Optimal. Leaf size=25 \[ \frac{1}{7 n \left (a-b x^n-c x^{2 n}\right )^7} \]
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Rubi [A] time = 0.0290118, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, 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 = 0.0680033, size = 23, normalized size = 0.92 \[ \frac{1}{7 n \left (a-x^n \left (b+c x^n\right )\right )^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 24, normalized size = 1. \begin{align*}{\frac{1}{7\,n \left ( -c \left ({x}^{n} \right ) ^{2}-b{x}^{n}+a \right ) ^{7}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.89127, size = 566, normalized size = 22.64 \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 = 1.51303, size = 851, normalized size = 34.04 \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 = 1.23779, size = 31, normalized size = 1.24 \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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