Optimal. Leaf size=176 \[ -\frac {b^{4/3} \log \left (\sqrt [3]{a} x^{-n/3}+\sqrt [3]{b}\right )}{a^{7/3} n}+\frac {b^{4/3} \log \left (a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}+b^{2/3}\right )}{2 a^{7/3} n}+\frac {\sqrt {3} b^{4/3} \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x^{-n/3}}{\sqrt {3} \sqrt [3]{b}}\right )}{a^{7/3} n}+\frac {3 b x^{-n/3}}{a^2 n}-\frac {3 x^{-4 n/3}}{4 a n} \]
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Rubi [A] time = 0.11, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {362, 345, 193, 321, 200, 31, 634, 617, 204, 628} \[ -\frac {b^{4/3} \log \left (\sqrt [3]{a} x^{-n/3}+\sqrt [3]{b}\right )}{a^{7/3} n}+\frac {b^{4/3} \log \left (a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}+b^{2/3}\right )}{2 a^{7/3} n}+\frac {\sqrt {3} b^{4/3} \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x^{-n/3}}{\sqrt {3} \sqrt [3]{b}}\right )}{a^{7/3} n}+\frac {3 b x^{-n/3}}{a^2 n}-\frac {3 x^{-4 n/3}}{4 a n} \]
Antiderivative was successfully verified.
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Rule 31
Rule 193
Rule 200
Rule 204
Rule 321
Rule 345
Rule 362
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {x^{-1-\frac {4 n}{3}}}{a+b x^n} \, dx &=-\frac {3 x^{-4 n/3}}{4 a n}-\frac {b \int \frac {x^{-1-\frac {n}{3}}}{a+b x^n} \, dx}{a}\\ &=-\frac {3 x^{-4 n/3}}{4 a n}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{a+\frac {b}{x^3}} \, dx,x,x^{-n/3}\right )}{a n}\\ &=-\frac {3 x^{-4 n/3}}{4 a n}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {x^3}{b+a x^3} \, dx,x,x^{-n/3}\right )}{a n}\\ &=-\frac {3 x^{-4 n/3}}{4 a n}+\frac {3 b x^{-n/3}}{a^2 n}-\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{b+a x^3} \, dx,x,x^{-n/3}\right )}{a^2 n}\\ &=-\frac {3 x^{-4 n/3}}{4 a n}+\frac {3 b x^{-n/3}}{a^2 n}-\frac {b^{4/3} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx,x,x^{-n/3}\right )}{a^2 n}-\frac {b^{4/3} \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^2 n}\\ &=-\frac {3 x^{-4 n/3}}{4 a n}+\frac {3 b x^{-n/3}}{a^2 n}-\frac {b^{4/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x^{-n/3}\right )}{a^{7/3} n}+\frac {b^{4/3} \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^{7/3} n}-\frac {\left (3 b^{5/3}\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^2 n}\\ &=-\frac {3 x^{-4 n/3}}{4 a n}+\frac {3 b x^{-n/3}}{a^2 n}-\frac {b^{4/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x^{-n/3}\right )}{a^{7/3} n}+\frac {b^{4/3} \log \left (b^{2/3}+a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}\right )}{2 a^{7/3} n}-\frac {\left (3 b^{4/3}\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^{7/3} n}\\ &=-\frac {3 x^{-4 n/3}}{4 a n}+\frac {3 b x^{-n/3}}{a^2 n}+\frac {\sqrt {3} b^{4/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{a} x^{-n/3}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{a^{7/3} n}-\frac {b^{4/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x^{-n/3}\right )}{a^{7/3} n}+\frac {b^{4/3} \log \left (b^{2/3}+a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}\right )}{2 a^{7/3} n}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 34, normalized size = 0.19 \[ -\frac {3 x^{-4 n/3} \, _2F_1\left (-\frac {4}{3},1;-\frac {1}{3};-\frac {b x^n}{a}\right )}{4 a n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 180, normalized size = 1.02 \[ -\frac {3 \, a x x^{-\frac {4}{3} \, n - 1} - 4 \, \sqrt {3} b \left (-\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x^{\frac {1}{4}} x^{-\frac {1}{3} \, n - \frac {1}{4}} \left (-\frac {b}{a}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) + 2 \, b \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (\frac {x^{\frac {3}{4}} x^{-\frac {1}{3} \, n - \frac {1}{4}} \left (-\frac {b}{a}\right )^{\frac {1}{3}} + x x^{-\frac {2}{3} \, n - \frac {1}{2}} + \sqrt {x} \left (-\frac {b}{a}\right )^{\frac {2}{3}}}{x}\right ) - 4 \, b \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (\frac {x x^{-\frac {1}{3} \, n - \frac {1}{4}} - x^{\frac {3}{4}} \left (-\frac {b}{a}\right )^{\frac {1}{3}}}{x}\right ) - 12 \, b x^{\frac {1}{4}} x^{-\frac {1}{3} \, n - \frac {1}{4}}}{4 \, a^{2} n} \]
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 \[ \int \frac {x^{-\frac {4}{3} \, n - 1}}{b x^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.08, size = 73, normalized size = 0.41 \[ \RootOf \left (a^{7} n^{3} \textit {\_Z}^{3}+b^{4}\right ) \ln \left (\frac {\RootOf \left (a^{7} n^{3} \textit {\_Z}^{3}+b^{4}\right )^{2} a^{5} n^{2}}{b^{3}}+x^{\frac {n}{3}}\right )-\frac {3 x^{-\frac {4 n}{3}}}{4 a n}+\frac {3 b \,x^{-\frac {n}{3}}}{a^{2} n} \]
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 \[ b^{2} \int \frac {x^{\frac {2}{3} \, n}}{a^{2} b x x^{n} + a^{3} x}\,{d x} + \frac {3 \, {\left (4 \, b x^{n} - a\right )}}{4 \, a^{2} n x^{\frac {4}{3} \, n}} \]
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 \[ \int \frac {1}{x^{\frac {4\,n}{3}+1}\,\left (a+b\,x^n\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 4.34, size = 230, normalized size = 1.31 \[ \frac {x^{- \frac {4 n}{3}} \Gamma \left (- \frac {4}{3}\right )}{a n \Gamma \left (- \frac {1}{3}\right )} - \frac {4 b x^{- \frac {n}{3}} \Gamma \left (- \frac {4}{3}\right )}{a^{2} n \Gamma \left (- \frac {1}{3}\right )} + \frac {4 b^{\frac {4}{3}} e^{- \frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} x^{\frac {n}{3}} e^{\frac {i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac {4}{3}\right )}{3 a^{\frac {7}{3}} n \Gamma \left (- \frac {1}{3}\right )} + \frac {4 b^{\frac {4}{3}} \log {\left (1 - \frac {\sqrt [3]{b} x^{\frac {n}{3}} e^{i \pi }}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac {4}{3}\right )}{3 a^{\frac {7}{3}} n \Gamma \left (- \frac {1}{3}\right )} + \frac {4 b^{\frac {4}{3}} e^{\frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} x^{\frac {n}{3}} e^{\frac {5 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac {4}{3}\right )}{3 a^{\frac {7}{3}} n \Gamma \left (- \frac {1}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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