Optimal. Leaf size=66 \[ \frac{4 c \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{b+2 c x}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]
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Rubi [A] time = 0.0345985, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {1585, 614, 618, 206} \[ \frac{4 c \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{b+2 c x}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]
Antiderivative was successfully verified.
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Rule 1585
Rule 614
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{x^4}{\left (a x^2+b x^3+c x^4\right )^2} \, dx &=\int \frac{1}{\left (a+b x+c x^2\right )^2} \, dx\\ &=-\frac{b+2 c x}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{(2 c) \int \frac{1}{a+b x+c x^2} \, dx}{b^2-4 a c}\\ &=-\frac{b+2 c x}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{(4 c) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{b^2-4 a c}\\ &=-\frac{b+2 c x}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{4 c \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0794278, size = 70, normalized size = 1.06 \[ -\frac{\frac{4 c \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+\frac{b+2 c x}{a+x (b+c x)}}{b^2-4 a c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 68, normalized size = 1. \begin{align*}{\frac{2\,cx+b}{ \left ( 4\,ac-{b}^{2} \right ) \left ( c{x}^{2}+bx+a \right ) }}+4\,{\frac{c}{ \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\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 = 1.55073, size = 745, normalized size = 11.29 \begin{align*} \left [-\frac{b^{3} - 4 \, a b c + 2 \,{\left (c^{2} x^{2} + b c x + a c\right )} \sqrt{b^{2} - 4 \, a c} \log \left (\frac{2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt{b^{2} - 4 \, a c}{\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x}{a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} +{\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{2} +{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x}, -\frac{b^{3} - 4 \, a b c - 4 \,{\left (c^{2} x^{2} + b c x + a c\right )} \sqrt{-b^{2} + 4 \, a c} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x}{a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} +{\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{2} +{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.721547, size = 265, normalized size = 4.02 \begin{align*} - 2 c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \log{\left (x + \frac{- 32 a^{2} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + 16 a b^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} - 2 b^{4} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + 2 b c}{4 c^{2}} \right )} + 2 c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \log{\left (x + \frac{32 a^{2} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} - 16 a b^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + 2 b^{4} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + 2 b c}{4 c^{2}} \right )} + \frac{b + 2 c x}{4 a^{2} c - a b^{2} + x^{2} \left (4 a c^{2} - b^{2} c\right ) + x \left (4 a b c - b^{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10605, size = 103, normalized size = 1.56 \begin{align*} -\frac{4 \, c \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} - 4 \, a c\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{2 \, c x + b}{{\left (c x^{2} + b x + a\right )}{\left (b^{2} - 4 \, a c\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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