Optimal. Leaf size=78 \[ -\frac {1}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {4 c \log \left (a+b x+c x^2\right )}{d \left (b^2-4 a c\right )^2}+\frac {8 c \log (b+2 c x)}{d \left (b^2-4 a c\right )^2} \]
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Rubi [A] time = 0.04, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {687, 681, 31, 628} \begin {gather*} -\frac {1}{d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {4 c \log \left (a+b x+c x^2\right )}{d \left (b^2-4 a c\right )^2}+\frac {8 c \log (b+2 c x)}{d \left (b^2-4 a c\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 628
Rule 681
Rule 687
Rubi steps
\begin {align*} \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )^2} \, dx &=-\frac {1}{\left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )}-\frac {(4 c) \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )} \, dx}{b^2-4 a c}\\ &=-\frac {1}{\left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )}-\frac {(4 c) \int \frac {b d+2 c d x}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^2 d^2}+\frac {\left (16 c^2\right ) \int \frac {1}{b+2 c x} \, dx}{\left (b^2-4 a c\right )^2 d}\\ &=-\frac {1}{\left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )}+\frac {8 c \log (b+2 c x)}{\left (b^2-4 a c\right )^2 d}-\frac {4 c \log \left (a+b x+c x^2\right )}{\left (b^2-4 a c\right )^2 d}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 59, normalized size = 0.76 \begin {gather*} \frac {-\frac {b^2-4 a c}{a+x (b+c x)}-4 c \log (a+x (b+c x))+8 c \log (b+2 c x)}{d \left (b^2-4 a c\right )^2} \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 {1}{(b d+2 c d x) \left (a+b x+c x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.40, size = 141, normalized size = 1.81 \begin {gather*} -\frac {b^{2} - 4 \, a c + 4 \, {\left (c^{2} x^{2} + b c x + a c\right )} \log \left (c x^{2} + b x + a\right ) - 8 \, {\left (c^{2} x^{2} + b c x + a c\right )} \log \left (2 \, c x + b\right )}{{\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d x^{2} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} d x + {\left (a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2}\right )} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 108, normalized size = 1.38 \begin {gather*} \frac {8 \, c^{2} \log \left ({\left | 2 \, c x + b \right |}\right )}{b^{4} c d - 8 \, a b^{2} c^{2} d + 16 \, a^{2} c^{3} d} - \frac {4 \, c \log \left (c x^{2} + b x + a\right )}{b^{4} d - 8 \, a b^{2} c d + 16 \, a^{2} c^{2} d} - \frac {1}{{\left (c x^{2} + b x + a\right )} {\left (b^{2} - 4 \, a c\right )} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 119, normalized size = 1.53 \begin {gather*} \frac {4 a c}{\left (4 a c -b^{2}\right )^{2} \left (c \,x^{2}+b x +a \right ) d}-\frac {b^{2}}{\left (4 a c -b^{2}\right )^{2} \left (c \,x^{2}+b x +a \right ) d}+\frac {8 c \ln \left (2 c x +b \right )}{\left (4 a c -b^{2}\right )^{2} d}-\frac {4 c \ln \left (c \,x^{2}+b x +a \right )}{\left (4 a c -b^{2}\right )^{2} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.45, size = 121, normalized size = 1.55 \begin {gather*} -\frac {4 \, c \log \left (c x^{2} + b x + a\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d} + \frac {8 \, c \log \left (2 \, c x + b\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d} - \frac {1}{{\left (b^{2} c - 4 \, a c^{2}\right )} d x^{2} + {\left (b^{3} - 4 \, a b c\right )} d x + {\left (a b^{2} - 4 \, a^{2} c\right )} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.58, size = 125, normalized size = 1.60 \begin {gather*} \frac {8\,c\,\ln \left (b+2\,c\,x\right )}{16\,d\,a^2\,c^2-8\,d\,a\,b^2\,c+d\,b^4}-\frac {4\,c\,\ln \left (c\,x^2+b\,x+a\right )}{16\,d\,a^2\,c^2-8\,d\,a\,b^2\,c+d\,b^4}-\frac {1}{-4\,d\,a^2\,c+d\,a\,b^2-4\,d\,a\,b\,c\,x-4\,d\,a\,c^2\,x^2+d\,b^3\,x+d\,b^2\,c\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.63, size = 102, normalized size = 1.31 \begin {gather*} \frac {8 c \log {\left (\frac {b}{2 c} + x \right )}}{d \left (4 a c - b^{2}\right )^{2}} - \frac {4 c \log {\left (\frac {a}{c} + \frac {b x}{c} + x^{2} \right )}}{d \left (4 a c - b^{2}\right )^{2}} + \frac {1}{4 a^{2} c d - a b^{2} d + x^{2} \left (4 a c^{2} d - b^{2} c d\right ) + x \left (4 a b c d - b^{3} d\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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