Optimal. Leaf size=37 \[ \frac{\left (a+b x+c x^2\right )^3}{3 d^7 \left (b^2-4 a c\right ) (b+2 c x)^6} \]
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Rubi [A] time = 0.0143671, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {682} \[ \frac{\left (a+b x+c x^2\right )^3}{3 d^7 \left (b^2-4 a c\right ) (b+2 c x)^6} \]
Antiderivative was successfully verified.
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Rule 682
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^7} \, dx &=\frac{\left (a+b x+c x^2\right )^3}{3 \left (b^2-4 a c\right ) d^7 (b+2 c x)^6}\\ \end{align*}
Mathematica [A] time = 0.0290149, size = 65, normalized size = 1.76 \[ -\frac{16 a^2 c^2-3 \left (b^2-4 a c\right ) (b+2 c x)^2-8 a b^2 c+b^4+3 (b+2 c x)^4}{192 c^3 d^7 (b+2 c x)^6} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.044, size = 74, normalized size = 2. \begin{align*}{\frac{1}{{d}^{7}} \left ( -{\frac{16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4}}{192\,{c}^{3} \left ( 2\,cx+b \right ) ^{6}}}-{\frac{1}{64\,{c}^{3} \left ( 2\,cx+b \right ) ^{2}}}-{\frac{4\,ac-{b}^{2}}{64\,{c}^{3} \left ( 2\,cx+b \right ) ^{4}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.18392, size = 221, normalized size = 5.97 \begin{align*} -\frac{48 \, c^{4} x^{4} + 96 \, b c^{3} x^{3} + b^{4} + 4 \, a b^{2} c + 16 \, a^{2} c^{2} + 12 \,{\left (5 \, b^{2} c^{2} + 4 \, a c^{3}\right )} x^{2} + 12 \,{\left (b^{3} c + 4 \, a b c^{2}\right )} x}{192 \,{\left (64 \, c^{9} d^{7} x^{6} + 192 \, b c^{8} d^{7} x^{5} + 240 \, b^{2} c^{7} d^{7} x^{4} + 160 \, b^{3} c^{6} d^{7} x^{3} + 60 \, b^{4} c^{5} d^{7} x^{2} + 12 \, b^{5} c^{4} d^{7} x + b^{6} c^{3} d^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.88119, size = 351, normalized size = 9.49 \begin{align*} -\frac{48 \, c^{4} x^{4} + 96 \, b c^{3} x^{3} + b^{4} + 4 \, a b^{2} c + 16 \, a^{2} c^{2} + 12 \,{\left (5 \, b^{2} c^{2} + 4 \, a c^{3}\right )} x^{2} + 12 \,{\left (b^{3} c + 4 \, a b c^{2}\right )} x}{192 \,{\left (64 \, c^{9} d^{7} x^{6} + 192 \, b c^{8} d^{7} x^{5} + 240 \, b^{2} c^{7} d^{7} x^{4} + 160 \, b^{3} c^{6} d^{7} x^{3} + 60 \, b^{4} c^{5} d^{7} x^{2} + 12 \, b^{5} c^{4} d^{7} x + b^{6} c^{3} d^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 3.89161, size = 172, normalized size = 4.65 \begin{align*} - \frac{16 a^{2} c^{2} + 4 a b^{2} c + b^{4} + 96 b c^{3} x^{3} + 48 c^{4} x^{4} + x^{2} \left (48 a c^{3} + 60 b^{2} c^{2}\right ) + x \left (48 a b c^{2} + 12 b^{3} c\right )}{192 b^{6} c^{3} d^{7} + 2304 b^{5} c^{4} d^{7} x + 11520 b^{4} c^{5} d^{7} x^{2} + 30720 b^{3} c^{6} d^{7} x^{3} + 46080 b^{2} c^{7} d^{7} x^{4} + 36864 b c^{8} d^{7} x^{5} + 12288 c^{9} d^{7} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.24655, size = 117, normalized size = 3.16 \begin{align*} -\frac{48 \, c^{4} x^{4} + 96 \, b c^{3} x^{3} + 60 \, b^{2} c^{2} x^{2} + 48 \, a c^{3} x^{2} + 12 \, b^{3} c x + 48 \, a b c^{2} x + b^{4} + 4 \, a b^{2} c + 16 \, a^{2} c^{2}}{192 \,{\left (2 \, c x + b\right )}^{6} c^{3} d^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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