Optimal. Leaf size=150 \[ -\frac{2 \left (6 a^2 c^2-6 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{3/2}}+\frac{2 x \left (b^2-3 a c\right )}{c^2 \left (b^2-4 a c\right )}+\frac{x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{b x^2}{c \left (b^2-4 a c\right )}-\frac{b \log \left (a+b x+c x^2\right )}{c^3} \]
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Rubi [A] time = 0.159277, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {1585, 738, 800, 634, 618, 206, 628} \[ -\frac{2 \left (6 a^2 c^2-6 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{3/2}}+\frac{2 x \left (b^2-3 a c\right )}{c^2 \left (b^2-4 a c\right )}+\frac{x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{b x^2}{c \left (b^2-4 a c\right )}-\frac{b \log \left (a+b x+c x^2\right )}{c^3} \]
Antiderivative was successfully verified.
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Rule 1585
Rule 738
Rule 800
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{x^8}{\left (a x^2+b x^3+c x^4\right )^2} \, dx &=\int \frac{x^4}{\left (a+b x+c x^2\right )^2} \, dx\\ &=\frac{x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{\int \frac{x^2 (6 a+2 b x)}{a+b x+c x^2} \, dx}{-b^2+4 a c}\\ &=\frac{x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{\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}{-b^2+4 a c}\\ &=\frac{2 \left (b^2-3 a c\right ) x}{c^2 \left (b^2-4 a c\right )}-\frac{b x^2}{c \left (b^2-4 a c\right )}+\frac{x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{2 \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}{c^2 \left (b^2-4 a c\right )}\\ &=\frac{2 \left (b^2-3 a c\right ) x}{c^2 \left (b^2-4 a c\right )}-\frac{b x^2}{c \left (b^2-4 a c\right )}+\frac{x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{b \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{c^3}+\frac{\left (b^4-6 a b^2 c+6 a^2 c^2\right ) \int \frac{1}{a+b x+c x^2} \, dx}{c^3 \left (b^2-4 a c\right )}\\ &=\frac{2 \left (b^2-3 a c\right ) x}{c^2 \left (b^2-4 a c\right )}-\frac{b x^2}{c \left (b^2-4 a c\right )}+\frac{x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{b \log \left (a+b x+c x^2\right )}{c^3}-\frac{\left (2 \left (b^4-6 a b^2 c+6 a^2 c^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c^3 \left (b^2-4 a c\right )}\\ &=\frac{2 \left (b^2-3 a c\right ) x}{c^2 \left (b^2-4 a c\right )}-\frac{b x^2}{c \left (b^2-4 a c\right )}+\frac{x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{2 \left (b^4-6 a b^2 c+6 a^2 c^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\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+c x^2\right )}{c^3}\\ \end{align*}
Mathematica [A] time = 0.192539, size = 132, normalized size = 0.88 \[ \frac{-\frac{2 \left (6 a^2 c^2-6 a b^2 c+b^4\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+\frac{a^2 c (3 b-2 c x)-a b^2 (b-4 c x)+b^4 (-x)}{\left (b^2-4 a c\right ) (a+x (b+c x))}-b \log (a+x (b+c x))+c x}{c^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.008, size = 352, normalized size = 2.4 \begin{align*}{\frac{x}{{c}^{2}}}+2\,{\frac{x{a}^{2}}{c \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-4\,{\frac{ax{b}^{2}}{{c}^{2} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{x{b}^{4}}{{c}^{3} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-3\,{\frac{{a}^{2}b}{{c}^{2} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{{b}^{3}a}{{c}^{3} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-4\,{\frac{\ln \left ( c{x}^{2}+bx+a \right ) ab}{ \left ( 4\,ac-{b}^{2} \right ){c}^{2}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ){b}^{3}}{{c}^{3} \left ( 4\,ac-{b}^{2} \right ) }}-12\,{\frac{{a}^{2}}{c \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+12\,{\frac{a{b}^{2}}{{c}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-2\,{\frac{{b}^{4}}{{c}^{3} \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 = 1.59795, size = 1760, normalized size = 11.73 \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 = 1.62455, size = 842, normalized size = 5.61 \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 = 1.11596, size = 217, normalized size = 1.45 \begin{align*} \frac{2 \,{\left (b^{4} - 6 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} \arctan \left (\frac{2 \, c x + 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}{c^{2}} - \frac{b \log \left (c x^{2} + b x + a\right )}{c^{3}} - \frac{\frac{{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} x}{c} + \frac{a b^{3} - 3 \, a^{2} b c}{c}}{{\left (c x^{2} + b x + a\right )}{\left (b^{2} - 4 \, a c\right )} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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