Optimal. Leaf size=89 \[ -\frac {\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^2 \sqrt {b^2-4 a c}}+\frac {b \log \left (a+b x^2+c x^4\right )}{4 a^2}-\frac {b \log (x)}{a^2}-\frac {1}{2 a x^2} \]
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Rubi [A] time = 0.13, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1585, 1114, 709, 800, 634, 618, 206, 628} \begin {gather*} -\frac {\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^2 \sqrt {b^2-4 a c}}+\frac {b \log \left (a+b x^2+c x^4\right )}{4 a^2}-\frac {b \log (x)}{a^2}-\frac {1}{2 a x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 628
Rule 634
Rule 709
Rule 800
Rule 1114
Rule 1585
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a x+b x^3+c x^5\right )} \, dx &=\int \frac {1}{x^3 \left (a+b x^2+c x^4\right )} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^2 \left (a+b x+c x^2\right )} \, dx,x,x^2\right )\\ &=-\frac {1}{2 a x^2}+\frac {\operatorname {Subst}\left (\int \frac {-b-c x}{x \left (a+b x+c x^2\right )} \, dx,x,x^2\right )}{2 a}\\ &=-\frac {1}{2 a x^2}+\frac {\operatorname {Subst}\left (\int \left (-\frac {b}{a x}+\frac {b^2-a c+b c x}{a \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )}{2 a}\\ &=-\frac {1}{2 a x^2}-\frac {b \log (x)}{a^2}+\frac {\operatorname {Subst}\left (\int \frac {b^2-a c+b c x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 a^2}\\ &=-\frac {1}{2 a x^2}-\frac {b \log (x)}{a^2}+\frac {b \operatorname {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^2}+\frac {\left (b^2-2 a c\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^2}\\ &=-\frac {1}{2 a x^2}-\frac {b \log (x)}{a^2}+\frac {b \log \left (a+b x^2+c x^4\right )}{4 a^2}-\frac {\left (b^2-2 a c\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 a^2}\\ &=-\frac {1}{2 a x^2}-\frac {\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^2 \sqrt {b^2-4 a c}}-\frac {b \log (x)}{a^2}+\frac {b \log \left (a+b x^2+c x^4\right )}{4 a^2}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 135, normalized size = 1.52 \begin {gather*} \frac {\frac {\left (b \sqrt {b^2-4 a c}-2 a c+b^2\right ) \log \left (-\sqrt {b^2-4 a c}+b+2 c x^2\right )}{\sqrt {b^2-4 a c}}+\frac {\left (b \sqrt {b^2-4 a c}+2 a c-b^2\right ) \log \left (\sqrt {b^2-4 a c}+b+2 c x^2\right )}{\sqrt {b^2-4 a c}}-\frac {2 a}{x^2}-4 b \log (x)}{4 a^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^2 \left (a x+b x^3+c x^5\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.37, size = 293, normalized size = 3.29 \begin {gather*} \left [-\frac {{\left (b^{2} - 2 \, a c\right )} \sqrt {b^{2} - 4 \, a c} x^{2} \log \left (\frac {2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c + {\left (2 \, c x^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) - {\left (b^{3} - 4 \, a b c\right )} x^{2} \log \left (c x^{4} + b x^{2} + a\right ) + 4 \, {\left (b^{3} - 4 \, a b c\right )} x^{2} \log \relax (x) + 2 \, a b^{2} - 8 \, a^{2} c}{4 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} x^{2}}, -\frac {2 \, {\left (b^{2} - 2 \, a c\right )} \sqrt {-b^{2} + 4 \, a c} x^{2} \arctan \left (-\frac {{\left (2 \, c x^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) - {\left (b^{3} - 4 \, a b c\right )} x^{2} \log \left (c x^{4} + b x^{2} + a\right ) + 4 \, {\left (b^{3} - 4 \, a b c\right )} x^{2} \log \relax (x) + 2 \, a b^{2} - 8 \, a^{2} c}{4 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} x^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 94, normalized size = 1.06 \begin {gather*} \frac {b \log \left (c x^{4} + b x^{2} + a\right )}{4 \, a^{2}} - \frac {b \log \left (x^{2}\right )}{2 \, a^{2}} + \frac {{\left (b^{2} - 2 \, a c\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt {-b^{2} + 4 \, a c} a^{2}} + \frac {b x^{2} - a}{2 \, a^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 119, normalized size = 1.34 \begin {gather*} -\frac {c \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, a}+\frac {b^{2} \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \sqrt {4 a c -b^{2}}\, a^{2}}-\frac {b \ln \relax (x )}{a^{2}}+\frac {b \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 a^{2}}-\frac {1}{2 a \,x^{2}} \end {gather*}
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*} -\frac {b \log \relax (x)}{a^{2}} + \frac {\frac {1}{4} \, b \log \left (c x^{4} + b x^{2} + a\right ) + \frac {{\left (b^{2} - 2 \, a c\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt {-b^{2} + 4 \, a c}}}{a^{2}} - \frac {1}{2 \, a x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.91, size = 2033, normalized size = 22.84
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 123.75, size = 345, normalized size = 3.88 \begin {gather*} \left (\frac {b}{4 a^{2}} - \frac {\sqrt {- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} \left (4 a c - b^{2}\right )}\right ) \log {\left (x^{2} + \frac {- 8 a^{3} c \left (\frac {b}{4 a^{2}} - \frac {\sqrt {- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} \left (4 a c - b^{2}\right )}\right ) + 2 a^{2} b^{2} \left (\frac {b}{4 a^{2}} - \frac {\sqrt {- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} \left (4 a c - b^{2}\right )}\right ) + 3 a b c - b^{3}}{2 a c^{2} - b^{2} c} \right )} + \left (\frac {b}{4 a^{2}} + \frac {\sqrt {- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} \left (4 a c - b^{2}\right )}\right ) \log {\left (x^{2} + \frac {- 8 a^{3} c \left (\frac {b}{4 a^{2}} + \frac {\sqrt {- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} \left (4 a c - b^{2}\right )}\right ) + 2 a^{2} b^{2} \left (\frac {b}{4 a^{2}} + \frac {\sqrt {- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 a^{2} \left (4 a c - b^{2}\right )}\right ) + 3 a b c - b^{3}}{2 a c^{2} - b^{2} c} \right )} - \frac {1}{2 a x^{2}} - \frac {b \log {\relax (x )}}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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