Optimal. Leaf size=103 \[ \frac{\left (b^2-4 a c\right )^3}{384 c^4 d^4 (b+2 c x)^3}-\frac{3 \left (b^2-4 a c\right )^2}{128 c^4 d^4 (b+2 c x)}-\frac{x \left (b^2-6 a c\right )}{32 c^3 d^4}+\frac{b x^2}{32 c^2 d^4}+\frac{x^3}{48 c d^4} \]
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Rubi [A] time = 0.0941266, antiderivative size = 103, 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} \[ \frac{\left (b^2-4 a c\right )^3}{384 c^4 d^4 (b+2 c x)^3}-\frac{3 \left (b^2-4 a c\right )^2}{128 c^4 d^4 (b+2 c x)}-\frac{x \left (b^2-6 a c\right )}{32 c^3 d^4}+\frac{b x^2}{32 c^2 d^4}+\frac{x^3}{48 c d^4} \]
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)^4} \, dx &=\int \left (\frac{-b^2+6 a c}{32 c^3 d^4}+\frac{b x}{16 c^2 d^4}+\frac{x^2}{16 c d^4}+\frac{\left (-b^2+4 a c\right )^3}{64 c^3 d^4 (b+2 c x)^4}+\frac{3 \left (-b^2+4 a c\right )^2}{64 c^3 d^4 (b+2 c x)^2}\right ) \, dx\\ &=-\frac{\left (b^2-6 a c\right ) x}{32 c^3 d^4}+\frac{b x^2}{32 c^2 d^4}+\frac{x^3}{48 c d^4}+\frac{\left (b^2-4 a c\right )^3}{384 c^4 d^4 (b+2 c x)^3}-\frac{3 \left (b^2-4 a c\right )^2}{128 c^4 d^4 (b+2 c x)}\\ \end{align*}
Mathematica [A] time = 0.064124, size = 81, normalized size = 0.79 \[ \frac{\frac{\left (b^2-4 a c\right )^3}{(b+2 c x)^3}-\frac{9 \left (b^2-4 a c\right )^2}{b+2 c x}+12 c x \left (6 a c-b^2\right )+12 b c^2 x^2+8 c^3 x^3}{384 c^4 d^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 116, normalized size = 1.1 \begin{align*}{\frac{1}{{d}^{4}} \left ({\frac{1}{32\,{c}^{3}} \left ({\frac{2\,{x}^{3}{c}^{2}}{3}}+bc{x}^{2}+6\,acx-{b}^{2}x \right ) }-{\frac{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}{384\,{c}^{4} \left ( 2\,cx+b \right ) ^{3}}}-{\frac{48\,{a}^{2}{c}^{2}-24\,ac{b}^{2}+3\,{b}^{4}}{128\,{c}^{4} \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 = 1.1684, size = 236, normalized size = 2.29 \begin{align*} -\frac{2 \, b^{6} - 15 \, a b^{4} c + 24 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3} + 9 \,{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + 9 \,{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x}{96 \,{\left (8 \, c^{7} d^{4} x^{3} + 12 \, b c^{6} d^{4} x^{2} + 6 \, b^{2} c^{5} d^{4} x + b^{3} c^{4} d^{4}\right )}} + \frac{2 \, c^{2} x^{3} + 3 \, b c x^{2} - 3 \,{\left (b^{2} - 6 \, a c\right )} x}{96 \, c^{3} d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.92167, size = 417, normalized size = 4.05 \begin{align*} \frac{16 \, c^{6} x^{6} + 48 \, b c^{5} x^{5} - 2 \, b^{6} + 15 \, a b^{4} c - 24 \, a^{2} b^{2} c^{2} - 16 \, a^{3} c^{3} + 24 \,{\left (b^{2} c^{4} + 6 \, a c^{5}\right )} x^{4} - 8 \,{\left (2 \, b^{3} c^{3} - 27 \, a b c^{4}\right )} x^{3} - 12 \,{\left (2 \, b^{4} c^{2} - 15 \, a b^{2} c^{3} + 12 \, a^{2} c^{4}\right )} x^{2} - 6 \,{\left (2 \, b^{5} c - 15 \, a b^{3} c^{2} + 24 \, a^{2} b c^{3}\right )} x}{96 \,{\left (8 \, c^{7} d^{4} x^{3} + 12 \, b c^{6} d^{4} x^{2} + 6 \, b^{2} c^{5} d^{4} x + b^{3} c^{4} d^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.40521, size = 185, normalized size = 1.8 \begin{align*} \frac{b x^{2}}{32 c^{2} d^{4}} - \frac{16 a^{3} c^{3} + 24 a^{2} b^{2} c^{2} - 15 a b^{4} c + 2 b^{6} + x^{2} \left (144 a^{2} c^{4} - 72 a b^{2} c^{3} + 9 b^{4} c^{2}\right ) + x \left (144 a^{2} b c^{3} - 72 a b^{3} c^{2} + 9 b^{5} c\right )}{96 b^{3} c^{4} d^{4} + 576 b^{2} c^{5} d^{4} x + 1152 b c^{6} d^{4} x^{2} + 768 c^{7} d^{4} x^{3}} + \frac{x^{3}}{48 c d^{4}} + \frac{x \left (6 a c - b^{2}\right )}{32 c^{3} d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.201, size = 221, normalized size = 2.15 \begin{align*} -\frac{9 \, b^{4} c^{2} x^{2} - 72 \, a b^{2} c^{3} x^{2} + 144 \, a^{2} c^{4} x^{2} + 9 \, b^{5} c x - 72 \, a b^{3} c^{2} x + 144 \, a^{2} b c^{3} x + 2 \, b^{6} - 15 \, a b^{4} c + 24 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3}}{96 \,{\left (2 \, c x + b\right )}^{3} c^{4} d^{4}} + \frac{2 \, c^{11} d^{8} x^{3} + 3 \, b c^{10} d^{8} x^{2} - 3 \, b^{2} c^{9} d^{8} x + 18 \, a c^{10} d^{8} x}{96 \, c^{12} d^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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