Optimal. Leaf size=72 \[ -\frac{\left (b^2-4 a c\right )^2}{128 c^3 d^5 (b+2 c x)^4}+\frac{b^2-4 a c}{32 c^3 d^5 (b+2 c x)^2}+\frac{\log (b+2 c x)}{32 c^3 d^5} \]
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Rubi [A] time = 0.0576435, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {683} \[ -\frac{\left (b^2-4 a c\right )^2}{128 c^3 d^5 (b+2 c x)^4}+\frac{b^2-4 a c}{32 c^3 d^5 (b+2 c x)^2}+\frac{\log (b+2 c x)}{32 c^3 d^5} \]
Antiderivative was successfully verified.
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Rule 683
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^5} \, dx &=\int \left (\frac{\left (-b^2+4 a c\right )^2}{16 c^2 d^5 (b+2 c x)^5}+\frac{-b^2+4 a c}{8 c^2 d^5 (b+2 c x)^3}+\frac{1}{16 c^2 d^5 (b+2 c x)}\right ) \, dx\\ &=-\frac{\left (b^2-4 a c\right )^2}{128 c^3 d^5 (b+2 c x)^4}+\frac{b^2-4 a c}{32 c^3 d^5 (b+2 c x)^2}+\frac{\log (b+2 c x)}{32 c^3 d^5}\\ \end{align*}
Mathematica [A] time = 0.0348702, size = 59, normalized size = 0.82 \[ \frac{\frac{\left (b^2-4 a c\right ) \left (4 c \left (a+4 c x^2\right )+3 b^2+16 b c x\right )}{(b+2 c x)^4}+4 \log (b+2 c x)}{128 c^3 d^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 111, normalized size = 1.5 \begin{align*} -{\frac{a}{8\,{c}^{2}{d}^{5} \left ( 2\,cx+b \right ) ^{2}}}+{\frac{{b}^{2}}{32\,{c}^{3}{d}^{5} \left ( 2\,cx+b \right ) ^{2}}}-{\frac{{a}^{2}}{8\,{d}^{5}c \left ( 2\,cx+b \right ) ^{4}}}+{\frac{{b}^{2}a}{16\,{c}^{2}{d}^{5} \left ( 2\,cx+b \right ) ^{4}}}-{\frac{{b}^{4}}{128\,{c}^{3}{d}^{5} \left ( 2\,cx+b \right ) ^{4}}}+{\frac{\ln \left ( 2\,cx+b \right ) }{32\,{c}^{3}{d}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.20123, size = 184, normalized size = 2.56 \begin{align*} \frac{3 \, b^{4} - 8 \, a b^{2} c - 16 \, a^{2} c^{2} + 16 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} + 16 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} x}{128 \,{\left (16 \, c^{7} d^{5} x^{4} + 32 \, b c^{6} d^{5} x^{3} + 24 \, b^{2} c^{5} d^{5} x^{2} + 8 \, b^{3} c^{4} d^{5} x + b^{4} c^{3} d^{5}\right )}} + \frac{\log \left (2 \, c x + b\right )}{32 \, c^{3} d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.05401, size = 360, normalized size = 5. \begin{align*} \frac{3 \, b^{4} - 8 \, a b^{2} c - 16 \, a^{2} c^{2} + 16 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} + 16 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} x + 4 \,{\left (16 \, c^{4} x^{4} + 32 \, b c^{3} x^{3} + 24 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x + b^{4}\right )} \log \left (2 \, c x + b\right )}{128 \,{\left (16 \, c^{7} d^{5} x^{4} + 32 \, b c^{6} d^{5} x^{3} + 24 \, b^{2} c^{5} d^{5} x^{2} + 8 \, b^{3} c^{4} d^{5} x + b^{4} c^{3} d^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.07894, size = 139, normalized size = 1.93 \begin{align*} - \frac{16 a^{2} c^{2} + 8 a b^{2} c - 3 b^{4} + x^{2} \left (64 a c^{3} - 16 b^{2} c^{2}\right ) + x \left (64 a b c^{2} - 16 b^{3} c\right )}{128 b^{4} c^{3} d^{5} + 1024 b^{3} c^{4} d^{5} x + 3072 b^{2} c^{5} d^{5} x^{2} + 4096 b c^{6} d^{5} x^{3} + 2048 c^{7} d^{5} x^{4}} + \frac{\log{\left (b + 2 c x \right )}}{32 c^{3} d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17544, size = 197, normalized size = 2.74 \begin{align*} -\frac{\log \left (\frac{1}{4 \,{\left (2 \, c d x + b d\right )}^{2} c^{2} d^{2}}\right )}{64 \, c^{3} d^{5}} - \frac{\frac{b^{4} c^{3} d^{9}}{{\left (2 \, c d x + b d\right )}^{4}} - \frac{8 \, a b^{2} c^{4} d^{9}}{{\left (2 \, c d x + b d\right )}^{4}} + \frac{16 \, a^{2} c^{5} d^{9}}{{\left (2 \, c d x + b d\right )}^{4}} - \frac{4 \, b^{2} c^{3} d^{7}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac{16 \, a c^{4} d^{7}}{{\left (2 \, c d x + b d\right )}^{2}}}{128 \, c^{6} d^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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