Optimal. Leaf size=158 \[ -\frac {\sqrt [3]{b} \log \left (a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}+b^{2/3}\right )}{2 a^{4/3} n}+\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a} x^{-n/3}+\sqrt [3]{b}\right )}{a^{4/3} n}-\frac {\sqrt {3} \sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x^{-n/3}}{\sqrt {3} \sqrt [3]{b}}\right )}{a^{4/3} n}-\frac {3 x^{-n/3}}{a n} \]
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Rubi [A] time = 0.11, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {345, 193, 321, 200, 31, 634, 617, 204, 628} \[ -\frac {\sqrt [3]{b} \log \left (a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}+b^{2/3}\right )}{2 a^{4/3} n}+\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a} x^{-n/3}+\sqrt [3]{b}\right )}{a^{4/3} n}-\frac {\sqrt {3} \sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x^{-n/3}}{\sqrt {3} \sqrt [3]{b}}\right )}{a^{4/3} n}-\frac {3 x^{-n/3}}{a n} \]
Antiderivative was successfully verified.
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Rule 31
Rule 193
Rule 200
Rule 204
Rule 321
Rule 345
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {x^{-1-\frac {n}{3}}}{a+b x^n} \, dx &=-\frac {3 \operatorname {Subst}\left (\int \frac {1}{a+\frac {b}{x^3}} \, dx,x,x^{-n/3}\right )}{n}\\ &=-\frac {3 \operatorname {Subst}\left (\int \frac {x^3}{b+a x^3} \, dx,x,x^{-n/3}\right )}{n}\\ &=-\frac {3 x^{-n/3}}{a n}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{b+a x^3} \, dx,x,x^{-n/3}\right )}{a n}\\ &=-\frac {3 x^{-n/3}}{a n}+\frac {\sqrt [3]{b} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx,x,x^{-n/3}\right )}{a n}+\frac {\sqrt [3]{b} \operatorname {Subst}\left (\int \frac {2 \sqrt [3]{b}-\sqrt [3]{a} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx,x,x^{-n/3}\right )}{a n}\\ &=-\frac {3 x^{-n/3}}{a n}+\frac {\sqrt [3]{b} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x^{-n/3}\right )}{a^{4/3} n}-\frac {\sqrt [3]{b} \operatorname {Subst}\left (\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 a^{2/3} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx,x,x^{-n/3}\right )}{2 a^{4/3} n}+\frac {\left (3 b^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx,x,x^{-n/3}\right )}{2 a n}\\ &=-\frac {3 x^{-n/3}}{a n}+\frac {\sqrt [3]{b} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x^{-n/3}\right )}{a^{4/3} n}-\frac {\sqrt [3]{b} \log \left (b^{2/3}+a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}\right )}{2 a^{4/3} n}+\frac {\left (3 \sqrt [3]{b}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{a} x^{-n/3}}{\sqrt [3]{b}}\right )}{a^{4/3} n}\\ &=-\frac {3 x^{-n/3}}{a n}-\frac {\sqrt {3} \sqrt [3]{b} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{a} x^{-n/3}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{a^{4/3} n}+\frac {\sqrt [3]{b} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x^{-n/3}\right )}{a^{4/3} n}-\frac {\sqrt [3]{b} \log \left (b^{2/3}+a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}\right )}{2 a^{4/3} n}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 32, normalized size = 0.20 \[ -\frac {3 x^{-n/3} \, _2F_1\left (-\frac {1}{3},1;\frac {2}{3};-\frac {b x^n}{a}\right )}{a n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 145, normalized size = 0.92 \[ -\frac {6 \, x x^{-\frac {1}{3} \, n - 1} - 2 \, \sqrt {3} \left (\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x x^{-\frac {1}{3} \, n - 1} \left (\frac {b}{a}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) - 2 \, \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (\frac {x x^{-\frac {1}{3} \, n - 1} + \left (\frac {b}{a}\right )^{\frac {1}{3}}}{x}\right ) + \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (\frac {x^{2} x^{-\frac {2}{3} \, n - 2} - x x^{-\frac {1}{3} \, n - 1} \left (\frac {b}{a}\right )^{\frac {1}{3}} + \left (\frac {b}{a}\right )^{\frac {2}{3}}}{x^{2}}\right )}{2 \, a n} \]
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 \[ \int \frac {x^{-\frac {1}{3} \, n - 1}}{b x^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.07, size = 57, normalized size = 0.36 \[ \RootOf \left (a^{4} n^{3} \textit {\_Z}^{3}-b \right ) \ln \left (\frac {\RootOf \left (a^{4} n^{3} \textit {\_Z}^{3}-b \right )^{2} a^{3} n^{2}}{b}+x^{\frac {n}{3}}\right )-\frac {3 x^{-\frac {n}{3}}}{a n} \]
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 \[ -b \int \frac {x^{\frac {2}{3} \, n}}{a b x x^{n} + a^{2} x}\,{d x} - \frac {3}{a n x^{\frac {1}{3} \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^{\frac {n}{3}+1}\,\left (a+b\,x^n\right )} \,d x \]
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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