Optimal. Leaf size=72 \[ -\frac {\left (b^2-4 a c\right )^2}{128 c^3 d^5 (b+2 c x)^4}+\frac {b^2-4 a c}{32 c^3 d^5 (b+2 c x)^2}+\frac {\log (b+2 c x)}{32 c^3 d^5} \]
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Rubi [A] time = 0.06, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {683} \begin {gather*} -\frac {\left (b^2-4 a c\right )^2}{128 c^3 d^5 (b+2 c x)^4}+\frac {b^2-4 a c}{32 c^3 d^5 (b+2 c x)^2}+\frac {\log (b+2 c x)}{32 c^3 d^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 683
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^5} \, dx &=\int \left (\frac {\left (-b^2+4 a c\right )^2}{16 c^2 d^5 (b+2 c x)^5}+\frac {-b^2+4 a c}{8 c^2 d^5 (b+2 c x)^3}+\frac {1}{16 c^2 d^5 (b+2 c x)}\right ) \, dx\\ &=-\frac {\left (b^2-4 a c\right )^2}{128 c^3 d^5 (b+2 c x)^4}+\frac {b^2-4 a c}{32 c^3 d^5 (b+2 c x)^2}+\frac {\log (b+2 c x)}{32 c^3 d^5}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 59, normalized size = 0.82 \begin {gather*} \frac {\frac {\left (b^2-4 a c\right ) \left (4 c \left (a+4 c x^2\right )+3 b^2+16 b c x\right )}{(b+2 c x)^4}+4 \log (b+2 c x)}{128 c^3 d^5} \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 {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^5} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.41, size = 168, normalized size = 2.33 \begin {gather*} \frac {3 \, b^{4} - 8 \, a b^{2} c - 16 \, a^{2} c^{2} + 16 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} + 16 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} x + 4 \, {\left (16 \, c^{4} x^{4} + 32 \, b c^{3} x^{3} + 24 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x + b^{4}\right )} \log \left (2 \, c x + b\right )}{128 \, {\left (16 \, c^{7} d^{5} x^{4} + 32 \, b c^{6} d^{5} x^{3} + 24 \, b^{2} c^{5} d^{5} x^{2} + 8 \, b^{3} c^{4} d^{5} x + b^{4} c^{3} d^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 146, normalized size = 2.03 \begin {gather*} -\frac {\log \left (\frac {1}{4 \, {\left (2 \, c d x + b d\right )}^{2} c^{2} d^{2}}\right )}{64 \, c^{3} d^{5}} - \frac {\frac {b^{4} c^{3} d^{9}}{{\left (2 \, c d x + b d\right )}^{4}} - \frac {8 \, a b^{2} c^{4} d^{9}}{{\left (2 \, c d x + b d\right )}^{4}} + \frac {16 \, a^{2} c^{5} d^{9}}{{\left (2 \, c d x + b d\right )}^{4}} - \frac {4 \, b^{2} c^{3} d^{7}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac {16 \, a c^{4} d^{7}}{{\left (2 \, c d x + b d\right )}^{2}}}{128 \, c^{6} d^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 111, normalized size = 1.54 \begin {gather*} -\frac {a^{2}}{8 \left (2 c x +b \right )^{4} c \,d^{5}}+\frac {a \,b^{2}}{16 \left (2 c x +b \right )^{4} c^{2} d^{5}}-\frac {b^{4}}{128 \left (2 c x +b \right )^{4} c^{3} d^{5}}-\frac {a}{8 \left (2 c x +b \right )^{2} c^{2} d^{5}}+\frac {b^{2}}{32 \left (2 c x +b \right )^{2} c^{3} d^{5}}+\frac {\ln \left (2 c x +b \right )}{32 c^{3} d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.46, size = 136, normalized size = 1.89 \begin {gather*} \frac {3 \, b^{4} - 8 \, a b^{2} c - 16 \, a^{2} c^{2} + 16 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} + 16 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} x}{128 \, {\left (16 \, c^{7} d^{5} x^{4} + 32 \, b c^{6} d^{5} x^{3} + 24 \, b^{2} c^{5} d^{5} x^{2} + 8 \, b^{3} c^{4} d^{5} x + b^{4} c^{3} d^{5}\right )}} + \frac {\log \left (2 \, c x + b\right )}{32 \, c^{3} d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.49, size = 135, normalized size = 1.88 \begin {gather*} \frac {\ln \left (b+2\,c\,x\right )}{32\,c^3\,d^5}-\frac {\frac {16\,a^2\,c^2+8\,a\,b^2\,c-3\,b^4}{128\,c^3}-\frac {x\,\left (b^3-4\,a\,b\,c\right )}{8\,c^2}+\frac {x^2\,\left (4\,a\,c-b^2\right )}{8\,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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.54, size = 139, normalized size = 1.93 \begin {gather*} \frac {- 16 a^{2} c^{2} - 8 a b^{2} c + 3 b^{4} + x^{2} \left (- 64 a c^{3} + 16 b^{2} c^{2}\right ) + x \left (- 64 a b c^{2} + 16 b^{3} c\right )}{128 b^{4} c^{3} d^{5} + 1024 b^{3} c^{4} d^{5} x + 3072 b^{2} c^{5} d^{5} x^{2} + 4096 b c^{6} d^{5} x^{3} + 2048 c^{7} d^{5} x^{4}} + \frac {\log {\left (b + 2 c x \right )}}{32 c^{3} d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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