Optimal. Leaf size=645 \[ \frac{\left (9 a^{2/3} c^{4/3}+12 \sqrt [3]{-1} \sqrt [3]{a} b c^{2/3}+2 (-1)^{2/3} b^2\right ) \tan ^{-1}\left (\frac{3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{81 \sqrt{3} \left (1+\sqrt [3]{-1}\right )^2 a^{23/6} c^{2/3} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}+\frac{\left (9 a^{2/3} c^{4/3}-12 \sqrt [3]{a} b c^{2/3}+2 b^2\right ) \tan ^{-1}\left (\frac{3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{243 \sqrt{3} a^{23/6} c^{2/3} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}}}+\frac{(-1)^{2/3} \left (9 (-1)^{2/3} a^{2/3} c^{4/3}+12 \sqrt [3]{-1} \sqrt [3]{a} b c^{2/3}+2 b^2\right ) \tan ^{-1}\left (\frac{3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt{3} \sqrt{a} \sqrt{3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}\right )}{81 \sqrt{3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{23/6} c^{2/3} \sqrt{3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}-\frac{\left (2 b-3 \sqrt [3]{a} c^{2/3}\right ) \log \left (3 a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{486 a^{11/3} \sqrt [3]{c}}+\frac{\left (2 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}\right ) \log \left (-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{162 \left (1+\sqrt [3]{-1}\right )^2 a^{11/3} \sqrt [3]{c}}+\frac{\sqrt [3]{-1} \left (3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+2 b\right ) \log \left (3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{486 a^{11/3} \sqrt [3]{c}}-\frac{1}{27 a^3 x} \]
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Rubi [A] time = 1.37957, antiderivative size = 640, normalized size of antiderivative = 0.99, 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} \[ \frac{\left (9 a^{2/3} c^{4/3}+12 \sqrt [3]{-1} \sqrt [3]{a} b c^{2/3}+2 (-1)^{2/3} b^2\right ) \tan ^{-1}\left (\frac{3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{81 \sqrt{3} \left (1+\sqrt [3]{-1}\right )^2 a^{23/6} c^{2/3} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}+\frac{\left (9 a^{2/3} c^{4/3}-12 \sqrt [3]{a} b c^{2/3}+2 b^2\right ) \tan ^{-1}\left (\frac{3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{243 \sqrt{3} a^{23/6} c^{2/3} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}}}+\frac{\left (-9 \sqrt [3]{-1} a^{2/3} c^{4/3}-12 \sqrt [3]{a} b c^{2/3}+2 (-1)^{2/3} b^2\right ) \tan ^{-1}\left (\frac{3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt{3} \sqrt{a} \sqrt{3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}\right )}{81 \sqrt{3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{23/6} c^{2/3} \sqrt{3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}-\frac{\left (2 b-3 \sqrt [3]{a} c^{2/3}\right ) \log \left (3 a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{486 a^{11/3} \sqrt [3]{c}}+\frac{\left (2 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}\right ) \log \left (-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{162 \left (1+\sqrt [3]{-1}\right )^2 a^{11/3} \sqrt [3]{c}}+\frac{\sqrt [3]{-1} \left (3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+2 b\right ) \log \left (3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{486 a^{11/3} \sqrt [3]{c}}-\frac{1}{27 a^3 x} \]
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^2 \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^2}+\frac{\sqrt [3]{a} \left (b^2-9 \sqrt [3]{a} b c^{2/3}+9 a^{2/3} c^{4/3}\right )-b \left (2 b-3 \sqrt [3]{a} c^{2/3}\right ) \sqrt [3]{c} x}{1594323 \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{29/3} c^{2/3} \left (3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2\right )}+\frac{-\sqrt [3]{a} \left ((-1)^{2/3} b^2+9 \sqrt [3]{-1} \sqrt [3]{a} b c^{2/3}+9 a^{2/3} c^{4/3}\right )+b \left (2 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}\right ) \sqrt [3]{c} x}{1594323 \left (1+\sqrt [3]{-1}\right )^2 a^{29/3} c^{2/3} \left (3 a-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+b x^2\right )}+\frac{\sqrt [3]{a} \left ((-1)^{2/3} b^2-9 \sqrt [3]{a} b c^{2/3}-9 \sqrt [3]{-1} a^{2/3} c^{4/3}\right )+\sqrt [3]{-1} b \left (2 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}\right ) \sqrt [3]{c} x}{1594323 \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{29/3} c^{2/3} \left (3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2\right )}\right ) \, dx\\ &=-\frac{1}{27 a^3 x}+\frac{\int \frac{\sqrt [3]{a} \left (b^2-9 \sqrt [3]{a} b c^{2/3}+9 a^{2/3} c^{4/3}\right )-b \left (2 b-3 \sqrt [3]{a} c^{2/3}\right ) \sqrt [3]{c} x}{3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{243 a^{11/3} c^{2/3}}+\frac{\int \frac{\sqrt [3]{a} \left ((-1)^{2/3} b^2-9 \sqrt [3]{a} b c^{2/3}-9 \sqrt [3]{-1} a^{2/3} c^{4/3}\right )+\sqrt [3]{-1} b \left (2 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}\right ) \sqrt [3]{c} x}{3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{243 a^{11/3} c^{2/3}}+\frac{\int \frac{-\sqrt [3]{a} \left ((-1)^{2/3} b^2+9 \sqrt [3]{-1} \sqrt [3]{a} b c^{2/3}+9 a^{2/3} c^{4/3}\right )+b \left (2 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}\right ) \sqrt [3]{c} 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^{11/3} c^{2/3}}\\ &=-\frac{1}{27 a^3 x}-\frac{\left (2 b-3 \sqrt [3]{a} 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^{11/3} \sqrt [3]{c}}+\frac{\left (\sqrt [3]{-1} \left (2 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}\right )\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^{11/3} \sqrt [3]{c}}+\frac{\left (2 b-3 (-1)^{2/3} \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}{162 \left (1+\sqrt [3]{-1}\right )^2 a^{11/3} \sqrt [3]{c}}+\frac{\left (2 b^2-12 \sqrt [3]{a} b c^{2/3}+9 a^{2/3} c^{4/3}\right ) \int \frac{1}{3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{486 a^{10/3} c^{2/3}}-\frac{\left (2 (-1)^{2/3} b^2+12 \sqrt [3]{-1} \sqrt [3]{a} b c^{2/3}+9 a^{2/3} c^{4/3}\right ) \int \frac{1}{3 a-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{162 \left (1+\sqrt [3]{-1}\right )^2 a^{10/3} c^{2/3}}+\frac{\left (2 (-1)^{2/3} b^2-12 \sqrt [3]{a} b c^{2/3}-9 \sqrt [3]{-1} a^{2/3} c^{4/3}\right ) \int \frac{1}{3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{486 a^{10/3} c^{2/3}}\\ &=-\frac{1}{27 a^3 x}-\frac{\left (2 b-3 \sqrt [3]{a} c^{2/3}\right ) \log \left (3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2\right )}{486 a^{11/3} \sqrt [3]{c}}+\frac{\left (2 b-3 (-1)^{2/3} \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 )}{162 \left (1+\sqrt [3]{-1}\right )^2 a^{11/3} \sqrt [3]{c}}+\frac{\sqrt [3]{-1} \left (2 b+3 \sqrt [3]{-1} \sqrt [3]{a} 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^{11/3} \sqrt [3]{c}}-\frac{\left (2 b^2-12 \sqrt [3]{a} b c^{2/3}+9 a^{2/3} c^{4/3}\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 )}{243 a^{10/3} c^{2/3}}+\frac{\left (2 (-1)^{2/3} b^2+12 \sqrt [3]{-1} \sqrt [3]{a} b c^{2/3}+9 a^{2/3} c^{4/3}\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 )}{81 \left (1+\sqrt [3]{-1}\right )^2 a^{10/3} c^{2/3}}-\frac{\left (2 (-1)^{2/3} b^2-12 \sqrt [3]{a} b c^{2/3}-9 \sqrt [3]{-1} a^{2/3} c^{4/3}\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 )}{243 a^{10/3} c^{2/3}}\\ &=-\frac{1}{27 a^3 x}+\frac{\left (2 (-1)^{2/3} b^2+12 \sqrt [3]{-1} \sqrt [3]{a} b c^{2/3}+9 a^{2/3} c^{4/3}\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 )}{81 \sqrt{3} \left (1+\sqrt [3]{-1}\right )^2 a^{23/6} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}} c^{2/3}}+\frac{\left (2 b^2-12 \sqrt [3]{a} b c^{2/3}+9 a^{2/3} c^{4/3}\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 )}{243 \sqrt{3} a^{23/6} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}} c^{2/3}}+\frac{\left (2 (-1)^{2/3} b^2-12 \sqrt [3]{a} b c^{2/3}-9 \sqrt [3]{-1} a^{2/3} c^{4/3}\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 )}{243 \sqrt{3} a^{23/6} \sqrt{4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}} c^{2/3}}-\frac{\left (2 b-3 \sqrt [3]{a} c^{2/3}\right ) \log \left (3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2\right )}{486 a^{11/3} \sqrt [3]{c}}+\frac{\left (2 b-3 (-1)^{2/3} \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 )}{162 \left (1+\sqrt [3]{-1}\right )^2 a^{11/3} \sqrt [3]{c}}+\frac{\sqrt [3]{-1} \left (2 b+3 \sqrt [3]{-1} \sqrt [3]{a} 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^{11/3} \sqrt [3]{c}}\\ \end{align*}
Mathematica [C] time = 0.119681, size = 163, normalized size = 0.25 \[ -\frac{x \text{RootSum}\left [27 \text{$\#$1}^2 a^2 b+27 \text{$\#$1}^3 a^2 c+9 \text{$\#$1}^4 a b^2+\text{$\#$1}^6 b^3+27 a^3\& ,\frac{9 \text{$\#$1}^2 a b^2 \log (x-\text{$\#$1})+\text{$\#$1}^4 b^3 \log (x-\text{$\#$1})+27 a^2 b \log (x-\text{$\#$1})+27 \text{$\#$1} a^2 c \log (x-\text{$\#$1})}{27 \text{$\#$1}^2 a^2 c+12 \text{$\#$1}^3 a b^2+2 \text{$\#$1}^5 b^3+18 \text{$\#$1} a^2 b}\& \right ]+3}{81 a^3 x} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.007, size = 133, 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}}^{4}{b}^{3}-9\,{{\it \_R}}^{2}a{b}^{2}-27\,{\it \_R}\,{a}^{2}c-27\,{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{1}{27\,{a}^{3}x}} \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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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