Optimal. Leaf size=61 \[ d^5 \left (b^2-4 a c\right )^2 \log \left (a+b x+c x^2\right )+d^5 \left (b^2-4 a c\right ) (b+2 c x)^2+\frac{1}{2} d^5 (b+2 c x)^4 \]
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Rubi [A] time = 0.0426205, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {692, 628} \[ d^5 \left (b^2-4 a c\right )^2 \log \left (a+b x+c x^2\right )+d^5 \left (b^2-4 a c\right ) (b+2 c x)^2+\frac{1}{2} d^5 (b+2 c x)^4 \]
Antiderivative was successfully verified.
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Rule 692
Rule 628
Rubi steps
\begin{align*} \int \frac{(b d+2 c d x)^5}{a+b x+c x^2} \, dx &=\frac{1}{2} d^5 (b+2 c x)^4+\left (\left (b^2-4 a c\right ) d^2\right ) \int \frac{(b d+2 c d x)^3}{a+b x+c x^2} \, dx\\ &=\left (b^2-4 a c\right ) d^5 (b+2 c x)^2+\frac{1}{2} d^5 (b+2 c x)^4+\left (\left (b^2-4 a c\right )^2 d^4\right ) \int \frac{b d+2 c d x}{a+b x+c x^2} \, dx\\ &=\left (b^2-4 a c\right ) d^5 (b+2 c x)^2+\frac{1}{2} d^5 (b+2 c x)^4+\left (b^2-4 a c\right )^2 d^5 \log \left (a+b x+c x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0232375, size = 54, normalized size = 0.89 \[ d^5 \left (8 c x (b+c x) \left (c \left (c x^2-2 a\right )+b^2+b c x\right )+\left (b^2-4 a c\right )^2 \log (a+x (b+c x))\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.041, size = 133, normalized size = 2.2 \begin{align*} 8\,{x}^{4}{c}^{4}{d}^{5}+16\,{x}^{3}b{c}^{3}{d}^{5}-16\,{x}^{2}a{c}^{3}{d}^{5}+16\,{x}^{2}{b}^{2}{c}^{2}{d}^{5}+16\,\ln \left ( c{x}^{2}+bx+a \right ){a}^{2}{c}^{2}{d}^{5}-8\,\ln \left ( c{x}^{2}+bx+a \right ) a{b}^{2}c{d}^{5}+\ln \left ( c{x}^{2}+bx+a \right ){b}^{4}{d}^{5}-16\,xab{c}^{2}{d}^{5}+8\,x{b}^{3}c{d}^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.58318, size = 134, normalized size = 2.2 \begin{align*} 8 \, c^{4} d^{5} x^{4} + 16 \, b c^{3} d^{5} x^{3} + 16 \,{\left (b^{2} c^{2} - a c^{3}\right )} d^{5} x^{2} + 8 \,{\left (b^{3} c - 2 \, a b c^{2}\right )} d^{5} x +{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{5} \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 = 1.69205, size = 207, normalized size = 3.39 \begin{align*} 8 \, c^{4} d^{5} x^{4} + 16 \, b c^{3} d^{5} x^{3} + 16 \,{\left (b^{2} c^{2} - a c^{3}\right )} d^{5} x^{2} + 8 \,{\left (b^{3} c - 2 \, a b c^{2}\right )} d^{5} x +{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{5} \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 = 0.874252, size = 99, normalized size = 1.62 \begin{align*} 16 b c^{3} d^{5} x^{3} + 8 c^{4} d^{5} x^{4} + d^{5} \left (4 a c - b^{2}\right )^{2} \log{\left (a + b x + c x^{2} \right )} + x^{2} \left (- 16 a c^{3} d^{5} + 16 b^{2} c^{2} d^{5}\right ) + x \left (- 16 a b c^{2} d^{5} + 8 b^{3} c d^{5}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20101, size = 159, normalized size = 2.61 \begin{align*}{\left (b^{4} d^{5} - 8 \, a b^{2} c d^{5} + 16 \, a^{2} c^{2} d^{5}\right )} \log \left (c x^{2} + b x + a\right ) + \frac{8 \,{\left (c^{8} d^{5} x^{4} + 2 \, b c^{7} d^{5} x^{3} + 2 \, b^{2} c^{6} d^{5} x^{2} - 2 \, a c^{7} d^{5} x^{2} + b^{3} c^{5} d^{5} x - 2 \, a b c^{6} d^{5} x\right )}}{c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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