Optimal. Leaf size=107 \[ \frac{\left (b^2-4 a c\right )^3}{512 c^4 d^5 (b+2 c x)^4}-\frac{3 \left (b^2-4 a c\right )^2}{256 c^4 d^5 (b+2 c x)^2}-\frac{3 \left (b^2-4 a c\right ) \log (b+2 c x)}{128 c^4 d^5}+\frac{b x}{64 c^3 d^5}+\frac{x^2}{64 c^2 d^5} \]
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Rubi [A] time = 0.0982241, antiderivative size = 107, 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} \[ \frac{\left (b^2-4 a c\right )^3}{512 c^4 d^5 (b+2 c x)^4}-\frac{3 \left (b^2-4 a c\right )^2}{256 c^4 d^5 (b+2 c x)^2}-\frac{3 \left (b^2-4 a c\right ) \log (b+2 c x)}{128 c^4 d^5}+\frac{b x}{64 c^3 d^5}+\frac{x^2}{64 c^2 d^5} \]
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)^5} \, dx &=\int \left (\frac{b}{64 c^3 d^5}+\frac{x}{32 c^2 d^5}+\frac{\left (-b^2+4 a c\right )^3}{64 c^3 d^5 (b+2 c x)^5}+\frac{3 \left (-b^2+4 a c\right )^2}{64 c^3 d^5 (b+2 c x)^3}+\frac{3 \left (-b^2+4 a c\right )}{64 c^3 d^5 (b+2 c x)}\right ) \, dx\\ &=\frac{b x}{64 c^3 d^5}+\frac{x^2}{64 c^2 d^5}+\frac{\left (b^2-4 a c\right )^3}{512 c^4 d^5 (b+2 c x)^4}-\frac{3 \left (b^2-4 a c\right )^2}{256 c^4 d^5 (b+2 c x)^2}-\frac{3 \left (b^2-4 a c\right ) \log (b+2 c x)}{128 c^4 d^5}\\ \end{align*}
Mathematica [A] time = 0.0427481, size = 80, normalized size = 0.75 \[ \frac{\frac{\left (b^2-4 a c\right )^3}{(b+2 c x)^4}-\frac{6 \left (b^2-4 a c\right )^2}{(b+2 c x)^2}-12 \left (b^2-4 a c\right ) \log (b+2 c x)+8 b c x+8 c^2 x^2}{512 c^4 d^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 195, normalized size = 1.8 \begin{align*}{\frac{{x}^{2}}{64\,{c}^{2}{d}^{5}}}+{\frac{bx}{64\,{c}^{3}{d}^{5}}}-{\frac{3\,{a}^{2}}{16\,{c}^{2}{d}^{5} \left ( 2\,cx+b \right ) ^{2}}}+{\frac{3\,{b}^{2}a}{32\,{c}^{3}{d}^{5} \left ( 2\,cx+b \right ) ^{2}}}-{\frac{3\,{b}^{4}}{256\,{c}^{4}{d}^{5} \left ( 2\,cx+b \right ) ^{2}}}-{\frac{{a}^{3}}{8\,{d}^{5}c \left ( 2\,cx+b \right ) ^{4}}}+{\frac{3\,{b}^{2}{a}^{2}}{32\,{c}^{2}{d}^{5} \left ( 2\,cx+b \right ) ^{4}}}-{\frac{3\,a{b}^{4}}{128\,{c}^{3}{d}^{5} \left ( 2\,cx+b \right ) ^{4}}}+{\frac{{b}^{6}}{512\,{c}^{4}{d}^{5} \left ( 2\,cx+b \right ) ^{4}}}+{\frac{3\,\ln \left ( 2\,cx+b \right ) a}{32\,{c}^{3}{d}^{5}}}-{\frac{3\,\ln \left ( 2\,cx+b \right ){b}^{2}}{128\,{c}^{4}{d}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13375, size = 262, normalized size = 2.45 \begin{align*} -\frac{5 \, b^{6} - 36 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} + 64 \, a^{3} c^{3} + 24 \,{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + 24 \,{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x}{512 \,{\left (16 \, c^{8} d^{5} x^{4} + 32 \, b c^{7} d^{5} x^{3} + 24 \, b^{2} c^{6} d^{5} x^{2} + 8 \, b^{3} c^{5} d^{5} x + b^{4} c^{4} d^{5}\right )}} + \frac{c x^{2} + b x}{64 \, c^{3} d^{5}} - \frac{3 \,{\left (b^{2} - 4 \, a c\right )} \log \left (2 \, c x + b\right )}{128 \, c^{4} d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.07979, size = 624, normalized size = 5.83 \begin{align*} \frac{128 \, c^{6} x^{6} + 384 \, b c^{5} x^{5} + 448 \, b^{2} c^{4} x^{4} + 256 \, b^{3} c^{3} x^{3} - 5 \, b^{6} + 36 \, a b^{4} c - 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3} + 48 \,{\left (b^{4} c^{2} + 4 \, a b^{2} c^{3} - 8 \, a^{2} c^{4}\right )} x^{2} - 16 \,{\left (b^{5} c - 12 \, a b^{3} c^{2} + 24 \, a^{2} b c^{3}\right )} x - 12 \,{\left (b^{6} - 4 \, a b^{4} c + 16 \,{\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} + 32 \,{\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{3} + 24 \,{\left (b^{4} c^{2} - 4 \, a b^{2} c^{3}\right )} x^{2} + 8 \,{\left (b^{5} c - 4 \, a b^{3} c^{2}\right )} x\right )} \log \left (2 \, c x + b\right )}{512 \,{\left (16 \, c^{8} d^{5} x^{4} + 32 \, b c^{7} d^{5} x^{3} + 24 \, b^{2} c^{6} d^{5} x^{2} + 8 \, b^{3} c^{5} d^{5} x + b^{4} c^{4} d^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.38392, size = 209, normalized size = 1.95 \begin{align*} \frac{b x}{64 c^{3} d^{5}} - \frac{64 a^{3} c^{3} + 48 a^{2} b^{2} c^{2} - 36 a b^{4} c + 5 b^{6} + x^{2} \left (384 a^{2} c^{4} - 192 a b^{2} c^{3} + 24 b^{4} c^{2}\right ) + x \left (384 a^{2} b c^{3} - 192 a b^{3} c^{2} + 24 b^{5} c\right )}{512 b^{4} c^{4} d^{5} + 4096 b^{3} c^{5} d^{5} x + 12288 b^{2} c^{6} d^{5} x^{2} + 16384 b c^{7} d^{5} x^{3} + 8192 c^{8} d^{5} x^{4}} + \frac{x^{2}}{64 c^{2} d^{5}} + \frac{3 \left (4 a c - b^{2}\right ) \log{\left (b + 2 c x \right )}}{128 c^{4} d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.21066, size = 354, normalized size = 3.31 \begin{align*} \frac{3 \,{\left (b^{2} - 4 \, a c\right )} \log \left (\frac{1}{4 \,{\left (2 \, c d x + b d\right )}^{2} c^{2} d^{2}}\right )}{256 \, c^{4} d^{5}} - \frac{{\left (2 \, c d x + b d\right )}^{2}{\left (\frac{3 \, b^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} - \frac{12 \, a c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} - 1\right )}}{256 \, c^{4} d^{7}} + \frac{\frac{b^{6} c^{8} d^{17}}{{\left (2 \, c d x + b d\right )}^{4}} - \frac{12 \, a b^{4} c^{9} d^{17}}{{\left (2 \, c d x + b d\right )}^{4}} + \frac{48 \, a^{2} b^{2} c^{10} d^{17}}{{\left (2 \, c d x + b d\right )}^{4}} - \frac{64 \, a^{3} c^{11} d^{17}}{{\left (2 \, c d x + b d\right )}^{4}} - \frac{6 \, b^{4} c^{8} d^{15}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac{48 \, a b^{2} c^{9} d^{15}}{{\left (2 \, c d x + b d\right )}^{2}} - \frac{96 \, a^{2} c^{10} d^{15}}{{\left (2 \, c d x + b d\right )}^{2}}}{512 \, c^{12} d^{18}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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