Optimal. Leaf size=534 \[ -\frac{\left (2 A \left (6 a^2 c^2-6 a b^2 c+b^4\right )-a b C \left (b^2-6 a c\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^3 \left (b^2-4 a c\right )^{3/2}}-\frac{-6 a A c-a b C+2 A b^2}{2 a^2 x^2 \left (b^2-4 a c\right )}+\frac{(2 A b-a C) \log \left (a+b x^2+c x^4\right )}{4 a^3}-\frac{\log (x) (2 A b-a C)}{a^3}-\frac{B \left (3 b^2-10 a c\right )}{2 a^2 x \left (b^2-4 a c\right )}-\frac{B \sqrt{c} \left (\left (3 b^2-10 a c\right ) \sqrt{b^2-4 a c}-16 a b c+3 b^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{B \sqrt{c} \left (-\left (3 b^2-10 a c\right ) \sqrt{b^2-4 a c}-16 a b c+3 b^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{A \left (b^2-2 a c\right )+c x^2 (A b-2 a C)-a b C}{2 a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{B \left (-2 a c+b^2+b c x^2\right )}{2 a x \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]
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Rubi [A] time = 1.99236, antiderivative size = 534, 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, 1251, 822, 800, 634, 618, 206, 628, 12, 1121, 1281, 1166, 205} \[ -\frac{\left (2 A \left (6 a^2 c^2-6 a b^2 c+b^4\right )-a b C \left (b^2-6 a c\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^3 \left (b^2-4 a c\right )^{3/2}}-\frac{-6 a A c-a b C+2 A b^2}{2 a^2 x^2 \left (b^2-4 a c\right )}+\frac{(2 A b-a C) \log \left (a+b x^2+c x^4\right )}{4 a^3}-\frac{\log (x) (2 A b-a C)}{a^3}-\frac{B \left (3 b^2-10 a c\right )}{2 a^2 x \left (b^2-4 a c\right )}-\frac{B \sqrt{c} \left (\left (3 b^2-10 a c\right ) \sqrt{b^2-4 a c}-16 a b c+3 b^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{B \sqrt{c} \left (-\left (3 b^2-10 a c\right ) \sqrt{b^2-4 a c}-16 a b c+3 b^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} a^2 \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{A \left (b^2-2 a c\right )+c x^2 (A b-2 a C)-a b C}{2 a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{B \left (-2 a c+b^2+b c x^2\right )}{2 a x \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]
Antiderivative was successfully verified.
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Rule 1662
Rule 1251
Rule 822
Rule 800
Rule 634
Rule 618
Rule 206
Rule 628
Rule 12
Rule 1121
Rule 1281
Rule 1166
Rule 205
Rubi steps
\begin{align*} \int \frac{A+B x+C x^2}{x^3 \left (a+b x^2+c x^4\right )^2} \, dx &=\int \frac{B}{x^2 \left (a+b x^2+c x^4\right )^2} \, dx+\int \frac{A+C x^2}{x^3 \left (a+b x^2+c x^4\right )^2} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{A+C x}{x^2 \left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )+B \int \frac{1}{x^2 \left (a+b x^2+c x^4\right )^2} \, dx\\ &=\frac{B \left (b^2-2 a c+b c x^2\right )}{2 a \left (b^2-4 a c\right ) x \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^2 \left (a+b x^2+c x^4\right )}-\frac{\operatorname{Subst}\left (\int \frac{-2 A b^2+6 a A c+a b C-2 c (A b-2 a C) x}{x^2 \left (a+b x+c x^2\right )} \, dx,x,x^2\right )}{2 a \left (b^2-4 a c\right )}-\frac{B \int \frac{-3 b^2+10 a c-3 b 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{B \left (3 b^2-10 a c\right )}{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 ) x \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^2 \left (a+b x^2+c x^4\right )}-\frac{\operatorname{Subst}\left (\int \left (\frac{-2 A b^2+6 a A c+a b C}{a x^2}+\frac{\left (-b^2+4 a c\right ) (-2 A b+a C)}{a^2 x}+\frac{-2 A \left (b^4-5 a b^2 c+3 a^2 c^2\right )+a b \left (b^2-5 a c\right ) C-c \left (b^2-4 a c\right ) (2 A b-a C) x}{a^2 \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )}{2 a \left (b^2-4 a c\right )}+\frac{B \int \frac{-b \left (3 b^2-13 a c\right )-c \left (3 b^2-10 a c\right ) x^2}{a+b x^2+c x^4} \, dx}{2 a^2 \left (b^2-4 a c\right )}\\ &=-\frac{2 A b^2-6 a A c-a b C}{2 a^2 \left (b^2-4 a c\right ) x^2}-\frac{B \left (3 b^2-10 a c\right )}{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 ) x \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^2 \left (a+b x^2+c x^4\right )}-\frac{(2 A b-a C) \log (x)}{a^3}-\frac{\operatorname{Subst}\left (\int \frac{-2 A \left (b^4-5 a b^2 c+3 a^2 c^2\right )+a b \left (b^2-5 a c\right ) C-c \left (b^2-4 a c\right ) (2 A b-a C) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 a^3 \left (b^2-4 a c\right )}-\frac{\left (B c \left (3 b^2-10 a c+\frac{3 b^3}{\sqrt{b^2-4 a c}}-\frac{16 a b 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{\left (B c \left (3 b^2-10 a c-\frac{3 b^3}{\sqrt{b^2-4 a c}}+\frac{16 a b 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{2 A b^2-6 a A c-a b C}{2 a^2 \left (b^2-4 a c\right ) x^2}-\frac{B \left (3 b^2-10 a c\right )}{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 ) x \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^2 \left (a+b x^2+c x^4\right )}-\frac{B \sqrt{c} \left (3 b^2-10 a c+\frac{3 b^3}{\sqrt{b^2-4 a c}}-\frac{16 a b 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 \sqrt{c} \left (3 b^2-10 a c-\frac{3 b^3}{\sqrt{b^2-4 a c}}+\frac{16 a b 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{(2 A b-a C) \log (x)}{a^3}+\frac{(2 A b-a C) \operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^3}+\frac{\left (2 A \left (b^4-6 a b^2 c+6 a^2 c^2\right )-a b \left (b^2-6 a c\right ) C\right ) \operatorname{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 )}\\ &=-\frac{2 A b^2-6 a A c-a b C}{2 a^2 \left (b^2-4 a c\right ) x^2}-\frac{B \left (3 b^2-10 a c\right )}{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 ) x \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^2 \left (a+b x^2+c x^4\right )}-\frac{B \sqrt{c} \left (3 b^2-10 a c+\frac{3 b^3}{\sqrt{b^2-4 a c}}-\frac{16 a b 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 \sqrt{c} \left (3 b^2-10 a c-\frac{3 b^3}{\sqrt{b^2-4 a c}}+\frac{16 a b 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{(2 A b-a C) \log (x)}{a^3}+\frac{(2 A b-a C) \log \left (a+b x^2+c x^4\right )}{4 a^3}-\frac{\left (2 A \left (b^4-6 a b^2 c+6 a^2 c^2\right )-a b \left (b^2-6 a c\right ) 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^3 \left (b^2-4 a c\right )}\\ &=-\frac{2 A b^2-6 a A c-a b C}{2 a^2 \left (b^2-4 a c\right ) x^2}-\frac{B \left (3 b^2-10 a c\right )}{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 ) x \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^2 \left (a+b x^2+c x^4\right )}-\frac{B \sqrt{c} \left (3 b^2-10 a c+\frac{3 b^3}{\sqrt{b^2-4 a c}}-\frac{16 a b 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 \sqrt{c} \left (3 b^2-10 a c-\frac{3 b^3}{\sqrt{b^2-4 a c}}+\frac{16 a b 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{\left (2 A \left (b^4-6 a b^2 c+6 a^2 c^2\right )-a b \left (b^2-6 a c\right ) C\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 )^{3/2}}-\frac{(2 A b-a C) \log (x)}{a^3}+\frac{(2 A b-a C) \log \left (a+b x^2+c x^4\right )}{4 a^3}\\ \end{align*}
Mathematica [A] time = 2.76088, size = 655, normalized size = 1.23 \[ \frac{-\frac{2 a \left (2 a^2 c C+A \left (-3 a b c-2 a c^2 x^2+b^2 c x^2+b^3\right )-a \left (b^2 C+b c x (3 B+C x)+2 B c^2 x^3\right )+b^2 B x \left (b+c x^2\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\left (2 A \left (6 a^2 c^2+b^3 \sqrt{b^2-4 a c}-6 a b^2 c-4 a b c \sqrt{b^2-4 a c}+b^4\right )+a C \left (-b^2 \sqrt{b^2-4 a c}+4 a c \sqrt{b^2-4 a c}+6 a b c-b^3\right )\right ) \log \left (\sqrt{b^2-4 a c}-b-2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac{\left (2 A \left (-6 a^2 c^2+b^3 \sqrt{b^2-4 a c}+6 a b^2 c-4 a b c \sqrt{b^2-4 a c}-b^4\right )+a C \left (-b^2 \sqrt{b^2-4 a c}+4 a c \sqrt{b^2-4 a c}-6 a b c+b^3\right )\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}+4 \log (x) (a C-2 A b)-\frac{2 a A}{x^2}+\frac{\sqrt{2} a B \sqrt{c} \left (-3 b^2 \sqrt{b^2-4 a c}+10 a c \sqrt{b^2-4 a c}+16 a b c-3 b^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} a B \sqrt{c} \left (-3 b^2 \sqrt{b^2-4 a c}+10 a c \sqrt{b^2-4 a c}-16 a b c+3 b^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{4 a B}{x}}{4 a^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.062, size = 2512, 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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