Optimal. Leaf size=115 \[ \frac{x (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{x \left (2 a c+b^2\right )+3 a b}{\left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{2 \left (2 a c+b^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}} \]
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Rubi [A] time = 0.0682724, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {1354, 738, 638, 618, 206} \[ \frac{x (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{x \left (2 a c+b^2\right )+3 a b}{\left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{2 \left (2 a c+b^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 1354
Rule 738
Rule 638
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\left (c+\frac{a}{x^2}+\frac{b}{x}\right )^3 x^4} \, dx &=\int \frac{x^2}{\left (a+b x+c x^2\right )^3} \, dx\\ &=\frac{x (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{\int \frac{2 a-2 b x}{\left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right )}\\ &=\frac{x (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{3 a b+\left (b^2+2 a c\right ) x}{\left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac{\left (b^2+2 a c\right ) \int \frac{1}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^2}\\ &=\frac{x (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{3 a b+\left (b^2+2 a c\right ) x}{\left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{\left (2 \left (b^2+2 a c\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{\left (b^2-4 a c\right )^2}\\ &=\frac{x (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{3 a b+\left (b^2+2 a c\right ) x}{\left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{2 \left (b^2+2 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.14782, size = 131, normalized size = 1.14 \[ \frac{1}{2} \left (\frac{\left (2 a c+b^2\right ) (b+2 c x)}{c \left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac{a (b-2 c x)+b^2 x}{c \left (4 a c-b^2\right ) (a+x (b+c x))^2}+\frac{4 \left (2 a c+b^2\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.009, size = 262, normalized size = 2.3 \begin{align*}{\frac{1}{ \left ( c{x}^{2}+bx+a \right ) ^{2}} \left ({\frac{c \left ( 2\,ac+{b}^{2} \right ){x}^{3}}{16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4}}}+{\frac{3\,b \left ( 2\,ac+{b}^{2} \right ){x}^{2}}{32\,{a}^{2}{c}^{2}-16\,a{b}^{2}c+2\,{b}^{4}}}-{\frac{a \left ( 2\,ac-5\,{b}^{2} \right ) x}{16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4}}}+3\,{\frac{{a}^{2}b}{16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4}}} \right ) }+4\,{\frac{ac}{ \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+2\,{\frac{{b}^{2}}{ \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.79526, size = 1893, normalized size = 16.46 \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 = 1.4503, size = 570, normalized size = 4.96 \begin{align*} - \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (2 a c + b^{2}\right ) \log{\left (x + \frac{- 64 a^{3} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (2 a c + b^{2}\right ) + 48 a^{2} b^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (2 a c + b^{2}\right ) - 12 a b^{4} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (2 a c + b^{2}\right ) + 2 a b c + b^{6} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (2 a c + b^{2}\right ) + b^{3}}{4 a c^{2} + 2 b^{2} c} \right )} + \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (2 a c + b^{2}\right ) \log{\left (x + \frac{64 a^{3} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (2 a c + b^{2}\right ) - 48 a^{2} b^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (2 a c + b^{2}\right ) + 12 a b^{4} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (2 a c + b^{2}\right ) + 2 a b c - b^{6} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (2 a c + b^{2}\right ) + b^{3}}{4 a c^{2} + 2 b^{2} c} \right )} + \frac{6 a^{2} b + x^{3} \left (4 a c^{2} + 2 b^{2} c\right ) + x^{2} \left (6 a b c + 3 b^{3}\right ) + x \left (- 4 a^{2} c + 10 a b^{2}\right )}{32 a^{4} c^{2} - 16 a^{3} b^{2} c + 2 a^{2} b^{4} + x^{4} \left (32 a^{2} c^{4} - 16 a b^{2} c^{3} + 2 b^{4} c^{2}\right ) + x^{3} \left (64 a^{2} b c^{3} - 32 a b^{3} c^{2} + 4 b^{5} c\right ) + x^{2} \left (64 a^{3} c^{3} - 12 a b^{4} c + 2 b^{6}\right ) + x \left (64 a^{3} b c^{2} - 32 a^{2} b^{3} c + 4 a b^{5}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15729, size = 208, normalized size = 1.81 \begin{align*} \frac{2 \,{\left (b^{2} + 2 \, a c\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{2 \, b^{2} c x^{3} + 4 \, a c^{2} x^{3} + 3 \, b^{3} x^{2} + 6 \, a b c x^{2} + 10 \, a b^{2} x - 4 \, a^{2} c x + 6 \, a^{2} b}{2 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}{\left (c x^{2} + b x + a\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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