Optimal. Leaf size=73 \[ -\frac{\left (b^2-4 a c\right )^2}{320 c^3 d^{11} (b+2 c x)^{10}}+\frac{b^2-4 a c}{128 c^3 d^{11} (b+2 c x)^8}-\frac{1}{192 c^3 d^{11} (b+2 c x)^6} \]
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Rubi [A] time = 0.0542056, 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} \[ -\frac{\left (b^2-4 a c\right )^2}{320 c^3 d^{11} (b+2 c x)^{10}}+\frac{b^2-4 a c}{128 c^3 d^{11} (b+2 c x)^8}-\frac{1}{192 c^3 d^{11} (b+2 c x)^6} \]
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)^{11}} \, dx &=\int \left (\frac{\left (-b^2+4 a c\right )^2}{16 c^2 d^{11} (b+2 c x)^{11}}+\frac{-b^2+4 a c}{8 c^2 d^{11} (b+2 c x)^9}+\frac{1}{16 c^2 d^{11} (b+2 c x)^7}\right ) \, dx\\ &=-\frac{\left (b^2-4 a c\right )^2}{320 c^3 d^{11} (b+2 c x)^{10}}+\frac{b^2-4 a c}{128 c^3 d^{11} (b+2 c x)^8}-\frac{1}{192 c^3 d^{11} (b+2 c x)^6}\\ \end{align*}
Mathematica [A] time = 0.0324032, size = 59, normalized size = 0.81 \[ \frac{15 \left (b^2-4 a c\right ) (b+2 c x)^2-6 \left (b^2-4 a c\right )^2-10 (b+2 c x)^4}{1920 c^3 d^{11} (b+2 c x)^{10}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 74, normalized size = 1. \begin{align*}{\frac{1}{{d}^{11}} \left ( -{\frac{1}{192\,{c}^{3} \left ( 2\,cx+b \right ) ^{6}}}-{\frac{16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4}}{320\,{c}^{3} \left ( 2\,cx+b \right ) ^{10}}}-{\frac{4\,ac-{b}^{2}}{128\,{c}^{3} \left ( 2\,cx+b \right ) ^{8}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.21255, size = 297, normalized size = 4.07 \begin{align*} -\frac{160 \, c^{4} x^{4} + 320 \, b c^{3} x^{3} + b^{4} + 12 \, a b^{2} c + 96 \, a^{2} c^{2} + 60 \,{\left (3 \, b^{2} c^{2} + 4 \, a c^{3}\right )} x^{2} + 20 \,{\left (b^{3} c + 12 \, a b c^{2}\right )} x}{1920 \,{\left (1024 \, c^{13} d^{11} x^{10} + 5120 \, b c^{12} d^{11} x^{9} + 11520 \, b^{2} c^{11} d^{11} x^{8} + 15360 \, b^{3} c^{10} d^{11} x^{7} + 13440 \, b^{4} c^{9} d^{11} x^{6} + 8064 \, b^{5} c^{8} d^{11} x^{5} + 3360 \, b^{6} c^{7} d^{11} x^{4} + 960 \, b^{7} c^{6} d^{11} x^{3} + 180 \, b^{8} c^{5} d^{11} x^{2} + 20 \, b^{9} c^{4} d^{11} x + b^{10} c^{3} d^{11}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.97898, size = 516, normalized size = 7.07 \begin{align*} -\frac{160 \, c^{4} x^{4} + 320 \, b c^{3} x^{3} + b^{4} + 12 \, a b^{2} c + 96 \, a^{2} c^{2} + 60 \,{\left (3 \, b^{2} c^{2} + 4 \, a c^{3}\right )} x^{2} + 20 \,{\left (b^{3} c + 12 \, a b c^{2}\right )} x}{1920 \,{\left (1024 \, c^{13} d^{11} x^{10} + 5120 \, b c^{12} d^{11} x^{9} + 11520 \, b^{2} c^{11} d^{11} x^{8} + 15360 \, b^{3} c^{10} d^{11} x^{7} + 13440 \, b^{4} c^{9} d^{11} x^{6} + 8064 \, b^{5} c^{8} d^{11} x^{5} + 3360 \, b^{6} c^{7} d^{11} x^{4} + 960 \, b^{7} c^{6} d^{11} x^{3} + 180 \, b^{8} c^{5} d^{11} x^{2} + 20 \, b^{9} c^{4} d^{11} x + b^{10} c^{3} d^{11}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 9.00974, size = 233, normalized size = 3.19 \begin{align*} - \frac{96 a^{2} c^{2} + 12 a b^{2} c + b^{4} + 320 b c^{3} x^{3} + 160 c^{4} x^{4} + x^{2} \left (240 a c^{3} + 180 b^{2} c^{2}\right ) + x \left (240 a b c^{2} + 20 b^{3} c\right )}{1920 b^{10} c^{3} d^{11} + 38400 b^{9} c^{4} d^{11} x + 345600 b^{8} c^{5} d^{11} x^{2} + 1843200 b^{7} c^{6} d^{11} x^{3} + 6451200 b^{6} c^{7} d^{11} x^{4} + 15482880 b^{5} c^{8} d^{11} x^{5} + 25804800 b^{4} c^{9} d^{11} x^{6} + 29491200 b^{3} c^{10} d^{11} x^{7} + 22118400 b^{2} c^{11} d^{11} x^{8} + 9830400 b c^{12} d^{11} x^{9} + 1966080 c^{13} d^{11} x^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17609, size = 117, normalized size = 1.6 \begin{align*} -\frac{160 \, c^{4} x^{4} + 320 \, b c^{3} x^{3} + 180 \, b^{2} c^{2} x^{2} + 240 \, a c^{3} x^{2} + 20 \, b^{3} c x + 240 \, a b c^{2} x + b^{4} + 12 \, a b^{2} c + 96 \, a^{2} c^{2}}{1920 \,{\left (2 \, c x + b\right )}^{10} c^{3} d^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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