Optimal. Leaf size=36 \[ d^3 \left (b^2-4 a c\right ) \log \left (a+b x+c x^2\right )+d^3 (b+2 c x)^2 \]
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Rubi [A] time = 0.0228052, antiderivative size = 36, 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 = {692, 628} \[ d^3 \left (b^2-4 a c\right ) \log \left (a+b x+c x^2\right )+d^3 (b+2 c x)^2 \]
Antiderivative was successfully verified.
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Rule 692
Rule 628
Rubi steps
\begin{align*} \int \frac{(b d+2 c d x)^3}{a+b x+c x^2} \, dx &=d^3 (b+2 c x)^2+\left (\left (b^2-4 a c\right ) d^2\right ) \int \frac{b d+2 c d x}{a+b x+c x^2} \, dx\\ &=d^3 (b+2 c x)^2+\left (b^2-4 a c\right ) d^3 \log \left (a+b x+c x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0123265, size = 33, normalized size = 0.92 \[ d^3 \left (\left (b^2-4 a c\right ) \log (a+x (b+c x))+4 c x (b+c x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 57, normalized size = 1.6 \begin{align*} 4\,{x}^{2}{c}^{2}{d}^{3}-4\,\ln \left ( c{x}^{2}+bx+a \right ) ac{d}^{3}+\ln \left ( c{x}^{2}+bx+a \right ){b}^{2}{d}^{3}+4\,xbc{d}^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14233, size = 58, normalized size = 1.61 \begin{align*} 4 \, c^{2} d^{3} x^{2} + 4 \, b c d^{3} x +{\left (b^{2} - 4 \, a c\right )} d^{3} \log \left (c x^{2} + b x + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67158, size = 95, normalized size = 2.64 \begin{align*} 4 \, c^{2} d^{3} x^{2} + 4 \, b c d^{3} x +{\left (b^{2} - 4 \, a c\right )} d^{3} \log \left (c x^{2} + b x + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.624678, size = 44, normalized size = 1.22 \begin{align*} 4 b c d^{3} x + 4 c^{2} d^{3} x^{2} - d^{3} \left (4 a c - b^{2}\right ) \log{\left (a + b x + c x^{2} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12149, size = 72, normalized size = 2. \begin{align*}{\left (b^{2} d^{3} - 4 \, a c d^{3}\right )} \log \left (c x^{2} + b x + a\right ) + \frac{4 \,{\left (c^{4} d^{3} x^{2} + b c^{3} d^{3} x\right )}}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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