Optimal. Leaf size=47 \[ \frac{C \tan ^{-1}\left (\frac{1-\frac{4 x}{\sqrt [3]{-a}}}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{1}{4} C \log \left (\sqrt [3]{-a}+2 x\right ) \]
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Rubi [A] time = 0.061084, antiderivative size = 47, normalized size of antiderivative = 1., 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} \[ \frac{C \tan ^{-1}\left (\frac{1-\frac{4 x}{\sqrt [3]{-a}}}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{1}{4} C \log \left (\sqrt [3]{-a}+2 x\right ) \]
Antiderivative was successfully verified.
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Rule 1864
Rule 31
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{(-a)^{2/3} C+2 C x^2}{a-8 x^3} \, dx &=-\left (\frac{1}{4} C \int \frac{1}{\frac{\sqrt [3]{-a}}{2}+x} \, dx\right )-\frac{1}{8} \left (\sqrt [3]{-a} C\right ) \int \frac{1}{\frac{1}{4} (-a)^{2/3}-\frac{1}{2} \sqrt [3]{-a} x+x^2} \, dx\\ &=-\frac{1}{4} C \log \left (\sqrt [3]{-a}+2 x\right )-\frac{1}{2} C \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{4 x}{\sqrt [3]{-a}}\right )\\ &=\frac{C \tan ^{-1}\left (\frac{1-\frac{4 x}{\sqrt [3]{-a}}}{\sqrt{3}}\right )}{2 \sqrt{3}}-\frac{1}{4} C \log \left (\sqrt [3]{-a}+2 x\right )\\ \end{align*}
Mathematica [B] time = 0.0390344, size = 106, normalized size = 2.26 \[ \frac{C \left ((-a)^{2/3} \log \left (a^{2/3}+2 \sqrt [3]{a} x+4 x^2\right )-a^{2/3} \log \left (8 x^3-a\right )-2 (-a)^{2/3} \log \left (\sqrt [3]{a}-2 x\right )+2 \sqrt{3} (-a)^{2/3} \tan ^{-1}\left (\frac{\frac{4 x}{\sqrt [3]{a}}+1}{\sqrt{3}}\right )\right )}{12 a^{2/3}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 110, normalized size = 2.3 \begin{align*} -{\frac{C{8}^{{\frac{2}{3}}}}{24} \left ( -a \right ) ^{{\frac{2}{3}}}\ln \left ( x-{\frac{{8}^{{\frac{2}{3}}}}{8}\sqrt [3]{a}} \right ){a}^{-{\frac{2}{3}}}}+{\frac{C{8}^{{\frac{2}{3}}}}{48} \left ( -a \right ) ^{{\frac{2}{3}}}\ln \left ({x}^{2}+{\frac{{8}^{{\frac{2}{3}}}x}{8}\sqrt [3]{a}}+{\frac{\sqrt [3]{8}}{8}{a}^{{\frac{2}{3}}}} \right ){a}^{-{\frac{2}{3}}}}+{\frac{C{8}^{{\frac{2}{3}}}\sqrt{3}}{24} \left ( -a \right ) ^{{\frac{2}{3}}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{\sqrt [3]{8}x}{\sqrt [3]{a}}}+1 \right ) } \right ){a}^{-{\frac{2}{3}}}}-{\frac{C\ln \left ( 8\,{x}^{3}-a \right ) }{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.02475, size = 130, normalized size = 2.77 \begin{align*} \frac{1}{6} \, \sqrt{3} C \arctan \left (\frac{4 \, \sqrt{3} \left (-a\right )^{\frac{2}{3}} x + \sqrt{3} a}{3 \, a}\right ) - \frac{1}{4} \, C \log \left (2 \, x + \left (-a\right )^{\frac{1}{3}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.467872, size = 95, normalized size = 2.02 \begin{align*} - C \left (\frac{\log{\left (- \frac{a}{2 \left (- a\right )^{\frac{2}{3}}} + x \right )}}{4} + \frac{\sqrt{3} i \log{\left (\frac{a}{4 \left (- a\right )^{\frac{2}{3}}} - \frac{\sqrt{3} i a}{4 \left (- a\right )^{\frac{2}{3}}} + x \right )}}{12} - \frac{\sqrt{3} i \log{\left (\frac{a}{4 \left (- a\right )^{\frac{2}{3}}} + \frac{\sqrt{3} i a}{4 \left (- a\right )^{\frac{2}{3}}} + x \right )}}{12}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13767, size = 132, normalized size = 2.81 \begin{align*} \frac{\sqrt{3}{\left (\sqrt{3} i{\left | a \right |} - a\right )} C \arctan \left (\frac{\sqrt{3}{\left (4 \, x + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right )}{12 \, a} + \frac{{\left (\sqrt{3} i{\left | a \right |} - 3 \, a\right )} C \log \left (x^{2} + \frac{1}{2} \, a^{\frac{1}{3}} x + \frac{1}{4} \, a^{\frac{2}{3}}\right )}{24 \, a} - \frac{{\left (2 \, C \left (-a\right )^{\frac{2}{3}} + C a^{\frac{2}{3}}\right )} \log \left ({\left | x - \frac{1}{2} \, a^{\frac{1}{3}} \right |}\right )}{12 \, a^{\frac{2}{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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