Optimal. Leaf size=563 \[ -\frac{\left (3 \sqrt [3]{a}-\frac{b}{c^{2/3}}\right ) \log \left (3 a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{486 a^{10/3}}-\frac{\left (6 \sqrt [3]{a} c^{2/3}+i \sqrt{3} b+b\right ) \log \left (-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{972 a^{10/3} c^{2/3}}-\frac{\left (3 \sqrt [3]{a}-\frac{(-1)^{2/3} b}{c^{2/3}}\right ) \log \left (3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{486 a^{10/3}}+\frac{\left (b-(-1)^{2/3} \sqrt [3]{a} c^{2/3}\right ) \tan ^{-1}\left (\frac{3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{9 \sqrt{3} \left (1+\sqrt [3]{-1}\right )^2 a^{19/6} \sqrt [3]{c} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}+\frac{\left (b-\sqrt [3]{a} c^{2/3}\right ) \tan ^{-1}\left (\frac{3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{27 \sqrt{3} a^{19/6} \sqrt [3]{c} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}}}+\frac{(-1)^{2/3} \left ((-1)^{2/3} b-\sqrt [3]{a} c^{2/3}\right ) \tan ^{-1}\left (\frac{3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt{3} \sqrt{a} \sqrt{3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}\right )}{9 \sqrt{3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{19/6} \sqrt [3]{c} \sqrt{3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}+\frac{\log (x)}{27 a^3} \]
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Rubi [A] time = 1.15653, antiderivative size = 563, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 5, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.109, Rules used = {2097, 634, 618, 204, 628} \[ -\frac{\left (3 \sqrt [3]{a}-\frac{b}{c^{2/3}}\right ) \log \left (3 a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{486 a^{10/3}}-\frac{\left (6 \sqrt [3]{a} c^{2/3}+i \sqrt{3} b+b\right ) \log \left (-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{972 a^{10/3} c^{2/3}}-\frac{\left (3 \sqrt [3]{a}-\frac{(-1)^{2/3} b}{c^{2/3}}\right ) \log \left (3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{486 a^{10/3}}+\frac{\left (b-(-1)^{2/3} \sqrt [3]{a} c^{2/3}\right ) \tan ^{-1}\left (\frac{3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{9 \sqrt{3} \left (1+\sqrt [3]{-1}\right )^2 a^{19/6} \sqrt [3]{c} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}+\frac{\left (b-\sqrt [3]{a} c^{2/3}\right ) \tan ^{-1}\left (\frac{3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{27 \sqrt{3} a^{19/6} \sqrt [3]{c} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}}}+\frac{(-1)^{2/3} \left ((-1)^{2/3} b-\sqrt [3]{a} c^{2/3}\right ) \tan ^{-1}\left (\frac{3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt{3} \sqrt{a} \sqrt{3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}\right )}{9 \sqrt{3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{19/6} \sqrt [3]{c} \sqrt{3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}+\frac{\log (x)}{27 a^3} \]
Antiderivative was successfully verified.
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Rule 2097
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{x \left (27 a^3+27 a^2 b x^2+27 a^2 c x^3+9 a b^2 x^4+b^3 x^6\right )} \, dx &=\left (19683 a^6\right ) \int \left (\frac{1}{531441 a^9 x}+\frac{3 a^{2/3} \left (2 b-3 \sqrt [3]{a} c^{2/3}\right ) \sqrt [3]{c}+b \left (b-3 \sqrt [3]{a} c^{2/3}\right ) x}{4782969 a^{28/3} c^{2/3} \left (3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2\right )}+\frac{-3 a^{2/3} \left (2 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}\right ) \sqrt [3]{c}-\sqrt [3]{-1} b \left (\sqrt [3]{-1} b+3 \sqrt [3]{a} c^{2/3}\right ) x}{1594323 \left (1+\sqrt [3]{-1}\right )^2 a^{28/3} c^{2/3} \left (3 a-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+b x^2\right )}+\frac{(-1)^{2/3} \left (3 a^{2/3} \left (2 (-1)^{2/3} b-3 \sqrt [3]{a} c^{2/3}\right ) \sqrt [3]{c}+b \left (b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}\right ) x\right )}{1594323 \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{28/3} c^{2/3} \left (3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2\right )}\right ) \, dx\\ &=\frac{\log (x)}{27 a^3}+\frac{\int \frac{3 a^{2/3} \left (2 b-3 \sqrt [3]{a} c^{2/3}\right ) \sqrt [3]{c}+b \left (b-3 \sqrt [3]{a} c^{2/3}\right ) x}{3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{243 a^{10/3} c^{2/3}}+\frac{(-1)^{2/3} \int \frac{3 a^{2/3} \left (2 (-1)^{2/3} b-3 \sqrt [3]{a} c^{2/3}\right ) \sqrt [3]{c}+b \left (b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}\right ) x}{3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{243 a^{10/3} c^{2/3}}+\frac{\int \frac{-3 a^{2/3} \left (2 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}\right ) \sqrt [3]{c}-\sqrt [3]{-1} b \left (\sqrt [3]{-1} b+3 \sqrt [3]{a} c^{2/3}\right ) x}{3 a-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{81 \left (1+\sqrt [3]{-1}\right )^2 a^{10/3} c^{2/3}}\\ &=\frac{\log (x)}{27 a^3}-\frac{\left (3 \sqrt [3]{a}-\frac{b}{c^{2/3}}\right ) \int \frac{3 a^{2/3} \sqrt [3]{c}+2 b x}{3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{486 a^{10/3}}-\frac{\left (3 \sqrt [3]{a}-\frac{(-1)^{2/3} b}{c^{2/3}}\right ) \int \frac{3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{486 a^{10/3}}-\frac{\left (b+i \sqrt{3} b+6 \sqrt [3]{a} c^{2/3}\right ) \int \frac{-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}+2 b x}{3 a-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{972 a^{10/3} c^{2/3}}+\frac{\left (b-\sqrt [3]{a} c^{2/3}\right ) \int \frac{1}{3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{54 a^{8/3} \sqrt [3]{c}}+\frac{\left ((-1)^{2/3} \left ((-1)^{2/3} b-\sqrt [3]{a} c^{2/3}\right )\right ) \int \frac{1}{3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{54 a^{8/3} \sqrt [3]{c}}-\frac{\left (b-(-1)^{2/3} \sqrt [3]{a} c^{2/3}\right ) \int \frac{1}{3 a-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{18 \left (1+\sqrt [3]{-1}\right )^2 a^{8/3} \sqrt [3]{c}}\\ &=\frac{\log (x)}{27 a^3}-\frac{\left (3 \sqrt [3]{a}-\frac{b}{c^{2/3}}\right ) \log \left (3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2\right )}{486 a^{10/3}}-\frac{\left (b+i \sqrt{3} b+6 \sqrt [3]{a} c^{2/3}\right ) \log \left (3 a-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{972 a^{10/3} c^{2/3}}-\frac{\left (3 \sqrt [3]{a}-\frac{(-1)^{2/3} b}{c^{2/3}}\right ) \log \left (3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{486 a^{10/3}}-\frac{\left (b-\sqrt [3]{a} c^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{-3 a \left (4 b-3 \sqrt [3]{a} c^{2/3}\right )-x^2} \, dx,x,3 a^{2/3} \sqrt [3]{c}+2 b x\right )}{27 a^{8/3} \sqrt [3]{c}}-\frac{\left ((-1)^{2/3} \left ((-1)^{2/3} b-\sqrt [3]{a} c^{2/3}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-3 a \left (4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}\right )-x^2} \, dx,x,3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x\right )}{27 a^{8/3} \sqrt [3]{c}}+\frac{\left (b-(-1)^{2/3} \sqrt [3]{a} c^{2/3}\right ) \operatorname{Subst}\left (\int \frac{1}{-3 a \left (4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}\right )-x^2} \, dx,x,-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}+2 b x\right )}{9 \left (1+\sqrt [3]{-1}\right )^2 a^{8/3} \sqrt [3]{c}}\\ &=\frac{\left (b-(-1)^{2/3} \sqrt [3]{a} c^{2/3}\right ) \tan ^{-1}\left (\frac{3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{9 \sqrt{3} \left (1+\sqrt [3]{-1}\right )^2 a^{19/6} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}} \sqrt [3]{c}}+\frac{\left (b-\sqrt [3]{a} c^{2/3}\right ) \tan ^{-1}\left (\frac{3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{27 \sqrt{3} a^{19/6} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}} \sqrt [3]{c}}+\frac{(-1)^{2/3} \left ((-1)^{2/3} b-\sqrt [3]{a} c^{2/3}\right ) \tan ^{-1}\left (\frac{3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}}}\right )}{27 \sqrt{3} a^{19/6} \sqrt{4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}} \sqrt [3]{c}}+\frac{\log (x)}{27 a^3}-\frac{\left (3 \sqrt [3]{a}-\frac{b}{c^{2/3}}\right ) \log \left (3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2\right )}{486 a^{10/3}}-\frac{\left (b+i \sqrt{3} b+6 \sqrt [3]{a} c^{2/3}\right ) \log \left (3 a-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{972 a^{10/3} c^{2/3}}-\frac{\left (3 \sqrt [3]{a}-\frac{(-1)^{2/3} b}{c^{2/3}}\right ) \log \left (3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{486 a^{10/3}}\\ \end{align*}
Mathematica [C] time = 0.0962968, size = 157, normalized size = 0.28 \[ -\frac{\text{RootSum}\left [27 \text{$\#$1}^2 a^2 b+27 \text{$\#$1}^3 a^2 c+9 \text{$\#$1}^4 a b^2+\text{$\#$1}^6 b^3+27 a^3\& ,\frac{9 \text{$\#$1}^2 a b^2 \log (x-\text{$\#$1})+\text{$\#$1}^4 b^3 \log (x-\text{$\#$1})+27 a^2 b \log (x-\text{$\#$1})+27 \text{$\#$1} a^2 c \log (x-\text{$\#$1})}{12 \text{$\#$1}^2 a b^2+2 \text{$\#$1}^4 b^3+27 \text{$\#$1} a^2 c+18 a^2 b}\& \right ]-3 \log (x)}{81 a^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.008, size = 134, normalized size = 0.2 \begin{align*} -{\frac{1}{81\,{a}^{3}}\sum _{{\it \_R}={\it RootOf} \left ({b}^{3}{{\it \_Z}}^{6}+9\,a{b}^{2}{{\it \_Z}}^{4}+27\,{a}^{2}c{{\it \_Z}}^{3}+27\,{a}^{2}b{{\it \_Z}}^{2}+27\,{a}^{3} \right ) }{\frac{ \left ({{\it \_R}}^{5}{b}^{3}+9\,{{\it \_R}}^{3}a{b}^{2}+27\,{{\it \_R}}^{2}{a}^{2}c+27\,{\it \_R}\,{a}^{2}b \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{5}{b}^{3}+12\,{{\it \_R}}^{3}a{b}^{2}+27\,{{\it \_R}}^{2}{a}^{2}c+18\,{\it \_R}\,{a}^{2}b}}}+{\frac{\ln \left ( x \right ) }{27\,{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b^{3} x^{6} + 9 \, a b^{2} x^{4} + 27 \, a^{2} c x^{3} + 27 \, a^{2} b x^{2} + 27 \, a^{3}\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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