Optimal. Leaf size=101 \[ \frac{\left (b^2-4 a c\right )^3}{1408 c^4 d^{12} (b+2 c x)^{11}}-\frac{\left (b^2-4 a c\right )^2}{384 c^4 d^{12} (b+2 c x)^9}+\frac{3 \left (b^2-4 a c\right )}{896 c^4 d^{12} (b+2 c x)^7}-\frac{1}{640 c^4 d^{12} (b+2 c x)^5} \]
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Rubi [A] time = 0.0846991, 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} \[ \frac{\left (b^2-4 a c\right )^3}{1408 c^4 d^{12} (b+2 c x)^{11}}-\frac{\left (b^2-4 a c\right )^2}{384 c^4 d^{12} (b+2 c x)^9}+\frac{3 \left (b^2-4 a c\right )}{896 c^4 d^{12} (b+2 c x)^7}-\frac{1}{640 c^4 d^{12} (b+2 c x)^5} \]
Antiderivative was successfully verified.
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Rule 683
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{12}} \, dx &=\int \left (\frac{\left (-b^2+4 a c\right )^3}{64 c^3 d^{12} (b+2 c x)^{12}}+\frac{3 \left (-b^2+4 a c\right )^2}{64 c^3 d^{12} (b+2 c x)^{10}}+\frac{3 \left (-b^2+4 a c\right )}{64 c^3 d^{12} (b+2 c x)^8}+\frac{1}{64 c^3 d^{12} (b+2 c x)^6}\right ) \, dx\\ &=\frac{\left (b^2-4 a c\right )^3}{1408 c^4 d^{12} (b+2 c x)^{11}}-\frac{\left (b^2-4 a c\right )^2}{384 c^4 d^{12} (b+2 c x)^9}+\frac{3 \left (b^2-4 a c\right )}{896 c^4 d^{12} (b+2 c x)^7}-\frac{1}{640 c^4 d^{12} (b+2 c x)^5}\\ \end{align*}
Mathematica [A] time = 0.0626427, size = 79, normalized size = 0.78 \[ \frac{495 \left (b^2-4 a c\right ) (b+2 c x)^4-385 \left (b^2-4 a c\right )^2 (b+2 c x)^2+105 \left (b^2-4 a c\right )^3-231 (b+2 c x)^6}{147840 c^4 d^{12} (b+2 c x)^{11}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 121, normalized size = 1.2 \begin{align*}{\frac{1}{{d}^{12}} \left ( -{\frac{1}{640\,{c}^{4} \left ( 2\,cx+b \right ) ^{5}}}-{\frac{12\,ac-3\,{b}^{2}}{896\,{c}^{4} \left ( 2\,cx+b \right ) ^{7}}}-{\frac{48\,{a}^{2}{c}^{2}-24\,ac{b}^{2}+3\,{b}^{4}}{1152\,{c}^{4} \left ( 2\,cx+b \right ) ^{9}}}-{\frac{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}{1408\,{c}^{4} \left ( 2\,cx+b \right ) ^{11}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.36038, size = 413, normalized size = 4.09 \begin{align*} -\frac{924 \, c^{6} x^{6} + 2772 \, b c^{5} x^{5} + b^{6} + 10 \, a b^{4} c + 70 \, a^{2} b^{2} c^{2} + 420 \, a^{3} c^{3} + 990 \,{\left (3 \, b^{2} c^{4} + 2 \, a c^{5}\right )} x^{4} + 1320 \,{\left (b^{3} c^{3} + 3 \, a b c^{4}\right )} x^{3} + 220 \,{\left (b^{4} c^{2} + 10 \, a b^{2} c^{3} + 7 \, a^{2} c^{4}\right )} x^{2} + 22 \,{\left (b^{5} c + 10 \, a b^{3} c^{2} + 70 \, a^{2} b c^{3}\right )} x}{9240 \,{\left (2048 \, c^{15} d^{12} x^{11} + 11264 \, b c^{14} d^{12} x^{10} + 28160 \, b^{2} c^{13} d^{12} x^{9} + 42240 \, b^{3} c^{12} d^{12} x^{8} + 42240 \, b^{4} c^{11} d^{12} x^{7} + 29568 \, b^{5} c^{10} d^{12} x^{6} + 14784 \, b^{6} c^{9} d^{12} x^{5} + 5280 \, b^{7} c^{8} d^{12} x^{4} + 1320 \, b^{8} c^{7} d^{12} x^{3} + 220 \, b^{9} c^{6} d^{12} x^{2} + 22 \, b^{10} c^{5} d^{12} x + b^{11} c^{4} d^{12}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.07111, size = 717, normalized size = 7.1 \begin{align*} -\frac{924 \, c^{6} x^{6} + 2772 \, b c^{5} x^{5} + b^{6} + 10 \, a b^{4} c + 70 \, a^{2} b^{2} c^{2} + 420 \, a^{3} c^{3} + 990 \,{\left (3 \, b^{2} c^{4} + 2 \, a c^{5}\right )} x^{4} + 1320 \,{\left (b^{3} c^{3} + 3 \, a b c^{4}\right )} x^{3} + 220 \,{\left (b^{4} c^{2} + 10 \, a b^{2} c^{3} + 7 \, a^{2} c^{4}\right )} x^{2} + 22 \,{\left (b^{5} c + 10 \, a b^{3} c^{2} + 70 \, a^{2} b c^{3}\right )} x}{9240 \,{\left (2048 \, c^{15} d^{12} x^{11} + 11264 \, b c^{14} d^{12} x^{10} + 28160 \, b^{2} c^{13} d^{12} x^{9} + 42240 \, b^{3} c^{12} d^{12} x^{8} + 42240 \, b^{4} c^{11} d^{12} x^{7} + 29568 \, b^{5} c^{10} d^{12} x^{6} + 14784 \, b^{6} c^{9} d^{12} x^{5} + 5280 \, b^{7} c^{8} d^{12} x^{4} + 1320 \, b^{8} c^{7} d^{12} x^{3} + 220 \, b^{9} c^{6} d^{12} x^{2} + 22 \, b^{10} c^{5} d^{12} x + b^{11} c^{4} d^{12}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 20.5477, size = 323, normalized size = 3.2 \begin{align*} - \frac{420 a^{3} c^{3} + 70 a^{2} b^{2} c^{2} + 10 a b^{4} c + b^{6} + 2772 b c^{5} x^{5} + 924 c^{6} x^{6} + x^{4} \left (1980 a c^{5} + 2970 b^{2} c^{4}\right ) + x^{3} \left (3960 a b c^{4} + 1320 b^{3} c^{3}\right ) + x^{2} \left (1540 a^{2} c^{4} + 2200 a b^{2} c^{3} + 220 b^{4} c^{2}\right ) + x \left (1540 a^{2} b c^{3} + 220 a b^{3} c^{2} + 22 b^{5} c\right )}{9240 b^{11} c^{4} d^{12} + 203280 b^{10} c^{5} d^{12} x + 2032800 b^{9} c^{6} d^{12} x^{2} + 12196800 b^{8} c^{7} d^{12} x^{3} + 48787200 b^{7} c^{8} d^{12} x^{4} + 136604160 b^{6} c^{9} d^{12} x^{5} + 273208320 b^{5} c^{10} d^{12} x^{6} + 390297600 b^{4} c^{11} d^{12} x^{7} + 390297600 b^{3} c^{12} d^{12} x^{8} + 260198400 b^{2} c^{13} d^{12} x^{9} + 104079360 b c^{14} d^{12} x^{10} + 18923520 c^{15} d^{12} x^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17052, size = 223, normalized size = 2.21 \begin{align*} -\frac{924 \, c^{6} x^{6} + 2772 \, b c^{5} x^{5} + 2970 \, b^{2} c^{4} x^{4} + 1980 \, a c^{5} x^{4} + 1320 \, b^{3} c^{3} x^{3} + 3960 \, a b c^{4} x^{3} + 220 \, b^{4} c^{2} x^{2} + 2200 \, a b^{2} c^{3} x^{2} + 1540 \, a^{2} c^{4} x^{2} + 22 \, b^{5} c x + 220 \, a b^{3} c^{2} x + 1540 \, a^{2} b c^{3} x + b^{6} + 10 \, a b^{4} c + 70 \, a^{2} b^{2} c^{2} + 420 \, a^{3} c^{3}}{9240 \,{\left (2 \, c x + b\right )}^{11} c^{4} d^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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