Optimal. Leaf size=72 \[ -\frac{x \left (b^2-8 a c\right )}{16 c^2 d^2}-\frac{\left (b^2-4 a c\right )^2}{32 c^3 d^2 (b+2 c x)}+\frac{b x^2}{8 c d^2}+\frac{x^3}{12 d^2} \]
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Rubi [A] time = 0.0598793, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {683} \[ -\frac{x \left (b^2-8 a c\right )}{16 c^2 d^2}-\frac{\left (b^2-4 a c\right )^2}{32 c^3 d^2 (b+2 c x)}+\frac{b x^2}{8 c d^2}+\frac{x^3}{12 d^2} \]
Antiderivative was successfully verified.
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Rule 683
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^2} \, dx &=\int \left (\frac{-b^2+8 a c}{16 c^2 d^2}+\frac{b x}{4 c d^2}+\frac{x^2}{4 d^2}+\frac{\left (-b^2+4 a c\right )^2}{16 c^2 d^2 (b+2 c x)^2}\right ) \, dx\\ &=-\frac{\left (b^2-8 a c\right ) x}{16 c^2 d^2}+\frac{b x^2}{8 c d^2}+\frac{x^3}{12 d^2}-\frac{\left (b^2-4 a c\right )^2}{32 c^3 d^2 (b+2 c x)}\\ \end{align*}
Mathematica [A] time = 0.0519079, size = 59, normalized size = 0.82 \[ \frac{-\frac{6 x \left (b^2-8 a c\right )}{c^2}-\frac{3 \left (b^2-4 a c\right )^2}{c^3 (b+2 c x)}+\frac{12 b x^2}{c}+8 x^3}{96 d^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 70, normalized size = 1. \begin{align*}{\frac{1}{{d}^{2}} \left ({\frac{1}{16\,{c}^{2}} \left ({\frac{4\,{x}^{3}{c}^{2}}{3}}+2\,bc{x}^{2}+8\,acx-{b}^{2}x \right ) }-{\frac{16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4}}{32\,{c}^{3} \left ( 2\,cx+b \right ) }} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.92837, size = 104, normalized size = 1.44 \begin{align*} -\frac{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}{32 \,{\left (2 \, c^{4} d^{2} x + b c^{3} d^{2}\right )}} + \frac{4 \, c^{2} x^{3} + 6 \, b c x^{2} - 3 \,{\left (b^{2} - 8 \, a c\right )} x}{48 \, c^{2} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8754, size = 182, normalized size = 2.53 \begin{align*} \frac{16 \, c^{4} x^{4} + 32 \, b c^{3} x^{3} + 96 \, a c^{3} x^{2} - 3 \, b^{4} + 24 \, a b^{2} c - 48 \, a^{2} c^{2} - 6 \,{\left (b^{3} c - 8 \, a b c^{2}\right )} x}{96 \,{\left (2 \, c^{4} d^{2} x + b c^{3} d^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.6431, size = 78, normalized size = 1.08 \begin{align*} \frac{b x^{2}}{8 c d^{2}} - \frac{16 a^{2} c^{2} - 8 a b^{2} c + b^{4}}{32 b c^{3} d^{2} + 64 c^{4} d^{2} x} + \frac{x^{3}}{12 d^{2}} + \frac{x \left (8 a c - b^{2}\right )}{16 c^{2} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.2094, size = 181, normalized size = 2.51 \begin{align*} -\frac{{\left (2 \, c d x + b d\right )}^{3}{\left (\frac{6 \, b^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} - \frac{24 \, a c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} - 1\right )}}{96 \, c^{3} d^{5}} - \frac{\frac{b^{4} c^{3} d^{7}}{2 \, c d x + b d} - \frac{8 \, a b^{2} c^{4} d^{7}}{2 \, c d x + b d} + \frac{16 \, a^{2} c^{5} d^{7}}{2 \, c d x + b d}}{32 \, c^{6} d^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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