Optimal. Leaf size=188 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [3]{b}+\frac {2 (b+c x)}{\sqrt [3]{b^2-3 a c}}}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} b^{2/3} \left (b^2-3 a c\right )^{2/3}}+\frac {\log \left (b-\sqrt [3]{b} \sqrt [3]{b^2-3 a c}+c x\right )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}-\frac {\log \left (b^{2/3} \left (b^2-3 a c\right )^{2/3}+\sqrt [3]{b} c \sqrt [3]{b^2-3 a c} \left (\frac {b}{c}+x\right )+c^2 \left (\frac {b}{c}+x\right )^2\right )}{6 b^{2/3} \left (b^2-3 a c\right )^{2/3}} \]
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Rubi [A]
time = 0.23, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2092, 206, 31,
648, 631, 210, 642} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\frac {2 (b+c x)}{\sqrt [3]{b^2-3 a c}}+\sqrt [3]{b}}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} b^{2/3} \left (b^2-3 a c\right )^{2/3}}-\frac {\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 )}{6 b^{2/3} \left (b^2-3 a c\right )^{2/3}}+\frac {\log \left (-\sqrt [3]{b} \sqrt [3]{b^2-3 a c}+b+c x\right )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 206
Rule 210
Rule 631
Rule 642
Rule 648
Rule 2092
Rubi steps
\begin {align*} \int \frac {1}{3 a b+3 b^2 x+3 b c x^2+c^2 x^3} \, dx &=\text {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 )\\ &=\frac {c^{2/3} \text {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 )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}+\frac {c^{2/3} \text {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 )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}\\ &=\frac {\log \left (\sqrt [3]{b} \left (b^{2/3}-\sqrt [3]{b^2-3 a c}\right )+c x\right )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}-\frac {\text {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 )}{6 b^{2/3} \left (b^2-3 a c\right )^{2/3}}-\frac {\sqrt [3]{c} \text {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 )}{2 \sqrt [3]{b} \sqrt [3]{b^2-3 a c}}\\ &=\frac {\log \left (\sqrt [3]{b} \left (b^{2/3}-\sqrt [3]{b^2-3 a c}\right )+c x\right )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}-\frac {\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 )}{6 b^{2/3} \left (b^2-3 a c\right )^{2/3}}+\frac {\text {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 )}{b^{2/3} \left (b^2-3 a c\right )^{2/3}}\\ &=-\frac {\tan ^{-1}\left (\frac {1+\frac {2 (b+c x)}{\sqrt [3]{b} \sqrt [3]{b^2-3 a c}}}{\sqrt {3}}\right )}{\sqrt {3} b^{2/3} \left (b^2-3 a c\right )^{2/3}}+\frac {\log \left (\sqrt [3]{b} \left (b^{2/3}-\sqrt [3]{b^2-3 a c}\right )+c x\right )}{3 b^{2/3} \left (b^2-3 a c\right )^{2/3}}-\frac {\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 )}{6 b^{2/3} \left (b^2-3 a c\right )^{2/3}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.01, size = 63, normalized size = 0.34 \begin {gather*} \frac {1}{3} \text {RootSum}\left [3 a b+3 b^2 \text {$\#$1}+3 b c \text {$\#$1}^2+c^2 \text {$\#$1}^3\&,\frac {\log (x-\text {$\#$1})}{b^2+2 b c \text {$\#$1}+c^2 \text {$\#$1}^2}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.02, size = 57, normalized size = 0.30
method | result | size |
default | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (c^{2} \textit {\_Z}^{3}+3 b \,\textit {\_Z}^{2} c +3 b^{2} \textit {\_Z} +3 a b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2} c^{2}+2 \textit {\_R} b c +b^{2}}\right )}{3}\) | \(57\) |
risch | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (c^{2} \textit {\_Z}^{3}+3 b \,\textit {\_Z}^{2} c +3 b^{2} \textit {\_Z} +3 a b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2} c^{2}+2 \textit {\_R} b c +b^{2}}\right )}{3}\) | \(57\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 387 vs.
\(2 (151) = 302\).
time = 0.39, size = 387, normalized size = 2.06 \begin {gather*} -\frac {2 \, \sqrt {3} {\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}^{\frac {1}{6}} {\left (b^{3} - 3 \, a b c\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}} \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}} \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 )}{6 \, {\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.19, size = 53, normalized size = 0.28 \begin {gather*} \operatorname {RootSum} {\left (t^{3} \cdot \left (243 a^{2} b^{2} c^{2} - 162 a b^{4} c + 27 b^{6}\right ) - 1, \left ( t \mapsto t \log {\left (x + \frac {9 t a b c - 3 t b^{3} + b}{c} \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.90, size = 212, normalized size = 1.13 \begin {gather*} \frac {\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 )}{3 \, {\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 )}{6 \, {\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}^{\frac {1}{3}}} + \frac {\log \left ({\left | c x + b + {\left (-b^{3} + 3 \, a b c\right )}^{\frac {1}{3}} \right |}\right )}{3 \, {\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}^{\frac {1}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.49, size = 174, normalized size = 0.93 \begin {gather*} \frac {\ln \left (b+b^{1/3}\,{\left (3\,a\,c-b^2\right )}^{1/3}+c\,x\right )}{3\,b^{2/3}\,{\left (3\,a\,c-b^2\right )}^{2/3}}+\frac {\ln \left (3\,b\,c^3+3\,c^4\,x+\frac {3\,b^{1/3}\,c^3\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (3\,a\,c-b^2\right )}^{1/3}}{2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,b^{2/3}\,{\left (3\,a\,c-b^2\right )}^{2/3}}-\frac {\ln \left (3\,b\,c^3+3\,c^4\,x-\frac {3\,b^{1/3}\,c^3\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (3\,a\,c-b^2\right )}^{1/3}}{2}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,b^{2/3}\,{\left (3\,a\,c-b^2\right )}^{2/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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