Optimal. Leaf size=37 \[ \frac {\left (a+b x+c x^2\right )^2}{2 d^5 \left (b^2-4 a c\right ) (b+2 c x)^4} \]
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Rubi [A] time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {682} \[ \frac {\left (a+b x+c x^2\right )^2}{2 d^5 \left (b^2-4 a c\right ) (b+2 c x)^4} \]
Antiderivative was successfully verified.
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Rule 682
Rubi steps
\begin {align*} \int \frac {a+b x+c x^2}{(b d+2 c d x)^5} \, dx &=\frac {\left (a+b x+c x^2\right )^2}{2 \left (b^2-4 a c\right ) d^5 (b+2 c x)^4}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 38, normalized size = 1.03 \[ -\frac {4 c \left (a+2 c x^2\right )+b^2+8 b c x}{32 c^2 d^5 (b+2 c x)^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.02, size = 85, normalized size = 2.30 \[ -\frac {8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c}{32 \, {\left (16 \, c^{6} d^{5} x^{4} + 32 \, b c^{5} d^{5} x^{3} + 24 \, b^{2} c^{4} d^{5} x^{2} + 8 \, b^{3} c^{3} d^{5} x + b^{4} c^{2} d^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 60, normalized size = 1.62 \[ \frac {\frac {b^{2}}{{\left (2 \, c d x + b d\right )}^{4} c^{2}} - \frac {4 \, a}{{\left (2 \, c d x + b d\right )}^{4} c} - \frac {2}{{\left (2 \, c d x + b d\right )}^{2} c^{2} d^{2}}}{32 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 42, normalized size = 1.14 \[ \frac {-\frac {4 a c -b^{2}}{32 \left (2 c x +b \right )^{4} c^{2}}-\frac {1}{16 \left (2 c x +b \right )^{2} c^{2}}}{d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.36, size = 85, normalized size = 2.30 \[ -\frac {8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c}{32 \, {\left (16 \, c^{6} d^{5} x^{4} + 32 \, b c^{5} d^{5} x^{3} + 24 \, b^{2} c^{4} d^{5} x^{2} + 8 \, b^{3} c^{3} d^{5} x + b^{4} c^{2} d^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 85, normalized size = 2.30 \[ -\frac {\frac {b^2+4\,a\,c}{32\,c^2}+\frac {x^2}{4}+\frac {b\,x}{4\,c}}{b^4\,d^5+8\,b^3\,c\,d^5\,x+24\,b^2\,c^2\,d^5\,x^2+32\,b\,c^3\,d^5\,x^3+16\,c^4\,d^5\,x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.63, size = 90, normalized size = 2.43 \[ \frac {- 4 a c - b^{2} - 8 b c x - 8 c^{2} x^{2}}{32 b^{4} c^{2} d^{5} + 256 b^{3} c^{3} d^{5} x + 768 b^{2} c^{4} d^{5} x^{2} + 1024 b c^{5} d^{5} x^{3} + 512 c^{6} d^{5} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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