Optimal. Leaf size=45 \[ \frac {b^2-4 a c}{56 c^2 d^8 (b+2 c x)^7}-\frac {1}{40 c^2 d^8 (b+2 c x)^5} \]
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Rubi [A] time = 0.03, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {683} \[ \frac {b^2-4 a c}{56 c^2 d^8 (b+2 c x)^7}-\frac {1}{40 c^2 d^8 (b+2 c x)^5} \]
Antiderivative was successfully verified.
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Rule 683
Rubi steps
\begin {align*} \int \frac {a+b x+c x^2}{(b d+2 c d x)^8} \, dx &=\int \left (\frac {-b^2+4 a c}{4 c d^8 (b+2 c x)^8}+\frac {1}{4 c d^8 (b+2 c x)^6}\right ) \, dx\\ &=\frac {b^2-4 a c}{56 c^2 d^8 (b+2 c x)^7}-\frac {1}{40 c^2 d^8 (b+2 c x)^5}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 39, normalized size = 0.87 \[ \frac {5 \left (b^2-4 a c\right )-7 (b+2 c x)^2}{280 c^2 d^8 (b+2 c x)^7} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.17, size = 127, normalized size = 2.82 \[ -\frac {14 \, c^{2} x^{2} + 14 \, b c x + b^{2} + 10 \, a c}{140 \, {\left (128 \, c^{9} d^{8} x^{7} + 448 \, b c^{8} d^{8} x^{6} + 672 \, b^{2} c^{7} d^{8} x^{5} + 560 \, b^{3} c^{6} d^{8} x^{4} + 280 \, b^{4} c^{5} d^{8} x^{3} + 84 \, b^{5} c^{4} d^{8} x^{2} + 14 \, b^{6} c^{3} d^{8} x + b^{7} c^{2} d^{8}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 37, normalized size = 0.82 \[ -\frac {14 \, c^{2} x^{2} + 14 \, b c x + b^{2} + 10 \, a c}{140 \, {\left (2 \, c x + b\right )}^{7} c^{2} d^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 42, normalized size = 0.93 \[ \frac {-\frac {4 a c -b^{2}}{56 \left (2 c x +b \right )^{7} c^{2}}-\frac {1}{40 \left (2 c x +b \right )^{5} c^{2}}}{d^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.38, size = 127, normalized size = 2.82 \[ -\frac {14 \, c^{2} x^{2} + 14 \, b c x + b^{2} + 10 \, a c}{140 \, {\left (128 \, c^{9} d^{8} x^{7} + 448 \, b c^{8} d^{8} x^{6} + 672 \, b^{2} c^{7} d^{8} x^{5} + 560 \, b^{3} c^{6} d^{8} x^{4} + 280 \, b^{4} c^{5} d^{8} x^{3} + 84 \, b^{5} c^{4} d^{8} x^{2} + 14 \, b^{6} c^{3} d^{8} x + b^{7} c^{2} d^{8}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.49, size = 127, normalized size = 2.82 \[ -\frac {\frac {b^2+10\,a\,c}{140\,c^2}+\frac {x^2}{10}+\frac {b\,x}{10\,c}}{b^7\,d^8+14\,b^6\,c\,d^8\,x+84\,b^5\,c^2\,d^8\,x^2+280\,b^4\,c^3\,d^8\,x^3+560\,b^3\,c^4\,d^8\,x^4+672\,b^2\,c^5\,d^8\,x^5+448\,b\,c^6\,d^8\,x^6+128\,c^7\,d^8\,x^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.93, size = 136, normalized size = 3.02 \[ \frac {- 10 a c - b^{2} - 14 b c x - 14 c^{2} x^{2}}{140 b^{7} c^{2} d^{8} + 1960 b^{6} c^{3} d^{8} x + 11760 b^{5} c^{4} d^{8} x^{2} + 39200 b^{4} c^{5} d^{8} x^{3} + 78400 b^{3} c^{6} d^{8} x^{4} + 94080 b^{2} c^{7} d^{8} x^{5} + 62720 b c^{8} d^{8} x^{6} + 17920 c^{9} d^{8} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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