Optimal. Leaf size=54 \[ \frac {C \log \left (\sqrt [3]{-\frac {a}{b}}-x\right )}{b}-\frac {2 C \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{-\frac {a}{b}}}+1}{\sqrt {3}}\right )}{\sqrt {3} b} \]
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Rubi [A] time = 0.06, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1869, 31, 617, 204} \[ \frac {C \log \left (\sqrt [3]{-\frac {a}{b}}-x\right )}{b}-\frac {2 C \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{-\frac {a}{b}}}+1}{\sqrt {3}}\right )}{\sqrt {3} b} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 617
Rule 1869
Rubi steps
\begin {align*} \int \frac {2 \left (-\frac {a}{b}\right )^{2/3} C+C x^2}{a+b x^3} \, dx &=-\frac {C \int \frac {1}{\sqrt [3]{-\frac {a}{b}}-x} \, dx}{b}-\frac {\left (\sqrt [3]{-\frac {a}{b}} C\right ) \int \frac {1}{\left (-\frac {a}{b}\right )^{2/3}+\sqrt [3]{-\frac {a}{b}} x+x^2} \, dx}{b}\\ &=\frac {C \log \left (\sqrt [3]{-\frac {a}{b}}-x\right )}{b}+\frac {(2 C) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x}{\sqrt [3]{-\frac {a}{b}}}\right )}{b}\\ &=-\frac {2 C \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-\frac {a}{b}}}}{\sqrt {3}}\right )}{\sqrt {3} b}+\frac {C \log \left (\sqrt [3]{-\frac {a}{b}}-x\right )}{b}\\ \end {align*}
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Mathematica [B] time = 0.07, size = 149, normalized size = 2.76 \[ \frac {C \left (-b^{2/3} \left (-\frac {a}{b}\right )^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+a^{2/3} \log \left (a+b x^3\right )+2 b^{2/3} \left (-\frac {a}{b}\right )^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 \sqrt {3} b^{2/3} \left (-\frac {a}{b}\right )^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )\right )}{3 a^{2/3} b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 56, normalized size = 1.04 \[ \frac {2 \, \sqrt {3} C \arctan \left (\frac {2 \, \sqrt {3} b x \left (-\frac {a}{b}\right )^{\frac {2}{3}} - \sqrt {3} a}{3 \, a}\right ) + 3 \, C \log \left (x - \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 91, normalized size = 1.69 \[ -\frac {2 \, \sqrt {3} C \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b} - \frac {{\left (C b^{2} \left (-\frac {a}{b}\right )^{\frac {2}{3}} + 2 \, \left (-a b^{2}\right )^{\frac {2}{3}} C\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 132, normalized size = 2.44 \[ \frac {2 \left (-\frac {a}{b}\right )^{\frac {2}{3}} \sqrt {3}\, C \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b}+\frac {2 \left (-\frac {a}{b}\right )^{\frac {2}{3}} C \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b}-\frac {\left (-\frac {a}{b}\right )^{\frac {2}{3}} C \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b}+\frac {C \ln \left (b \,x^{3}+a \right )}{3 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 3.15, size = 168, normalized size = 3.11 \[ -\frac {2 \, \sqrt {3} {\left (C a - {\left (3 \, C \left (\frac {a}{b}\right )^{\frac {1}{3}} \left (-\frac {a}{b}\right )^{\frac {2}{3}} + \frac {C a}{b}\right )} b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a b} + \frac {{\left (C \left (\frac {a}{b}\right )^{\frac {2}{3}} - C \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{3 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (C \left (\frac {a}{b}\right )^{\frac {2}{3}} + 2 \, C \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.27, size = 173, normalized size = 3.20 \[ \sum _{k=1}^3\ln \left (-\frac {\left (C-\mathrm {root}\left (27\,a^2\,b^3\,z^3-27\,C\,a^2\,b^2\,z^2+9\,C^2\,a^2\,b\,z-9\,C^3\,a^2,z,k\right )\,b\,3\right )\,\left (-C\,a+\mathrm {root}\left (27\,a^2\,b^3\,z^3-27\,C\,a^2\,b^2\,z^2+9\,C^2\,a^2\,b\,z-9\,C^3\,a^2,z,k\right )\,a\,b\,3+2\,C\,b\,x\,{\left (-\frac {a}{b}\right )}^{2/3}\right )}{b^3}\right )\,\mathrm {root}\left (27\,a^2\,b^3\,z^3-27\,C\,a^2\,b^2\,z^2+9\,C^2\,a^2\,b\,z-9\,C^3\,a^2,z,k\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.77, size = 109, normalized size = 2.02 \[ \frac {C \left (\log {\left (\frac {a}{b \left (- \frac {a}{b}\right )^{\frac {2}{3}}} + x \right )} - \frac {\sqrt {3} i \log {\left (- \frac {a}{2 b \left (- \frac {a}{b}\right )^{\frac {2}{3}}} - \frac {\sqrt {3} i a}{2 b \left (- \frac {a}{b}\right )^{\frac {2}{3}}} + x \right )}}{3} + \frac {\sqrt {3} i \log {\left (- \frac {a}{2 b \left (- \frac {a}{b}\right )^{\frac {2}{3}}} + \frac {\sqrt {3} i a}{2 b \left (- \frac {a}{b}\right )^{\frac {2}{3}}} + x \right )}}{3}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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