Optimal. Leaf size=98 \[ -\frac{1}{d^2 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )}-\frac{12 c}{d^2 \left (b^2-4 a c\right )^2 (b+2 c x)}+\frac{12 c \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{d^2 \left (b^2-4 a c\right )^{5/2}} \]
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Rubi [A] time = 0.0575797, antiderivative size = 98, 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 = {687, 693, 618, 206} \[ -\frac{1}{d^2 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )}-\frac{12 c}{d^2 \left (b^2-4 a c\right )^2 (b+2 c x)}+\frac{12 c \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{d^2 \left (b^2-4 a c\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 687
Rule 693
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^2} \, dx &=-\frac{1}{\left (b^2-4 a c\right ) d^2 (b+2 c x) \left (a+b x+c x^2\right )}-\frac{(6 c) \int \frac{1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )} \, dx}{b^2-4 a c}\\ &=-\frac{12 c}{\left (b^2-4 a c\right )^2 d^2 (b+2 c x)}-\frac{1}{\left (b^2-4 a c\right ) d^2 (b+2 c x) \left (a+b x+c x^2\right )}-\frac{(6 c) \int \frac{1}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^2 d^2}\\ &=-\frac{12 c}{\left (b^2-4 a c\right )^2 d^2 (b+2 c x)}-\frac{1}{\left (b^2-4 a c\right ) d^2 (b+2 c x) \left (a+b x+c x^2\right )}+\frac{(12 c) \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 d^2}\\ &=-\frac{12 c}{\left (b^2-4 a c\right )^2 d^2 (b+2 c x)}-\frac{1}{\left (b^2-4 a c\right ) d^2 (b+2 c x) \left (a+b x+c x^2\right )}+\frac{12 c \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} d^2}\\ \end{align*}
Mathematica [A] time = 0.121909, size = 84, normalized size = 0.86 \[ -\frac{\frac{12 c \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+\frac{b+2 c x}{a+x (b+c x)}+\frac{8 c}{b+2 c x}}{d^2 \left (b^2-4 a c\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.158, size = 127, normalized size = 1.3 \begin{align*} -2\,{\frac{cx}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) }}-{\frac{b}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) }}-12\,{\frac{c}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{5/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-8\,{\frac{c}{{d}^{2} \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( 2\,cx+b \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 = 2.15535, size = 1384, normalized size = 14.12 \begin{align*} \left [-\frac{b^{4} + 4 \, a b^{2} c - 32 \, a^{2} c^{2} + 12 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} - 6 \,{\left (2 \, c^{3} x^{3} + 3 \, b c^{2} x^{2} + a b c +{\left (b^{2} c + 2 \, a c^{2}\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 ) + 12 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} x}{2 \,{\left (b^{6} c^{2} - 12 \, a b^{4} c^{3} + 48 \, a^{2} b^{2} c^{4} - 64 \, a^{3} c^{5}\right )} d^{2} x^{3} + 3 \,{\left (b^{7} c - 12 \, a b^{5} c^{2} + 48 \, a^{2} b^{3} c^{3} - 64 \, a^{3} b c^{4}\right )} d^{2} x^{2} +{\left (b^{8} - 10 \, a b^{6} c + 24 \, a^{2} b^{4} c^{2} + 32 \, a^{3} b^{2} c^{3} - 128 \, a^{4} c^{4}\right )} d^{2} x +{\left (a b^{7} - 12 \, a^{2} b^{5} c + 48 \, a^{3} b^{3} c^{2} - 64 \, a^{4} b c^{3}\right )} d^{2}}, -\frac{b^{4} + 4 \, a b^{2} c - 32 \, a^{2} c^{2} + 12 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} - 12 \,{\left (2 \, c^{3} x^{3} + 3 \, b c^{2} x^{2} + a b c +{\left (b^{2} c + 2 \, a c^{2}\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 ) + 12 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} x}{2 \,{\left (b^{6} c^{2} - 12 \, a b^{4} c^{3} + 48 \, a^{2} b^{2} c^{4} - 64 \, a^{3} c^{5}\right )} d^{2} x^{3} + 3 \,{\left (b^{7} c - 12 \, a b^{5} c^{2} + 48 \, a^{2} b^{3} c^{3} - 64 \, a^{3} b c^{4}\right )} d^{2} x^{2} +{\left (b^{8} - 10 \, a b^{6} c + 24 \, a^{2} b^{4} c^{2} + 32 \, a^{3} b^{2} c^{3} - 128 \, a^{4} c^{4}\right )} d^{2} x +{\left (a b^{7} - 12 \, a^{2} b^{5} c + 48 \, a^{3} b^{3} c^{2} - 64 \, a^{4} b c^{3}\right )} d^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.676, size = 457, normalized size = 4.66 \begin{align*} \frac{6 c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \log{\left (x + \frac{- 384 a^{3} c^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 288 a^{2} b^{2} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} - 72 a b^{4} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 6 b^{6} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 6 b c}{12 c^{2}} \right )}}{d^{2}} - \frac{6 c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \log{\left (x + \frac{384 a^{3} c^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} - 288 a^{2} b^{2} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 72 a b^{4} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} - 6 b^{6} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 6 b c}{12 c^{2}} \right )}}{d^{2}} - \frac{8 a c + b^{2} + 12 b c x + 12 c^{2} x^{2}}{16 a^{3} b c^{2} d^{2} - 8 a^{2} b^{3} c d^{2} + a b^{5} d^{2} + x^{3} \left (32 a^{2} c^{4} d^{2} - 16 a b^{2} c^{3} d^{2} + 2 b^{4} c^{2} d^{2}\right ) + x^{2} \left (48 a^{2} b c^{3} d^{2} - 24 a b^{3} c^{2} d^{2} + 3 b^{5} c d^{2}\right ) + x \left (32 a^{3} c^{3} d^{2} - 6 a b^{4} c d^{2} + b^{6} d^{2}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.23668, size = 297, normalized size = 3.03 \begin{align*} -\frac{8 \, c^{5} d^{7}}{{\left (b^{4} c^{4} d^{8} - 8 \, a b^{2} c^{5} d^{8} + 16 \, a^{2} c^{6} d^{8}\right )}{\left (2 \, c d x + b d\right )}} - \frac{12 \, c \arctan \left (\frac{\frac{b^{2} d}{2 \, c d x + b d} - \frac{4 \, a c d}{2 \, c d x + b d}}{\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} d^{2}} + \frac{4 \, c}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}{\left (2 \, c d x + b d\right )}{\left (\frac{b^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} - \frac{4 \, a c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} - 1\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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