Optimal. Leaf size=514 \[ -\frac{-10 a A c-a b C+3 A b^2}{2 a^2 x \left (b^2-4 a c\right )}-\frac{\sqrt{c} \left (A \left (3 b^2 \sqrt{b^2-4 a c}-10 a c \sqrt{b^2-4 a c}-16 a b c+3 b^3\right )-a C \left (b \sqrt{b^2-4 a c}-12 a c+b^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left (-\frac{A \left (3 b^3-16 a b c\right )-a C \left (b^2-12 a c\right )}{\sqrt{b^2-4 a c}}-10 a A c-a b C+3 A b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} a^2 \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{b B \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^{3/2}}-\frac{B \log \left (a+b x^2+c x^4\right )}{4 a^2}+\frac{B \log (x)}{a^2}+\frac{A \left (b^2-2 a c\right )+c x^2 (A b-2 a C)-a b C}{2 a x \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{B \left (-2 a c+b^2+b c x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]
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Rubi [A] time = 1.48571, antiderivative size = 514, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 13, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.464, Rules used = {1662, 1277, 1281, 1166, 205, 12, 1114, 740, 800, 634, 618, 206, 628} \[ -\frac{-10 a A c-a b C+3 A b^2}{2 a^2 x \left (b^2-4 a c\right )}-\frac{\sqrt{c} \left (A \left (3 b^2 \sqrt{b^2-4 a c}-10 a c \sqrt{b^2-4 a c}-16 a b c+3 b^3\right )-a C \left (b \sqrt{b^2-4 a c}-12 a c+b^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left (-\frac{A \left (3 b^3-16 a b c\right )-a C \left (b^2-12 a c\right )}{\sqrt{b^2-4 a c}}-10 a A c-a b C+3 A b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} a^2 \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{b B \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^{3/2}}-\frac{B \log \left (a+b x^2+c x^4\right )}{4 a^2}+\frac{B \log (x)}{a^2}+\frac{A \left (b^2-2 a c\right )+c x^2 (A b-2 a C)-a b C}{2 a x \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{B \left (-2 a c+b^2+b c x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]
Antiderivative was successfully verified.
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Rule 1662
Rule 1277
Rule 1281
Rule 1166
Rule 205
Rule 12
Rule 1114
Rule 740
Rule 800
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{A+B x+C x^2}{x^2 \left (a+b x^2+c x^4\right )^2} \, dx &=\int \frac{B}{x \left (a+b x^2+c x^4\right )^2} \, dx+\int \frac{A+C x^2}{x^2 \left (a+b x^2+c x^4\right )^2} \, dx\\ &=\frac{A \left (b^2-2 a c\right )-a b C+c (A b-2 a C) x^2}{2 a \left (b^2-4 a c\right ) x \left (a+b x^2+c x^4\right )}+B \int \frac{1}{x \left (a+b x^2+c x^4\right )^2} \, dx-\frac{\int \frac{-3 A b^2+10 a A c+a b C-3 c (A b-2 a C) x^2}{x^2 \left (a+b x^2+c x^4\right )} \, dx}{2 a \left (b^2-4 a c\right )}\\ &=-\frac{3 A b^2-10 a A c-a b C}{2 a^2 \left (b^2-4 a c\right ) x}+\frac{A \left (b^2-2 a c\right )-a b C+c (A b-2 a C) x^2}{2 a \left (b^2-4 a c\right ) x \left (a+b x^2+c x^4\right )}+\frac{1}{2} B \operatorname{Subst}\left (\int \frac{1}{x \left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )+\frac{\int \frac{-A \left (3 b^3-13 a b c\right )+a \left (b^2-6 a c\right ) C-c \left (3 A b^2-10 a A c-a b C\right ) x^2}{a+b x^2+c x^4} \, dx}{2 a^2 \left (b^2-4 a c\right )}\\ &=-\frac{3 A b^2-10 a A c-a b C}{2 a^2 \left (b^2-4 a c\right ) x}+\frac{B \left (b^2-2 a c+b c x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{A \left (b^2-2 a c\right )-a b C+c (A b-2 a C) x^2}{2 a \left (b^2-4 a c\right ) x \left (a+b x^2+c x^4\right )}-\frac{B \operatorname{Subst}\left (\int \frac{-b^2+4 a c-b c x}{x \left (a+b x+c x^2\right )} \, dx,x,x^2\right )}{2 a \left (b^2-4 a c\right )}-\frac{\left (c \left (A \left (3 b^3-16 a b c+3 b^2 \sqrt{b^2-4 a c}-10 a c \sqrt{b^2-4 a c}\right )-a \left (b^2-12 a c+b \sqrt{b^2-4 a c}\right ) C\right )\right ) \int \frac{1}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{4 a^2 \left (b^2-4 a c\right )^{3/2}}-\frac{\left (c \left (3 A b^2-10 a A c-a b C-\frac{A \left (3 b^3-16 a b c\right )-a \left (b^2-12 a c\right ) C}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{1}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{4 a^2 \left (b^2-4 a c\right )}\\ &=-\frac{3 A b^2-10 a A c-a b C}{2 a^2 \left (b^2-4 a c\right ) x}+\frac{B \left (b^2-2 a c+b c x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{A \left (b^2-2 a c\right )-a b C+c (A b-2 a C) x^2}{2 a \left (b^2-4 a c\right ) x \left (a+b x^2+c x^4\right )}-\frac{\sqrt{c} \left (A \left (3 b^3-16 a b c+3 b^2 \sqrt{b^2-4 a c}-10 a c \sqrt{b^2-4 a c}\right )-a \left (b^2-12 a c+b \sqrt{b^2-4 a c}\right ) C\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left (3 A b^2-10 a A c-a b C-\frac{A \left (3 b^3-16 a b c\right )-a \left (b^2-12 a c\right ) C}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a^2 \left (b^2-4 a c\right ) \sqrt{b+\sqrt{b^2-4 a c}}}-\frac{B \operatorname{Subst}\left (\int \left (\frac{-b^2+4 a c}{a x}+\frac{b \left (b^2-5 a c\right )+c \left (b^2-4 a c\right ) x}{a \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 A b^2-10 a A c-a b C}{2 a^2 \left (b^2-4 a c\right ) x}+\frac{B \left (b^2-2 a c+b c x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{A \left (b^2-2 a c\right )-a b C+c (A b-2 a C) x^2}{2 a \left (b^2-4 a c\right ) x \left (a+b x^2+c x^4\right )}-\frac{\sqrt{c} \left (A \left (3 b^3-16 a b c+3 b^2 \sqrt{b^2-4 a c}-10 a c \sqrt{b^2-4 a c}\right )-a \left (b^2-12 a c+b \sqrt{b^2-4 a c}\right ) C\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left (3 A b^2-10 a A c-a b C-\frac{A \left (3 b^3-16 a b c\right )-a \left (b^2-12 a c\right ) C}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a^2 \left (b^2-4 a c\right ) \sqrt{b+\sqrt{b^2-4 a c}}}+\frac{B \log (x)}{a^2}-\frac{B \operatorname{Subst}\left (\int \frac{b \left (b^2-5 a c\right )+c \left (b^2-4 a c\right ) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 a^2 \left (b^2-4 a c\right )}\\ &=-\frac{3 A b^2-10 a A c-a b C}{2 a^2 \left (b^2-4 a c\right ) x}+\frac{B \left (b^2-2 a c+b c x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{A \left (b^2-2 a c\right )-a b C+c (A b-2 a C) x^2}{2 a \left (b^2-4 a c\right ) x \left (a+b x^2+c x^4\right )}-\frac{\sqrt{c} \left (A \left (3 b^3-16 a b c+3 b^2 \sqrt{b^2-4 a c}-10 a c \sqrt{b^2-4 a c}\right )-a \left (b^2-12 a c+b \sqrt{b^2-4 a c}\right ) C\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left (3 A b^2-10 a A c-a b C-\frac{A \left (3 b^3-16 a b c\right )-a \left (b^2-12 a c\right ) C}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a^2 \left (b^2-4 a c\right ) \sqrt{b+\sqrt{b^2-4 a c}}}+\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 B \left (b^2-6 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^2 \left (b^2-4 a c\right )}\\ &=-\frac{3 A b^2-10 a A c-a b C}{2 a^2 \left (b^2-4 a c\right ) x}+\frac{B \left (b^2-2 a c+b c x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{A \left (b^2-2 a c\right )-a b C+c (A b-2 a C) x^2}{2 a \left (b^2-4 a c\right ) x \left (a+b x^2+c x^4\right )}-\frac{\sqrt{c} \left (A \left (3 b^3-16 a b c+3 b^2 \sqrt{b^2-4 a c}-10 a c \sqrt{b^2-4 a c}\right )-a \left (b^2-12 a c+b \sqrt{b^2-4 a c}\right ) C\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left (3 A b^2-10 a A c-a b C-\frac{A \left (3 b^3-16 a b c\right )-a \left (b^2-12 a c\right ) C}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a^2 \left (b^2-4 a c\right ) \sqrt{b+\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}+\frac{\left (b B \left (b^2-6 a c\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^2 \left (b^2-4 a c\right )}\\ &=-\frac{3 A b^2-10 a A c-a b C}{2 a^2 \left (b^2-4 a c\right ) x}+\frac{B \left (b^2-2 a c+b c x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{A \left (b^2-2 a c\right )-a b C+c (A b-2 a C) x^2}{2 a \left (b^2-4 a c\right ) x \left (a+b x^2+c x^4\right )}-\frac{\sqrt{c} \left (A \left (3 b^3-16 a b c+3 b^2 \sqrt{b^2-4 a c}-10 a c \sqrt{b^2-4 a c}\right )-a \left (b^2-12 a c+b \sqrt{b^2-4 a c}\right ) C\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{c} \left (3 A b^2-10 a A c-a b C-\frac{A \left (3 b^3-16 a b c\right )-a \left (b^2-12 a c\right ) C}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a^2 \left (b^2-4 a c\right ) \sqrt{b+\sqrt{b^2-4 a c}}}+\frac{b B \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^{3/2}}+\frac{B \log (x)}{a^2}-\frac{B \log \left (a+b x^2+c x^4\right )}{4 a^2}\\ \end{align*}
Mathematica [A] time = 2.27809, size = 559, normalized size = 1.09 \[ \frac{\frac{-4 a^2 c (B+C x)+2 a \left (b c x (3 A+x (B+C x))+2 A c^2 x^3+b^2 (B+C x)\right )-2 A b^2 x \left (b+c x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\sqrt{2} \sqrt{c} \left (A \left (-3 b^2 \sqrt{b^2-4 a c}+10 a c \sqrt{b^2-4 a c}+16 a b c-3 b^3\right )+a C \left (b \sqrt{b^2-4 a c}-12 a c+b^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \left (A \left (-3 b^2 \sqrt{b^2-4 a c}+10 a c \sqrt{b^2-4 a c}-16 a b c+3 b^3\right )+a C \left (b \sqrt{b^2-4 a c}+12 a c-b^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{B \left (b^2 \sqrt{b^2-4 a c}-4 a c \sqrt{b^2-4 a c}-6 a b c+b^3\right ) \log \left (\sqrt{b^2-4 a c}-b-2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{B \left (b^2 \sqrt{b^2-4 a c}-4 a c \sqrt{b^2-4 a c}+6 a b c-b^3\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{4 A}{x}+4 B \log (x)}{4 a^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.057, size = 2398, normalized size = 4.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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