Optimal. Leaf size=57 \[ \frac {2 C \tan ^{-1}\left (\frac {1-\sqrt [3]{-b} x}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{-b}}-\frac {C \log \left (\sqrt [3]{-b} x+2\right )}{\sqrt [3]{-b}} \]
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Rubi [A] time = 0.07, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1864, 31, 617, 204} \[ \frac {2 C \tan ^{-1}\left (\frac {1-\sqrt [3]{-b} x}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{-b}}-\frac {C \log \left (\sqrt [3]{-b} x+2\right )}{\sqrt [3]{-b}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 617
Rule 1864
Rubi steps
\begin {align*} \int \frac {8 C+(-b)^{2/3} C x^2}{-8+b x^3} \, dx &=-\frac {(2 C) \int \frac {1}{\frac {4}{(-b)^{2/3}}-\frac {2 x}{\sqrt [3]{-b}}+x^2} \, dx}{(-b)^{2/3}}-\frac {C \int \frac {1}{\frac {2}{\sqrt [3]{-b}}+x} \, dx}{\sqrt [3]{-b}}\\ &=-\frac {C \log \left (2+\sqrt [3]{-b} x\right )}{\sqrt [3]{-b}}-\frac {(2 C) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\sqrt [3]{-b} x\right )}{\sqrt [3]{-b}}\\ &=\frac {2 C \tan ^{-1}\left (\frac {1-\sqrt [3]{-b} x}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{-b}}-\frac {C \log \left (2+\sqrt [3]{-b} x\right )}{\sqrt [3]{-b}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 99, normalized size = 1.74 \[ \frac {C \left (-b^{2/3} \log \left (b^{2/3} x^2+2 \sqrt [3]{b} x+4\right )+2 b^{2/3} \log \left (2-\sqrt [3]{b} x\right )-2 \sqrt {3} b^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{b} x+1}{\sqrt {3}}\right )+(-b)^{2/3} \log \left (8-b x^3\right )\right )}{3 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 182, normalized size = 3.19 \[ \left [\frac {\sqrt {\frac {1}{3}} C b \sqrt {\frac {\left (-b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {b x^{3} - 6 \, \sqrt {\frac {1}{3}} {\left (b x^{2} - \left (-b\right )^{\frac {2}{3}} x + 2 \, \left (-b\right )^{\frac {1}{3}}\right )} \sqrt {\frac {\left (-b\right )^{\frac {1}{3}}}{b}} + 6 \, \left (-b\right )^{\frac {1}{3}} x + 4}{b x^{3} - 8}\right ) + C \left (-b\right )^{\frac {2}{3}} \log \left (b x - 2 \, \left (-b\right )^{\frac {2}{3}}\right )}{b}, -\frac {2 \, \sqrt {\frac {1}{3}} C b \sqrt {-\frac {\left (-b\right )^{\frac {1}{3}}}{b}} \arctan \left (\sqrt {\frac {1}{3}} {\left (\left (-b\right )^{\frac {2}{3}} x - \left (-b\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-b\right )^{\frac {1}{3}}}{b}}\right ) - C \left (-b\right )^{\frac {2}{3}} \log \left (b x - 2 \, \left (-b\right )^{\frac {2}{3}}\right )}{b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.31, size = 91, normalized size = 1.60 \[ -\frac {2 \, \sqrt {3} C {\left | b \right |}^{\frac {2}{3}} \arctan \left (\frac {1}{3} \, \sqrt {3} b^{\frac {1}{3}} {\left (x + \frac {1}{b^{\frac {1}{3}}}\right )}\right )}{3 \, b} + \frac {1}{3} \, {\left (\frac {C \left (-b\right )^{\frac {2}{3}}}{b} - \frac {C}{b^{\frac {1}{3}}}\right )} \log \left (x^{2} + \frac {2 \, x}{b^{\frac {1}{3}}} + \frac {4}{b^{\frac {2}{3}}}\right ) + \frac {{\left (2 \, C + \frac {C \left (-b\right )^{\frac {2}{3}}}{b^{\frac {2}{3}}}\right )} \log \left ({\left | x - \frac {2}{b^{\frac {1}{3}}} \right |}\right )}{3 \, b^{\frac {1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 122, normalized size = 2.14 \[ -\frac {8^{\frac {1}{3}} \sqrt {3}\, C \arctan \left (\frac {\sqrt {3}\, \left (\frac {8^{\frac {2}{3}} x}{4 \left (\frac {1}{b}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{3 \left (\frac {1}{b}\right )^{\frac {2}{3}} b}+\frac {8^{\frac {1}{3}} C \ln \left (x -8^{\frac {1}{3}} \left (\frac {1}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {1}{b}\right )^{\frac {2}{3}} b}-\frac {8^{\frac {1}{3}} C \ln \left (x^{2}+8^{\frac {1}{3}} \left (\frac {1}{b}\right )^{\frac {1}{3}} x +8^{\frac {2}{3}} \left (\frac {1}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {1}{b}\right )^{\frac {2}{3}} b}+\frac {\left (-b \right )^{\frac {2}{3}} C \ln \left (b \,x^{3}-8\right )}{3 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.94, size = 122, normalized size = 2.14 \[ \frac {{\left (C \left (-b\right )^{\frac {2}{3}} - C b^{\frac {2}{3}}\right )} \log \left (b^{\frac {2}{3}} x^{2} + 2 \, b^{\frac {1}{3}} x + 4\right )}{3 \, b} + \frac {{\left (C \left (-b\right )^{\frac {2}{3}} + 2 \, C b^{\frac {2}{3}}\right )} \log \left (\frac {b^{\frac {1}{3}} x - 2}{b^{\frac {1}{3}}}\right )}{3 \, b} + \frac {2 \, \sqrt {3} {\left (C \left (-b\right )^{\frac {2}{3}} b^{\frac {4}{3}} - {\left (C \left (-b\right )^{\frac {2}{3}} b^{\frac {1}{3}} + 3 \, C b\right )} b\right )} \arctan \left (\frac {\sqrt {3} {\left (b^{\frac {2}{3}} x + b^{\frac {1}{3}}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{9 \, b^{\frac {7}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.27, size = 176, normalized size = 3.09 \[ \sum _{k=1}^3\ln \left (\frac {8\,C^2}{{\left (-b\right )}^{5/3}}+\mathrm {root}\left (27\,b^3\,z^3-27\,C\,{\left (-b\right )}^{8/3}\,z^2-9\,C^2\,{\left (-b\right )}^{7/3}\,z-9\,C^3\,b^2,z,k\right )\,\left (-\frac {\mathrm {root}\left (27\,b^3\,z^3-27\,C\,{\left (-b\right )}^{8/3}\,z^2-9\,C^2\,{\left (-b\right )}^{7/3}\,z-9\,C^3\,b^2,z,k\right )\,72}{b}+\frac {48\,C}{{\left (-b\right )}^{4/3}}+\frac {24\,C\,x}{b}\right )-\frac {8\,C^2\,x}{{\left (-b\right )}^{4/3}}\right )\,\mathrm {root}\left (27\,b^3\,z^3-27\,C\,{\left (-b\right )}^{8/3}\,z^2-9\,C^2\,{\left (-b\right )}^{7/3}\,z-9\,C^3\,b^2,z,k\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.99, size = 58, normalized size = 1.02 \[ \operatorname {RootSum} {\left (3 t^{3} b^{2} - 3 t^{2} C b \left (- b\right )^{\frac {2}{3}} + t C^{2} \left (- b\right )^{\frac {4}{3}} - C^{3} b, \left (t \mapsto t \log {\left (- \frac {3 t}{C} + x + \frac {\left (- b\right )^{\frac {2}{3}}}{b} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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