Optimal. Leaf size=246 \[ -\frac {b^{4/3} x^{4 n/3} (c x)^{-4 n/3} \log \left (\sqrt [3]{a} x^{-n/3}+\sqrt [3]{b}\right )}{a^{7/3} c n}+\frac {b^{4/3} x^{4 n/3} (c x)^{-4 n/3} \log \left (a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}+b^{2/3}\right )}{2 a^{7/3} c n}+\frac {\sqrt {3} b^{4/3} x^{4 n/3} (c x)^{-4 n/3} \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x^{-n/3}}{\sqrt {3} \sqrt [3]{b}}\right )}{a^{7/3} c n}+\frac {3 b x^n (c x)^{-4 n/3}}{a^2 c n}-\frac {3 (c x)^{-4 n/3}}{4 a c n} \]
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Rubi [A] time = 0.15, antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {363, 362, 345, 193, 321, 200, 31, 634, 617, 204, 628} \[ -\frac {b^{4/3} x^{4 n/3} (c x)^{-4 n/3} \log \left (\sqrt [3]{a} x^{-n/3}+\sqrt [3]{b}\right )}{a^{7/3} c n}+\frac {b^{4/3} x^{4 n/3} (c x)^{-4 n/3} \log \left (a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}+b^{2/3}\right )}{2 a^{7/3} c n}+\frac {\sqrt {3} b^{4/3} x^{4 n/3} (c x)^{-4 n/3} \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x^{-n/3}}{\sqrt {3} \sqrt [3]{b}}\right )}{a^{7/3} c n}+\frac {3 b x^n (c x)^{-4 n/3}}{a^2 c n}-\frac {3 (c x)^{-4 n/3}}{4 a c n} \]
Antiderivative was successfully verified.
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Rule 31
Rule 193
Rule 200
Rule 204
Rule 321
Rule 345
Rule 362
Rule 363
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {(c x)^{-1-\frac {4 n}{3}}}{a+b x^n} \, dx &=\frac {\left (x^{4 n/3} (c x)^{-4 n/3}\right ) \int \frac {x^{-1-\frac {4 n}{3}}}{a+b x^n} \, dx}{c}\\ &=-\frac {3 (c x)^{-4 n/3}}{4 a c n}-\frac {\left (b x^{4 n/3} (c x)^{-4 n/3}\right ) \int \frac {x^{-1-\frac {n}{3}}}{a+b x^n} \, dx}{a c}\\ &=-\frac {3 (c x)^{-4 n/3}}{4 a c n}+\frac {\left (3 b x^{4 n/3} (c x)^{-4 n/3}\right ) \operatorname {Subst}\left (\int \frac {1}{a+\frac {b}{x^3}} \, dx,x,x^{-n/3}\right )}{a c n}\\ &=-\frac {3 (c x)^{-4 n/3}}{4 a c n}+\frac {\left (3 b x^{4 n/3} (c x)^{-4 n/3}\right ) \operatorname {Subst}\left (\int \frac {x^3}{b+a x^3} \, dx,x,x^{-n/3}\right )}{a c n}\\ &=-\frac {3 (c x)^{-4 n/3}}{4 a c n}+\frac {3 b x^n (c x)^{-4 n/3}}{a^2 c n}-\frac {\left (3 b^2 x^{4 n/3} (c x)^{-4 n/3}\right ) \operatorname {Subst}\left (\int \frac {1}{b+a x^3} \, dx,x,x^{-n/3}\right )}{a^2 c n}\\ &=-\frac {3 (c x)^{-4 n/3}}{4 a c n}+\frac {3 b x^n (c x)^{-4 n/3}}{a^2 c n}-\frac {\left (b^{4/3} x^{4 n/3} (c x)^{-4 n/3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx,x,x^{-n/3}\right )}{a^2 c n}-\frac {\left (b^{4/3} x^{4 n/3} (c x)^{-4 n/3}\right ) \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 c n}\\ &=-\frac {3 (c x)^{-4 n/3}}{4 a c n}+\frac {3 b x^n (c x)^{-4 n/3}}{a^2 c n}-\frac {b^{4/3} x^{4 n/3} (c x)^{-4 n/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x^{-n/3}\right )}{a^{7/3} c n}+\frac {\left (b^{4/3} x^{4 n/3} (c x)^{-4 n/3}\right ) \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} c n}-\frac {\left (3 b^{5/3} x^{4 n/3} (c x)^{-4 n/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 c n}\\ &=-\frac {3 (c x)^{-4 n/3}}{4 a c n}+\frac {3 b x^n (c x)^{-4 n/3}}{a^2 c n}-\frac {b^{4/3} x^{4 n/3} (c x)^{-4 n/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x^{-n/3}\right )}{a^{7/3} c n}+\frac {b^{4/3} x^{4 n/3} (c x)^{-4 n/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} c n}-\frac {\left (3 b^{4/3} x^{4 n/3} (c x)^{-4 n/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} c n}\\ &=-\frac {3 (c x)^{-4 n/3}}{4 a c n}+\frac {3 b x^n (c x)^{-4 n/3}}{a^2 c n}+\frac {\sqrt {3} b^{4/3} x^{4 n/3} (c x)^{-4 n/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{a} x^{-n/3}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{a^{7/3} c n}-\frac {b^{4/3} x^{4 n/3} (c x)^{-4 n/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x^{-n/3}\right )}{a^{7/3} c n}+\frac {b^{4/3} x^{4 n/3} (c x)^{-4 n/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} c n}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 39, normalized size = 0.16 \[ -\frac {3 x (c x)^{-\frac {4 n}{3}-1} \, _2F_1\left (-\frac {4}{3},1;-\frac {1}{3};-\frac {b x^n}{a}\right )}{4 a n} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.26, size = 351, normalized size = 1.43 \[ \frac {4 \, \sqrt {3} b c^{-n - \frac {3}{4}} \left (-\frac {b c^{-n - \frac {3}{4}}}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a \left (-\frac {b c^{-n - \frac {3}{4}}}{a}\right )^{\frac {2}{3}} x^{\frac {1}{4}} e^{\left (-\frac {1}{12} \, {\left (4 \, n + 3\right )} \log \relax (c) - \frac {1}{12} \, {\left (4 \, n + 3\right )} \log \relax (x)\right )} - \sqrt {3} b c^{-n - \frac {3}{4}}}{3 \, b c^{-n - \frac {3}{4}}}\right ) - 2 \, b c^{-n - \frac {3}{4}} \left (-\frac {b c^{-n - \frac {3}{4}}}{a}\right )^{\frac {1}{3}} \log \left (\frac {\left (-\frac {b c^{-n - \frac {3}{4}}}{a}\right )^{\frac {1}{3}} x^{\frac {3}{4}} e^{\left (-\frac {1}{12} \, {\left (4 \, n + 3\right )} \log \relax (c) - \frac {1}{12} \, {\left (4 \, n + 3\right )} \log \relax (x)\right )} + x e^{\left (-\frac {1}{6} \, {\left (4 \, n + 3\right )} \log \relax (c) - \frac {1}{6} \, {\left (4 \, n + 3\right )} \log \relax (x)\right )} + \left (-\frac {b c^{-n - \frac {3}{4}}}{a}\right )^{\frac {2}{3}} \sqrt {x}}{x}\right ) + 4 \, b c^{-n - \frac {3}{4}} \left (-\frac {b c^{-n - \frac {3}{4}}}{a}\right )^{\frac {1}{3}} \log \left (\frac {x e^{\left (-\frac {1}{12} \, {\left (4 \, n + 3\right )} \log \relax (c) - \frac {1}{12} \, {\left (4 \, n + 3\right )} \log \relax (x)\right )} - \left (-\frac {b c^{-n - \frac {3}{4}}}{a}\right )^{\frac {1}{3}} x^{\frac {3}{4}}}{x}\right ) + 12 \, b c^{-n - \frac {3}{4}} x^{\frac {1}{4}} e^{\left (-\frac {1}{12} \, {\left (4 \, n + 3\right )} \log \relax (c) - \frac {1}{12} \, {\left (4 \, n + 3\right )} \log \relax (x)\right )} - 3 \, a x e^{\left (-\frac {1}{3} \, {\left (4 \, n + 3\right )} \log \relax (c) - \frac {1}{3} \, {\left (4 \, n + 3\right )} \log \relax (x)\right )}}{4 \, a^{2} n} \]
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 \[ \int \frac {\left (c x\right )^{-\frac {4}{3} \, n - 1}}{b x^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.21, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x \right )^{-\frac {4 n}{3}-1}}{b \,x^{n}+a}\, dx \]
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 \[ b^{2} \int \frac {x^{\frac {2}{3} \, n}}{a^{2} b c^{\frac {4}{3} \, n + 1} x x^{n} + a^{3} c^{\frac {4}{3} \, n + 1} x}\,{d x} + \frac {3 \, {\left (4 \, b x^{n} - a\right )} c^{-\frac {4}{3} \, n - 1}}{4 \, a^{2} n x^{\frac {4}{3} \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (c\,x\right )}^{\frac {4\,n}{3}+1}\,\left (a+b\,x^n\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 5.17, size = 272, normalized size = 1.11 \[ \frac {c^{- \frac {4 n}{3}} x^{- \frac {4 n}{3}} \Gamma \left (- \frac {4}{3}\right )}{a c n \Gamma \left (- \frac {1}{3}\right )} - \frac {4 b c^{- \frac {4 n}{3}} x^{- \frac {n}{3}} \Gamma \left (- \frac {4}{3}\right )}{a^{2} c n \Gamma \left (- \frac {1}{3}\right )} + \frac {4 b^{\frac {4}{3}} c^{- \frac {4 n}{3}} e^{- \frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} x^{\frac {n}{3}} e^{\frac {i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac {4}{3}\right )}{3 a^{\frac {7}{3}} c n \Gamma \left (- \frac {1}{3}\right )} + \frac {4 b^{\frac {4}{3}} c^{- \frac {4 n}{3}} \log {\left (1 - \frac {\sqrt [3]{b} x^{\frac {n}{3}} e^{i \pi }}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac {4}{3}\right )}{3 a^{\frac {7}{3}} c n \Gamma \left (- \frac {1}{3}\right )} + \frac {4 b^{\frac {4}{3}} c^{- \frac {4 n}{3}} e^{\frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} x^{\frac {n}{3}} e^{\frac {5 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac {4}{3}\right )}{3 a^{\frac {7}{3}} c n \Gamma \left (- \frac {1}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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