Optimal. Leaf size=94 \[ \frac{\left (b^2-4 a c\right )^3}{640 c^4 d^6 (b+2 c x)^5}-\frac{\left (b^2-4 a c\right )^2}{128 c^4 d^6 (b+2 c x)^3}+\frac{3 \left (b^2-4 a c\right )}{128 c^4 d^6 (b+2 c x)}+\frac{x}{64 c^3 d^6} \]
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Rubi [A] time = 0.0854048, antiderivative size = 94, 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} \[ \frac{\left (b^2-4 a c\right )^3}{640 c^4 d^6 (b+2 c x)^5}-\frac{\left (b^2-4 a c\right )^2}{128 c^4 d^6 (b+2 c x)^3}+\frac{3 \left (b^2-4 a c\right )}{128 c^4 d^6 (b+2 c x)}+\frac{x}{64 c^3 d^6} \]
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)^6} \, dx &=\int \left (\frac{1}{64 c^3 d^6}+\frac{\left (-b^2+4 a c\right )^3}{64 c^3 d^6 (b+2 c x)^6}+\frac{3 \left (-b^2+4 a c\right )^2}{64 c^3 d^6 (b+2 c x)^4}+\frac{3 \left (-b^2+4 a c\right )}{64 c^3 d^6 (b+2 c x)^2}\right ) \, dx\\ &=\frac{x}{64 c^3 d^6}+\frac{\left (b^2-4 a c\right )^3}{640 c^4 d^6 (b+2 c x)^5}-\frac{\left (b^2-4 a c\right )^2}{128 c^4 d^6 (b+2 c x)^3}+\frac{3 \left (b^2-4 a c\right )}{128 c^4 d^6 (b+2 c x)}\\ \end{align*}
Mathematica [A] time = 0.0629908, size = 72, normalized size = 0.77 \[ \frac{\frac{\left (b^2-4 a c\right )^3}{(b+2 c x)^5}-\frac{5 \left (b^2-4 a c\right )^2}{(b+2 c x)^3}+\frac{15 \left (b^2-4 a c\right )}{b+2 c x}+10 c x}{640 c^4 d^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 114, normalized size = 1.2 \begin{align*}{\frac{1}{{d}^{6}} \left ({\frac{x}{64\,{c}^{3}}}-{\frac{48\,{a}^{2}{c}^{2}-24\,ac{b}^{2}+3\,{b}^{4}}{384\,{c}^{4} \left ( 2\,cx+b \right ) ^{3}}}-{\frac{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}{640\,{c}^{4} \left ( 2\,cx+b \right ) ^{5}}}-{\frac{12\,ac-3\,{b}^{2}}{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 [B] time = 1.23413, size = 294, normalized size = 3.13 \begin{align*} \frac{11 \, b^{6} - 32 \, a b^{4} c - 32 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3} + 240 \,{\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} + 480 \,{\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{3} + 20 \,{\left (17 \, b^{4} c^{2} - 64 \, a b^{2} c^{3} - 16 \, a^{2} c^{4}\right )} x^{2} + 20 \,{\left (5 \, b^{5} c - 16 \, a b^{3} c^{2} - 16 \, a^{2} b c^{3}\right )} x}{640 \,{\left (32 \, c^{9} d^{6} x^{5} + 80 \, b c^{8} d^{6} x^{4} + 80 \, b^{2} c^{7} d^{6} x^{3} + 40 \, b^{3} c^{6} d^{6} x^{2} + 10 \, b^{4} c^{5} d^{6} x + b^{5} c^{4} d^{6}\right )}} + \frac{x}{64 \, c^{3} d^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.93359, size = 493, normalized size = 5.24 \begin{align*} \frac{320 \, c^{6} x^{6} + 800 \, b c^{5} x^{5} + 11 \, b^{6} - 32 \, a b^{4} c - 32 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3} + 80 \,{\left (13 \, b^{2} c^{4} - 12 \, a c^{5}\right )} x^{4} + 80 \,{\left (11 \, b^{3} c^{3} - 24 \, a b c^{4}\right )} x^{3} + 40 \,{\left (11 \, b^{4} c^{2} - 32 \, a b^{2} c^{3} - 8 \, a^{2} c^{4}\right )} x^{2} + 10 \,{\left (11 \, b^{5} c - 32 \, a b^{3} c^{2} - 32 \, a^{2} b c^{3}\right )} x}{640 \,{\left (32 \, c^{9} d^{6} x^{5} + 80 \, b c^{8} d^{6} x^{4} + 80 \, b^{2} c^{7} d^{6} x^{3} + 40 \, b^{3} c^{6} d^{6} x^{2} + 10 \, b^{4} c^{5} d^{6} x + b^{5} c^{4} d^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 5.23528, size = 223, normalized size = 2.37 \begin{align*} - \frac{64 a^{3} c^{3} + 32 a^{2} b^{2} c^{2} + 32 a b^{4} c - 11 b^{6} + x^{4} \left (960 a c^{5} - 240 b^{2} c^{4}\right ) + x^{3} \left (1920 a b c^{4} - 480 b^{3} c^{3}\right ) + x^{2} \left (320 a^{2} c^{4} + 1280 a b^{2} c^{3} - 340 b^{4} c^{2}\right ) + x \left (320 a^{2} b c^{3} + 320 a b^{3} c^{2} - 100 b^{5} c\right )}{640 b^{5} c^{4} d^{6} + 6400 b^{4} c^{5} d^{6} x + 25600 b^{3} c^{6} d^{6} x^{2} + 51200 b^{2} c^{7} d^{6} x^{3} + 51200 b c^{8} d^{6} x^{4} + 20480 c^{9} d^{6} x^{5}} + \frac{x}{64 c^{3} d^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18641, size = 216, normalized size = 2.3 \begin{align*} \frac{x}{64 \, c^{3} d^{6}} + \frac{240 \, b^{2} c^{4} x^{4} - 960 \, a c^{5} x^{4} + 480 \, b^{3} c^{3} x^{3} - 1920 \, a b c^{4} x^{3} + 340 \, b^{4} c^{2} x^{2} - 1280 \, a b^{2} c^{3} x^{2} - 320 \, a^{2} c^{4} x^{2} + 100 \, b^{5} c x - 320 \, a b^{3} c^{2} x - 320 \, a^{2} b c^{3} x + 11 \, b^{6} - 32 \, a b^{4} c - 32 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}{640 \,{\left (2 \, c x + b\right )}^{5} c^{4} d^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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