Optimal. Leaf size=110 \[ -\frac{1}{d^3 \left (b^2-4 a c\right ) (b+2 c x)^2 \left (a+b x+c x^2\right )}-\frac{8 c \log \left (a+b x+c x^2\right )}{d^3 \left (b^2-4 a c\right )^3}-\frac{8 c}{d^3 \left (b^2-4 a c\right )^2 (b+2 c x)^2}+\frac{16 c \log (b+2 c x)}{d^3 \left (b^2-4 a c\right )^3} \]
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Rubi [A] time = 0.0621794, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {687, 693, 681, 31, 628} \[ -\frac{1}{d^3 \left (b^2-4 a c\right ) (b+2 c x)^2 \left (a+b x+c x^2\right )}-\frac{8 c \log \left (a+b x+c x^2\right )}{d^3 \left (b^2-4 a c\right )^3}-\frac{8 c}{d^3 \left (b^2-4 a c\right )^2 (b+2 c x)^2}+\frac{16 c \log (b+2 c x)}{d^3 \left (b^2-4 a c\right )^3} \]
Antiderivative was successfully verified.
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Rule 687
Rule 693
Rule 681
Rule 31
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )^2} \, dx &=-\frac{1}{\left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )}-\frac{(8 c) \int \frac{1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )} \, dx}{b^2-4 a c}\\ &=-\frac{8 c}{\left (b^2-4 a c\right )^2 d^3 (b+2 c x)^2}-\frac{1}{\left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )}-\frac{(8 c) \int \frac{1}{(b d+2 c d x) \left (a+b x+c x^2\right )} \, dx}{\left (b^2-4 a c\right )^2 d^2}\\ &=-\frac{8 c}{\left (b^2-4 a c\right )^2 d^3 (b+2 c x)^2}-\frac{1}{\left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )}-\frac{(8 c) \int \frac{b d+2 c d x}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^3 d^4}+\frac{\left (32 c^2\right ) \int \frac{1}{b+2 c x} \, dx}{\left (b^2-4 a c\right )^3 d^3}\\ &=-\frac{8 c}{\left (b^2-4 a c\right )^2 d^3 (b+2 c x)^2}-\frac{1}{\left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )}+\frac{16 c \log (b+2 c x)}{\left (b^2-4 a c\right )^3 d^3}-\frac{8 c \log \left (a+b x+c x^2\right )}{\left (b^2-4 a c\right )^3 d^3}\\ \end{align*}
Mathematica [A] time = 0.0902598, size = 79, normalized size = 0.72 \[ \frac{-\frac{4 c \left (b^2-4 a c\right )}{(b+2 c x)^2}+\frac{4 a c-b^2}{a+x (b+c x)}-8 c \log (a+x (b+c x))+16 c \log (b+2 c x)}{d^3 \left (b^2-4 a c\right )^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 144, normalized size = 1.3 \begin{align*} -4\,{\frac{ac}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( c{x}^{2}+bx+a \right ) }}+{\frac{{b}^{2}}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( c{x}^{2}+bx+a \right ) }}+8\,{\frac{c\ln \left ( c{x}^{2}+bx+a \right ) }{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{3}}}-16\,{\frac{c\ln \left ( 2\,cx+b \right ) }{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{3}}}-4\,{\frac{c}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( 2\,cx+b \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.12979, size = 400, normalized size = 3.64 \begin{align*} -\frac{8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c}{4 \,{\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{3} x^{4} + 8 \,{\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} d^{3} x^{3} +{\left (5 \, b^{6} c - 36 \, a b^{4} c^{2} + 48 \, a^{2} b^{2} c^{3} + 64 \, a^{3} c^{4}\right )} d^{3} x^{2} +{\left (b^{7} - 4 \, a b^{5} c - 16 \, a^{2} b^{3} c^{2} + 64 \, a^{3} b c^{3}\right )} d^{3} x +{\left (a b^{6} - 8 \, a^{2} b^{4} c + 16 \, a^{3} b^{2} c^{2}\right )} d^{3}} - \frac{8 \, c \log \left (c x^{2} + b x + a\right )}{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{3}} + \frac{16 \, c \log \left (2 \, c x + b\right )}{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.05137, size = 864, normalized size = 7.85 \begin{align*} -\frac{b^{4} - 16 \, a^{2} c^{2} + 8 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} + 8 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} x + 8 \,{\left (4 \, c^{4} x^{4} + 8 \, b c^{3} x^{3} + a b^{2} c +{\left (5 \, b^{2} c^{2} + 4 \, a c^{3}\right )} x^{2} +{\left (b^{3} c + 4 \, a b c^{2}\right )} x\right )} \log \left (c x^{2} + b x + a\right ) - 16 \,{\left (4 \, c^{4} x^{4} + 8 \, b c^{3} x^{3} + a b^{2} c +{\left (5 \, b^{2} c^{2} + 4 \, a c^{3}\right )} x^{2} +{\left (b^{3} c + 4 \, a b c^{2}\right )} x\right )} \log \left (2 \, c x + b\right )}{4 \,{\left (b^{6} c^{3} - 12 \, a b^{4} c^{4} + 48 \, a^{2} b^{2} c^{5} - 64 \, a^{3} c^{6}\right )} d^{3} x^{4} + 8 \,{\left (b^{7} c^{2} - 12 \, a b^{5} c^{3} + 48 \, a^{2} b^{3} c^{4} - 64 \, a^{3} b c^{5}\right )} d^{3} x^{3} +{\left (5 \, b^{8} c - 56 \, a b^{6} c^{2} + 192 \, a^{2} b^{4} c^{3} - 128 \, a^{3} b^{2} c^{4} - 256 \, a^{4} c^{5}\right )} d^{3} x^{2} +{\left (b^{9} - 8 \, a b^{7} c + 128 \, a^{3} b^{3} c^{3} - 256 \, a^{4} b c^{4}\right )} d^{3} x +{\left (a b^{8} - 12 \, a^{2} b^{6} c + 48 \, a^{3} b^{4} c^{2} - 64 \, a^{4} b^{2} c^{3}\right )} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 6.44781, size = 303, normalized size = 2.75 \begin{align*} - \frac{16 c \log{\left (\frac{b}{2 c} + x \right )}}{d^{3} \left (4 a c - b^{2}\right )^{3}} + \frac{8 c \log{\left (\frac{a}{c} + \frac{b x}{c} + x^{2} \right )}}{d^{3} \left (4 a c - b^{2}\right )^{3}} - \frac{4 a c + b^{2} + 8 b c x + 8 c^{2} x^{2}}{16 a^{3} b^{2} c^{2} d^{3} - 8 a^{2} b^{4} c d^{3} + a b^{6} d^{3} + x^{4} \left (64 a^{2} c^{5} d^{3} - 32 a b^{2} c^{4} d^{3} + 4 b^{4} c^{3} d^{3}\right ) + x^{3} \left (128 a^{2} b c^{4} d^{3} - 64 a b^{3} c^{3} d^{3} + 8 b^{5} c^{2} d^{3}\right ) + x^{2} \left (64 a^{3} c^{4} d^{3} + 48 a^{2} b^{2} c^{3} d^{3} - 36 a b^{4} c^{2} d^{3} + 5 b^{6} c d^{3}\right ) + x \left (64 a^{3} b c^{3} d^{3} - 16 a^{2} b^{3} c^{2} d^{3} - 4 a b^{5} c d^{3} + b^{7} d^{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17857, size = 274, normalized size = 2.49 \begin{align*} \frac{16 \, c^{2} \log \left ({\left | 2 \, c x + b \right |}\right )}{b^{6} c d^{3} - 12 \, a b^{4} c^{2} d^{3} + 48 \, a^{2} b^{2} c^{3} d^{3} - 64 \, a^{3} c^{4} d^{3}} - \frac{8 \, c \log \left (c x^{2} + b x + a\right )}{b^{6} d^{3} - 12 \, a b^{4} c d^{3} + 48 \, a^{2} b^{2} c^{2} d^{3} - 64 \, a^{3} c^{3} d^{3}} - \frac{b^{4} - 16 \, a^{2} c^{2} + 8 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} + 8 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} x}{{\left (c x^{2} + b x + a\right )}{\left (b^{2} - 4 \, a c\right )}^{3}{\left (2 \, c x + b\right )}^{2} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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