Optimal. Leaf size=23 \[ -\frac{1}{7 n \left (a+b x^n+c x^{2 n}\right )^7} \]
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Rubi [A] time = 0.0266705, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {1468, 629} \[ -\frac{1}{7 n \left (a+b x^n+c x^{2 n}\right )^7} \]
Antiderivative was successfully verified.
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Rule 1468
Rule 629
Rubi steps
\begin{align*} \int \frac{x^{-1+n} \left (b+2 c x^n\right )}{\left (a+b x^n+c x^{2 n}\right )^8} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{b+2 c x}{\left (a+b x+c x^2\right )^8} \, dx,x,x^n\right )}{n}\\ &=-\frac{1}{7 n \left (a+b x^n+c x^{2 n}\right )^7}\\ \end{align*}
Mathematica [A] time = 0.0621506, size = 22, normalized size = 0.96 \[ -\frac{1}{7 n \left (a+x^n \left (b+c x^n\right )\right )^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 22, normalized size = 1. \begin{align*} -{\frac{1}{7\,n \left ( a+b{x}^{n}+c \left ({x}^{n} \right ) ^{2} \right ) ^{7}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 3.20277, size = 562, normalized size = 24.43 \begin{align*} -\frac{1}{7 \,{\left (c^{7} n x^{14 \, n} + 7 \, b c^{6} n x^{13 \, n} + 7 \, a^{6} b n x^{n} + a^{7} n + 7 \,{\left (3 \, b^{2} c^{5} n + a c^{6} n\right )} x^{12 \, n} + 7 \,{\left (5 \, b^{3} c^{4} n + 6 \, a b c^{5} n\right )} x^{11 \, n} + 7 \,{\left (5 \, b^{4} c^{3} n + 15 \, a b^{2} c^{4} n + 3 \, a^{2} c^{5} n\right )} x^{10 \, n} + 7 \,{\left (3 \, b^{5} c^{2} n + 20 \, a b^{3} c^{3} n + 15 \, a^{2} b c^{4} n\right )} x^{9 \, n} + 7 \,{\left (b^{6} c n + 15 \, a b^{4} c^{2} n + 30 \, a^{2} b^{2} c^{3} n + 5 \, a^{3} c^{4} n\right )} x^{8 \, n} +{\left (b^{7} n + 42 \, a b^{5} c n + 210 \, a^{2} b^{3} c^{2} n + 140 \, a^{3} b c^{3} n\right )} x^{7 \, n} + 7 \,{\left (a b^{6} n + 15 \, a^{2} b^{4} c n + 30 \, a^{3} b^{2} c^{2} n + 5 \, a^{4} c^{3} n\right )} x^{6 \, n} + 7 \,{\left (3 \, a^{2} b^{5} n + 20 \, a^{3} b^{3} c n + 15 \, a^{4} b c^{2} n\right )} x^{5 \, n} + 7 \,{\left (5 \, a^{3} b^{4} n + 15 \, a^{4} b^{2} c n + 3 \, a^{5} c^{2} n\right )} x^{4 \, n} + 7 \,{\left (5 \, a^{4} b^{3} n + 6 \, a^{5} b c n\right )} x^{3 \, n} + 7 \,{\left (3 \, a^{5} b^{2} n + a^{6} c n\right )} x^{2 \, n}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.35582, size = 851, normalized size = 37. \begin{align*} -\frac{1}{7 \,{\left (c^{7} n x^{14 \, n} + 7 \, b c^{6} n x^{13 \, n} + 7 \, a^{6} b n x^{n} + a^{7} n + 7 \,{\left (3 \, b^{2} c^{5} + a c^{6}\right )} n x^{12 \, n} + 7 \,{\left (5 \, b^{3} c^{4} + 6 \, a b c^{5}\right )} n x^{11 \, n} + 7 \,{\left (5 \, b^{4} c^{3} + 15 \, a b^{2} c^{4} + 3 \, a^{2} c^{5}\right )} n x^{10 \, n} + 7 \,{\left (3 \, b^{5} c^{2} + 20 \, a b^{3} c^{3} + 15 \, a^{2} b c^{4}\right )} n x^{9 \, n} + 7 \,{\left (b^{6} c + 15 \, a b^{4} c^{2} + 30 \, a^{2} b^{2} c^{3} + 5 \, a^{3} c^{4}\right )} n x^{8 \, n} +{\left (b^{7} + 42 \, a b^{5} c + 210 \, a^{2} b^{3} c^{2} + 140 \, a^{3} b c^{3}\right )} n x^{7 \, n} + 7 \,{\left (a b^{6} + 15 \, a^{2} b^{4} c + 30 \, a^{3} b^{2} c^{2} + 5 \, a^{4} c^{3}\right )} n x^{6 \, n} + 7 \,{\left (3 \, a^{2} b^{5} + 20 \, a^{3} b^{3} c + 15 \, a^{4} b c^{2}\right )} n x^{5 \, n} + 7 \,{\left (5 \, a^{3} b^{4} + 15 \, a^{4} b^{2} c + 3 \, a^{5} c^{2}\right )} n x^{4 \, n} + 7 \,{\left (5 \, a^{4} b^{3} + 6 \, a^{5} b c\right )} n x^{3 \, n} + 7 \,{\left (3 \, a^{5} b^{2} + a^{6} c\right )} n x^{2 \, n}\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 = 1.18251, size = 28, normalized size = 1.22 \begin{align*} -\frac{1}{7 \,{\left (c x^{2 \, n} + b x^{n} + a\right )}^{7} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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