Optimal. Leaf size=110 \[ -\frac {\sqrt [3]{a+b x^3}}{3 a x^3}+\frac {2 b \tan ^{-1}\left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3}}+\frac {b \log (x)}{3 a^{5/3}}-\frac {b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{3 a^{5/3}} \]
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Rubi [A]
time = 0.05, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {272, 44, 59,
631, 210, 31} \begin {gather*} \frac {2 b \text {ArcTan}\left (\frac {2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3}}-\frac {b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{3 a^{5/3}}+\frac {b \log (x)}{3 a^{5/3}}-\frac {\sqrt [3]{a+b x^3}}{3 a x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 44
Rule 59
Rule 210
Rule 272
Rule 631
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (a+b x^3\right )^{2/3}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{x^2 (a+b x)^{2/3}} \, dx,x,x^3\right )\\ &=-\frac {\sqrt [3]{a+b x^3}}{3 a x^3}-\frac {(2 b) \text {Subst}\left (\int \frac {1}{x (a+b x)^{2/3}} \, dx,x,x^3\right )}{9 a}\\ &=-\frac {\sqrt [3]{a+b x^3}}{3 a x^3}+\frac {b \log (x)}{3 a^{5/3}}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{3 a^{5/3}}+\frac {b \text {Subst}\left (\int \frac {1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x^3}\right )}{3 a^{4/3}}\\ &=-\frac {\sqrt [3]{a+b x^3}}{3 a x^3}+\frac {b \log (x)}{3 a^{5/3}}-\frac {b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{3 a^{5/3}}-\frac {(2 b) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}\right )}{3 a^{5/3}}\\ &=-\frac {\sqrt [3]{a+b x^3}}{3 a x^3}+\frac {2 b \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{3 \sqrt {3} a^{5/3}}+\frac {b \log (x)}{3 a^{5/3}}-\frac {b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{3 a^{5/3}}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 135, normalized size = 1.23 \begin {gather*} \frac {-3 a^{2/3} \sqrt [3]{a+b x^3}+2 \sqrt {3} b x^3 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 b x^3 \log \left (-\sqrt [3]{a}+\sqrt [3]{a+b x^3}\right )+b x^3 \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{9 a^{5/3} x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{4} \left (b \,x^{3}+a \right )^{\frac {2}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 118, normalized size = 1.07 \begin {gather*} \frac {2 \, \sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{9 \, a^{\frac {5}{3}}} - \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} b}{3 \, {\left ({\left (b x^{3} + a\right )} a - a^{2}\right )}} + \frac {b \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{9 \, a^{\frac {5}{3}}} - \frac {2 \, b \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{9 \, a^{\frac {5}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 182 vs.
\(2 (81) = 162\).
time = 0.37, size = 182, normalized size = 1.65 \begin {gather*} \frac {2 \, \sqrt {3} a b x^{3} \sqrt {-\left (-a^{2}\right )^{\frac {1}{3}}} \arctan \left (-\frac {{\left (\sqrt {3} \left (-a^{2}\right )^{\frac {1}{3}} a - 2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-a^{2}\right )^{\frac {2}{3}}\right )} \sqrt {-\left (-a^{2}\right )^{\frac {1}{3}}}}{3 \, a^{2}}\right ) + \left (-a^{2}\right )^{\frac {2}{3}} b x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} a - \left (-a^{2}\right )^{\frac {1}{3}} a + {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-a^{2}\right )^{\frac {2}{3}}\right ) - 2 \, \left (-a^{2}\right )^{\frac {2}{3}} b x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} a - \left (-a^{2}\right )^{\frac {2}{3}}\right ) - 3 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{2}}{9 \, a^{3} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.68, size = 39, normalized size = 0.35 \begin {gather*} - \frac {\Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{3}}} \right )}}{3 b^{\frac {2}{3}} x^{5} \Gamma \left (\frac {8}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.73, size = 118, normalized size = 1.07 \begin {gather*} \frac {\frac {2 \, \sqrt {3} b^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {5}{3}}} + \frac {b^{2} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{a^{\frac {5}{3}}} - \frac {2 \, b^{2} \log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{a^{\frac {5}{3}}} - \frac {3 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} b}{a x^{3}}}{9 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.32, size = 128, normalized size = 1.16 \begin {gather*} \frac {\ln \left (\frac {b-\sqrt {3}\,b\,1{}\mathrm {i}}{a^{2/3}}+\frac {2\,b\,{\left (b\,x^3+a\right )}^{1/3}}{a}\right )\,\left (b-\sqrt {3}\,b\,1{}\mathrm {i}\right )}{9\,a^{5/3}}+\frac {\ln \left (\frac {b+\sqrt {3}\,b\,1{}\mathrm {i}}{a^{2/3}}+\frac {2\,b\,{\left (b\,x^3+a\right )}^{1/3}}{a}\right )\,\left (b+\sqrt {3}\,b\,1{}\mathrm {i}\right )}{9\,a^{5/3}}-\frac {2\,b\,\ln \left ({\left (b\,x^3+a\right )}^{1/3}-a^{1/3}\right )}{9\,a^{5/3}}-\frac {{\left (b\,x^3+a\right )}^{1/3}}{3\,a\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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