Optimal. Leaf size=18 \[ -\frac{1}{14 \left (a+b x^2+c x^4\right )^7} \]
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Rubi [A] time = 0.019708, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, 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 = 0.0135346, size = 18, 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 = 0.002, size = 17, normalized size = 0.9 \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 = 1.67348, size = 475, normalized size = 26.39 \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 = 1.24354, size = 748, normalized size = 41.56 \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 = 50.3184, size = 22, normalized size = 1.22 \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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