Optimal. Leaf size=65 \[ 8 c d^5 \left (b^2-4 a c\right ) \log \left (a+b x+c x^2\right )-\frac{d^5 (b+2 c x)^4}{a+b x+c x^2}+8 c d^5 (b+2 c x)^2 \]
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Rubi [A] time = 0.039496, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {686, 692, 628} \[ 8 c d^5 \left (b^2-4 a c\right ) \log \left (a+b x+c x^2\right )-\frac{d^5 (b+2 c x)^4}{a+b x+c x^2}+8 c d^5 (b+2 c x)^2 \]
Antiderivative was successfully verified.
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Rule 686
Rule 692
Rule 628
Rubi steps
\begin{align*} \int \frac{(b d+2 c d x)^5}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac{d^5 (b+2 c x)^4}{a+b x+c x^2}+\left (8 c d^2\right ) \int \frac{(b d+2 c d x)^3}{a+b x+c x^2} \, dx\\ &=8 c d^5 (b+2 c x)^2-\frac{d^5 (b+2 c x)^4}{a+b x+c x^2}+\left (8 c \left (b^2-4 a c\right ) d^4\right ) \int \frac{b d+2 c d x}{a+b x+c x^2} \, dx\\ &=8 c d^5 (b+2 c x)^2-\frac{d^5 (b+2 c x)^4}{a+b x+c x^2}+8 c \left (b^2-4 a c\right ) d^5 \log \left (a+b x+c x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0342878, size = 64, normalized size = 0.98 \[ d^5 \left (-\frac{\left (b^2-4 a c\right )^2}{a+x (b+c x)}+8 c \left (b^2-4 a c\right ) \log (a+x (b+c x))+16 b c^2 x+16 c^3 x^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 128, normalized size = 2. \begin{align*} 16\,{d}^{5}{c}^{3}{x}^{2}+16\,{d}^{5}b{c}^{2}x-16\,{\frac{{d}^{5}{a}^{2}{c}^{2}}{c{x}^{2}+bx+a}}+8\,{\frac{{d}^{5}ac{b}^{2}}{c{x}^{2}+bx+a}}-{\frac{{d}^{5}{b}^{4}}{c{x}^{2}+bx+a}}-32\,{d}^{5}\ln \left ( c{x}^{2}+bx+a \right ) a{c}^{2}+8\,{d}^{5}\ln \left ( c{x}^{2}+bx+a \right ){b}^{2}c \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1619, size = 116, normalized size = 1.78 \begin{align*} 16 \, c^{3} d^{5} x^{2} + 16 \, b c^{2} d^{5} x + 8 \,{\left (b^{2} c - 4 \, a c^{2}\right )} d^{5} \log \left (c x^{2} + b x + a\right ) - \frac{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{5}}{c x^{2} + b x + a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.96973, size = 342, normalized size = 5.26 \begin{align*} \frac{16 \, c^{4} d^{5} x^{4} + 32 \, b c^{3} d^{5} x^{3} + 16 \, a b c^{2} d^{5} x + 16 \,{\left (b^{2} c^{2} + a c^{3}\right )} d^{5} x^{2} -{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{5} + 8 \,{\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{5} x^{2} +{\left (b^{3} c - 4 \, a b c^{2}\right )} d^{5} x +{\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} d^{5}\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 = 2.30193, size = 90, normalized size = 1.38 \begin{align*} 16 b c^{2} d^{5} x + 16 c^{3} d^{5} x^{2} - 8 c d^{5} \left (4 a c - b^{2}\right ) \log{\left (a + b x + c x^{2} \right )} - \frac{16 a^{2} c^{2} d^{5} - 8 a b^{2} c d^{5} + b^{4} d^{5}}{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 = 1.16529, size = 135, normalized size = 2.08 \begin{align*} 8 \,{\left (b^{2} c d^{5} - 4 \, a c^{2} d^{5}\right )} \log \left (c x^{2} + b x + a\right ) - \frac{b^{4} d^{5} - 8 \, a b^{2} c d^{5} + 16 \, a^{2} c^{2} d^{5}}{c x^{2} + b x + a} + \frac{16 \,{\left (c^{7} d^{5} x^{2} + 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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