Optimal. Leaf size=147 \[ \frac{3 n x \left (a+b x^n\right )^{-\frac{1}{n}-2}}{a^2 \left (6 n^2+5 n+1\right )}+\frac{6 n^3 x \left (a+b x^n\right )^{-1/n}}{a^4 (n+1) (2 n+1) (3 n+1)}+\frac{6 n^2 x \left (a+b x^n\right )^{-\frac{n+1}{n}}}{a^3 (n+1) (2 n+1) (3 n+1)}+\frac{x \left (a+b x^n\right )^{-\frac{1}{n}-3}}{a (3 n+1)} \]
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Rubi [A] time = 0.10286, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {192, 191} \[ \frac{3 n x \left (a+b x^n\right )^{-\frac{1}{n}-2}}{a^2 \left (6 n^2+5 n+1\right )}+\frac{6 n^3 x \left (a+b x^n\right )^{-1/n}}{a^4 (n+1) (2 n+1) (3 n+1)}+\frac{6 n^2 x \left (a+b x^n\right )^{-\frac{n+1}{n}}}{a^3 (n+1) (2 n+1) (3 n+1)}+\frac{x \left (a+b x^n\right )^{-\frac{1}{n}-3}}{a (3 n+1)} \]
Antiderivative was successfully verified.
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Rule 192
Rule 191
Rubi steps
\begin{align*} \int \left (a+b x^n\right )^{-\frac{1+4 n}{n}} \, dx &=\frac{x \left (a+b x^n\right )^{-3-\frac{1}{n}}}{a (1+3 n)}+\frac{(3 n) \int \left (a+b x^n\right )^{1-\frac{1+4 n}{n}} \, dx}{a (1+3 n)}\\ &=\frac{x \left (a+b x^n\right )^{-3-\frac{1}{n}}}{a (1+3 n)}+\frac{3 n x \left (a+b x^n\right )^{-2-\frac{1}{n}}}{a^2 \left (1+5 n+6 n^2\right )}+\frac{\left (6 n^2\right ) \int \left (a+b x^n\right )^{2-\frac{1+4 n}{n}} \, dx}{a^2 \left (1+5 n+6 n^2\right )}\\ &=\frac{x \left (a+b x^n\right )^{-3-\frac{1}{n}}}{a (1+3 n)}+\frac{3 n x \left (a+b x^n\right )^{-2-\frac{1}{n}}}{a^2 \left (1+5 n+6 n^2\right )}+\frac{6 n^2 x \left (a+b x^n\right )^{-\frac{1+n}{n}}}{a^3 (1+n) \left (1+5 n+6 n^2\right )}+\frac{\left (6 n^3\right ) \int \left (a+b x^n\right )^{3-\frac{1+4 n}{n}} \, dx}{a^3 (1+n) \left (1+5 n+6 n^2\right )}\\ &=\frac{x \left (a+b x^n\right )^{-3-\frac{1}{n}}}{a (1+3 n)}+\frac{3 n x \left (a+b x^n\right )^{-2-\frac{1}{n}}}{a^2 \left (1+5 n+6 n^2\right )}+\frac{6 n^3 x \left (a+b x^n\right )^{-1/n}}{a^4 (1+n) \left (1+5 n+6 n^2\right )}+\frac{6 n^2 x \left (a+b x^n\right )^{-\frac{1+n}{n}}}{a^3 (1+n) \left (1+5 n+6 n^2\right )}\\ \end{align*}
Mathematica [C] time = 0.0341706, size = 55, normalized size = 0.37 \[ \frac{x \left (a+b x^n\right )^{-1/n} \left (\frac{b x^n}{a}+1\right )^{\frac{1}{n}} \, _2F_1\left (4+\frac{1}{n},\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^4} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.117, size = 0, normalized size = 0. \begin{align*} \int \left ( \left ( a+b{x}^{n} \right ) ^{{\frac{1+4\,n}{n}}} \right ) ^{-1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{n} + a\right )}^{\frac{4 \, n + 1}{n}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.38964, size = 397, normalized size = 2.7 \begin{align*} \frac{6 \, b^{4} n^{3} x x^{4 \, n} + 6 \,{\left (4 \, a b^{3} n^{3} + a b^{3} n^{2}\right )} x x^{3 \, n} + 3 \,{\left (12 \, a^{2} b^{2} n^{3} + 7 \, a^{2} b^{2} n^{2} + a^{2} b^{2} n\right )} x x^{2 \, n} +{\left (24 \, a^{3} b n^{3} + 26 \, a^{3} b n^{2} + 9 \, a^{3} b n + a^{3} b\right )} x x^{n} +{\left (6 \, a^{4} n^{3} + 11 \, a^{4} n^{2} + 6 \, a^{4} n + a^{4}\right )} x}{{\left (6 \, a^{4} n^{3} + 11 \, a^{4} n^{2} + 6 \, a^{4} n + a^{4}\right )}{\left (b x^{n} + a\right )}^{\frac{4 \, n + 1}{n}}} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{n} + a\right )}^{\frac{4 \, n + 1}{n}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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