Optimal. Leaf size=44 \[ \frac {b^2-4 a c}{16 c^2 d^3 (b+2 c x)^2}+\frac {\log (b+2 c x)}{8 c^2 d^3} \]
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Rubi [A] time = 0.03, antiderivative size = 44, 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}{16 c^2 d^3 (b+2 c x)^2}+\frac {\log (b+2 c x)}{8 c^2 d^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)^3} \, dx &=\int \left (\frac {-b^2+4 a c}{4 c d^3 (b+2 c x)^3}+\frac {1}{4 c d^3 (b+2 c x)}\right ) \, dx\\ &=\frac {b^2-4 a c}{16 c^2 d^3 (b+2 c x)^2}+\frac {\log (b+2 c x)}{8 c^2 d^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 37, normalized size = 0.84 \begin {gather*} \frac {\frac {b^2-4 a c}{(b+2 c x)^2}+2 \log (b+2 c x)}{16 c^2 d^3} \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)^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 70, normalized size = 1.59 \begin {gather*} \frac {b^{2} - 4 \, a c + 2 \, {\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )} \log \left (2 \, c x + b\right )}{16 \, {\left (4 \, c^{4} d^{3} x^{2} + 4 \, b c^{3} d^{3} x + b^{2} c^{2} d^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 41, normalized size = 0.93 \begin {gather*} \frac {\log \left ({\left | 2 \, c x + b \right |}\right )}{8 \, c^{2} d^{3}} + \frac {b^{2} - 4 \, a c}{16 \, {\left (2 \, c x + b\right )}^{2} c^{2} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 53, normalized size = 1.20 \begin {gather*} -\frac {a}{4 \left (2 c x +b \right )^{2} c \,d^{3}}+\frac {b^{2}}{16 \left (2 c x +b \right )^{2} c^{2} d^{3}}+\frac {\ln \left (2 c x +b \right )}{8 c^{2} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 60, normalized size = 1.36 \begin {gather*} \frac {b^{2} - 4 \, a c}{16 \, {\left (4 \, c^{4} d^{3} x^{2} + 4 \, b c^{3} d^{3} x + b^{2} c^{2} d^{3}\right )}} + \frac {\log \left (2 \, c x + b\right )}{8 \, c^{2} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.45, size = 42, normalized size = 0.95 \begin {gather*} \frac {\ln \left (b+2\,c\,x\right )}{8\,c^2\,d^3}-\frac {\frac {a\,c}{4}-\frac {b^2}{16}}{c^2\,d^3\,{\left (b+2\,c\,x\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.38, size = 60, normalized size = 1.36 \begin {gather*} \frac {- 4 a c + b^{2}}{16 b^{2} c^{2} d^{3} + 64 b c^{3} d^{3} x + 64 c^{4} d^{3} x^{2}} + \frac {\log {\left (b + 2 c x \right )}}{8 c^{2} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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