Optimal. Leaf size=101 \[ \frac{\left (b^2-4 a c\right )^3}{1280 c^4 d^{11} (b+2 c x)^{10}}-\frac{3 \left (b^2-4 a c\right )^2}{1024 c^4 d^{11} (b+2 c x)^8}+\frac{b^2-4 a c}{256 c^4 d^{11} (b+2 c x)^6}-\frac{1}{512 c^4 d^{11} (b+2 c x)^4} \]
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Rubi [A] time = 0.0814115, antiderivative size = 101, 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} \[ \frac{\left (b^2-4 a c\right )^3}{1280 c^4 d^{11} (b+2 c x)^{10}}-\frac{3 \left (b^2-4 a c\right )^2}{1024 c^4 d^{11} (b+2 c x)^8}+\frac{b^2-4 a c}{256 c^4 d^{11} (b+2 c x)^6}-\frac{1}{512 c^4 d^{11} (b+2 c x)^4} \]
Antiderivative was successfully verified.
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Rule 683
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{11}} \, dx &=\int \left (\frac{\left (-b^2+4 a c\right )^3}{64 c^3 d^{11} (b+2 c x)^{11}}+\frac{3 \left (-b^2+4 a c\right )^2}{64 c^3 d^{11} (b+2 c x)^9}+\frac{3 \left (-b^2+4 a c\right )}{64 c^3 d^{11} (b+2 c x)^7}+\frac{1}{64 c^3 d^{11} (b+2 c x)^5}\right ) \, dx\\ &=\frac{\left (b^2-4 a c\right )^3}{1280 c^4 d^{11} (b+2 c x)^{10}}-\frac{3 \left (b^2-4 a c\right )^2}{1024 c^4 d^{11} (b+2 c x)^8}+\frac{b^2-4 a c}{256 c^4 d^{11} (b+2 c x)^6}-\frac{1}{512 c^4 d^{11} (b+2 c x)^4}\\ \end{align*}
Mathematica [A] time = 0.0567742, size = 79, normalized size = 0.78 \[ \frac{20 \left (b^2-4 a c\right ) (b+2 c x)^4-15 \left (b^2-4 a c\right )^2 (b+2 c x)^2+4 \left (b^2-4 a c\right )^3-10 (b+2 c x)^6}{5120 c^4 d^{11} (b+2 c x)^{10}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 121, normalized size = 1.2 \begin{align*}{\frac{1}{{d}^{11}} \left ( -{\frac{12\,ac-3\,{b}^{2}}{768\,{c}^{4} \left ( 2\,cx+b \right ) ^{6}}}-{\frac{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}{1280\,{c}^{4} \left ( 2\,cx+b \right ) ^{10}}}-{\frac{48\,{a}^{2}{c}^{2}-24\,ac{b}^{2}+3\,{b}^{4}}{1024\,{c}^{4} \left ( 2\,cx+b \right ) ^{8}}}-{\frac{1}{512\,{c}^{4} \left ( 2\,cx+b \right ) ^{4}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.35701, size = 397, normalized size = 3.93 \begin{align*} -\frac{640 \, c^{6} x^{6} + 1920 \, b c^{5} x^{5} + b^{6} + 8 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} + 256 \, a^{3} c^{3} + 160 \,{\left (13 \, b^{2} c^{4} + 8 \, a c^{5}\right )} x^{4} + 320 \,{\left (3 \, b^{3} c^{3} + 8 \, a b c^{4}\right )} x^{3} + 60 \,{\left (3 \, b^{4} c^{2} + 24 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + 20 \,{\left (b^{5} c + 8 \, a b^{3} c^{2} + 48 \, a^{2} b c^{3}\right )} x}{5120 \,{\left (1024 \, c^{14} d^{11} x^{10} + 5120 \, b c^{13} d^{11} x^{9} + 11520 \, b^{2} c^{12} d^{11} x^{8} + 15360 \, b^{3} c^{11} d^{11} x^{7} + 13440 \, b^{4} c^{10} d^{11} x^{6} + 8064 \, b^{5} c^{9} d^{11} x^{5} + 3360 \, b^{6} c^{8} d^{11} x^{4} + 960 \, b^{7} c^{7} d^{11} x^{3} + 180 \, b^{8} c^{6} d^{11} x^{2} + 20 \, b^{9} c^{5} d^{11} x + b^{10} c^{4} d^{11}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.05677, size = 678, normalized size = 6.71 \begin{align*} -\frac{640 \, c^{6} x^{6} + 1920 \, b c^{5} x^{5} + b^{6} + 8 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} + 256 \, a^{3} c^{3} + 160 \,{\left (13 \, b^{2} c^{4} + 8 \, a c^{5}\right )} x^{4} + 320 \,{\left (3 \, b^{3} c^{3} + 8 \, a b c^{4}\right )} x^{3} + 60 \,{\left (3 \, b^{4} c^{2} + 24 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + 20 \,{\left (b^{5} c + 8 \, a b^{3} c^{2} + 48 \, a^{2} b c^{3}\right )} x}{5120 \,{\left (1024 \, c^{14} d^{11} x^{10} + 5120 \, b c^{13} d^{11} x^{9} + 11520 \, b^{2} c^{12} d^{11} x^{8} + 15360 \, b^{3} c^{11} d^{11} x^{7} + 13440 \, b^{4} c^{10} d^{11} x^{6} + 8064 \, b^{5} c^{9} d^{11} x^{5} + 3360 \, b^{6} c^{8} d^{11} x^{4} + 960 \, b^{7} c^{7} d^{11} x^{3} + 180 \, b^{8} c^{6} d^{11} x^{2} + 20 \, b^{9} c^{5} d^{11} x + b^{10} c^{4} d^{11}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 42.6112, size = 308, normalized size = 3.05 \begin{align*} - \frac{256 a^{3} c^{3} + 48 a^{2} b^{2} c^{2} + 8 a b^{4} c + b^{6} + 1920 b c^{5} x^{5} + 640 c^{6} x^{6} + x^{4} \left (1280 a c^{5} + 2080 b^{2} c^{4}\right ) + x^{3} \left (2560 a b c^{4} + 960 b^{3} c^{3}\right ) + x^{2} \left (960 a^{2} c^{4} + 1440 a b^{2} c^{3} + 180 b^{4} c^{2}\right ) + x \left (960 a^{2} b c^{3} + 160 a b^{3} c^{2} + 20 b^{5} c\right )}{5120 b^{10} c^{4} d^{11} + 102400 b^{9} c^{5} d^{11} x + 921600 b^{8} c^{6} d^{11} x^{2} + 4915200 b^{7} c^{7} d^{11} x^{3} + 17203200 b^{6} c^{8} d^{11} x^{4} + 41287680 b^{5} c^{9} d^{11} x^{5} + 68812800 b^{4} c^{10} d^{11} x^{6} + 78643200 b^{3} c^{11} d^{11} x^{7} + 58982400 b^{2} c^{12} d^{11} x^{8} + 26214400 b c^{13} d^{11} x^{9} + 5242880 c^{14} d^{11} x^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17969, size = 223, normalized size = 2.21 \begin{align*} -\frac{640 \, c^{6} x^{6} + 1920 \, b c^{5} x^{5} + 2080 \, b^{2} c^{4} x^{4} + 1280 \, a c^{5} x^{4} + 960 \, b^{3} c^{3} x^{3} + 2560 \, a b c^{4} x^{3} + 180 \, b^{4} c^{2} x^{2} + 1440 \, a b^{2} c^{3} x^{2} + 960 \, a^{2} c^{4} x^{2} + 20 \, b^{5} c x + 160 \, a b^{3} c^{2} x + 960 \, a^{2} b c^{3} x + b^{6} + 8 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} + 256 \, a^{3} c^{3}}{5120 \,{\left (2 \, c x + b\right )}^{10} c^{4} d^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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