Optimal. Leaf size=107 \[ \frac {b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{6 a^{2/3}}-\frac {b \tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{2/3}}-\frac {b \log (x)}{6 a^{2/3}}-\frac {\sqrt [3]{a+b x^3}}{3 x^3} \]
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Rubi [A] time = 0.06, 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 = {266, 47, 57, 617, 204, 31} \[ \frac {b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{6 a^{2/3}}-\frac {b \tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{2/3}}-\frac {b \log (x)}{6 a^{2/3}}-\frac {\sqrt [3]{a+b x^3}}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 31
Rule 47
Rule 57
Rule 204
Rule 266
Rule 617
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{a+b x^3}}{x^4} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt [3]{a+b x}}{x^2} \, dx,x,x^3\right )\\ &=-\frac {\sqrt [3]{a+b x^3}}{3 x^3}+\frac {1}{9} b \operatorname {Subst}\left (\int \frac {1}{x (a+b x)^{2/3}} \, dx,x,x^3\right )\\ &=-\frac {\sqrt [3]{a+b x^3}}{3 x^3}-\frac {b \log (x)}{6 a^{2/3}}-\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{6 a^{2/3}}-\frac {b \operatorname {Subst}\left (\int \frac {1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x^3}\right )}{6 \sqrt [3]{a}}\\ &=-\frac {\sqrt [3]{a+b x^3}}{3 x^3}-\frac {b \log (x)}{6 a^{2/3}}+\frac {b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{6 a^{2/3}}+\frac {b \operatorname {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^{2/3}}\\ &=-\frac {\sqrt [3]{a+b x^3}}{3 x^3}-\frac {b \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{3 \sqrt {3} a^{2/3}}-\frac {b \log (x)}{6 a^{2/3}}+\frac {b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{6 a^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 37, normalized size = 0.35 \[ \frac {b \left (a+b x^3\right )^{4/3} \, _2F_1\left (\frac {4}{3},2;\frac {7}{3};\frac {b x^3}{a}+1\right )}{4 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 155, normalized size = 1.45 \[ -\frac {2 \, \sqrt {3} {\left (a^{2}\right )}^{\frac {1}{6}} a b x^{3} \arctan \left (\frac {{\left (a^{2}\right )}^{\frac {1}{6}} {\left (\sqrt {3} {\left (a^{2}\right )}^{\frac {1}{3}} a + 2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (a^{2}\right )}^{\frac {2}{3}}\right )}}{3 \, a^{2}}\right ) + {\left (a^{2}\right )}^{\frac {2}{3}} b x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} a + {\left (a^{2}\right )}^{\frac {1}{3}} a + {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (a^{2}\right )}^{\frac {2}{3}}\right ) - 2 \, {\left (a^{2}\right )}^{\frac {2}{3}} b x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} a - {\left (a^{2}\right )}^{\frac {2}{3}}\right ) + 6 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{2}}{18 \, a^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.69, size = 115, normalized size = 1.07 \[ -\frac {\frac {2 \, \sqrt {3} b^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {2}{3}}} + \frac {b^{2} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{a^{\frac {2}{3}}} - \frac {2 \, b^{2} \log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{a^{\frac {2}{3}}} + \frac {6 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} b}{x^{3}}}{18 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.97, size = 103, normalized size = 0.96 \[ -\frac {\sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{9 \, a^{\frac {2}{3}}} - \frac {b \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{18 \, a^{\frac {2}{3}}} + \frac {b \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{9 \, a^{\frac {2}{3}}} - \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.21, size = 122, normalized size = 1.14 \[ \frac {b\,\ln \left (b\,{\left (b\,x^3+a\right )}^{1/3}-a^{1/3}\,b\right )}{9\,a^{2/3}}-\frac {{\left (b\,x^3+a\right )}^{1/3}}{3\,x^3}-\frac {\ln \left (\frac {a^{1/3}\,\left (b-\sqrt {3}\,b\,1{}\mathrm {i}\right )}{2}+b\,{\left (b\,x^3+a\right )}^{1/3}\right )\,\left (b-\sqrt {3}\,b\,1{}\mathrm {i}\right )}{18\,a^{2/3}}-\frac {\ln \left (\frac {a^{1/3}\,\left (b+\sqrt {3}\,b\,1{}\mathrm {i}\right )}{2}+b\,{\left (b\,x^3+a\right )}^{1/3}\right )\,\left (b+\sqrt {3}\,b\,1{}\mathrm {i}\right )}{18\,a^{2/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.62, size = 41, normalized size = 0.38 \[ - \frac {\sqrt [3]{b} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{3}}} \right )}}{3 x^{2} \Gamma \left (\frac {5}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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