Optimal. Leaf size=43 \[ 4 c d^3 \log \left (a+b x+c x^2\right )-\frac{d^3 (b+2 c x)^2}{a+b x+c x^2} \]
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Rubi [A] time = 0.0202037, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {686, 628} \[ 4 c d^3 \log \left (a+b x+c x^2\right )-\frac{d^3 (b+2 c x)^2}{a+b x+c x^2} \]
Antiderivative was successfully verified.
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Rule 686
Rule 628
Rubi steps
\begin{align*} \int \frac{(b d+2 c d x)^3}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac{d^3 (b+2 c x)^2}{a+b x+c x^2}+\left (4 c d^2\right ) \int \frac{b d+2 c d x}{a+b x+c x^2} \, dx\\ &=-\frac{d^3 (b+2 c x)^2}{a+b x+c x^2}+4 c d^3 \log \left (a+b x+c x^2\right )\\ \end{align*}
Mathematica [A] time = 0.020575, size = 42, normalized size = 0.98 \[ d^3 \left (\frac{4 a c-b^2}{a+b x+c x^2}+4 c \log \left (a+b x+c x^2\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 58, normalized size = 1.4 \begin{align*} 4\,{\frac{{d}^{3}ac}{c{x}^{2}+bx+a}}-{\frac{{d}^{3}{b}^{2}}{c{x}^{2}+bx+a}}+4\,c{d}^{3}\ln \left ( c{x}^{2}+bx+a \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01148, size = 58, normalized size = 1.35 \begin{align*} 4 \, c d^{3} \log \left (c x^{2} + b x + a\right ) - \frac{{\left (b^{2} - 4 \, a c\right )} d^{3}}{c x^{2} + b x + a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01536, size = 136, normalized size = 3.16 \begin{align*} -\frac{{\left (b^{2} - 4 \, a c\right )} d^{3} - 4 \,{\left (c^{2} d^{3} x^{2} + b c d^{3} x + a c d^{3}\right )} \log \left (c x^{2} + b x + a\right )}{c x^{2} + b x + a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.21328, size = 42, normalized size = 0.98 \begin{align*} 4 c d^{3} \log{\left (a + b x + c x^{2} \right )} + \frac{4 a c d^{3} - b^{2} d^{3}}{a + b x + c x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14961, size = 63, normalized size = 1.47 \begin{align*} 4 \, c d^{3} \log \left (c x^{2} + b x + a\right ) - \frac{b^{2} d^{3} - 4 \, a c d^{3}}{c x^{2} + b x + a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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