Optimal. Leaf size=522 \[ -\frac{\sqrt [3]{-1} \left (3 \sqrt [3]{a} c^{2/3}+2 \sqrt [3]{-1} b\right ) \tan ^{-1}\left (\frac{3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{27 \sqrt{3} \left (1+\sqrt [3]{-1}\right )^2 a^{17/6} c^{2/3} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}-\frac{\left (2 b-3 \sqrt [3]{a} c^{2/3}\right ) \tan ^{-1}\left (\frac{3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{81 \sqrt{3} a^{17/6} c^{2/3} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}}}-\frac{\left (2 (-1)^{2/3} b-3 \sqrt [3]{a} c^{2/3}\right ) \tan ^{-1}\left (\frac{3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt{3} \sqrt{a} \sqrt{3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}\right )}{27 \sqrt{3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{17/6} c^{2/3} \sqrt{3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}+\frac{\log \left (3 a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{162 a^{8/3} \sqrt [3]{c}}-\frac{\log \left (-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{54 \left (1+\sqrt [3]{-1}\right )^2 a^{8/3} \sqrt [3]{c}}-\frac{\sqrt [3]{-1} \log \left (3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{162 a^{8/3} \sqrt [3]{c}} \]
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Rubi [A] time = 0.860841, antiderivative size = 522, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 5, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.119, Rules used = {2070, 634, 618, 204, 628} \[ -\frac{\sqrt [3]{-1} \left (3 \sqrt [3]{a} c^{2/3}+2 \sqrt [3]{-1} b\right ) \tan ^{-1}\left (\frac{3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{27 \sqrt{3} \left (1+\sqrt [3]{-1}\right )^2 a^{17/6} c^{2/3} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}-\frac{\left (2 b-3 \sqrt [3]{a} c^{2/3}\right ) \tan ^{-1}\left (\frac{3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt{3} \sqrt{a} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{81 \sqrt{3} a^{17/6} c^{2/3} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}}}-\frac{\left (2 (-1)^{2/3} b-3 \sqrt [3]{a} c^{2/3}\right ) \tan ^{-1}\left (\frac{3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt{3} \sqrt{a} \sqrt{3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}\right )}{27 \sqrt{3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{17/6} c^{2/3} \sqrt{3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}+\frac{\log \left (3 a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{162 a^{8/3} \sqrt [3]{c}}-\frac{\log \left (-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{54 \left (1+\sqrt [3]{-1}\right )^2 a^{8/3} \sqrt [3]{c}}-\frac{\sqrt [3]{-1} \log \left (3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{162 a^{8/3} \sqrt [3]{c}} \]
Antiderivative was successfully verified.
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Rule 2070
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{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} \, dx &=\left (19683 a^6\right ) \int \left (\frac{-(-1)^{2/3} \sqrt [3]{a} b-3 \sqrt [3]{-1} a^{2/3} c^{2/3}+b \sqrt [3]{c} x}{531441 \left (1+\sqrt [3]{-1}\right )^2 a^{26/3} 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} b+3 a^{2/3} c^{2/3}+b \sqrt [3]{c} x}{1594323 a^{26/3} c^{2/3} \left (3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2\right )}+\frac{(-1)^{2/3} \sqrt [3]{a} b-3 a^{2/3} c^{2/3}+\sqrt [3]{-1} b \sqrt [3]{c} x}{531441 \left (-1+\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{26/3} c^{2/3} \left (3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{-\sqrt [3]{a} b+3 a^{2/3} c^{2/3}+b \sqrt [3]{c} x}{3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{81 a^{8/3} c^{2/3}}-\frac{\int \frac{(-1)^{2/3} \sqrt [3]{a} b-3 a^{2/3} c^{2/3}+\sqrt [3]{-1} b \sqrt [3]{c} x}{3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{81 a^{8/3} c^{2/3}}+\frac{\int \frac{-(-1)^{2/3} \sqrt [3]{a} b-3 \sqrt [3]{-1} a^{2/3} c^{2/3}+b \sqrt [3]{c} x}{-3 a+3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x-b x^2} \, dx}{27 \left (1+\sqrt [3]{-1}\right )^2 a^{8/3} c^{2/3}}\\ &=-\frac{\left (2 b-3 \sqrt [3]{a} c^{2/3}\right ) \int \frac{1}{3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{162 a^{7/3} c^{2/3}}-\frac{\left (2 (-1)^{2/3} b-3 \sqrt [3]{a} c^{2/3}\right ) \int \frac{1}{3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{162 a^{7/3} c^{2/3}}-\frac{\left (-2 b \left (-(-1)^{2/3} \sqrt [3]{a} b-3 \sqrt [3]{-1} a^{2/3} c^{2/3}\right )-3 \sqrt [3]{-1} a^{2/3} b c^{2/3}\right ) \int \frac{1}{-3 a+3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x-b x^2} \, dx}{54 \left (1+\sqrt [3]{-1}\right )^2 a^{8/3} b c^{2/3}}+\frac{\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}{162 a^{8/3} \sqrt [3]{c}}-\frac{\sqrt [3]{-1} \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}{162 a^{8/3} \sqrt [3]{c}}-\frac{\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}{54 \left (1+\sqrt [3]{-1}\right )^2 a^{8/3} \sqrt [3]{c}}\\ &=\frac{\log \left (3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2\right )}{162 a^{8/3} \sqrt [3]{c}}-\frac{\log \left (3 a-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{54 \left (1+\sqrt [3]{-1}\right )^2 a^{8/3} \sqrt [3]{c}}-\frac{\sqrt [3]{-1} \log \left (3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{162 a^{8/3} \sqrt [3]{c}}+\frac{\left (2 b-3 \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 )}{81 a^{7/3} c^{2/3}}+\frac{\left (2 (-1)^{2/3} b-3 \sqrt [3]{a} c^{2/3}\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 )}{81 a^{7/3} c^{2/3}}+\frac{\left (-2 b \left (-(-1)^{2/3} \sqrt [3]{a} b-3 \sqrt [3]{-1} a^{2/3} c^{2/3}\right )-3 \sqrt [3]{-1} a^{2/3} b 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 )}{27 \left (1+\sqrt [3]{-1}\right )^2 a^{8/3} b c^{2/3}}\\ &=-\frac{\left (2 (-1)^{2/3} b+3 \sqrt [3]{-1} \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 )}{27 \sqrt{3} \left (1+\sqrt [3]{-1}\right )^2 a^{17/6} \sqrt{4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}} c^{2/3}}-\frac{\left (2 b-3 \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 )}{81 \sqrt{3} a^{17/6} \sqrt{4 b-3 \sqrt [3]{a} c^{2/3}} c^{2/3}}-\frac{\left (2 (-1)^{2/3} b-3 \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 )}{81 \sqrt{3} a^{17/6} \sqrt{4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}} c^{2/3}}+\frac{\log \left (3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2\right )}{162 a^{8/3} \sqrt [3]{c}}-\frac{\log \left (3 a-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{54 \left (1+\sqrt [3]{-1}\right )^2 a^{8/3} \sqrt [3]{c}}-\frac{\sqrt [3]{-1} \log \left (3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{162 a^{8/3} \sqrt [3]{c}}\\ \end{align*}
Mathematica [C] time = 0.0591077, size = 99, normalized size = 0.19 \[ \frac{1}{3} \text{RootSum}\left [27 \text{$\#$1}^2 a^2 b+27 \text{$\#$1}^3 a^2 c+9 \text{$\#$1}^4 a b^2+\text{$\#$1}^6 b^3+27 a^3\& ,\frac{\log (x-\text{$\#$1})}{27 \text{$\#$1}^2 a^2 c+12 \text{$\#$1}^3 a b^2+2 \text{$\#$1}^5 b^3+18 \text{$\#$1} a^2 b}\& \right ] \]
Antiderivative was successfully verified.
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Maple [C] time = 0.003, size = 90, normalized size = 0.2 \begin{align*}{\frac{1}{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{\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}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{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}}\,{d x} \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}{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}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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