Optimal. Leaf size=114 \[ \frac{b \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^2 \left (b^2-4 a c\right )^{3/2}}+\frac{x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{b x}{c \left (b^2-4 a c\right )}+\frac{\log \left (a+b x+c x^2\right )}{2 c^2} \]
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Rubi [A] time = 0.103282, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {1585, 738, 773, 634, 618, 206, 628} \[ \frac{b \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^2 \left (b^2-4 a c\right )^{3/2}}+\frac{x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{b x}{c \left (b^2-4 a c\right )}+\frac{\log \left (a+b x+c x^2\right )}{2 c^2} \]
Antiderivative was successfully verified.
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Rule 1585
Rule 738
Rule 773
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{x^7}{\left (a x^2+b x^3+c x^4\right )^2} \, dx &=\int \frac{x^3}{\left (a+b x+c x^2\right )^2} \, dx\\ &=\frac{x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{\int \frac{x (4 a+b x)}{a+b x+c x^2} \, dx}{-b^2+4 a c}\\ &=-\frac{b x}{c \left (b^2-4 a c\right )}+\frac{x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{\int \frac{-a b+\left (-b^2+4 a c\right ) x}{a+b x+c x^2} \, dx}{c \left (b^2-4 a c\right )}\\ &=-\frac{b x}{c \left (b^2-4 a c\right )}+\frac{x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{\int \frac{b+2 c x}{a+b x+c x^2} \, dx}{2 c^2}-\frac{\left (b \left (b^2-6 a c\right )\right ) \int \frac{1}{a+b x+c x^2} \, dx}{2 c^2 \left (b^2-4 a c\right )}\\ &=-\frac{b x}{c \left (b^2-4 a c\right )}+\frac{x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{\log \left (a+b x+c x^2\right )}{2 c^2}+\frac{\left (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\right )}{c^2 \left (b^2-4 a c\right )}\\ &=-\frac{b x}{c \left (b^2-4 a c\right )}+\frac{x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{b \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^2 \left (b^2-4 a c\right )^{3/2}}+\frac{\log \left (a+b x+c x^2\right )}{2 c^2}\\ \end{align*}
Mathematica [A] time = 0.153361, size = 109, normalized size = 0.96 \[ \frac{\frac{2 \left (-2 a^2 c+a b (b-3 c x)+b^3 x\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\frac{2 b \left (b^2-6 a c\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+\log (a+x (b+c x))}{2 c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 209, normalized size = 1.8 \begin{align*}{\frac{1}{c{x}^{2}+bx+a} \left ({\frac{b \left ( 3\,ac-{b}^{2} \right ) x}{{c}^{2} \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{a \left ( 2\,ac-{b}^{2} \right ) }{{c}^{2} \left ( 4\,ac-{b}^{2} \right ) }} \right ) }+2\,{\frac{\ln \left ( c{x}^{2}+bx+a \right ) a}{c \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ){b}^{2}}{2\,{c}^{2} \left ( 4\,ac-{b}^{2} \right ) }}-6\,{\frac{ab}{c \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{{b}^{3}}{{c}^{2}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ) \left ( 4\,ac-{b}^{2} \right ) ^{-{\frac{3}{2}}}} \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 = 1.65751, size = 1374, normalized size = 12.05 \begin{align*} \left [\frac{2 \, a b^{4} - 12 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} +{\left (a b^{3} - 6 \, a^{2} b c +{\left (b^{3} c - 6 \, a b c^{2}\right )} x^{2} +{\left (b^{4} - 6 \, a b^{2} c\right )} x\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^{5} - 7 \, a b^{3} c + 12 \, a^{2} b c^{2}\right )} x +{\left (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 )} \log \left (c x^{2} + b x + a\right )}{2 \,{\left (a b^{4} c^{2} - 8 \, a^{2} b^{2} c^{3} + 16 \, a^{3} c^{4} +{\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} x^{2} +{\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} x\right )}}, \frac{2 \, a b^{4} - 12 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + 2 \,{\left (a b^{3} - 6 \, a^{2} b c +{\left (b^{3} c - 6 \, a b c^{2}\right )} x^{2} +{\left (b^{4} - 6 \, a b^{2} c\right )} x\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^{5} - 7 \, a b^{3} c + 12 \, a^{2} b c^{2}\right )} x +{\left (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 )} \log \left (c x^{2} + b x + a\right )}{2 \,{\left (a b^{4} c^{2} - 8 \, a^{2} b^{2} c^{3} + 16 \, a^{3} c^{4} +{\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} x^{2} +{\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} x\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.20897, size = 729, normalized size = 6.39 \begin{align*} \left (- \frac{b \sqrt{- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{2 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac{1}{2 c^{2}}\right ) \log{\left (x + \frac{- 16 a^{2} c^{3} \left (- \frac{b \sqrt{- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{2 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac{1}{2 c^{2}}\right ) + 8 a^{2} c + 8 a b^{2} c^{2} \left (- \frac{b \sqrt{- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{2 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac{1}{2 c^{2}}\right ) - a b^{2} - b^{4} c \left (- \frac{b \sqrt{- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{2 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac{1}{2 c^{2}}\right )}{6 a b c - b^{3}} \right )} + \left (\frac{b \sqrt{- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{2 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac{1}{2 c^{2}}\right ) \log{\left (x + \frac{- 16 a^{2} c^{3} \left (\frac{b \sqrt{- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{2 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac{1}{2 c^{2}}\right ) + 8 a^{2} c + 8 a b^{2} c^{2} \left (\frac{b \sqrt{- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{2 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac{1}{2 c^{2}}\right ) - a b^{2} - b^{4} c \left (\frac{b \sqrt{- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{2 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac{1}{2 c^{2}}\right )}{6 a b c - b^{3}} \right )} + \frac{2 a^{2} c - a b^{2} + x \left (3 a b c - b^{3}\right )}{4 a^{2} c^{3} - a b^{2} c^{2} + x^{2} \left (4 a c^{4} - b^{2} c^{3}\right ) + x \left (4 a b c^{3} - b^{3} c^{2}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09743, size = 169, normalized size = 1.48 \begin{align*} -\frac{{\left (b^{3} - 6 \, a b c\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{\log \left (c x^{2} + b x + a\right )}{2 \, c^{2}} + \frac{a b^{2} - 2 \, a^{2} c +{\left (b^{3} - 3 \, a b c\right )} x}{{\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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