Optimal. Leaf size=469 \[ -\frac {\log \left (3 a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{162 a^{7/3} c^{2/3}}+\frac {(-1)^{2/3} \log \left (-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{54 \left (1+\sqrt [3]{-1}\right )^2 a^{7/3} c^{2/3}}-\frac {(-1)^{2/3} \log \left (3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{162 a^{7/3} c^{2/3}}-\frac {\tan ^{-1}\left (\frac {3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{9 \sqrt {3} \left (1+\sqrt [3]{-1}\right )^2 a^{13/6} \sqrt [3]{c} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}-\frac {\tan ^{-1}\left (\frac {3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{27 \sqrt {3} a^{13/6} \sqrt [3]{c} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}}}+\frac {\sqrt [3]{-1} \tan ^{-1}\left (\frac {3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}\right )}{9 \sqrt {3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{13/6} \sqrt [3]{c} \sqrt {3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}} \]
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Rubi [A] time = 0.68, antiderivative size = 469, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 5, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {2097, 634, 618, 204, 628} \[ -\frac {\log \left (3 a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{162 a^{7/3} c^{2/3}}+\frac {(-1)^{2/3} \log \left (-3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{54 \left (1+\sqrt [3]{-1}\right )^2 a^{7/3} c^{2/3}}-\frac {(-1)^{2/3} \log \left (3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+3 a+b x^2\right )}{162 a^{7/3} c^{2/3}}-\frac {\tan ^{-1}\left (\frac {3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c}-2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}\right )}{9 \sqrt {3} \left (1+\sqrt [3]{-1}\right )^2 a^{13/6} \sqrt [3]{c} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}}}-\frac {\tan ^{-1}\left (\frac {3 a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}}}\right )}{27 \sqrt {3} a^{13/6} \sqrt [3]{c} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}}}+\frac {\sqrt [3]{-1} \tan ^{-1}\left (\frac {3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+2 b x}{\sqrt {3} \sqrt {a} \sqrt {3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}}\right )}{9 \sqrt {3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{13/6} \sqrt [3]{c} \sqrt {3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}+4 b}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 2097
Rubi steps
\begin {align*} \int \frac {x}{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 {-3 a^{2/3} \sqrt [3]{c}-(-1)^{2/3} b x}{531441 \left (1+\sqrt [3]{-1}\right )^2 a^{25/3} c^{2/3} \left (-3 a+3 \sqrt [3]{-1} a^{2/3} \sqrt [3]{c} x-b x^2\right )}+\frac {-3 a^{2/3} \sqrt [3]{c}-b x}{1594323 a^{25/3} c^{2/3} \left (3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2\right )}+\frac {(-1)^{2/3} \left (3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+b x\right )}{531441 \left (-1+\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 a^{25/3} 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 {-3 a^{2/3} \sqrt [3]{c}-b x}{3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{81 a^{7/3} c^{2/3}}-\frac {(-1)^{2/3} \int \frac {3 (-1)^{2/3} a^{2/3} \sqrt [3]{c}+b x}{3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{81 a^{7/3} c^{2/3}}+\frac {\int \frac {-3 a^{2/3} \sqrt [3]{c}-(-1)^{2/3} b 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^{7/3} 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^{7/3} c^{2/3}}-\frac {(-1)^{2/3} \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^{7/3} c^{2/3}}+\frac {(-1)^{2/3} \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^{7/3} c^{2/3}}-\frac {\int \frac {1}{3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{54 a^{5/3} \sqrt [3]{c}}+\frac {\sqrt [3]{-1} \int \frac {1}{3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2} \, dx}{54 a^{5/3} \sqrt [3]{c}}-\frac {\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^{5/3} \sqrt [3]{c}}\\ &=-\frac {\log \left (3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2\right )}{162 a^{7/3} c^{2/3}}+\frac {(-1)^{2/3} \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^{7/3} c^{2/3}}-\frac {(-1)^{2/3} \log \left (3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{162 a^{7/3} c^{2/3}}+\frac {\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^{5/3} \sqrt [3]{c}}-\frac {\sqrt [3]{-1} \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^{5/3} \sqrt [3]{c}}+\frac {\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^{5/3} \sqrt [3]{c}}\\ &=-\frac {\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^{13/6} \sqrt {4 b-3 (-1)^{2/3} \sqrt [3]{a} c^{2/3}} \sqrt [3]{c}}-\frac {\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^{13/6} \sqrt {4 b-3 \sqrt [3]{a} c^{2/3}} \sqrt [3]{c}}+\frac {\sqrt [3]{-1} \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^{13/6} \sqrt {4 b+3 \sqrt [3]{-1} \sqrt [3]{a} c^{2/3}} \sqrt [3]{c}}-\frac {\log \left (3 a+3 a^{2/3} \sqrt [3]{c} x+b x^2\right )}{162 a^{7/3} c^{2/3}}+\frac {(-1)^{2/3} \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^{7/3} c^{2/3}}-\frac {(-1)^{2/3} \log \left (3 a+3 (-1)^{2/3} a^{2/3} \sqrt [3]{c} x+b x^2\right )}{162 a^{7/3} c^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 95, normalized size = 0.20 \[ \frac {1}{3} \text {RootSum}\left [\text {$\#$1}^6 b^3+9 \text {$\#$1}^4 a b^2+27 \text {$\#$1}^3 a^2 c+27 \text {$\#$1}^2 a^2 b+27 a^3\& ,\frac {\log (x-\text {$\#$1})}{2 \text {$\#$1}^4 b^3+12 \text {$\#$1}^2 a b^2+27 \text {$\#$1} a^2 c+18 a^2 b}\& \right ] \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{b^{3} x^{6} + 9 \, a b^{2} x^{4} + 27 \, a^{2} c x^{3} + 27 \, a^{2} b x^{2} + 27 \, a^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.00, size = 91, normalized size = 0.19 \[ \frac {\RootOf \left (b^{3} \textit {\_Z}^{6}+9 b^{2} a \,\textit {\_Z}^{4}+27 a^{2} c \,\textit {\_Z}^{3}+27 b \,a^{2} \textit {\_Z}^{2}+27 a^{3}\right ) \ln \left (-\RootOf \left (b^{3} \textit {\_Z}^{6}+9 b^{2} a \,\textit {\_Z}^{4}+27 a^{2} c \,\textit {\_Z}^{3}+27 b \,a^{2} \textit {\_Z}^{2}+27 a^{3}\right )+x \right )}{6 \RootOf \left (b^{3} \textit {\_Z}^{6}+9 b^{2} a \,\textit {\_Z}^{4}+27 a^{2} c \,\textit {\_Z}^{3}+27 b \,a^{2} \textit {\_Z}^{2}+27 a^{3}\right )^{5} b^{3}+36 \RootOf \left (b^{3} \textit {\_Z}^{6}+9 b^{2} a \,\textit {\_Z}^{4}+27 a^{2} c \,\textit {\_Z}^{3}+27 b \,a^{2} \textit {\_Z}^{2}+27 a^{3}\right )^{3} a \,b^{2}+81 \RootOf \left (b^{3} \textit {\_Z}^{6}+9 b^{2} a \,\textit {\_Z}^{4}+27 a^{2} c \,\textit {\_Z}^{3}+27 b \,a^{2} \textit {\_Z}^{2}+27 a^{3}\right )^{2} a^{2} c +54 \RootOf \left (b^{3} \textit {\_Z}^{6}+9 b^{2} a \,\textit {\_Z}^{4}+27 a^{2} c \,\textit {\_Z}^{3}+27 b \,a^{2} \textit {\_Z}^{2}+27 a^{3}\right ) a^{2} b} \]
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 \[ \int \frac {x}{b^{3} x^{6} + 9 \, a b^{2} x^{4} + 27 \, a^{2} c x^{3} + 27 \, a^{2} b x^{2} + 27 \, a^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.90, size = 1057, normalized size = 2.25 \[ \sum _{k=1}^6\ln \left (b^{12}\,x+{\mathrm {root}\left (18075490334784\,a^{14}\,b^3\,c^4\,z^6-7625597484987\,a^{15}\,c^6\,z^6+1162261467\,a^{10}\,b\,c^4\,z^4+8503056\,a^7\,b^3\,c^2\,z^3-14348907\,a^8\,c^4\,z^3+177147\,a^5\,b^2\,c^2\,z^2+b^3,z,k\right )}^4\,a^{10}\,b^{11}\,c^3\,1033121304+{\mathrm {root}\left (18075490334784\,a^{14}\,b^3\,c^4\,z^6-7625597484987\,a^{15}\,c^6\,z^6+1162261467\,a^{10}\,b\,c^4\,z^4+8503056\,a^7\,b^3\,c^2\,z^3-14348907\,a^8\,c^4\,z^3+177147\,a^5\,b^2\,c^2\,z^2+b^3,z,k\right )}^5\,a^{12}\,b^{12}\,c^3\,167365651248-{\mathrm {root}\left (18075490334784\,a^{14}\,b^3\,c^4\,z^6-7625597484987\,a^{15}\,c^6\,z^6+1162261467\,a^{10}\,b\,c^4\,z^4+8503056\,a^7\,b^3\,c^2\,z^3-14348907\,a^8\,c^4\,z^3+177147\,a^5\,b^2\,c^2\,z^2+b^3,z,k\right )}^5\,a^{13}\,b^9\,c^5\,94143178827+\mathrm {root}\left (18075490334784\,a^{14}\,b^3\,c^4\,z^6-7625597484987\,a^{15}\,c^6\,z^6+1162261467\,a^{10}\,b\,c^4\,z^4+8503056\,a^7\,b^3\,c^2\,z^3-14348907\,a^8\,c^4\,z^3+177147\,a^5\,b^2\,c^2\,z^2+b^3,z,k\right )\,a^2\,b^{13}\,x\,54+{\mathrm {root}\left (18075490334784\,a^{14}\,b^3\,c^4\,z^6-7625597484987\,a^{15}\,c^6\,z^6+1162261467\,a^{10}\,b\,c^4\,z^4+8503056\,a^7\,b^3\,c^2\,z^3-14348907\,a^8\,c^4\,z^3+177147\,a^5\,b^2\,c^2\,z^2+b^3,z,k\right )}^2\,a^5\,b^{11}\,c^2\,x\,177147+{\mathrm {root}\left (18075490334784\,a^{14}\,b^3\,c^4\,z^6-7625597484987\,a^{15}\,c^6\,z^6+1162261467\,a^{10}\,b\,c^4\,z^4+8503056\,a^7\,b^3\,c^2\,z^3-14348907\,a^8\,c^4\,z^3+177147\,a^5\,b^2\,c^2\,z^2+b^3,z,k\right )}^3\,a^7\,b^{12}\,c^2\,x\,17006112-{\mathrm {root}\left (18075490334784\,a^{14}\,b^3\,c^4\,z^6-7625597484987\,a^{15}\,c^6\,z^6+1162261467\,a^{10}\,b\,c^4\,z^4+8503056\,a^7\,b^3\,c^2\,z^3-14348907\,a^8\,c^4\,z^3+177147\,a^5\,b^2\,c^2\,z^2+b^3,z,k\right )}^3\,a^8\,b^9\,c^4\,x\,14348907+{\mathrm {root}\left (18075490334784\,a^{14}\,b^3\,c^4\,z^6-7625597484987\,a^{15}\,c^6\,z^6+1162261467\,a^{10}\,b\,c^4\,z^4+8503056\,a^7\,b^3\,c^2\,z^3-14348907\,a^8\,c^4\,z^3+177147\,a^5\,b^2\,c^2\,z^2+b^3,z,k\right )}^4\,a^9\,b^{13}\,c^2\,x\,229582512+{\mathrm {root}\left (18075490334784\,a^{14}\,b^3\,c^4\,z^6-7625597484987\,a^{15}\,c^6\,z^6+1162261467\,a^{10}\,b\,c^4\,z^4+8503056\,a^7\,b^3\,c^2\,z^3-14348907\,a^8\,c^4\,z^3+177147\,a^5\,b^2\,c^2\,z^2+b^3,z,k\right )}^4\,a^{10}\,b^{10}\,c^4\,x\,387420489-{\mathrm {root}\left (18075490334784\,a^{14}\,b^3\,c^4\,z^6-7625597484987\,a^{15}\,c^6\,z^6+1162261467\,a^{10}\,b\,c^4\,z^4+8503056\,a^7\,b^3\,c^2\,z^3-14348907\,a^8\,c^4\,z^3+177147\,a^5\,b^2\,c^2\,z^2+b^3,z,k\right )}^5\,a^{12}\,b^{11}\,c^4\,x\,20920706406\right )\,\mathrm {root}\left (18075490334784\,a^{14}\,b^3\,c^4\,z^6-7625597484987\,a^{15}\,c^6\,z^6+1162261467\,a^{10}\,b\,c^4\,z^4+8503056\,a^7\,b^3\,c^2\,z^3-14348907\,a^8\,c^4\,z^3+177147\,a^5\,b^2\,c^2\,z^2+b^3,z,k\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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