normalized size = 0.80 \begin {gather*} - \frac {a^{4} \sqrt [3]{1 + \frac {b}{a^{3} x^{3}}} \Gamma \left (- \frac {4}{3}\right )}{3 \Gamma \left (- \frac {1}{3}\right )} + \frac {a^{3} \sqrt [3]{b} \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {\frac {a^{3} x^{3} e^{i \pi }}{b}} \right )}}{3 x \Gamma \left (\frac {2}{3}\right )} - \frac {a b \sqrt [3]{1 + \frac {b}{a^{3} x^{3}}} \Gamma \left (- \frac {4}{3}\right )}{3 x^{3} \Gamma \left (- \frac {1}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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