Optimal. Leaf size=37 \[ \frac{\left (a+b x+c x^2\right )^4}{4 d^9 \left (b^2-4 a c\right ) (b+2 c x)^8} \]
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Rubi [A] time = 0.0133028, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {682} \[ \frac{\left (a+b x+c x^2\right )^4}{4 d^9 \left (b^2-4 a c\right ) (b+2 c x)^8} \]
Antiderivative was successfully verified.
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Rule 682
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^9} \, dx &=\frac{\left (a+b x+c x^2\right )^4}{4 \left (b^2-4 a c\right ) d^9 (b+2 c x)^8}\\ \end{align*}
Mathematica [B] time = 0.0524216, size = 96, normalized size = 2.59 \[ \frac{48 a^2 b^2 c^2-64 a^3 c^3+6 \left (b^2-4 a c\right ) (b+2 c x)^4-4 \left (b^2-4 a c\right )^2 (b+2 c x)^2-12 a b^4 c+b^6-4 (b+2 c x)^6}{1024 c^4 d^9 (b+2 c x)^8} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.046, size = 121, normalized size = 3.3 \begin{align*}{\frac{1}{{d}^{9}} \left ( -{\frac{48\,{a}^{2}{c}^{2}-24\,ac{b}^{2}+3\,{b}^{4}}{768\,{c}^{4} \left ( 2\,cx+b \right ) ^{6}}}-{\frac{1}{256\,{c}^{4} \left ( 2\,cx+b \right ) ^{2}}}-{\frac{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}{1024\,{c}^{4} \left ( 2\,cx+b \right ) ^{8}}}-{\frac{12\,ac-3\,{b}^{2}}{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 = 1.43157, size = 359, normalized size = 9.7 \begin{align*} -\frac{256 \, c^{6} x^{6} + 768 \, b c^{5} x^{5} + b^{6} + 4 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2} + 64 \, a^{3} c^{3} + 96 \,{\left (9 \, b^{2} c^{4} + 4 \, a c^{5}\right )} x^{4} + 64 \,{\left (7 \, b^{3} c^{3} + 12 \, a b c^{4}\right )} x^{3} + 16 \,{\left (7 \, b^{4} c^{2} + 28 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + 16 \,{\left (b^{5} c + 4 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x}{1024 \,{\left (256 \, c^{12} d^{9} x^{8} + 1024 \, b c^{11} d^{9} x^{7} + 1792 \, b^{2} c^{10} d^{9} x^{6} + 1792 \, b^{3} c^{9} d^{9} x^{5} + 1120 \, b^{4} c^{8} d^{9} x^{4} + 448 \, b^{5} c^{7} d^{9} x^{3} + 112 \, b^{6} c^{6} d^{9} x^{2} + 16 \, b^{7} c^{5} d^{9} x + b^{8} c^{4} d^{9}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.98935, size = 585, normalized size = 15.81 \begin{align*} -\frac{256 \, c^{6} x^{6} + 768 \, b c^{5} x^{5} + b^{6} + 4 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2} + 64 \, a^{3} c^{3} + 96 \,{\left (9 \, b^{2} c^{4} + 4 \, a c^{5}\right )} x^{4} + 64 \,{\left (7 \, b^{3} c^{3} + 12 \, a b c^{4}\right )} x^{3} + 16 \,{\left (7 \, b^{4} c^{2} + 28 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + 16 \,{\left (b^{5} c + 4 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x}{1024 \,{\left (256 \, c^{12} d^{9} x^{8} + 1024 \, b c^{11} d^{9} x^{7} + 1792 \, b^{2} c^{10} d^{9} x^{6} + 1792 \, b^{3} c^{9} d^{9} x^{5} + 1120 \, b^{4} c^{8} d^{9} x^{4} + 448 \, b^{5} c^{7} d^{9} x^{3} + 112 \, b^{6} c^{6} d^{9} x^{2} + 16 \, b^{7} c^{5} d^{9} x + b^{8} c^{4} d^{9}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 22.5226, size = 277, normalized size = 7.49 \begin{align*} - \frac{64 a^{3} c^{3} + 16 a^{2} b^{2} c^{2} + 4 a b^{4} c + b^{6} + 768 b c^{5} x^{5} + 256 c^{6} x^{6} + x^{4} \left (384 a c^{5} + 864 b^{2} c^{4}\right ) + x^{3} \left (768 a b c^{4} + 448 b^{3} c^{3}\right ) + x^{2} \left (256 a^{2} c^{4} + 448 a b^{2} c^{3} + 112 b^{4} c^{2}\right ) + x \left (256 a^{2} b c^{3} + 64 a b^{3} c^{2} + 16 b^{5} c\right )}{1024 b^{8} c^{4} d^{9} + 16384 b^{7} c^{5} d^{9} x + 114688 b^{6} c^{6} d^{9} x^{2} + 458752 b^{5} c^{7} d^{9} x^{3} + 1146880 b^{4} c^{8} d^{9} x^{4} + 1835008 b^{3} c^{9} d^{9} x^{5} + 1835008 b^{2} c^{10} d^{9} x^{6} + 1048576 b c^{11} d^{9} x^{7} + 262144 c^{12} d^{9} x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17934, size = 223, normalized size = 6.03 \begin{align*} -\frac{256 \, c^{6} x^{6} + 768 \, b c^{5} x^{5} + 864 \, b^{2} c^{4} x^{4} + 384 \, a c^{5} x^{4} + 448 \, b^{3} c^{3} x^{3} + 768 \, a b c^{4} x^{3} + 112 \, b^{4} c^{2} x^{2} + 448 \, a b^{2} c^{3} x^{2} + 256 \, a^{2} c^{4} x^{2} + 16 \, b^{5} c x + 64 \, a b^{3} c^{2} x + 256 \, a^{2} b c^{3} x + b^{6} + 4 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2} + 64 \, a^{3} c^{3}}{1024 \,{\left (2 \, c x + b\right )}^{8} c^{4} d^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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