Optimal. Leaf size=166 \[ -\frac{\left (6 a^2 c^2-6 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{3/2}}+\frac{x^2 \left (b^2-3 a c\right )}{c^2 \left (b^2-4 a c\right )}+\frac{x^6 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{b x^4}{2 c \left (b^2-4 a c\right )}-\frac{b \log \left (a+b x^2+c x^4\right )}{2 c^3} \]
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Rubi [A] time = 0.222667, antiderivative size = 166, 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, 738, 800, 634, 618, 206, 628} \[ -\frac{\left (6 a^2 c^2-6 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{3/2}}+\frac{x^2 \left (b^2-3 a c\right )}{c^2 \left (b^2-4 a c\right )}+\frac{x^6 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{b x^4}{2 c \left (b^2-4 a c\right )}-\frac{b \log \left (a+b x^2+c x^4\right )}{2 c^3} \]
Antiderivative was successfully verified.
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Rule 1585
Rule 1114
Rule 738
Rule 800
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{x^{11}}{\left (a x+b x^3+c x^5\right )^2} \, dx &=\int \frac{x^9}{\left (a+b x^2+c x^4\right )^2} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^4}{\left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac{x^6 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\operatorname{Subst}\left (\int \frac{x^2 (6 a+2 b x)}{a+b x+c x^2} \, dx,x,x^2\right )}{2 \left (b^2-4 a c\right )}\\ &=\frac{x^6 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\operatorname{Subst}\left (\int \left (-\frac{2 \left (b^2-3 a c\right )}{c^2}+\frac{2 b x}{c}+\frac{2 \left (a \left (b^2-3 a c\right )+b \left (b^2-4 a c\right ) x\right )}{c^2 \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )}{2 \left (b^2-4 a c\right )}\\ &=\frac{\left (b^2-3 a c\right ) x^2}{c^2 \left (b^2-4 a c\right )}-\frac{b x^4}{2 c \left (b^2-4 a c\right )}+\frac{x^6 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\operatorname{Subst}\left (\int \frac{a \left (b^2-3 a c\right )+b \left (b^2-4 a c\right ) x}{a+b x+c x^2} \, dx,x,x^2\right )}{c^2 \left (b^2-4 a c\right )}\\ &=\frac{\left (b^2-3 a c\right ) x^2}{c^2 \left (b^2-4 a c\right )}-\frac{b x^4}{2 c \left (b^2-4 a c\right )}+\frac{x^6 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{b \operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 c^3}+\frac{\left (b^4-6 a b^2 c+6 a^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{2 c^3 \left (b^2-4 a c\right )}\\ &=\frac{\left (b^2-3 a c\right ) x^2}{c^2 \left (b^2-4 a c\right )}-\frac{b x^4}{2 c \left (b^2-4 a c\right )}+\frac{x^6 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{b \log \left (a+b x^2+c x^4\right )}{2 c^3}-\frac{\left (b^4-6 a b^2 c+6 a^2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{c^3 \left (b^2-4 a c\right )}\\ &=\frac{\left (b^2-3 a c\right ) x^2}{c^2 \left (b^2-4 a c\right )}-\frac{b x^4}{2 c \left (b^2-4 a c\right )}+\frac{x^6 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\left (b^4-6 a b^2 c+6 a^2 c^2\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{3/2}}-\frac{b \log \left (a+b x^2+c x^4\right )}{2 c^3}\\ \end{align*}
Mathematica [A] time = 0.20864, size = 151, normalized size = 0.91 \[ \frac{-\frac{2 \left (6 a^2 c^2-6 a b^2 c+b^4\right ) \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+\frac{a^2 c \left (3 b-2 c x^2\right )-a b^2 \left (b-4 c x^2\right )+b^4 \left (-x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-b \log \left (a+b x^2+c x^4\right )+c x^2}{2 c^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 383, normalized size = 2.3 \begin{align*}{\frac{{x}^{2}}{2\,{c}^{2}}}+{\frac{{a}^{2}{x}^{2}}{c \left ( c{x}^{4}+b{x}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-2\,{\frac{a{x}^{2}{b}^{2}}{{c}^{2} \left ( c{x}^{4}+b{x}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{{x}^{2}{b}^{4}}{2\,{c}^{3} \left ( c{x}^{4}+b{x}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{3\,{a}^{2}b}{2\,{c}^{2} \left ( c{x}^{4}+b{x}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{a{b}^{3}}{2\,{c}^{3} \left ( c{x}^{4}+b{x}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-2\,{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) ab}{ \left ( 4\,ac-{b}^{2} \right ){c}^{2}}}+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{3}}{2\,{c}^{3} \left ( 4\,ac-{b}^{2} \right ) }}-6\,{\frac{{a}^{2}}{c \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,c{x}^{2}+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+6\,{\frac{a{b}^{2}}{{c}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,c{x}^{2}+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{{b}^{4}}{{c}^{3}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ) \left ( 4\,ac-{b}^{2} \right ) ^{-{\frac{3}{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{a b^{3} - 3 \, a^{2} b c +{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} x^{2}}{2 \,{\left (a b^{2} c^{3} - 4 \, a^{2} c^{4} +{\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} +{\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{2}\right )}} + \frac{x^{2}}{2 \, c^{2}} + \frac{-2 \, \int \frac{{\left (b^{3} - 4 \, a b c\right )} x^{3} +{\left (a b^{2} - 3 \, a^{2} c\right )} x}{c x^{4} + b x^{2} + a}\,{d x}}{b^{2} c^{2} - 4 \, a c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.67139, size = 1806, normalized size = 10.88 \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 = 4.75116, size = 877, normalized size = 5.28 \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 = 21.8737, size = 217, normalized size = 1.31 \begin{align*} \frac{{\left (b^{4} - 6 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{x^{2}}{2 \, c^{2}} + \frac{b^{3} x^{4} - 4 \, a b c x^{4} - 2 \, a^{2} c x^{2} - a^{2} b}{2 \,{\left (c x^{4} + b x^{2} + a\right )}{\left (b^{2} c^{2} - 4 \, a c^{3}\right )}} - \frac{b \log \left (c x^{4} + b x^{2} + a\right )}{2 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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