Optimal. Leaf size=73 \[ -\frac{\left (b^2-4 a c\right )^2}{160 c^3 d^6 (b+2 c x)^5}+\frac{b^2-4 a c}{48 c^3 d^6 (b+2 c x)^3}-\frac{1}{32 c^3 d^6 (b+2 c x)} \]
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Rubi [A] time = 0.0569732, antiderivative size = 73, 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} \[ -\frac{\left (b^2-4 a c\right )^2}{160 c^3 d^6 (b+2 c x)^5}+\frac{b^2-4 a c}{48 c^3 d^6 (b+2 c x)^3}-\frac{1}{32 c^3 d^6 (b+2 c x)} \]
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)^6} \, dx &=\int \left (\frac{\left (-b^2+4 a c\right )^2}{16 c^2 d^6 (b+2 c x)^6}+\frac{-b^2+4 a c}{8 c^2 d^6 (b+2 c x)^4}+\frac{1}{16 c^2 d^6 (b+2 c x)^2}\right ) \, dx\\ &=-\frac{\left (b^2-4 a c\right )^2}{160 c^3 d^6 (b+2 c x)^5}+\frac{b^2-4 a c}{48 c^3 d^6 (b+2 c x)^3}-\frac{1}{32 c^3 d^6 (b+2 c x)}\\ \end{align*}
Mathematica [A] time = 0.034111, size = 59, normalized size = 0.81 \[ \frac{10 \left (b^2-4 a c\right ) (b+2 c x)^2-3 \left (b^2-4 a c\right )^2-15 (b+2 c x)^4}{480 c^3 d^6 (b+2 c x)^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 74, normalized size = 1. \begin{align*}{\frac{1}{{d}^{6}} \left ( -{\frac{4\,ac-{b}^{2}}{48\,{c}^{3} \left ( 2\,cx+b \right ) ^{3}}}-{\frac{1}{32\,{c}^{3} \left ( 2\,cx+b \right ) }}-{\frac{16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4}}{160\,{c}^{3} \left ( 2\,cx+b \right ) ^{5}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.28222, size = 201, normalized size = 2.75 \begin{align*} -\frac{30 \, c^{4} x^{4} + 60 \, b c^{3} x^{3} + b^{4} + 2 \, a b^{2} c + 6 \, a^{2} c^{2} + 20 \,{\left (2 \, b^{2} c^{2} + a c^{3}\right )} x^{2} + 10 \,{\left (b^{3} c + 2 \, a b c^{2}\right )} x}{60 \,{\left (32 \, c^{8} d^{6} x^{5} + 80 \, b c^{7} d^{6} x^{4} + 80 \, b^{2} c^{6} d^{6} x^{3} + 40 \, b^{3} c^{5} d^{6} x^{2} + 10 \, b^{4} c^{4} d^{6} x + b^{5} c^{3} d^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.98415, size = 313, normalized size = 4.29 \begin{align*} -\frac{30 \, c^{4} x^{4} + 60 \, b c^{3} x^{3} + b^{4} + 2 \, a b^{2} c + 6 \, a^{2} c^{2} + 20 \,{\left (2 \, b^{2} c^{2} + a c^{3}\right )} x^{2} + 10 \,{\left (b^{3} c + 2 \, a b c^{2}\right )} x}{60 \,{\left (32 \, c^{8} d^{6} x^{5} + 80 \, b c^{7} d^{6} x^{4} + 80 \, b^{2} c^{6} d^{6} x^{3} + 40 \, b^{3} c^{5} d^{6} x^{2} + 10 \, b^{4} c^{4} d^{6} x + b^{5} c^{3} d^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.89372, size = 156, normalized size = 2.14 \begin{align*} - \frac{6 a^{2} c^{2} + 2 a b^{2} c + b^{4} + 60 b c^{3} x^{3} + 30 c^{4} x^{4} + x^{2} \left (20 a c^{3} + 40 b^{2} c^{2}\right ) + x \left (20 a b c^{2} + 10 b^{3} c\right )}{60 b^{5} c^{3} d^{6} + 600 b^{4} c^{4} d^{6} x + 2400 b^{3} c^{5} d^{6} x^{2} + 4800 b^{2} c^{6} d^{6} x^{3} + 4800 b c^{7} d^{6} x^{4} + 1920 c^{8} d^{6} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19, size = 117, normalized size = 1.6 \begin{align*} -\frac{30 \, c^{4} x^{4} + 60 \, b c^{3} x^{3} + 40 \, b^{2} c^{2} x^{2} + 20 \, a c^{3} x^{2} + 10 \, b^{3} c x + 20 \, a b c^{2} x + b^{4} + 2 \, a b^{2} c + 6 \, a^{2} c^{2}}{60 \,{\left (2 \, c x + b\right )}^{5} c^{3} d^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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