Optimal. Leaf size=176 \[ -\frac {c^{4/3} \log \left (\sqrt [3]{b} x^{-n/3}+\sqrt [3]{c}\right )}{b^{7/3} n}+\frac {c^{4/3} \log \left (b^{2/3} x^{-2 n/3}-\sqrt [3]{b} \sqrt [3]{c} x^{-n/3}+c^{2/3}\right )}{2 b^{7/3} n}+\frac {\sqrt {3} c^{4/3} \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{b} x^{-n/3}}{\sqrt {3} \sqrt [3]{c}}\right )}{b^{7/3} n}+\frac {3 c x^{-n/3}}{b^2 n}-\frac {3 x^{-4 n/3}}{4 b n} \]
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Rubi [A] time = 0.14, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {1584, 362, 345, 193, 321, 200, 31, 634, 617, 204, 628} \[ -\frac {c^{4/3} \log \left (\sqrt [3]{b} x^{-n/3}+\sqrt [3]{c}\right )}{b^{7/3} n}+\frac {c^{4/3} \log \left (b^{2/3} x^{-2 n/3}-\sqrt [3]{b} \sqrt [3]{c} x^{-n/3}+c^{2/3}\right )}{2 b^{7/3} n}+\frac {\sqrt {3} c^{4/3} \tan ^{-1}\left (\frac {\sqrt [3]{c}-2 \sqrt [3]{b} x^{-n/3}}{\sqrt {3} \sqrt [3]{c}}\right )}{b^{7/3} n}+\frac {3 c x^{-n/3}}{b^2 n}-\frac {3 x^{-4 n/3}}{4 b n} \]
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
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^{-1-\frac {n}{3}}}{b x^n+c x^{2 n}} \, dx &=\int \frac {x^{-1-\frac {4 n}{3}}}{b+c x^n} \, dx\\ &=-\frac {3 x^{-4 n/3}}{4 b n}-\frac {c \int \frac {x^{-1-\frac {n}{3}}}{b+c x^n} \, dx}{b}\\ &=-\frac {3 x^{-4 n/3}}{4 b n}+\frac {(3 c) \operatorname {Subst}\left (\int \frac {1}{b+\frac {c}{x^3}} \, dx,x,x^{-n/3}\right )}{b n}\\ &=-\frac {3 x^{-4 n/3}}{4 b n}+\frac {(3 c) \operatorname {Subst}\left (\int \frac {x^3}{c+b x^3} \, dx,x,x^{-n/3}\right )}{b n}\\ &=-\frac {3 x^{-4 n/3}}{4 b n}+\frac {3 c x^{-n/3}}{b^2 n}-\frac {\left (3 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{c+b x^3} \, dx,x,x^{-n/3}\right )}{b^2 n}\\ &=-\frac {3 x^{-4 n/3}}{4 b n}+\frac {3 c x^{-n/3}}{b^2 n}-\frac {c^{4/3} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{c}+\sqrt [3]{b} x} \, dx,x,x^{-n/3}\right )}{b^2 n}-\frac {c^{4/3} \operatorname {Subst}\left (\int \frac {2 \sqrt [3]{c}-\sqrt [3]{b} x}{c^{2/3}-\sqrt [3]{b} \sqrt [3]{c} x+b^{2/3} x^2} \, dx,x,x^{-n/3}\right )}{b^2 n}\\ &=-\frac {3 x^{-4 n/3}}{4 b n}+\frac {3 c x^{-n/3}}{b^2 n}-\frac {c^{4/3} \log \left (\sqrt [3]{c}+\sqrt [3]{b} x^{-n/3}\right )}{b^{7/3} n}+\frac {c^{4/3} \operatorname {Subst}\left (\int \frac {-\sqrt [3]{b} \sqrt [3]{c}+2 b^{2/3} x}{c^{2/3}-\sqrt [3]{b} \sqrt [3]{c} x+b^{2/3} x^2} \, dx,x,x^{-n/3}\right )}{2 b^{7/3} n}-\frac {\left (3 c^{5/3}\right ) \operatorname {Subst}\left (\int \frac {1}{c^{2/3}-\sqrt [3]{b} \sqrt [3]{c} x+b^{2/3} x^2} \, dx,x,x^{-n/3}\right )}{2 b^2 n}\\ &=-\frac {3 x^{-4 n/3}}{4 b n}+\frac {3 c x^{-n/3}}{b^2 n}-\frac {c^{4/3} \log \left (\sqrt [3]{c}+\sqrt [3]{b} x^{-n/3}\right )}{b^{7/3} n}+\frac {c^{4/3} \log \left (c^{2/3}+b^{2/3} x^{-2 n/3}-\sqrt [3]{b} \sqrt [3]{c} x^{-n/3}\right )}{2 b^{7/3} n}-\frac {\left (3 c^{4/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x^{-n/3}}{\sqrt [3]{c}}\right )}{b^{7/3} n}\\ &=-\frac {3 x^{-4 n/3}}{4 b n}+\frac {3 c x^{-n/3}}{b^2 n}+\frac {\sqrt {3} c^{4/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x^{-n/3}}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{b^{7/3} n}-\frac {c^{4/3} \log \left (\sqrt [3]{c}+\sqrt [3]{b} x^{-n/3}\right )}{b^{7/3} n}+\frac {c^{4/3} \log \left (c^{2/3}+b^{2/3} x^{-2 n/3}-\sqrt [3]{b} \sqrt [3]{c} x^{-n/3}\right )}{2 b^{7/3} n}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 34, normalized size = 0.19 \[ -\frac {3 x^{-4 n/3} \, _2F_1\left (-\frac {4}{3},1;-\frac {1}{3};-\frac {c x^n}{b}\right )}{4 b n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 171, normalized size = 0.97 \[ -\frac {3 \, b x^{4} x^{-\frac {4}{3} \, n - 4} - 12 \, c x x^{-\frac {1}{3} \, n - 1} - 4 \, \sqrt {3} c \left (-\frac {c}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x x^{-\frac {1}{3} \, n - 1} \left (-\frac {c}{b}\right )^{\frac {2}{3}} - \sqrt {3} c}{3 \, c}\right ) - 4 \, c \left (-\frac {c}{b}\right )^{\frac {1}{3}} \log \left (\frac {x x^{-\frac {1}{3} \, n - 1} - \left (-\frac {c}{b}\right )^{\frac {1}{3}}}{x}\right ) + 2 \, c \left (-\frac {c}{b}\right )^{\frac {1}{3}} \log \left (\frac {x^{2} x^{-\frac {2}{3} \, n - 2} + x x^{-\frac {1}{3} \, n - 1} \left (-\frac {c}{b}\right )^{\frac {1}{3}} + \left (-\frac {c}{b}\right )^{\frac {2}{3}}}{x^{2}}\right )}{4 \, b^{2} n} \]
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 \[ \int \frac {x^{-\frac {1}{3} \, n - 1}}{c x^{2 \, n} + b x^{n}}\,{d x} \]
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 \[ \RootOf \left (b^{7} n^{3} \textit {\_Z}^{3}+c^{4}\right ) \ln \left (\frac {\RootOf \left (b^{7} n^{3} \textit {\_Z}^{3}+c^{4}\right )^{2} b^{5} n^{2}}{c^{3}}+x^{\frac {n}{3}}\right )-\frac {3 x^{-\frac {4 n}{3}}}{4 b n}+\frac {3 c \,x^{-\frac {n}{3}}}{b^{2} n} \]
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 \[ c^{2} \int \frac {x^{\frac {2}{3} \, n}}{b^{2} c x x^{n} + b^{3} x}\,{d x} + \frac {3 \, {\left (4 \, c x^{n} - b\right )}}{4 \, b^{2} n x^{\frac {4}{3} \, n}} \]
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 \[ \int \frac {1}{x^{\frac {n}{3}+1}\,\left (b\,x^n+c\,x^{2\,n}\right )} \,d x \]
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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