Optimal. Leaf size=20 \[ \frac{1}{14 \left (a-b x^2-c x^4\right )^7} \]
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Rubi [A] time = 0.0195398, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {1247, 629} \[ \frac{1}{14 \left (a-b x^2-c x^4\right )^7} \]
Antiderivative was successfully verified.
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Rule 1247
Rule 629
Rubi steps
\begin{align*} \int \frac{x \left (b+2 c x^2\right )}{\left (-a+b x^2+c x^4\right )^8} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{b+2 c x}{\left (-a+b x+c x^2\right )^8} \, dx,x,x^2\right )\\ &=\frac{1}{14 \left (a-b x^2-c x^4\right )^7}\\ \end{align*}
Mathematica [A] time = 0.0169894, size = 20, normalized size = 1. \[ -\frac{1}{14 \left (-a+b x^2+c x^4\right )^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 19, normalized size = 1. \begin{align*} -{\frac{1}{14\, \left ( c{x}^{4}+b{x}^{2}-a \right ) ^{7}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.48109, size = 481, normalized size = 24.05 \begin{align*} -\frac{1}{14 \,{\left (c^{7} x^{28} + 7 \, b c^{6} x^{26} + 7 \,{\left (3 \, b^{2} c^{5} - a c^{6}\right )} x^{24} + 7 \,{\left (5 \, b^{3} c^{4} - 6 \, a b c^{5}\right )} x^{22} + 7 \,{\left (5 \, b^{4} c^{3} - 15 \, a b^{2} c^{4} + 3 \, a^{2} c^{5}\right )} x^{20} + 7 \,{\left (3 \, b^{5} c^{2} - 20 \, a b^{3} c^{3} + 15 \, a^{2} b c^{4}\right )} x^{18} + 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^{16} +{\left (b^{7} - 42 \, a b^{5} c + 210 \, a^{2} b^{3} c^{2} - 140 \, a^{3} b c^{3}\right )} x^{14} - 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^{12} + 7 \,{\left (3 \, a^{2} b^{5} - 20 \, a^{3} b^{3} c + 15 \, a^{4} b c^{2}\right )} x^{10} + 7 \, a^{6} b x^{2} - 7 \,{\left (5 \, a^{3} b^{4} - 15 \, a^{4} b^{2} c + 3 \, a^{5} c^{2}\right )} x^{8} - a^{7} + 7 \,{\left (5 \, a^{4} b^{3} - 6 \, a^{5} b c\right )} x^{6} - 7 \,{\left (3 \, a^{5} b^{2} - a^{6} c\right )} x^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.32395, size = 748, normalized size = 37.4 \begin{align*} -\frac{1}{14 \,{\left (c^{7} x^{28} + 7 \, b c^{6} x^{26} + 7 \,{\left (3 \, b^{2} c^{5} - a c^{6}\right )} x^{24} + 7 \,{\left (5 \, b^{3} c^{4} - 6 \, a b c^{5}\right )} x^{22} + 7 \,{\left (5 \, b^{4} c^{3} - 15 \, a b^{2} c^{4} + 3 \, a^{2} c^{5}\right )} x^{20} + 7 \,{\left (3 \, b^{5} c^{2} - 20 \, a b^{3} c^{3} + 15 \, a^{2} b c^{4}\right )} x^{18} + 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^{16} +{\left (b^{7} - 42 \, a b^{5} c + 210 \, a^{2} b^{3} c^{2} - 140 \, a^{3} b c^{3}\right )} x^{14} - 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^{12} + 7 \,{\left (3 \, a^{2} b^{5} - 20 \, a^{3} b^{3} c + 15 \, a^{4} b c^{2}\right )} x^{10} + 7 \, a^{6} b x^{2} - 7 \,{\left (5 \, a^{3} b^{4} - 15 \, a^{4} b^{2} c + 3 \, a^{5} c^{2}\right )} x^{8} - a^{7} + 7 \,{\left (5 \, a^{4} b^{3} - 6 \, a^{5} b c\right )} x^{6} - 7 \,{\left (3 \, a^{5} b^{2} - a^{6} c\right )} x^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 55.4883, size = 24, normalized size = 1.2 \begin{align*} -\frac{1}{14 \,{\left (c x^{4} + b x^{2} - a\right )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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