Optimal. Leaf size=100 \[ -\frac{3 \left (b^2-4 a c\right ) (b+2 c x)^4}{512 c^4 d}+\frac{3 \left (b^2-4 a c\right )^2 (b+2 c x)^2}{256 c^4 d}-\frac{\left (b^2-4 a c\right )^3 \log (b+2 c x)}{128 c^4 d}+\frac{(b+2 c x)^6}{768 c^4 d} \]
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Rubi [A] time = 0.117778, antiderivative size = 100, 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} \[ -\frac{3 \left (b^2-4 a c\right ) (b+2 c x)^4}{512 c^4 d}+\frac{3 \left (b^2-4 a c\right )^2 (b+2 c x)^2}{256 c^4 d}-\frac{\left (b^2-4 a c\right )^3 \log (b+2 c x)}{128 c^4 d}+\frac{(b+2 c x)^6}{768 c^4 d} \]
Antiderivative was successfully verified.
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Rule 683
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^3}{b d+2 c d x} \, dx &=\int \left (\frac{\left (-b^2+4 a c\right )^3}{64 c^3 (b d+2 c d x)}+\frac{3 \left (-b^2+4 a c\right )^2 (b d+2 c d x)}{64 c^3 d^2}+\frac{3 \left (-b^2+4 a c\right ) (b d+2 c d x)^3}{64 c^3 d^4}+\frac{(b d+2 c d x)^5}{64 c^3 d^6}\right ) \, dx\\ &=\frac{3 \left (b^2-4 a c\right )^2 (b+2 c x)^2}{256 c^4 d}-\frac{3 \left (b^2-4 a c\right ) (b+2 c x)^4}{512 c^4 d}+\frac{(b+2 c x)^6}{768 c^4 d}-\frac{\left (b^2-4 a c\right )^3 \log (b+2 c x)}{128 c^4 d}\\ \end{align*}
Mathematica [A] time = 0.0451798, size = 111, normalized size = 1.11 \[ \frac{2 c x (b+c x) \left (8 c^2 \left (18 a^2+9 a c x^2+2 c^2 x^4\right )+2 b^2 c \left (5 c x^2-18 a\right )+8 b c^2 x \left (9 a+4 c x^2\right )-6 b^3 c x+3 b^4\right )-3 \left (b^2-4 a c\right )^3 \log (b+2 c x)}{384 c^4 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.041, size = 222, normalized size = 2.2 \begin{align*}{\frac{{c}^{2}{x}^{6}}{12\,d}}+{\frac{bc{x}^{5}}{4\,d}}+{\frac{3\,a{x}^{4}c}{8\,d}}+{\frac{7\,{x}^{4}{b}^{2}}{32\,d}}+{\frac{3\,a{x}^{3}b}{4\,d}}+{\frac{{x}^{3}{b}^{3}}{48\,cd}}+{\frac{3\,{a}^{2}{x}^{2}}{4\,d}}+{\frac{3\,a{b}^{2}{x}^{2}}{16\,cd}}-{\frac{{x}^{2}{b}^{4}}{64\,{c}^{2}d}}+{\frac{3\,b{a}^{2}x}{4\,cd}}-{\frac{3\,a{b}^{3}x}{16\,{c}^{2}d}}+{\frac{{b}^{5}x}{64\,d{c}^{3}}}+{\frac{\ln \left ( 2\,cx+b \right ){a}^{3}}{2\,cd}}-{\frac{3\,\ln \left ( 2\,cx+b \right ){a}^{2}{b}^{2}}{8\,{c}^{2}d}}+{\frac{3\,\ln \left ( 2\,cx+b \right ) a{b}^{4}}{32\,d{c}^{3}}}-{\frac{\ln \left ( 2\,cx+b \right ){b}^{6}}{128\,d{c}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14785, size = 220, normalized size = 2.2 \begin{align*} \frac{16 \, c^{5} x^{6} + 48 \, b c^{4} x^{5} + 6 \,{\left (7 \, b^{2} c^{3} + 12 \, a c^{4}\right )} x^{4} + 4 \,{\left (b^{3} c^{2} + 36 \, a b c^{3}\right )} x^{3} - 3 \,{\left (b^{4} c - 12 \, a b^{2} c^{2} - 48 \, a^{2} c^{3}\right )} x^{2} + 3 \,{\left (b^{5} - 12 \, a b^{3} c + 48 \, a^{2} b c^{2}\right )} x}{192 \, c^{3} d} - \frac{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \log \left (2 \, c x + b\right )}{128 \, c^{4} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.97344, size = 356, normalized size = 3.56 \begin{align*} \frac{32 \, c^{6} x^{6} + 96 \, b c^{5} x^{5} + 12 \,{\left (7 \, b^{2} c^{4} + 12 \, a c^{5}\right )} x^{4} + 8 \,{\left (b^{3} c^{3} + 36 \, a b c^{4}\right )} x^{3} - 6 \,{\left (b^{4} c^{2} - 12 \, a b^{2} c^{3} - 48 \, a^{2} c^{4}\right )} x^{2} + 6 \,{\left (b^{5} c - 12 \, a b^{3} c^{2} + 48 \, a^{2} b c^{3}\right )} x - 3 \,{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \log \left (2 \, c x + b\right )}{384 \, c^{4} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.08956, size = 141, normalized size = 1.41 \begin{align*} \frac{b c x^{5}}{4 d} + \frac{c^{2} x^{6}}{12 d} + \frac{x^{4} \left (12 a c + 7 b^{2}\right )}{32 d} + \frac{x^{3} \left (36 a b c + b^{3}\right )}{48 c d} + \frac{x^{2} \left (48 a^{2} c^{2} + 12 a b^{2} c - b^{4}\right )}{64 c^{2} d} + \frac{x \left (48 a^{2} b c^{2} - 12 a b^{3} c + b^{5}\right )}{64 c^{3} d} + \frac{\left (4 a c - b^{2}\right )^{3} \log{\left (b + 2 c x \right )}}{128 c^{4} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.16337, size = 288, normalized size = 2.88 \begin{align*} -\frac{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \log \left ({\left | 2 \, c x + b \right |}\right )}{128 \, c^{4} d} + \frac{16 \, c^{8} d^{5} x^{6} + 48 \, b c^{7} d^{5} x^{5} + 42 \, b^{2} c^{6} d^{5} x^{4} + 72 \, a c^{7} d^{5} x^{4} + 4 \, b^{3} c^{5} d^{5} x^{3} + 144 \, a b c^{6} d^{5} x^{3} - 3 \, b^{4} c^{4} d^{5} x^{2} + 36 \, a b^{2} c^{5} d^{5} x^{2} + 144 \, a^{2} c^{6} d^{5} x^{2} + 3 \, b^{5} c^{3} d^{5} x - 36 \, a b^{3} c^{4} d^{5} x + 144 \, a^{2} b c^{5} d^{5} x}{192 \, c^{6} d^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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