Optimal. Leaf size=305 \[ \frac{5 c^2 \left (\frac{b}{c}+x\right )}{18 b^2 \left (b^2-3 a c\right )^2 \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )}-\frac{c \left (\frac{b}{c}+x\right )}{6 b \left (b^2-3 a c\right ) \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^2}+\frac{5 c^2 \log \left (-\sqrt [3]{b} \sqrt [3]{b^2-3 a c}+b+c x\right )}{27 b^{8/3} \left (b^2-3 a c\right )^{8/3}}-\frac{5 c^2 \log \left (\sqrt [3]{b} c \sqrt [3]{b^2-3 a c} \left (\frac{b}{c}+x\right )+b^{2/3} \left (b^2-3 a c\right )^{2/3}+c^2 \left (\frac{b}{c}+x\right )^2\right )}{54 b^{8/3} \left (b^2-3 a c\right )^{8/3}}-\frac{5 c^2 \tan ^{-1}\left (\frac{\frac{2 (b+c x)}{\sqrt [3]{b^2-3 a c}}+\sqrt [3]{b}}{\sqrt{3} \sqrt [3]{b}}\right )}{9 \sqrt{3} b^{8/3} \left (b^2-3 a c\right )^{8/3}} \]
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Rubi [A] time = 0.301938, antiderivative size = 305, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{5 c^2 \left (\frac{b}{c}+x\right )}{18 b^2 \left (b^2-3 a c\right )^2 \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )}-\frac{c \left (\frac{b}{c}+x\right )}{6 b \left (b^2-3 a c\right ) \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^2}+\frac{5 c^2 \log \left (-\sqrt [3]{b} \sqrt [3]{b^2-3 a c}+b+c x\right )}{27 b^{8/3} \left (b^2-3 a c\right )^{8/3}}-\frac{5 c^2 \log \left (\sqrt [3]{b} c \sqrt [3]{b^2-3 a c} \left (\frac{b}{c}+x\right )+b^{2/3} \left (b^2-3 a c\right )^{2/3}+c^2 \left (\frac{b}{c}+x\right )^2\right )}{54 b^{8/3} \left (b^2-3 a c\right )^{8/3}}-\frac{5 c^2 \tan ^{-1}\left (\frac{\frac{2 (b+c x)}{\sqrt [3]{b^2-3 a c}}+\sqrt [3]{b}}{\sqrt{3} \sqrt [3]{b}}\right )}{9 \sqrt{3} b^{8/3} \left (b^2-3 a c\right )^{8/3}} \]
Antiderivative was successfully verified.
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Rule 2067
Rule 199
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
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 )^3} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\left (b \left (3 a-\frac{b^2}{c}\right )+c^2 x^3\right )^3} \, dx,x,\frac{b}{c}+x\right )\\ &=-\frac{c \left (\frac{b}{c}+x\right )}{6 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 )^2}-\frac{(5 c) \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 )}{6 b \left (b^2-3 a c\right )}\\ &=-\frac{c \left (\frac{b}{c}+x\right )}{6 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 )^2}+\frac{5 c (b+c x)}{18 b^2 \left (b^2-3 a c\right )^2 \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )}+\frac{\left (5 c^2\right ) \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 )}{9 b^2 \left (b^2-3 a c\right )^2}\\ &=-\frac{c \left (\frac{b}{c}+x\right )}{6 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 )^2}+\frac{5 c (b+c x)}{18 b^2 \left (b^2-3 a c\right )^2 \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )}+\frac{\left (5 c^{8/3}\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 )}{27 b^{8/3} \left (b^2-3 a c\right )^{8/3}}+\frac{\left (5 c^{8/3}\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 )}{27 b^{8/3} \left (b^2-3 a c\right )^{8/3}}\\ &=-\frac{c \left (\frac{b}{c}+x\right )}{6 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 )^2}+\frac{5 c (b+c x)}{18 b^2 \left (b^2-3 a c\right )^2 \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )}+\frac{5 c^2 \log \left (\sqrt [3]{b} \left (b^{2/3}-\sqrt [3]{b^2-3 a c}\right )+c x\right )}{27 b^{8/3} \left (b^2-3 a c\right )^{8/3}}-\frac{\left (5 c^2\right ) \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 )}{54 b^{8/3} \left (b^2-3 a c\right )^{8/3}}-\frac{\left (5 c^{7/3}\right ) \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 )}{18 b^{7/3} \left (b^2-3 a c\right )^{7/3}}\\ &=-\frac{c \left (\frac{b}{c}+x\right )}{6 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 )^2}+\frac{5 c (b+c x)}{18 b^2 \left (b^2-3 a c\right )^2 \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )}+\frac{5 c^2 \log \left (\sqrt [3]{b} \left (b^{2/3}-\sqrt [3]{b^2-3 a c}\right )+c x\right )}{27 b^{8/3} \left (b^2-3 a c\right )^{8/3}}-\frac{5 c^2 \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 )}{54 b^{8/3} \left (b^2-3 a c\right )^{8/3}}+\frac{\left (5 c^2\right ) \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 )}{9 b^{8/3} \left (b^2-3 a c\right )^{8/3}}\\ &=-\frac{c \left (\frac{b}{c}+x\right )}{6 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 )^2}+\frac{5 c (b+c x)}{18 b^2 \left (b^2-3 a c\right )^2 \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )}-\frac{5 c^2 \tan ^{-1}\left (\frac{1+\frac{2 (b+c x)}{\sqrt [3]{b} \sqrt [3]{b^2-3 a c}}}{\sqrt{3}}\right )}{9 \sqrt{3} b^{8/3} \left (b^2-3 a c\right )^{8/3}}+\frac{5 c^2 \log \left (\sqrt [3]{b} \left (b^{2/3}-\sqrt [3]{b^2-3 a c}\right )+c x\right )}{27 b^{8/3} \left (b^2-3 a c\right )^{8/3}}-\frac{5 c^2 \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 )}{54 b^{8/3} \left (b^2-3 a c\right )^{8/3}}\\ \end{align*}
Mathematica [C] time = 0.079559, size = 149, normalized size = 0.49 \[ \frac{10 c^2 \text{RootSum}\left [3 \text{$\#$1}^2 b c+\text{$\#$1}^3 c^2+3 \text{$\#$1} b^2+3 a b\& ,\frac{\log (x-\text{$\#$1})}{\text{$\#$1}^2 c^2+2 \text{$\#$1} b c+b^2}\& \right ]-\frac{3 (b+c x) \left (-3 b c \left (8 a+5 c x^2\right )-15 b^2 c x+3 b^3-5 c^3 x^3\right )}{\left (3 a b+x \left (3 b^2+3 b c x+c^2 x^2\right )\right )^2}}{54 \left (b^3-3 a b c\right )^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.014, size = 276, normalized size = 0.9 \begin{align*}{\frac{1}{ \left ({c}^{2}{x}^{3}+3\,bc{x}^{2}+3\,{b}^{2}x+3\,ab \right ) ^{2}} \left ({\frac{5\,{c}^{4}{x}^{4}}{18\,{b}^{2} \left ( 9\,{a}^{2}{c}^{2}-6\,a{b}^{2}c+{b}^{4} \right ) }}+{\frac{10\,{c}^{3}{x}^{3}}{9\,b \left ( 9\,{a}^{2}{c}^{2}-6\,a{b}^{2}c+{b}^{4} \right ) }}+{\frac{5\,{c}^{2}{x}^{2}}{27\,{a}^{2}{c}^{2}-18\,a{b}^{2}c+3\,{b}^{4}}}+{\frac{ \left ( 4\,ac+2\,{b}^{2} \right ) cx}{3\,b \left ( 9\,{a}^{2}{c}^{2}-6\,a{b}^{2}c+{b}^{4} \right ) }}+{\frac{8\,ac-{b}^{2}}{54\,{a}^{2}{c}^{2}-36\,a{b}^{2}c+6\,{b}^{4}}} \right ) }+{\frac{5\,{c}^{2}}{27\,{b}^{2} \left ( 9\,{a}^{2}{c}^{2}-6\,a{b}^{2}c+{b}^{4} \right ) }\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{3}{c}^{2}+3\,{{\it \_Z}}^{2}bc+3\,{\it \_Z}\,{b}^{2}+3\,ab \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{2}{c}^{2}+2\,{\it \_R}\,bc+{b}^{2}}}} \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*} \frac{\frac{5}{6} \,{\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 ({\left (\sqrt{3} c x + \sqrt{3} b - \sqrt{3}{\left (-b^{3} + 3 \, a b c\right )}^{\frac{1}{3}}\right )}^{2} +{\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^{2}}{9 \,{\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}} + \frac{5 \, c^{4} x^{4} + 20 \, b c^{3} x^{3} + 30 \, b^{2} c^{2} x^{2} - 3 \, b^{4} + 24 \, a b^{2} c + 12 \,{\left (b^{3} c + 2 \, a b c^{2}\right )} x}{18 \,{\left (9 \, a^{2} b^{8} - 54 \, a^{3} b^{6} c + 81 \, a^{4} b^{4} c^{2} +{\left (b^{6} c^{4} - 6 \, a b^{4} c^{5} + 9 \, a^{2} b^{2} c^{6}\right )} x^{6} + 6 \,{\left (b^{7} c^{3} - 6 \, a b^{5} c^{4} + 9 \, a^{2} b^{3} c^{5}\right )} x^{5} + 15 \,{\left (b^{8} c^{2} - 6 \, a b^{6} c^{3} + 9 \, a^{2} b^{4} c^{4}\right )} x^{4} + 6 \,{\left (3 \, b^{9} c - 17 \, a b^{7} c^{2} + 21 \, a^{2} b^{5} c^{3} + 9 \, a^{3} b^{3} c^{4}\right )} x^{3} + 9 \,{\left (b^{10} - 4 \, a b^{8} c - 3 \, a^{2} b^{6} c^{2} + 18 \, a^{3} b^{4} c^{3}\right )} x^{2} + 18 \,{\left (a b^{9} - 6 \, a^{2} b^{7} c + 9 \, a^{3} b^{5} c^{2}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.46634, size = 2731, normalized size = 8.95 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.97428, size = 474, normalized size = 1.55 \begin{align*} \frac{24 a b^{2} c - 3 b^{4} + 30 b^{2} c^{2} x^{2} + 20 b c^{3} x^{3} + 5 c^{4} x^{4} + x \left (24 a b c^{2} + 12 b^{3} c\right )}{1458 a^{4} b^{4} c^{2} - 972 a^{3} b^{6} c + 162 a^{2} b^{8} + x^{6} \left (162 a^{2} b^{2} c^{6} - 108 a b^{4} c^{5} + 18 b^{6} c^{4}\right ) + x^{5} \left (972 a^{2} b^{3} c^{5} - 648 a b^{5} c^{4} + 108 b^{7} c^{3}\right ) + x^{4} \left (2430 a^{2} b^{4} c^{4} - 1620 a b^{6} c^{3} + 270 b^{8} c^{2}\right ) + x^{3} \left (972 a^{3} b^{3} c^{4} + 2268 a^{2} b^{5} c^{3} - 1836 a b^{7} c^{2} + 324 b^{9} c\right ) + x^{2} \left (2916 a^{3} b^{4} c^{3} - 486 a^{2} b^{6} c^{2} - 648 a b^{8} c + 162 b^{10}\right ) + x \left (2916 a^{3} b^{5} c^{2} - 1944 a^{2} b^{7} c + 324 a b^{9}\right )} + \operatorname{RootSum}{\left (t^{3} \left (129140163 a^{8} b^{8} c^{8} - 344373768 a^{7} b^{10} c^{7} + 401769396 a^{6} b^{12} c^{6} - 267846264 a^{5} b^{14} c^{5} + 111602610 a^{4} b^{16} c^{4} - 29760696 a^{3} b^{18} c^{3} + 4960116 a^{2} b^{20} c^{2} - 472392 a b^{22} c + 19683 b^{24}\right ) - 125 c^{6}, \left ( t \mapsto t \log{\left (x + \frac{729 t a^{3} b^{3} c^{3} - 729 t a^{2} b^{5} c^{2} + 243 t a b^{7} c - 27 t b^{9} + 5 b c^{2}}{5 c^{3}} \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.28468, size = 817, normalized size = 2.68 \begin{align*} \frac{5}{27} \, \sqrt{3} \left (\frac{c^{6}}{b^{24} - 24 \, a b^{22} c + 252 \, a^{2} b^{20} c^{2} - 1512 \, a^{3} b^{18} c^{3} + 5670 \, a^{4} b^{16} c^{4} - 13608 \, a^{5} b^{14} c^{5} + 20412 \, a^{6} b^{12} c^{6} - 17496 \, a^{7} b^{10} c^{7} + 6561 \, a^{8} b^{8} c^{8}}\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 ) - \frac{5}{54} \, \left (\frac{c^{6}}{b^{24} - 24 \, a b^{22} c + 252 \, a^{2} b^{20} c^{2} - 1512 \, a^{3} b^{18} c^{3} + 5670 \, a^{4} b^{16} c^{4} - 13608 \, a^{5} b^{14} c^{5} + 20412 \, a^{6} b^{12} c^{6} - 17496 \, a^{7} b^{10} c^{7} + 6561 \, a^{8} b^{8} c^{8}}\right )^{\frac{1}{3}} \log \left ({\left (\sqrt{3} c x + \sqrt{3} b - \sqrt{3}{\left (-b^{3} + 3 \, a b c\right )}^{\frac{1}{3}}\right )}^{2} +{\left (c x + b +{\left (-b^{3} + 3 \, a b c\right )}^{\frac{1}{3}}\right )}^{2}\right ) + \frac{5}{27} \, \left (\frac{c^{6}}{b^{24} - 24 \, a b^{22} c + 252 \, a^{2} b^{20} c^{2} - 1512 \, a^{3} b^{18} c^{3} + 5670 \, a^{4} b^{16} c^{4} - 13608 \, a^{5} b^{14} c^{5} + 20412 \, a^{6} b^{12} c^{6} - 17496 \, a^{7} b^{10} c^{7} + 6561 \, a^{8} b^{8} c^{8}}\right )^{\frac{1}{3}} \log \left ({\left | 9 \, b^{7} - 54 \, a b^{5} c + 81 \, a^{2} b^{3} c^{2} + 9 \,{\left (b^{6} c - 6 \, a b^{4} c^{2} + 9 \, a^{2} b^{2} c^{3}\right )} x + 9 \,{\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}{\left (-b^{3} + 3 \, a b c\right )}^{\frac{1}{3}} \right |}\right ) + \frac{5 \, c^{4} x^{4} + 20 \, b c^{3} x^{3} + 30 \, b^{2} c^{2} x^{2} + 12 \, b^{3} c x + 24 \, a b c^{2} x - 3 \, b^{4} + 24 \, a b^{2} c}{18 \,{\left (b^{6} - 6 \, a b^{4} c + 9 \, a^{2} b^{2} c^{2}\right )}{\left (c^{2} x^{3} + 3 \, b c x^{2} + 3 \, b^{2} x + 3 \, a b\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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