Optimal. Leaf size=76 \[ -12 c d^4 \sqrt{b^2-4 a c} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )-\frac{d^4 (b+2 c x)^3}{a+b x+c x^2}+12 c d^4 (b+2 c x) \]
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Rubi [A] time = 0.0507377, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {686, 692, 618, 206} \[ -12 c d^4 \sqrt{b^2-4 a c} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )-\frac{d^4 (b+2 c x)^3}{a+b x+c x^2}+12 c d^4 (b+2 c x) \]
Antiderivative was successfully verified.
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Rule 686
Rule 692
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac{d^4 (b+2 c x)^3}{a+b x+c x^2}+\left (6 c d^2\right ) \int \frac{(b d+2 c d x)^2}{a+b x+c x^2} \, dx\\ &=12 c d^4 (b+2 c x)-\frac{d^4 (b+2 c x)^3}{a+b x+c x^2}+\left (6 c \left (b^2-4 a c\right ) d^4\right ) \int \frac{1}{a+b x+c x^2} \, dx\\ &=12 c d^4 (b+2 c x)-\frac{d^4 (b+2 c x)^3}{a+b x+c x^2}-\left (12 c \left (b^2-4 a c\right ) d^4\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )\\ &=12 c d^4 (b+2 c x)-\frac{d^4 (b+2 c x)^3}{a+b x+c x^2}-12 c \sqrt{b^2-4 a c} d^4 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )\\ \end{align*}
Mathematica [A] time = 0.0517713, size = 77, normalized size = 1.01 \[ d^4 \left (-\frac{\left (b^2-4 a c\right ) (b+2 c x)}{a+x (b+c x)}-12 c \sqrt{4 a c-b^2} \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )+16 c^2 x\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.157, size = 133, normalized size = 1.8 \begin{align*} 16\,{d}^{4}{c}^{2}x+8\,{\frac{{d}^{4}a{c}^{2}x}{c{x}^{2}+bx+a}}-2\,{\frac{{d}^{4}{b}^{2}cx}{c{x}^{2}+bx+a}}+4\,{\frac{{d}^{4}abc}{c{x}^{2}+bx+a}}-{\frac{{d}^{4}{b}^{3}}{c{x}^{2}+bx+a}}-12\,{d}^{4}c\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 [A] time = 2.08437, size = 647, normalized size = 8.51 \begin{align*} \left [\frac{16 \, c^{3} d^{4} x^{3} + 16 \, b c^{2} d^{4} x^{2} - 2 \,{\left (b^{2} c - 12 \, a c^{2}\right )} d^{4} x -{\left (b^{3} - 4 \, a b c\right )} d^{4} + 6 \,{\left (c^{2} d^{4} x^{2} + b c d^{4} x + a c d^{4}\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 )}{c x^{2} + b x + a}, \frac{16 \, c^{3} d^{4} x^{3} + 16 \, b c^{2} d^{4} x^{2} - 2 \,{\left (b^{2} c - 12 \, a c^{2}\right )} d^{4} x -{\left (b^{3} - 4 \, a b c\right )} d^{4} - 12 \,{\left (c^{2} d^{4} x^{2} + b c d^{4} x + a c d^{4}\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 )}{c x^{2} + b x + a}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.65756, size = 173, normalized size = 2.28 \begin{align*} 16 c^{2} d^{4} x + c d^{4} \sqrt{- 144 a c + 36 b^{2}} \log{\left (x + \frac{6 b c d^{4} - c d^{4} \sqrt{- 144 a c + 36 b^{2}}}{12 c^{2} d^{4}} \right )} - c d^{4} \sqrt{- 144 a c + 36 b^{2}} \log{\left (x + \frac{6 b c d^{4} + c d^{4} \sqrt{- 144 a c + 36 b^{2}}}{12 c^{2} d^{4}} \right )} + \frac{4 a b c d^{4} - b^{3} d^{4} + x \left (8 a c^{2} d^{4} - 2 b^{2} c d^{4}\right )}{a + b x + c x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16134, size = 151, normalized size = 1.99 \begin{align*} 16 \, c^{2} d^{4} x + \frac{12 \,{\left (b^{2} c d^{4} - 4 \, a c^{2} d^{4}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c}} - \frac{2 \, b^{2} c d^{4} x - 8 \, a c^{2} d^{4} x + b^{3} d^{4} - 4 \, a b c d^{4}}{c x^{2} + b x + a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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