Optimal. Leaf size=107 \[ -\frac {\left (a+b x^3\right )^{2/3}}{3 x^3}+\frac {2 b \tan ^{-1}\left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} \sqrt [3]{a}}-\frac {b \log (x)}{3 \sqrt [3]{a}}+\frac {b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{3 \sqrt [3]{a}} \]
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Rubi [A]
time = 0.05, antiderivative size = 107, 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, 43, 57,
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} \sqrt [3]{a}}-\frac {\left (a+b x^3\right )^{2/3}}{3 x^3}+\frac {b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{3 \sqrt [3]{a}}-\frac {b \log (x)}{3 \sqrt [3]{a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 43
Rule 57
Rule 210
Rule 272
Rule 631
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^{2/3}}{x^4} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {(a+b x)^{2/3}}{x^2} \, dx,x,x^3\right )\\ &=-\frac {\left (a+b x^3\right )^{2/3}}{3 x^3}+\frac {1}{9} (2 b) \text {Subst}\left (\int \frac {1}{x \sqrt [3]{a+b x}} \, dx,x,x^3\right )\\ &=-\frac {\left (a+b x^3\right )^{2/3}}{3 x^3}-\frac {b \log (x)}{3 \sqrt [3]{a}}+\frac {1}{3} 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 )-\frac {b \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{3 \sqrt [3]{a}}\\ &=-\frac {\left (a+b x^3\right )^{2/3}}{3 x^3}-\frac {b \log (x)}{3 \sqrt [3]{a}}+\frac {b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{3 \sqrt [3]{a}}-\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 \sqrt [3]{a}}\\ &=-\frac {\left (a+b x^3\right )^{2/3}}{3 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} \sqrt [3]{a}}-\frac {b \log (x)}{3 \sqrt [3]{a}}+\frac {b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{3 \sqrt [3]{a}}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 136, normalized size = 1.27 \begin {gather*} \frac {-3 \sqrt [3]{a} \left (a+b x^3\right )^{2/3}+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 \sqrt [3]{a} x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{4}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 103, normalized size = 0.96 \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 {1}{3}}} - \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 {1}{3}}} + \frac {2 \, b \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{9 \, a^{\frac {1}{3}}} - \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 290, normalized size = 2.71 \begin {gather*} \left [\frac {3 \, \sqrt {\frac {1}{3}} a b x^{3} \sqrt {-\frac {1}{a^{\frac {2}{3}}}} \log \left (\frac {2 \, b x^{3} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} a^{\frac {2}{3}} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} a - a^{\frac {4}{3}}\right )} \sqrt {-\frac {1}{a^{\frac {2}{3}}}} - 3 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {2}{3}} + 3 \, a}{x^{3}}\right ) - a^{\frac {2}{3}} b x^{3} \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 ) + 2 \, a^{\frac {2}{3}} b x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) - 3 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} a}{9 \, a x^{3}}, \frac {6 \, \sqrt {\frac {1}{3}} a^{\frac {2}{3}} b x^{3} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{a^{\frac {1}{3}}}\right ) - a^{\frac {2}{3}} b x^{3} \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 ) + 2 \, a^{\frac {2}{3}} b x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) - 3 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} a}{9 \, a x^{3}}\right ] \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 = 0.61, size = 39, normalized size = 0.36 \begin {gather*} - \frac {b^{\frac {2}{3}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{3}}} \right )}}{3 x \Gamma \left (\frac {4}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.78, size = 116, normalized size = 1.08 \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 {1}{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 {1}{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 {1}{3}}} - \frac {3 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} b}{x^{3}}}{9 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.19, size = 137, normalized size = 1.28 \begin {gather*} \frac {2\,b\,\ln \left (\frac {4\,a^{1/3}\,b^2}{9}-\frac {4\,b^2\,{\left (b\,x^3+a\right )}^{1/3}}{9}\right )}{9\,a^{1/3}}-\frac {{\left (b\,x^3+a\right )}^{2/3}}{3\,x^3}-\frac {\ln \left (\frac {a^{1/3}\,{\left (b-\sqrt {3}\,b\,1{}\mathrm {i}\right )}^2}{9}-\frac {4\,b^2\,{\left (b\,x^3+a\right )}^{1/3}}{9}\right )\,\left (b-\sqrt {3}\,b\,1{}\mathrm {i}\right )}{9\,a^{1/3}}-\frac {\ln \left (\frac {a^{1/3}\,{\left (b+\sqrt {3}\,b\,1{}\mathrm {i}\right )}^2}{9}-\frac {4\,b^2\,{\left (b\,x^3+a\right )}^{1/3}}{9}\right )\,\left (b+\sqrt {3}\,b\,1{}\mathrm {i}\right )}{9\,a^{1/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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