Optimal. Leaf size=73 \[ -\frac {\left (b^2-4 a c\right )^2}{256 c^3 d^9 (b+2 c x)^8}+\frac {b^2-4 a c}{96 c^3 d^9 (b+2 c x)^6}-\frac {1}{128 c^3 d^9 (b+2 c x)^4} \]
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Rubi [A] time = 0.05, antiderivative size = 73, normalized size of antiderivative = 1.00, 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} \[ -\frac {\left (b^2-4 a c\right )^2}{256 c^3 d^9 (b+2 c x)^8}+\frac {b^2-4 a c}{96 c^3 d^9 (b+2 c x)^6}-\frac {1}{128 c^3 d^9 (b+2 c x)^4} \]
Antiderivative was successfully verified.
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Rule 683
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^9} \, dx &=\int \left (\frac {\left (-b^2+4 a c\right )^2}{16 c^2 d^9 (b+2 c x)^9}+\frac {-b^2+4 a c}{8 c^2 d^9 (b+2 c x)^7}+\frac {1}{16 c^2 d^9 (b+2 c x)^5}\right ) \, dx\\ &=-\frac {\left (b^2-4 a c\right )^2}{256 c^3 d^9 (b+2 c x)^8}+\frac {b^2-4 a c}{96 c^3 d^9 (b+2 c x)^6}-\frac {1}{128 c^3 d^9 (b+2 c x)^4}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 59, normalized size = 0.81 \[ \frac {8 \left (b^2-4 a c\right ) (b+2 c x)^2-3 \left (b^2-4 a c\right )^2-6 (b+2 c x)^4}{768 c^3 d^9 (b+2 c x)^8} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.02, size = 192, normalized size = 2.63 \[ -\frac {96 \, c^{4} x^{4} + 192 \, b c^{3} x^{3} + b^{4} + 8 \, a b^{2} c + 48 \, a^{2} c^{2} + 16 \, {\left (7 \, b^{2} c^{2} + 8 \, a c^{3}\right )} x^{2} + 16 \, {\left (b^{3} c + 8 \, a b c^{2}\right )} x}{768 \, {\left (256 \, c^{11} d^{9} x^{8} + 1024 \, b c^{10} d^{9} x^{7} + 1792 \, b^{2} c^{9} d^{9} x^{6} + 1792 \, b^{3} c^{8} d^{9} x^{5} + 1120 \, b^{4} c^{7} d^{9} x^{4} + 448 \, b^{5} c^{6} d^{9} x^{3} + 112 \, b^{6} c^{5} d^{9} x^{2} + 16 \, b^{7} c^{4} d^{9} x + b^{8} c^{3} d^{9}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 87, normalized size = 1.19 \[ -\frac {96 \, c^{4} x^{4} + 192 \, b c^{3} x^{3} + 112 \, b^{2} c^{2} x^{2} + 128 \, a c^{3} x^{2} + 16 \, b^{3} c x + 128 \, a b c^{2} x + b^{4} + 8 \, a b^{2} c + 48 \, a^{2} c^{2}}{768 \, {\left (2 \, c x + b\right )}^{8} c^{3} d^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 74, normalized size = 1.01 \[ \frac {-\frac {1}{128 \left (2 c x +b \right )^{4} c^{3}}-\frac {16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}{256 \left (2 c x +b \right )^{8} c^{3}}-\frac {4 a c -b^{2}}{96 \left (2 c x +b \right )^{6} c^{3}}}{d^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.54, size = 192, normalized size = 2.63 \[ -\frac {96 \, c^{4} x^{4} + 192 \, b c^{3} x^{3} + b^{4} + 8 \, a b^{2} c + 48 \, a^{2} c^{2} + 16 \, {\left (7 \, b^{2} c^{2} + 8 \, a c^{3}\right )} x^{2} + 16 \, {\left (b^{3} c + 8 \, a b c^{2}\right )} x}{768 \, {\left (256 \, c^{11} d^{9} x^{8} + 1024 \, b c^{10} d^{9} x^{7} + 1792 \, b^{2} c^{9} d^{9} x^{6} + 1792 \, b^{3} c^{8} d^{9} x^{5} + 1120 \, b^{4} c^{7} d^{9} x^{4} + 448 \, b^{5} c^{6} d^{9} x^{3} + 112 \, b^{6} c^{5} d^{9} x^{2} + 16 \, b^{7} c^{4} d^{9} x + b^{8} c^{3} d^{9}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.53, size = 185, normalized size = 2.53 \[ -\frac {\frac {48\,a^2\,c^2+8\,a\,b^2\,c+b^4}{768\,c^3}+\frac {b\,x^3}{4}+\frac {c\,x^4}{8}+\frac {x^2\,\left (7\,b^2+8\,a\,c\right )}{48\,c}+\frac {b\,x\,\left (b^2+8\,a\,c\right )}{48\,c^2}}{b^8\,d^9+16\,b^7\,c\,d^9\,x+112\,b^6\,c^2\,d^9\,x^2+448\,b^5\,c^3\,d^9\,x^3+1120\,b^4\,c^4\,d^9\,x^4+1792\,b^3\,c^5\,d^9\,x^5+1792\,b^2\,c^6\,d^9\,x^6+1024\,b\,c^7\,d^9\,x^7+256\,c^8\,d^9\,x^8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.55, size = 204, normalized size = 2.79 \[ \frac {- 48 a^{2} c^{2} - 8 a b^{2} c - b^{4} - 192 b c^{3} x^{3} - 96 c^{4} x^{4} + x^{2} \left (- 128 a c^{3} - 112 b^{2} c^{2}\right ) + x \left (- 128 a b c^{2} - 16 b^{3} c\right )}{768 b^{8} c^{3} d^{9} + 12288 b^{7} c^{4} d^{9} x + 86016 b^{6} c^{5} d^{9} x^{2} + 344064 b^{5} c^{6} d^{9} x^{3} + 860160 b^{4} c^{7} d^{9} x^{4} + 1376256 b^{3} c^{8} d^{9} x^{5} + 1376256 b^{2} c^{9} d^{9} x^{6} + 786432 b c^{10} d^{9} x^{7} + 196608 c^{11} d^{9} x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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