∫ 1_____ x(a+bx2+cx4)3 dx [879]

Optimal. Leaf size=200 \[ \frac {b^2-2 a c+b c x^2}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) x^2}{4 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {b \left (b^4-10 a b^2 c+30 a^2 c^2\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^3 \left (b^2-4 a c\right )^{5/2}}+\frac {\log (x)}{a^3}-\frac {\log \left (a+b x^2+c x^4\right )}{4 a^3} \]

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Rubi [A]
time = 0.20, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {1128, 754, 836, 814, 648, 632, 212, 642} \begin {gather*} -\frac {\log \left (a+b x^2+c x^4\right )}{4 a^3}+\frac {\log (x)}{a^3}+\frac {16 a^2 c^2+2 b c x^2 \left (b^2-7 a c\right )-15 a b^2 c+2 b^4}{4 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {b \left (30 a^2 c^2-10 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^3 \left (b^2-4 a c\right )^{5/2}}+\frac {-2 a c+b^2+b c x^2}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \end {gather*}

\begin {align*} \int \frac {1}{x \left (a+b x^2+c x^4\right )^3} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x \left (a+b x+c x^2\right )^3} \, dx,x,x^2\right )\\ &=\frac {b^2-2 a c+b c x^2}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {\text {Subst}\left (\int \frac {-2 \left (b^2-4 a c\right )-3 b c x}{x \left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )}{4 a \left (b^2-4 a c\right )}\\ &=\frac {b^2-2 a c+b c x^2}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) x^2}{4 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\text {Subst}\left (\int \frac {2 \left (b^2-4 a c\right )^2+2 b c \left (b^2-7 a c\right ) x}{x \left (a+b x+c x^2\right )} \, dx,x,x^2\right )}{4 a^2 \left (b^2-4 a c\right )^2}\\ &=\frac {b^2-2 a c+b c x^2}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) x^2}{4 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\text {Subst}\left (\int \left (\frac {2 \left (-b^2+4 a c\right )^2}{a x}+\frac {2 \left (-b \left (b^4-9 a b^2 c+23 a^2 c^2\right )-c \left (b^2-4 a c\right )^2 x\right )}{a \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )}{4 a^2 \left (b^2-4 a c\right )^2}\\ &=\frac {b^2-2 a c+b c x^2}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) x^2}{4 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\log (x)}{a^3}+\frac {\text {Subst}\left (\int \frac {-b \left (b^4-9 a b^2 c+23 a^2 c^2\right )-c \left (b^2-4 a c\right )^2 x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 a^3 \left (b^2-4 a c\right )^2}\\ &=\frac {b^2-2 a c+b c x^2}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) x^2}{4 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\log (x)}{a^3}-\frac {\text {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^3}-\frac {\left (b \left (b^4-10 a b^2 c+30 a^2 c^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^3 \left (b^2-4 a c\right )^2}\\ &=\frac {b^2-2 a c+b c x^2}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) x^2}{4 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\log (x)}{a^3}-\frac {\log \left (a+b x^2+c x^4\right )}{4 a^3}+\frac {\left (b \left (b^4-10 a b^2 c+30 a^2 c^2\right )\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 a^3 \left (b^2-4 a c\right )^2}\\ &=\frac {b^2-2 a c+b c x^2}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {2 b^4-15 a b^2 c+16 a^2 c^2+2 b c \left (b^2-7 a c\right ) x^2}{4 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {b \left (b^4-10 a b^2 c+30 a^2 c^2\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^3 \left (b^2-4 a c\right )^{5/2}}+\frac {\log (x)}{a^3}-\frac {\log \left (a+b x^2+c x^4\right )}{4 a^3}\\ \end {align*}

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Mathematica [A]
time = 0.33, size = 342, normalized size = 1.71 \begin {gather*} \frac {\frac {a^2 \left (b^2-2 a c+b c x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {a \left (2 b^4-15 a b^2 c+16 a^2 c^2+2 b^3 c x^2-14 a b c^2 x^2\right )}{\left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+4 \log (x)-\frac {\left (b^5-10 a b^3 c+30 a^2 b c^2+b^4 \sqrt {b^2-4 a c}-8 a b^2 c \sqrt {b^2-4 a c}+16 a^2 c^2 \sqrt {b^2-4 a c}\right ) \log \left (b-\sqrt {b^2-4 a c}+2 c x^2\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac {\left (b^5-10 a b^3 c+30 a^2 b c^2-b^4 \sqrt {b^2-4 a c}+8 a b^2 c \sqrt {b^2-4 a c}-16 a^2 c^2 \sqrt {b^2-4 a c}\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )}{\left (b^2-4 a c\right )^{5/2}}}{4 a^3} \end {gather*}

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method	result	size

default	\(-\frac {\frac {\frac {a b \,c^{2} \left (7 a c -b^{2}\right ) x^{6}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {c a \left (16 a^{2} c^{2}-29 a \,b^{2} c +4 b^{4}\right ) x^{4}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {a b \left (a^{2} c^{2}+6 a \,b^{2} c -b^{4}\right ) x^{2}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {3 a^{2} \left (8 a^{2} c^{2}-7 a \,b^{2} c +b^{4}\right )}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {\frac {\left (16 a^{2} c^{3}-8 a \,b^{2} c^{2}+b^{4} c \right ) \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2 c}+\frac {2 \left (23 a^{2} b \,c^{2}-9 a \,b^{3} c +b^{5}-\frac {\left (16 a^{2} c^{3}-8 a \,b^{2} c^{2}+b^{4} c \right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}}{2 a^{3}}+\frac {\ln \left (x \right )}{a^{3}}\)	\(360\)
risch	\(\frac {-\frac {b \,c^{2} \left (7 a c -b^{2}\right ) x^{6}}{2 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {c \left (16 a^{2} c^{2}-29 a \,b^{2} c +4 b^{4}\right ) x^{4}}{4 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) a^{2}}-\frac {b \left (a^{2} c^{2}+6 a \,b^{2} c -b^{4}\right ) x^{2}}{2 a^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {6 a^{2} c^{2}-\frac {21}{4} a \,b^{2} c +\frac {3}{4} b^{4}}{a \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {\ln \left (x \right )}{a^{3}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (1024 a^{8} c^{5}-1280 a^{7} b^{2} c^{4}+640 c^{3} b^{4} a^{6}-160 a^{5} b^{6} c^{2}+20 a^{4} b^{8} c -b^{10} a^{3}\right ) \textit {\_Z}^{2}+\left (1024 a^{5} c^{5}-1280 a^{4} b^{2} c^{4}+640 a^{3} b^{4} c^{3}-160 a^{2} b^{6} c^{2}+20 a \,b^{8} c -b^{10}\right ) \textit {\_Z} +256 a^{2} c^{5}-95 a \,b^{2} c^{4}+10 b^{4} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (2560 a^{9} c^{5}-3328 c^{4} b^{2} a^{8}+1728 a^{7} b^{4} c^{3}-448 a^{6} b^{6} c^{2}+58 a^{5} b^{8} c -3 a^{4} b^{10}\right ) \textit {\_R}^{2}+\left (1280 a^{6} c^{5}-1168 c^{4} b^{2} a^{5}+408 a^{4} b^{4} c^{3}-65 a^{3} b^{6} c^{2}+4 a^{2} b^{8} c \right ) \textit {\_R} +98 a^{2} b^{2} c^{4}-28 a \,b^{4} c^{3}+2 c^{2} b^{6}\right ) x^{2}+\left (-256 a^{9} b \,c^{4}+256 a^{8} b^{3} c^{3}-96 a^{7} b^{5} c^{2}+16 a^{6} b^{7} c -a^{5} b^{9}\right ) \textit {\_R}^{2}+\left (624 a^{6} b \,c^{4}-584 a^{5} b^{3} c^{3}+207 a^{4} b^{5} c^{2}-33 a^{3} b^{7} c +2 a^{2} b^{9}\right ) \textit {\_R} -224 a^{3} b \,c^{4}+144 a^{2} b^{3} c^{3}-30 a \,b^{5} c^{2}+2 b^{7} c \right )\right )}{2}\)	\(656\)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 997 vs. \(2 (188) = 376\).
time = 0.65, size = 2017, normalized size = 10.08 \begin {gather*} \text {Too large to display} \end {gather*}

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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Giac [A]
time = 6.43, size = 323, normalized size = 1.62 \begin {gather*} -\frac {{\left (b^{5} - 10 \, a b^{3} c + 30 \, a^{2} b c^{2}\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, {\left (a^{3} b^{4} - 8 \, a^{4} b^{2} c + 16 \, a^{5} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {3 \, b^{4} c^{2} x^{8} - 24 \, a b^{2} c^{3} x^{8} + 48 \, a^{2} c^{4} x^{8} + 6 \, b^{5} c x^{6} - 44 \, a b^{3} c^{2} x^{6} + 68 \, a^{2} b c^{3} x^{6} + 3 \, b^{6} x^{4} - 10 \, a b^{4} c x^{4} - 58 \, a^{2} b^{2} c^{2} x^{4} + 128 \, a^{3} c^{3} x^{4} + 10 \, a b^{5} x^{2} - 72 \, a^{2} b^{3} c x^{2} + 92 \, a^{3} b c^{2} x^{2} + 9 \, a^{2} b^{4} - 66 \, a^{3} b^{2} c + 96 \, a^{4} c^{2}}{8 \, {\left (a^{3} b^{4} - 8 \, a^{4} b^{2} c + 16 \, a^{5} c^{2}\right )} {\left (c x^{4} + b x^{2} + a\right )}^{2}} - \frac {\log \left (c x^{4} + b x^{2} + a\right )}{4 \, a^{3}} + \frac {\log \left (x^{2}\right )}{2 \, a^{3}} \end {gather*}

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Mupad [B]
time = 10.95, size = 2500, normalized size = 12.50 \begin {gather*} \text {Too large to display} \end {gather*}

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3.9.79 \(\int \frac {1}{x (a+b x^2+c x^4)^3} \, dx\) [879]