Optimal. Leaf size=18 \[ \frac{1}{7 \left (a-b x-c x^2\right )^7} \]
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Rubi [A] time = 0.004369, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {629} \[ \frac{1}{7 \left (a-b x-c x^2\right )^7} \]
Antiderivative was successfully verified.
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Rule 629
Rubi steps
\begin{align*} \int \frac{b+2 c x}{\left (-a+b x+c x^2\right )^8} \, dx &=\frac{1}{7 \left (a-b x-c x^2\right )^7}\\ \end{align*}
Mathematica [A] time = 0.0133975, size = 16, normalized size = 0.89 \[ \frac{1}{7 (a-x (b+c x))^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 17, normalized size = 0.9 \begin{align*} -{\frac{1}{7\, \left ( c{x}^{2}+bx-a \right ) ^{7}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00645, size = 22, normalized size = 1.22 \begin{align*} -\frac{1}{7 \,{\left (c x^{2} + b x - a\right )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.31757, size = 737, normalized size = 40.94 \begin{align*} -\frac{1}{7 \,{\left (c^{7} x^{14} + 7 \, b c^{6} x^{13} + 7 \,{\left (3 \, b^{2} c^{5} - a c^{6}\right )} x^{12} + 7 \,{\left (5 \, b^{3} c^{4} - 6 \, a b c^{5}\right )} x^{11} + 7 \,{\left (5 \, b^{4} c^{3} - 15 \, a b^{2} c^{4} + 3 \, a^{2} c^{5}\right )} x^{10} + 7 \,{\left (3 \, b^{5} c^{2} - 20 \, a b^{3} c^{3} + 15 \, a^{2} b c^{4}\right )} x^{9} + 7 \,{\left (b^{6} c - 15 \, a b^{4} c^{2} + 30 \, a^{2} b^{2} c^{3} - 5 \, a^{3} c^{4}\right )} x^{8} + 7 \, a^{6} b x +{\left (b^{7} - 42 \, a b^{5} c + 210 \, a^{2} b^{3} c^{2} - 140 \, a^{3} b c^{3}\right )} x^{7} - a^{7} - 7 \,{\left (a b^{6} - 15 \, a^{2} b^{4} c + 30 \, a^{3} b^{2} c^{2} - 5 \, a^{4} c^{3}\right )} x^{6} + 7 \,{\left (3 \, a^{2} b^{5} - 20 \, a^{3} b^{3} c + 15 \, a^{4} b c^{2}\right )} x^{5} - 7 \,{\left (5 \, a^{3} b^{4} - 15 \, a^{4} b^{2} c + 3 \, a^{5} c^{2}\right )} x^{4} + 7 \,{\left (5 \, a^{4} b^{3} - 6 \, a^{5} b c\right )} x^{3} - 7 \,{\left (3 \, a^{5} b^{2} - a^{6} c\right )} x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 55.2036, size = 359, normalized size = 19.94 \begin{align*} - \frac{1}{- 7 a^{7} + 49 a^{6} b x + 49 b c^{6} x^{13} + 7 c^{7} x^{14} + x^{12} \left (- 49 a c^{6} + 147 b^{2} c^{5}\right ) + x^{11} \left (- 294 a b c^{5} + 245 b^{3} c^{4}\right ) + x^{10} \left (147 a^{2} c^{5} - 735 a b^{2} c^{4} + 245 b^{4} c^{3}\right ) + x^{9} \left (735 a^{2} b c^{4} - 980 a b^{3} c^{3} + 147 b^{5} c^{2}\right ) + x^{8} \left (- 245 a^{3} c^{4} + 1470 a^{2} b^{2} c^{3} - 735 a b^{4} c^{2} + 49 b^{6} c\right ) + x^{7} \left (- 980 a^{3} b c^{3} + 1470 a^{2} b^{3} c^{2} - 294 a b^{5} c + 7 b^{7}\right ) + x^{6} \left (245 a^{4} c^{3} - 1470 a^{3} b^{2} c^{2} + 735 a^{2} b^{4} c - 49 a b^{6}\right ) + x^{5} \left (735 a^{4} b c^{2} - 980 a^{3} b^{3} c + 147 a^{2} b^{5}\right ) + x^{4} \left (- 147 a^{5} c^{2} + 735 a^{4} b^{2} c - 245 a^{3} b^{4}\right ) + x^{3} \left (- 294 a^{5} b c + 245 a^{4} b^{3}\right ) + x^{2} \left (49 a^{6} c - 147 a^{5} b^{2}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10916, size = 22, normalized size = 1.22 \begin{align*} -\frac{1}{7 \,{\left (c x^{2} + b x - a\right )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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