Optimal. Leaf size=45 \[ \frac {b^2-4 a c}{40 c^2 d^6 (b+2 c x)^5}-\frac {1}{24 c^2 d^6 (b+2 c x)^3} \]
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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} \begin {gather*} \frac {b^2-4 a c}{40 c^2 d^6 (b+2 c x)^5}-\frac {1}{24 c^2 d^6 (b+2 c x)^3} \end {gather*}
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)^6} \, dx &=\int \left (\frac {-b^2+4 a c}{4 c d^6 (b+2 c x)^6}+\frac {1}{4 c d^6 (b+2 c x)^4}\right ) \, dx\\ &=\frac {b^2-4 a c}{40 c^2 d^6 (b+2 c x)^5}-\frac {1}{24 c^2 d^6 (b+2 c x)^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 43, normalized size = 0.96 \begin {gather*} \frac {\frac {b^2-4 a c}{40 c^2 (b+2 c x)^5}-\frac {1}{24 c^2 (b+2 c x)^3}}{d^6} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b x+c x^2}{(b d+2 c d x)^6} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.41, size = 99, normalized size = 2.20 \begin {gather*} -\frac {10 \, c^{2} x^{2} + 10 \, b c x + b^{2} + 6 \, a c}{60 \, {\left (32 \, c^{7} d^{6} x^{5} + 80 \, b c^{6} d^{6} x^{4} + 80 \, b^{2} c^{5} d^{6} x^{3} + 40 \, b^{3} c^{4} d^{6} x^{2} + 10 \, b^{4} c^{3} d^{6} x + b^{5} c^{2} d^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 37, normalized size = 0.82 \begin {gather*} -\frac {10 \, c^{2} x^{2} + 10 \, b c x + b^{2} + 6 \, a c}{60 \, {\left (2 \, c x + b\right )}^{5} c^{2} d^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 42, normalized size = 0.93 \begin {gather*} \frac {-\frac {1}{24 \left (2 c x +b \right )^{3} c^{2}}-\frac {4 a c -b^{2}}{40 \left (2 c x +b \right )^{5} c^{2}}}{d^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.43, size = 99, normalized size = 2.20 \begin {gather*} -\frac {10 \, c^{2} x^{2} + 10 \, b c x + b^{2} + 6 \, a c}{60 \, {\left (32 \, c^{7} d^{6} x^{5} + 80 \, b c^{6} d^{6} x^{4} + 80 \, b^{2} c^{5} d^{6} x^{3} + 40 \, b^{3} c^{4} d^{6} x^{2} + 10 \, b^{4} c^{3} d^{6} x + b^{5} c^{2} d^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.45, size = 99, normalized size = 2.20 \begin {gather*} -\frac {\frac {b^2+6\,a\,c}{60\,c^2}+\frac {x^2}{6}+\frac {b\,x}{6\,c}}{b^5\,d^6+10\,b^4\,c\,d^6\,x+40\,b^3\,c^2\,d^6\,x^2+80\,b^2\,c^3\,d^6\,x^3+80\,b\,c^4\,d^6\,x^4+32\,c^5\,d^6\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.66, size = 105, normalized size = 2.33 \begin {gather*} \frac {- 6 a c - b^{2} - 10 b c x - 10 c^{2} x^{2}}{60 b^{5} c^{2} d^{6} + 600 b^{4} c^{3} d^{6} x + 2400 b^{3} c^{4} d^{6} x^{2} + 4800 b^{2} c^{5} d^{6} x^{3} + 4800 b c^{6} d^{6} x^{4} + 1920 c^{7} d^{6} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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