Optimal. Leaf size=245 \[ -\frac {c \left (\frac {b}{c}+x\right )}{3 b \left (b^2-3 a c\right ) \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )}+\frac {c \log \left (\sqrt [3]{b} c \sqrt [3]{b^2-3 a c} \left (\frac {b}{c}+x\right )+b^{2/3} \left (b^2-3 a c\right )^{2/3}+c^2 \left (\frac {b}{c}+x\right )^2\right )}{9 b^{5/3} \left (b^2-3 a c\right )^{5/3}}-\frac {2 c \log \left (-\sqrt [3]{b} \sqrt [3]{b^2-3 a c}+b+c x\right )}{9 b^{5/3} \left (b^2-3 a c\right )^{5/3}}+\frac {2 c \tan ^{-1}\left (\frac {\frac {2 (b+c x)}{\sqrt [3]{b^2-3 a c}}+\sqrt [3]{b}}{\sqrt {3} \sqrt [3]{b}}\right )}{3 \sqrt {3} b^{5/3} \left (b^2-3 a c\right )^{5/3}} \]
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Rubi [A] time = 0.25, antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2067, 199, 200, 31, 634, 617, 204, 628} \[ -\frac {c \left (\frac {b}{c}+x\right )}{3 b \left (b^2-3 a c\right ) \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )}+\frac {c \log \left (\sqrt [3]{b} c \sqrt [3]{b^2-3 a c} \left (\frac {b}{c}+x\right )+b^{2/3} \left (b^2-3 a c\right )^{2/3}+c^2 \left (\frac {b}{c}+x\right )^2\right )}{9 b^{5/3} \left (b^2-3 a c\right )^{5/3}}-\frac {2 c \log \left (-\sqrt [3]{b} \sqrt [3]{b^2-3 a c}+b+c x\right )}{9 b^{5/3} \left (b^2-3 a c\right )^{5/3}}+\frac {2 c \tan ^{-1}\left (\frac {\frac {2 (b+c x)}{\sqrt [3]{b^2-3 a c}}+\sqrt [3]{b}}{\sqrt {3} \sqrt [3]{b}}\right )}{3 \sqrt {3} b^{5/3} \left (b^2-3 a c\right )^{5/3}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 199
Rule 200
Rule 204
Rule 617
Rule 628
Rule 634
Rule 2067
Rubi steps
\begin {align*} \int \frac {1}{\left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^2} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\left (b \left (3 a-\frac {b^2}{c}\right )+c^2 x^3\right )^2} \, dx,x,\frac {b}{c}+x\right )\\ &=-\frac {c \left (\frac {b}{c}+x\right )}{3 b \left (b^2-3 a c\right ) \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )}-\frac {(2 c) \operatorname {Subst}\left (\int \frac {1}{b \left (3 a-\frac {b^2}{c}\right )+c^2 x^3} \, dx,x,\frac {b}{c}+x\right )}{3 b \left (b^2-3 a c\right )}\\ &=-\frac {c \left (\frac {b}{c}+x\right )}{3 b \left (b^2-3 a c\right ) \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )}-\frac {\left (2 c^{5/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {\sqrt [3]{b} \sqrt [3]{b^2-3 a c}}{\sqrt [3]{c}}+c^{2/3} x} \, dx,x,\frac {b}{c}+x\right )}{9 b^{5/3} \left (b^2-3 a c\right )^{5/3}}-\frac {\left (2 c^{5/3}\right ) \operatorname {Subst}\left (\int \frac {-\frac {2 \sqrt [3]{b} \sqrt [3]{b^2-3 a c}}{\sqrt [3]{c}}-c^{2/3} x}{\frac {b^{2/3} \left (b^2-3 a c\right )^{2/3}}{c^{2/3}}+\sqrt [3]{b} \sqrt [3]{c} \sqrt [3]{b^2-3 a c} x+c^{4/3} x^2} \, dx,x,\frac {b}{c}+x\right )}{9 b^{5/3} \left (b^2-3 a c\right )^{5/3}}\\ &=-\frac {c \left (\frac {b}{c}+x\right )}{3 b \left (b^2-3 a c\right ) \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )}-\frac {2 c \log \left (\sqrt [3]{b} \left (b^{2/3}-\sqrt [3]{b^2-3 a c}\right )+c x\right )}{9 b^{5/3} \left (b^2-3 a c\right )^{5/3}}+\frac {c \operatorname {Subst}\left (\int \frac {\sqrt [3]{b} \sqrt [3]{c} \sqrt [3]{b^2-3 a c}+2 c^{4/3} x}{\frac {b^{2/3} \left (b^2-3 a c\right )^{2/3}}{c^{2/3}}+\sqrt [3]{b} \sqrt [3]{c} \sqrt [3]{b^2-3 a c} x+c^{4/3} x^2} \, dx,x,\frac {b}{c}+x\right )}{9 b^{5/3} \left (b^2-3 a c\right )^{5/3}}+\frac {c^{4/3} \operatorname {Subst}\left (\int \frac {1}{\frac {b^{2/3} \left (b^2-3 a c\right )^{2/3}}{c^{2/3}}+\sqrt [3]{b} \sqrt [3]{c} \sqrt [3]{b^2-3 a c} x+c^{4/3} x^2} \, dx,x,\frac {b}{c}+x\right )}{3 b^{4/3} \left (b^2-3 a c\right )^{4/3}}\\ &=-\frac {c \left (\frac {b}{c}+x\right )}{3 b \left (b^2-3 a c\right ) \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )}-\frac {2 c \log \left (\sqrt [3]{b} \left (b^{2/3}-\sqrt [3]{b^2-3 a c}\right )+c x\right )}{9 b^{5/3} \left (b^2-3 a c\right )^{5/3}}+\frac {c \log \left (b^{2/3} \left (b^2-3 a c\right )^{2/3}+\sqrt [3]{b} \sqrt [3]{b^2-3 a c} (b+c x)+(b+c x)^2\right )}{9 b^{5/3} \left (b^2-3 a c\right )^{5/3}}-\frac {(2 c) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 c \left (\frac {b}{c}+x\right )}{\sqrt [3]{b} \sqrt [3]{b^2-3 a c}}\right )}{3 b^{5/3} \left (b^2-3 a c\right )^{5/3}}\\ &=-\frac {c \left (\frac {b}{c}+x\right )}{3 b \left (b^2-3 a c\right ) \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )}+\frac {2 c \tan ^{-1}\left (\frac {1+\frac {2 (b+c x)}{\sqrt [3]{b} \sqrt [3]{b^2-3 a c}}}{\sqrt {3}}\right )}{3 \sqrt {3} b^{5/3} \left (b^2-3 a c\right )^{5/3}}-\frac {2 c \log \left (\sqrt [3]{b} \left (b^{2/3}-\sqrt [3]{b^2-3 a c}\right )+c x\right )}{9 b^{5/3} \left (b^2-3 a c\right )^{5/3}}+\frac {c \log \left (b^{2/3} \left (b^2-3 a c\right )^{2/3}+\sqrt [3]{b} \sqrt [3]{b^2-3 a c} (b+c x)+(b+c x)^2\right )}{9 b^{5/3} \left (b^2-3 a c\right )^{5/3}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 112, normalized size = 0.46 \[ -\frac {2 c \text {RootSum}\left [\text {$\#$1}^3 c^2+3 \text {$\#$1}^2 b c+3 \text {$\#$1} b^2+3 a b\& ,\frac {\log (x-\text {$\#$1})}{\text {$\#$1}^2 c^2+2 \text {$\#$1} b c+b^2}\& \right ]+\frac {3 (b+c x)}{3 a b+x \left (3 b^2+3 b c x+c^2 x^2\right )}}{9 \left (b^3-3 a b c\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.82, size = 704, normalized size = 2.87 \[ -\frac {3 \, b^{7} - 18 \, a b^{5} c + 27 \, a^{2} b^{3} c^{2} - 2 \, \sqrt {3} {\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}^{\frac {1}{6}} {\left (3 \, a b^{4} c - 9 \, a^{2} b^{2} c^{2} + {\left (b^{3} c^{3} - 3 \, a b c^{4}\right )} x^{3} + 3 \, {\left (b^{4} c^{2} - 3 \, a b^{2} c^{3}\right )} x^{2} + 3 \, {\left (b^{5} c - 3 \, a b^{3} c^{2}\right )} x\right )} \arctan \left (\frac {2 \, \sqrt {3} {\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}^{\frac {2}{3}} {\left (c x + b\right )} + \sqrt {3} {\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}^{\frac {1}{3}} {\left (b^{3} - 3 \, a b c\right )}}{3 \, {\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}^{\frac {5}{6}}}\right ) - {\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}^{\frac {2}{3}} {\left (c^{3} x^{3} + 3 \, b c^{2} x^{2} + 3 \, b^{2} c x + 3 \, a b c\right )} \log \left (-b^{5} + 3 \, a b^{3} c - {\left (b^{3} c^{2} - 3 \, a b c^{3}\right )} x^{2} - 2 \, {\left (b^{4} c - 3 \, a b^{2} c^{2}\right )} x - {\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}^{\frac {2}{3}} {\left (c x + b\right )} - {\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}^{\frac {1}{3}} {\left (b^{3} - 3 \, a b c\right )}\right ) + 2 \, {\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}^{\frac {2}{3}} {\left (c^{3} x^{3} + 3 \, b c^{2} x^{2} + 3 \, b^{2} c x + 3 \, a b c\right )} \log \left (-b^{4} + 3 \, a b^{2} c - {\left (b^{3} c - 3 \, a b c^{2}\right )} x + {\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}^{\frac {2}{3}}\right ) + 3 \, {\left (b^{6} c - 6 \, a b^{4} c^{2} + 9 \, a^{2} b^{2} c^{3}\right )} x}{9 \, {\left (3 \, a b^{10} - 27 \, a^{2} b^{8} c + 81 \, a^{3} b^{6} c^{2} - 81 \, a^{4} b^{4} c^{3} + {\left (b^{9} c^{2} - 9 \, a b^{7} c^{3} + 27 \, a^{2} b^{5} c^{4} - 27 \, a^{3} b^{3} c^{5}\right )} x^{3} + 3 \, {\left (b^{10} c - 9 \, a b^{8} c^{2} + 27 \, a^{2} b^{6} c^{3} - 27 \, a^{3} b^{4} c^{4}\right )} x^{2} + 3 \, {\left (b^{11} - 9 \, a b^{9} c + 27 \, a^{2} b^{7} c^{2} - 27 \, a^{3} b^{5} c^{3}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 289, normalized size = 1.18 \[ -\frac {2 \, \sqrt {3} \left (\frac {c^{3}}{b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} c x + \sqrt {3} b - \sqrt {3} {\left (-b^{3} + 3 \, a b c\right )}^{\frac {1}{3}}}{c x + b + {\left (-b^{3} + 3 \, a b c\right )}^{\frac {1}{3}}}\right ) - \left (\frac {c^{3}}{b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}}\right )^{\frac {1}{3}} \log \left (4 \, {\left (\sqrt {3} c x + \sqrt {3} b - \sqrt {3} {\left (-b^{3} + 3 \, a b c\right )}^{\frac {1}{3}}\right )}^{2} + 4 \, {\left (c x + b + {\left (-b^{3} + 3 \, a b c\right )}^{\frac {1}{3}}\right )}^{2}\right ) + 2 \, \left (\frac {c^{3}}{b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}}\right )^{\frac {1}{3}} \log \left ({\left | c x + b + {\left (-b^{3} + 3 \, a b c\right )}^{\frac {1}{3}} \right |}\right )}{9 \, {\left (b^{3} - 3 \, a b c\right )}} - \frac {c x + b}{3 \, {\left (c^{2} x^{3} + 3 \, b c x^{2} + 3 \, b^{2} x + 3 \, a b\right )} {\left (b^{3} - 3 \, a b c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 136, normalized size = 0.56 \[ \frac {2 c \ln \left (-\RootOf \left (c^{2} \textit {\_Z}^{3}+3 b c \,\textit {\_Z}^{2}+3 b^{2} \textit {\_Z} +3 a b \right )+x \right )}{9 \left (3 a c -b^{2}\right ) b \left (\RootOf \left (c^{2} \textit {\_Z}^{3}+3 b c \,\textit {\_Z}^{2}+3 b^{2} \textit {\_Z} +3 a b \right )^{2} c^{2}+2 \RootOf \left (c^{2} \textit {\_Z}^{3}+3 b c \,\textit {\_Z}^{2}+3 b^{2} \textit {\_Z} +3 a b \right ) b c +b^{2}\right )}+\frac {\frac {c x}{3 \left (3 a c -b^{2}\right ) b}+\frac {1}{9 a c -3 b^{2}}}{c^{2} x^{3}+3 b c \,x^{2}+3 b^{2} x +3 a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {\frac {1}{3} \, {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} c x + \sqrt {3} b - \sqrt {3} {\left (-b^{3} + 3 \, a b c\right )}^{\frac {1}{3}}}{c x + b + {\left (-b^{3} + 3 \, a b c\right )}^{\frac {1}{3}}}\right )}{{\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}^{\frac {1}{3}}} - \frac {\log \left (4 \, {\left (\sqrt {3} c x + \sqrt {3} b - \sqrt {3} {\left (-b^{3} + 3 \, a b c\right )}^{\frac {1}{3}}\right )}^{2} + 4 \, {\left (c x + b + {\left (-b^{3} + 3 \, a b c\right )}^{\frac {1}{3}}\right )}^{2}\right )}{{\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}^{\frac {1}{3}}} + \frac {2 \, \log \left ({\left | c x + b + {\left (-b^{3} + 3 \, a b c\right )}^{\frac {1}{3}} \right |}\right )}{{\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}^{\frac {1}{3}}}\right )} c}{3 \, {\left (b^{3} - 3 \, a b c\right )}} - \frac {c x + b}{3 \, {\left (3 \, a b^{4} - 9 \, a^{2} b^{2} c + {\left (b^{3} c^{2} - 3 \, a b c^{3}\right )} x^{3} + 3 \, {\left (b^{4} c - 3 \, a b^{2} c^{2}\right )} x^{2} + 3 \, {\left (b^{5} - 3 \, a b^{3} c\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.65, size = 247, normalized size = 1.01 \[ \frac {\frac {1}{3\,\left (3\,a\,c-b^2\right )}+\frac {c\,x}{3\,b\,\left (3\,a\,c-b^2\right )}}{3\,b^2\,x+3\,b\,c\,x^2+3\,a\,b+c^2\,x^3}+\frac {2\,c\,\ln \left (b+b^{1/3}\,{\left (3\,a\,c-b^2\right )}^{1/3}+c\,x\right )}{9\,b^{5/3}\,{\left (3\,a\,c-b^2\right )}^{5/3}}-\frac {\ln \left (2\,b-b^{1/3}\,{\left (3\,a\,c-b^2\right )}^{1/3}+2\,c\,x-\sqrt {3}\,b^{1/3}\,{\left (3\,a\,c-b^2\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (c+\sqrt {3}\,c\,1{}\mathrm {i}\right )}{9\,b^{5/3}\,{\left (3\,a\,c-b^2\right )}^{5/3}}-\frac {\ln \left (2\,b-b^{1/3}\,{\left (3\,a\,c-b^2\right )}^{1/3}+2\,c\,x+\sqrt {3}\,b^{1/3}\,{\left (3\,a\,c-b^2\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (c-\sqrt {3}\,c\,1{}\mathrm {i}\right )}{9\,b^{5/3}\,{\left (3\,a\,c-b^2\right )}^{5/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.22, size = 192, normalized size = 0.78 \[ \frac {b + c x}{27 a^{2} b^{2} c - 9 a b^{4} + x^{3} \left (9 a b c^{3} - 3 b^{3} c^{2}\right ) + x^{2} \left (27 a b^{2} c^{2} - 9 b^{4} c\right ) + x \left (27 a b^{3} c - 9 b^{5}\right )} + \operatorname {RootSum} {\left (t^{3} \left (177147 a^{5} b^{5} c^{5} - 295245 a^{4} b^{7} c^{4} + 196830 a^{3} b^{9} c^{3} - 65610 a^{2} b^{11} c^{2} + 10935 a b^{13} c - 729 b^{15}\right ) - 8 c^{3}, \left (t \mapsto t \log {\left (x + \frac {81 t a^{2} b^{2} c^{2} - 54 t a b^{4} c + 9 t b^{6} + 2 b c}{2 c^{2}} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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