Optimal. Leaf size=219 \[ \frac{b \left (30 a^2 c^2-20 a b^2 c+3 b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^4 \left (b^2-4 a c\right )^{3/2}}+\frac{b \left (3 b^2-11 a c\right )}{2 a^3 x^2 \left (b^2-4 a c\right )}-\frac{3 b^2-8 a c}{4 a^2 x^4 \left (b^2-4 a c\right )}-\frac{\left (3 b^2-2 a c\right ) \log \left (a+b x^2+c x^4\right )}{4 a^4}+\frac{\log (x) \left (3 b^2-2 a c\right )}{a^4}+\frac{-2 a c+b^2+b c x^2}{2 a x^4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]
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Rubi [A] time = 0.312461, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {1585, 1114, 740, 800, 634, 618, 206, 628} \[ \frac{b \left (30 a^2 c^2-20 a b^2 c+3 b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^4 \left (b^2-4 a c\right )^{3/2}}+\frac{b \left (3 b^2-11 a c\right )}{2 a^3 x^2 \left (b^2-4 a c\right )}-\frac{3 b^2-8 a c}{4 a^2 x^4 \left (b^2-4 a c\right )}-\frac{\left (3 b^2-2 a c\right ) \log \left (a+b x^2+c x^4\right )}{4 a^4}+\frac{\log (x) \left (3 b^2-2 a c\right )}{a^4}+\frac{-2 a c+b^2+b c x^2}{2 a x^4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]
Antiderivative was successfully verified.
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Rule 1585
Rule 1114
Rule 740
Rule 800
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (a x+b x^3+c x^5\right )^2} \, dx &=\int \frac{1}{x^5 \left (a+b x^2+c x^4\right )^2} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^3 \left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac{b^2-2 a c+b c x^2}{2 a \left (b^2-4 a c\right ) x^4 \left (a+b x^2+c x^4\right )}-\frac{\operatorname{Subst}\left (\int \frac{-3 b^2+8 a c-3 b c x}{x^3 \left (a+b x+c x^2\right )} \, dx,x,x^2\right )}{2 a \left (b^2-4 a c\right )}\\ &=\frac{b^2-2 a c+b c x^2}{2 a \left (b^2-4 a c\right ) x^4 \left (a+b x^2+c x^4\right )}-\frac{\operatorname{Subst}\left (\int \left (\frac{-3 b^2+8 a c}{a x^3}+\frac{3 b^3-11 a b c}{a^2 x^2}+\frac{\left (b^2-4 a c\right ) \left (-3 b^2+2 a c\right )}{a^3 x}+\frac{b \left (3 b^4-17 a b^2 c+19 a^2 c^2\right )+c \left (b^2-4 a c\right ) \left (3 b^2-2 a c\right ) x}{a^3 \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )}{2 a \left (b^2-4 a c\right )}\\ &=-\frac{3 b^2-8 a c}{4 a^2 \left (b^2-4 a c\right ) x^4}+\frac{b \left (3 b^2-11 a c\right )}{2 a^3 \left (b^2-4 a c\right ) x^2}+\frac{b^2-2 a c+b c x^2}{2 a \left (b^2-4 a c\right ) x^4 \left (a+b x^2+c x^4\right )}+\frac{\left (3 b^2-2 a c\right ) \log (x)}{a^4}-\frac{\operatorname{Subst}\left (\int \frac{b \left (3 b^4-17 a b^2 c+19 a^2 c^2\right )+c \left (b^2-4 a c\right ) \left (3 b^2-2 a c\right ) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 a^4 \left (b^2-4 a c\right )}\\ &=-\frac{3 b^2-8 a c}{4 a^2 \left (b^2-4 a c\right ) x^4}+\frac{b \left (3 b^2-11 a c\right )}{2 a^3 \left (b^2-4 a c\right ) x^2}+\frac{b^2-2 a c+b c x^2}{2 a \left (b^2-4 a c\right ) x^4 \left (a+b x^2+c x^4\right )}+\frac{\left (3 b^2-2 a c\right ) \log (x)}{a^4}-\frac{\left (3 b^2-2 a c\right ) \operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^4}-\frac{\left (b \left (3 b^4-20 a b^2 c+30 a^2 c^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^4 \left (b^2-4 a c\right )}\\ &=-\frac{3 b^2-8 a c}{4 a^2 \left (b^2-4 a c\right ) x^4}+\frac{b \left (3 b^2-11 a c\right )}{2 a^3 \left (b^2-4 a c\right ) x^2}+\frac{b^2-2 a c+b c x^2}{2 a \left (b^2-4 a c\right ) x^4 \left (a+b x^2+c x^4\right )}+\frac{\left (3 b^2-2 a c\right ) \log (x)}{a^4}-\frac{\left (3 b^2-2 a c\right ) \log \left (a+b x^2+c x^4\right )}{4 a^4}+\frac{\left (b \left (3 b^4-20 a b^2 c+30 a^2 c^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 a^4 \left (b^2-4 a c\right )}\\ &=-\frac{3 b^2-8 a c}{4 a^2 \left (b^2-4 a c\right ) x^4}+\frac{b \left (3 b^2-11 a c\right )}{2 a^3 \left (b^2-4 a c\right ) x^2}+\frac{b^2-2 a c+b c x^2}{2 a \left (b^2-4 a c\right ) x^4 \left (a+b x^2+c x^4\right )}+\frac{b \left (3 b^4-20 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^4 \left (b^2-4 a c\right )^{3/2}}+\frac{\left (3 b^2-2 a c\right ) \log (x)}{a^4}-\frac{\left (3 b^2-2 a c\right ) \log \left (a+b x^2+c x^4\right )}{4 a^4}\\ \end{align*}
Mathematica [A] time = 0.365738, size = 328, normalized size = 1.5 \[ \frac{\frac{2 a \left (2 a^2 c^2-4 a b^2 c-3 a b c^2 x^2+b^3 c x^2+b^4\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\left (8 a^2 c^2 \sqrt{b^2-4 a c}+30 a^2 b c^2+3 b^4 \sqrt{b^2-4 a c}-20 a b^3 c-14 a b^2 c \sqrt{b^2-4 a c}+3 b^5\right ) \log \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac{\left (-8 a^2 c^2 \sqrt{b^2-4 a c}+30 a^2 b c^2-3 b^4 \sqrt{b^2-4 a c}-20 a b^3 c+14 a b^2 c \sqrt{b^2-4 a c}+3 b^5\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{a^2}{x^4}+4 \log (x) \left (3 b^2-2 a c\right )+\frac{4 a b}{x^2}}{4 a^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.019, size = 443, normalized size = 2. \begin{align*}{\frac{3\,{c}^{2}b{x}^{2}}{2\,{a}^{2} \left ( c{x}^{4}+b{x}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{{b}^{3}c{x}^{2}}{2\,{a}^{3} \left ( c{x}^{4}+b{x}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{{c}^{2}}{a \left ( c{x}^{4}+b{x}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+2\,{\frac{{b}^{2}c}{{a}^{2} \left ( c{x}^{4}+b{x}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{{b}^{4}}{2\,{a}^{3} \left ( c{x}^{4}+b{x}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+2\,{\frac{{c}^{2}\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) }{{a}^{2} \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{7\,c\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{2}}{2\,{a}^{3} \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{3\,\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{4}}{4\,{a}^{4} \left ( 4\,ac-{b}^{2} \right ) }}+15\,{\frac{{c}^{2}b}{{a}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,c{x}^{2}+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-10\,{\frac{{b}^{3}c}{{a}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,c{x}^{2}+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{3\,{b}^{5}}{2\,{a}^{4}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ) \left ( 4\,ac-{b}^{2} \right ) ^{-{\frac{3}{2}}}}-{\frac{1}{4\,{a}^{2}{x}^{4}}}-2\,{\frac{\ln \left ( x \right ) c}{{a}^{3}}}+3\,{\frac{\ln \left ( x \right ){b}^{2}}{{a}^{4}}}+{\frac{b}{{x}^{2}{a}^{3}}} \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{2 \,{\left (3 \, b^{3} c - 11 \, a b c^{2}\right )} x^{6} +{\left (6 \, b^{4} - 25 \, a b^{2} c + 8 \, a^{2} c^{2}\right )} x^{4} - a^{2} b^{2} + 4 \, a^{3} c + 3 \,{\left (a b^{3} - 4 \, a^{2} b c\right )} x^{2}}{4 \,{\left ({\left (a^{3} b^{2} c - 4 \, a^{4} c^{2}\right )} x^{8} +{\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} x^{6} +{\left (a^{4} b^{2} - 4 \, a^{5} c\right )} x^{4}\right )}} - \frac{\frac{1}{4} \,{\left (3 \, b^{4} - 14 \, a b^{2} c + 8 \, a^{2} c^{2}\right )} \log \left (c x^{4} + b x^{2} + a\right ) + \frac{{\left (3 \, b^{5} - 20 \, 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 \, \sqrt{-b^{2} + 4 \, a c}}}{a^{4} b^{2} - 4 \, a^{5} c} + \frac{{\left (3 \, b^{2} - 2 \, a c\right )} \log \left (x\right )}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.7237, size = 2627, normalized size = 12. \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 [B] time = 131.412, size = 1074, normalized size = 4.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 23.7638, size = 370, normalized size = 1.69 \begin{align*} -\frac{{\left (3 \, b^{5} - 20 \, 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^{4} b^{2} - 4 \, a^{5} c\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{3 \, b^{4} c x^{4} - 14 \, a b^{2} c^{2} x^{4} + 8 \, a^{2} c^{3} x^{4} + 3 \, b^{5} x^{2} - 12 \, a b^{3} c x^{2} + 2 \, a^{2} b c^{2} x^{2} + 5 \, a b^{4} - 22 \, a^{2} b^{2} c + 12 \, a^{3} c^{2}}{4 \,{\left (a^{4} b^{2} - 4 \, a^{5} c\right )}{\left (c x^{4} + b x^{2} + a\right )}} - \frac{{\left (3 \, b^{2} - 2 \, a c\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, a^{4}} + \frac{{\left (3 \, b^{2} - 2 \, a c\right )} \log \left (x^{2}\right )}{2 \, a^{4}} - \frac{9 \, b^{2} x^{4} - 6 \, a c x^{4} - 4 \, a b x^{2} + a^{2}}{4 \, a^{4} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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