Optimal. Leaf size=73 \[ 16 c^2 d^5 \log \left (a+b x+c x^2\right )-\frac{4 c d^5 (b+2 c x)^2}{a+b x+c x^2}-\frac{d^5 (b+2 c x)^4}{2 \left (a+b x+c x^2\right )^2} \]
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Rubi [A] time = 0.0379107, antiderivative size = 73, 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 = {686, 628} \[ 16 c^2 d^5 \log \left (a+b x+c x^2\right )-\frac{4 c d^5 (b+2 c x)^2}{a+b x+c x^2}-\frac{d^5 (b+2 c x)^4}{2 \left (a+b x+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 686
Rule 628
Rubi steps
\begin{align*} \int \frac{(b d+2 c d x)^5}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac{d^5 (b+2 c x)^4}{2 \left (a+b x+c x^2\right )^2}+\left (4 c d^2\right ) \int \frac{(b d+2 c d x)^3}{\left (a+b x+c x^2\right )^2} \, dx\\ &=-\frac{d^5 (b+2 c x)^4}{2 \left (a+b x+c x^2\right )^2}-\frac{4 c d^5 (b+2 c x)^2}{a+b x+c x^2}+\left (16 c^2 d^4\right ) \int \frac{b d+2 c d x}{a+b x+c x^2} \, dx\\ &=-\frac{d^5 (b+2 c x)^4}{2 \left (a+b x+c x^2\right )^2}-\frac{4 c d^5 (b+2 c x)^2}{a+b x+c x^2}+16 c^2 d^5 \log \left (a+b x+c x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0418596, size = 65, normalized size = 0.89 \[ d^5 \left (16 c^2 \log (a+x (b+c x))-\frac{\left (b^2-4 a c\right ) \left (4 c \left (3 a+4 c x^2\right )+b^2+16 b c x\right )}{2 (a+x (b+c x))^2}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.048, size = 181, normalized size = 2.5 \begin{align*} 32\,{\frac{{x}^{2}a{c}^{3}{d}^{5}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-8\,{\frac{{x}^{2}{b}^{2}{c}^{2}{d}^{5}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}+32\,{\frac{xab{c}^{2}{d}^{5}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-8\,{\frac{x{b}^{3}c{d}^{5}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}+24\,{\frac{{d}^{5}{a}^{2}{c}^{2}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-4\,{\frac{{d}^{5}ac{b}^{2}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-{\frac{{d}^{5}{b}^{4}}{2\, \left ( c{x}^{2}+bx+a \right ) ^{2}}}+16\,{c}^{2}{d}^{5}\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 = 1.2268, size = 167, normalized size = 2.29 \begin{align*} 16 \, c^{2} d^{5} \log \left (c x^{2} + b x + a\right ) - \frac{16 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{5} x^{2} + 16 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} d^{5} x +{\left (b^{4} + 8 \, a b^{2} c - 48 \, a^{2} c^{2}\right )} d^{5}}{2 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.35211, size = 385, normalized size = 5.27 \begin{align*} -\frac{16 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{5} x^{2} + 16 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} d^{5} x +{\left (b^{4} + 8 \, a b^{2} c - 48 \, a^{2} c^{2}\right )} d^{5} - 32 \,{\left (c^{4} d^{5} x^{4} + 2 \, b c^{3} d^{5} x^{3} + 2 \, a b c^{2} d^{5} x + a^{2} c^{2} d^{5} +{\left (b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} x^{2}\right )} \log \left (c x^{2} + b x + a\right )}{2 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 6.81381, size = 141, normalized size = 1.93 \begin{align*} 16 c^{2} d^{5} \log{\left (a + b x + c x^{2} \right )} + \frac{48 a^{2} c^{2} d^{5} - 8 a b^{2} c d^{5} - b^{4} d^{5} + x^{2} \left (64 a c^{3} d^{5} - 16 b^{2} c^{2} d^{5}\right ) + x \left (64 a b c^{2} d^{5} - 16 b^{3} c d^{5}\right )}{2 a^{2} + 4 a b x + 4 b c x^{3} + 2 c^{2} x^{4} + x^{2} \left (4 a c + 2 b^{2}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1679, size = 149, normalized size = 2.04 \begin{align*} 16 \, c^{2} d^{5} \log \left (c x^{2} + b x + a\right ) - \frac{b^{4} d^{5} + 8 \, a b^{2} c d^{5} - 48 \, a^{2} c^{2} d^{5} + 16 \,{\left (b^{2} c^{2} d^{5} - 4 \, a c^{3} d^{5}\right )} x^{2} + 16 \,{\left (b^{3} c d^{5} - 4 \, a b c^{2} d^{5}\right )} x}{2 \,{\left (c x^{2} + b x + a\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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