Optimal. Leaf size=112 \[ \frac {16 c^2 \log \left (a+b x+c x^2\right )}{d \left (b^2-4 a c\right )^3}-\frac {32 c^2 \log (b+2 c x)}{d \left (b^2-4 a c\right )^3}+\frac {4 c}{d \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {1}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]
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Rubi [A] time = 0.06, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 5, 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 {16 c^2 \log \left (a+b x+c x^2\right )}{d \left (b^2-4 a c\right )^3}-\frac {32 c^2 \log (b+2 c x)}{d \left (b^2-4 a c\right )^3}+\frac {4 c}{d \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {1}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\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 )^3} \, dx &=-\frac {1}{2 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^2}-\frac {(4 c) \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )^2} \, dx}{b^2-4 a c}\\ &=-\frac {1}{2 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^2}+\frac {4 c}{\left (b^2-4 a c\right )^2 d \left (a+b x+c x^2\right )}+\frac {\left (16 c^2\right ) \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )} \, dx}{\left (b^2-4 a c\right )^2}\\ &=-\frac {1}{2 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^2}+\frac {4 c}{\left (b^2-4 a c\right )^2 d \left (a+b x+c x^2\right )}+\frac {\left (16 c^2\right ) \int \frac {b d+2 c d x}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^3 d^2}-\frac {\left (64 c^3\right ) \int \frac {1}{b+2 c x} \, dx}{\left (b^2-4 a c\right )^3 d}\\ &=-\frac {1}{2 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^2}+\frac {4 c}{\left (b^2-4 a c\right )^2 d \left (a+b x+c x^2\right )}-\frac {32 c^2 \log (b+2 c x)}{\left (b^2-4 a c\right )^3 d}+\frac {16 c^2 \log \left (a+b x+c x^2\right )}{\left (b^2-4 a c\right )^3 d}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 90, normalized size = 0.80 \begin {gather*} \frac {\frac {8 c \left (b^2-4 a c\right )}{a+x (b+c x)}-\frac {\left (b^2-4 a c\right )^2}{(a+x (b+c x))^2}+32 c^2 \log (a+x (b+c x))-64 c^2 \log (b+2 c x)}{2 d \left (b^2-4 a c\right )^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 {1}{(b d+2 c d x) \left (a+b x+c x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.40, size = 386, normalized size = 3.45 \begin {gather*} -\frac {b^{4} - 16 \, a b^{2} c + 48 \, a^{2} c^{2} - 8 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} - 8 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} x - 32 \, {\left (c^{4} x^{4} + 2 \, b c^{3} x^{3} + 2 \, a b c^{2} x + a^{2} c^{2} + {\left (b^{2} c^{2} + 2 \, a c^{3}\right )} x^{2}\right )} \log \left (c x^{2} + b x + a\right ) + 64 \, {\left (c^{4} x^{4} + 2 \, b c^{3} x^{3} + 2 \, a b c^{2} x + a^{2} c^{2} + {\left (b^{2} c^{2} + 2 \, a c^{3}\right )} x^{2}\right )} \log \left (2 \, c x + b\right )}{2 \, {\left ({\left (b^{6} c^{2} - 12 \, a b^{4} c^{3} + 48 \, a^{2} b^{2} c^{4} - 64 \, a^{3} c^{5}\right )} d x^{4} + 2 \, {\left (b^{7} c - 12 \, a b^{5} c^{2} + 48 \, a^{2} b^{3} c^{3} - 64 \, a^{3} b c^{4}\right )} d x^{3} + {\left (b^{8} - 10 \, a b^{6} c + 24 \, a^{2} b^{4} c^{2} + 32 \, a^{3} b^{2} c^{3} - 128 \, a^{4} c^{4}\right )} d x^{2} + 2 \, {\left (a b^{7} - 12 \, a^{2} b^{5} c + 48 \, a^{3} b^{3} c^{2} - 64 \, a^{4} b c^{3}\right )} d x + {\left (a^{2} b^{6} - 12 \, a^{3} b^{4} c + 48 \, a^{4} b^{2} c^{2} - 64 \, a^{5} c^{3}\right )} d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 188, normalized size = 1.68 \begin {gather*} -\frac {32 \, c^{3} \log \left ({\left | 2 \, c x + b \right |}\right )}{b^{6} c d - 12 \, a b^{4} c^{2} d + 48 \, a^{2} b^{2} c^{3} d - 64 \, a^{3} c^{4} d} + \frac {16 \, c^{2} \log \left (c x^{2} + b x + a\right )}{b^{6} d - 12 \, a b^{4} c d + 48 \, a^{2} b^{2} c^{2} d - 64 \, a^{3} c^{3} d} - \frac {b^{4} - 16 \, a b^{2} c + 48 \, a^{2} c^{2} - 8 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} - 8 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} x}{2 \, {\left (c x^{2} + b x + a\right )}^{2} {\left (b^{2} - 4 \, a c\right )}^{3} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 304, normalized size = 2.71 \begin {gather*} \frac {16 a \,c^{3} x^{2}}{\left (4 a c -b^{2}\right )^{3} \left (c \,x^{2}+b x +a \right )^{2} d}-\frac {4 b^{2} c^{2} x^{2}}{\left (4 a c -b^{2}\right )^{3} \left (c \,x^{2}+b x +a \right )^{2} d}+\frac {16 a b \,c^{2} x}{\left (4 a c -b^{2}\right )^{3} \left (c \,x^{2}+b x +a \right )^{2} d}-\frac {4 b^{3} c x}{\left (4 a c -b^{2}\right )^{3} \left (c \,x^{2}+b x +a \right )^{2} d}+\frac {24 a^{2} c^{2}}{\left (4 a c -b^{2}\right )^{3} \left (c \,x^{2}+b x +a \right )^{2} d}-\frac {8 a \,b^{2} c}{\left (4 a c -b^{2}\right )^{3} \left (c \,x^{2}+b x +a \right )^{2} d}+\frac {b^{4}}{2 \left (4 a c -b^{2}\right )^{3} \left (c \,x^{2}+b x +a \right )^{2} d}+\frac {32 c^{2} \ln \left (2 c x +b \right )}{\left (4 a c -b^{2}\right )^{3} d}-\frac {16 c^{2} \ln \left (c \,x^{2}+b x +a \right )}{\left (4 a c -b^{2}\right )^{3} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.60, size = 266, normalized size = 2.38 \begin {gather*} \frac {16 \, c^{2} \log \left (c x^{2} + b x + a\right )}{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d} - \frac {32 \, c^{2} \log \left (2 \, c x + b\right )}{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d} + \frac {8 \, c^{2} x^{2} + 8 \, b c x - b^{2} + 12 \, a c}{2 \, {\left ({\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d x^{4} + 2 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d x^{3} + {\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} d x^{2} + 2 \, {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} d x + {\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2}\right )} d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.75, size = 239, normalized size = 2.13 \begin {gather*} \frac {\frac {12\,a\,c-b^2}{2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {4\,c^2\,x^2}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}+\frac {4\,b\,c\,x}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}}{a^2\,d+x^2\,\left (d\,b^2+2\,a\,c\,d\right )+c^2\,d\,x^4+2\,b\,c\,d\,x^3+2\,a\,b\,d\,x}-\frac {32\,c^2\,\ln \left (b+2\,c\,x\right )}{-64\,d\,a^3\,c^3+48\,d\,a^2\,b^2\,c^2-12\,d\,a\,b^4\,c+d\,b^6}+\frac {16\,c^2\,\ln \left (c\,x^2+b\,x+a\right )}{-64\,d\,a^3\,c^3+48\,d\,a^2\,b^2\,c^2-12\,d\,a\,b^4\,c+d\,b^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.17, size = 246, normalized size = 2.20 \begin {gather*} \frac {32 c^{2} \log {\left (\frac {b}{2 c} + x \right )}}{d \left (4 a c - b^{2}\right )^{3}} - \frac {16 c^{2} \log {\left (\frac {a}{c} + \frac {b x}{c} + x^{2} \right )}}{d \left (4 a c - b^{2}\right )^{3}} + \frac {12 a c - b^{2} + 8 b c x + 8 c^{2} x^{2}}{32 a^{4} c^{2} d - 16 a^{3} b^{2} c d + 2 a^{2} b^{4} d + x^{4} \left (32 a^{2} c^{4} d - 16 a b^{2} c^{3} d + 2 b^{4} c^{2} d\right ) + x^{3} \left (64 a^{2} b c^{3} d - 32 a b^{3} c^{2} d + 4 b^{5} c d\right ) + x^{2} \left (64 a^{3} c^{3} d - 12 a b^{4} c d + 2 b^{6} d\right ) + x \left (64 a^{3} b c^{2} d - 32 a^{2} b^{3} c d + 4 a b^{5} d\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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