Optimal. Leaf size=97 \[ \frac {2 n^2 x \left (a+b x^n\right )^{-1/n}}{a^3 (n+1) (2 n+1)}+\frac {2 n x \left (a+b x^n\right )^{-\frac {n+1}{n}}}{a^2 (n+1) (2 n+1)}+\frac {x \left (a+b x^n\right )^{-\frac {1}{n}-2}}{a (2 n+1)} \]
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Rubi [A] time = 0.05, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {192, 191} \[ \frac {2 n^2 x \left (a+b x^n\right )^{-1/n}}{a^3 (n+1) (2 n+1)}+\frac {2 n x \left (a+b x^n\right )^{-\frac {n+1}{n}}}{a^2 (n+1) (2 n+1)}+\frac {x \left (a+b x^n\right )^{-\frac {1}{n}-2}}{a (2 n+1)} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rubi steps
\begin {align*} \int \left (a+b x^n\right )^{-\frac {1+3 n}{n}} \, dx &=\frac {x \left (a+b x^n\right )^{-2-\frac {1}{n}}}{a (1+2 n)}+\frac {(2 n) \int \left (a+b x^n\right )^{1-\frac {1+3 n}{n}} \, dx}{a (1+2 n)}\\ &=\frac {x \left (a+b x^n\right )^{-2-\frac {1}{n}}}{a (1+2 n)}+\frac {2 n x \left (a+b x^n\right )^{-\frac {1+n}{n}}}{a^2 (1+n) (1+2 n)}+\frac {\left (2 n^2\right ) \int \left (a+b x^n\right )^{2-\frac {1+3 n}{n}} \, dx}{a^2 (1+n) (1+2 n)}\\ &=\frac {x \left (a+b x^n\right )^{-2-\frac {1}{n}}}{a (1+2 n)}+\frac {2 n^2 x \left (a+b x^n\right )^{-1/n}}{a^3 (1+n) (1+2 n)}+\frac {2 n x \left (a+b x^n\right )^{-\frac {1+n}{n}}}{a^2 (1+n) (1+2 n)}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 55, normalized size = 0.57 \[ \frac {x \left (a+b x^n\right )^{-1/n} \left (\frac {b x^n}{a}+1\right )^{\frac {1}{n}} \, _2F_1\left (3+\frac {1}{n},\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 126, normalized size = 1.30 \[ \frac {2 \, b^{3} n^{2} x x^{3 \, n} + 2 \, {\left (3 \, a b^{2} n^{2} + a b^{2} n\right )} x x^{2 \, n} + {\left (6 \, a^{2} b n^{2} + 5 \, a^{2} b n + a^{2} b\right )} x x^{n} + {\left (2 \, a^{3} n^{2} + 3 \, a^{3} n + a^{3}\right )} x}{{\left (2 \, a^{3} n^{2} + 3 \, a^{3} n + a^{3}\right )} {\left (b x^{n} + a\right )}^{\frac {3 \, n + 1}{n}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{n} + a\right )}^{\frac {3 \, n + 1}{n}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.22, size = 0, normalized size = 0.00 \[ \int \left (b \,x^{n}+a \right )^{-\frac {3 n +1}{n}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{n} + a\right )}^{\frac {3 \, n + 1}{n}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.34, size = 64, normalized size = 0.66 \[ -\frac {x^{1-3\,n}\,{\left (\frac {a}{b\,x^n}+1\right )}^{1/n}\,{{}}_2{\mathrm {F}}_1\left (3,\frac {1}{n}+3;\ 4;\ -\frac {a}{b\,x^n}\right )}{3\,b^3\,n\,{\left (a+b\,x^n\right )}^{1/n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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