Optimal. Leaf size=220 \[ -\frac {3 (c x)^{-n/3}}{a c n}-\frac {\sqrt {3} \sqrt [3]{b} x^{n/3} (c x)^{-n/3} \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x^{-n/3}}{\sqrt {3} \sqrt [3]{b}}\right )}{a^{4/3} c n}+\frac {\sqrt [3]{b} x^{n/3} (c x)^{-n/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x^{-n/3}\right )}{a^{4/3} c n}-\frac {\sqrt [3]{b} x^{n/3} (c x)^{-n/3} \log \left (b^{2/3}+a^{2/3} x^{-2 n/3}-\sqrt [3]{a} \sqrt [3]{b} x^{-n/3}\right )}{2 a^{4/3} c n} \]
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Rubi [A]
time = 0.09, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 10, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {354, 352,
199, 327, 206, 31, 648, 631, 210, 642} \begin {gather*} -\frac {\sqrt {3} \sqrt [3]{b} x^{n/3} (c x)^{-n/3} \text {ArcTan}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a} x^{-n/3}}{\sqrt {3} \sqrt [3]{b}}\right )}{a^{4/3} c n}-\frac {\sqrt [3]{b} x^{n/3} (c x)^{-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^{4/3} c n}+\frac {\sqrt [3]{b} x^{n/3} (c x)^{-n/3} \log \left (\sqrt [3]{a} x^{-n/3}+\sqrt [3]{b}\right )}{a^{4/3} c n}-\frac {3 (c x)^{-n/3}}{a c n} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 199
Rule 206
Rule 210
Rule 327
Rule 352
Rule 354
Rule 631
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {(c x)^{-1-\frac {n}{3}}}{a+b x^n} \, dx &=\frac {\left (x^{n/3} (c x)^{-n/3}\right ) \int \frac {x^{-1-\frac {n}{3}}}{a+b x^n} \, dx}{c}\\ &=-\frac {\left (3 x^{n/3} (c x)^{-n/3}\right ) \text {Subst}\left (\int \frac {1}{a+\frac {b}{x^3}} \, dx,x,x^{-n/3}\right )}{c n}\\ &=-\frac {\left (3 x^{n/3} (c x)^{-n/3}\right ) \text {Subst}\left (\int \frac {x^3}{b+a x^3} \, dx,x,x^{-n/3}\right )}{c n}\\ &=-\frac {3 (c x)^{-n/3}}{a c n}+\frac {\left (3 b x^{n/3} (c x)^{-n/3}\right ) \text {Subst}\left (\int \frac {1}{b+a x^3} \, dx,x,x^{-n/3}\right )}{a c n}\\ &=-\frac {3 (c x)^{-n/3}}{a c n}+\frac {\left (\sqrt [3]{b} x^{n/3} (c x)^{-n/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx,x,x^{-n/3}\right )}{a c n}+\frac {\left (\sqrt [3]{b} x^{n/3} (c x)^{-n/3}\right ) \text {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 c n}\\ &=-\frac {3 (c x)^{-n/3}}{a c n}+\frac {\sqrt [3]{b} x^{n/3} (c x)^{-n/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x^{-n/3}\right )}{a^{4/3} c n}-\frac {\left (\sqrt [3]{b} x^{n/3} (c x)^{-n/3}\right ) \text {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} c n}+\frac {\left (3 b^{2/3} x^{n/3} (c x)^{-n/3}\right ) \text {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 c n}\\ &=-\frac {3 (c x)^{-n/3}}{a c n}+\frac {\sqrt [3]{b} x^{n/3} (c x)^{-n/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x^{-n/3}\right )}{a^{4/3} c n}-\frac {\sqrt [3]{b} x^{n/3} (c x)^{-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^{4/3} c n}+\frac {\left (3 \sqrt [3]{b} x^{n/3} (c x)^{-n/3}\right ) \text {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} c n}\\ &=-\frac {3 (c x)^{-n/3}}{a c n}-\frac {\sqrt {3} \sqrt [3]{b} x^{n/3} (c x)^{-n/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{a} x^{-n/3}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{a^{4/3} c n}+\frac {\sqrt [3]{b} x^{n/3} (c x)^{-n/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x^{-n/3}\right )}{a^{4/3} c n}-\frac {\sqrt [3]{b} x^{n/3} (c x)^{-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^{4/3} c n}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.01, size = 37, normalized size = 0.17 \begin {gather*} -\frac {3 x (c x)^{-1-\frac {n}{3}} \, _2F_1\left (-\frac {1}{3},1;\frac {2}{3};-\frac {b x^n}{a}\right )}{a n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {\left (c x \right )^{-1-\frac {n}{3}}}{a +b \,x^{n}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 255, normalized size = 1.16 \begin {gather*} -\frac {6 \, x e^{\left (-\frac {1}{3} \, {\left (n + 3\right )} \log \left (c\right ) - \frac {1}{3} \, {\left (n + 3\right )} \log \left (x\right )\right )} - 2 \, \sqrt {3} \left (\frac {b c^{-n - 3}}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a \left (\frac {b c^{-n - 3}}{a}\right )^{\frac {2}{3}} x e^{\left (-\frac {1}{3} \, {\left (n + 3\right )} \log \left (c\right ) - \frac {1}{3} \, {\left (n + 3\right )} \log \left (x\right )\right )} - \sqrt {3} b c^{-n - 3}}{3 \, b c^{-n - 3}}\right ) - 2 \, \left (\frac {b c^{-n - 3}}{a}\right )^{\frac {1}{3}} \log \left (\frac {x e^{\left (-\frac {1}{3} \, {\left (n + 3\right )} \log \left (c\right ) - \frac {1}{3} \, {\left (n + 3\right )} \log \left (x\right )\right )} + \left (\frac {b c^{-n - 3}}{a}\right )^{\frac {1}{3}}}{x}\right ) + \left (\frac {b c^{-n - 3}}{a}\right )^{\frac {1}{3}} \log \left (\frac {x^{2} e^{\left (-\frac {2}{3} \, {\left (n + 3\right )} \log \left (c\right ) - \frac {2}{3} \, {\left (n + 3\right )} \log \left (x\right )\right )} - \left (\frac {b c^{-n - 3}}{a}\right )^{\frac {1}{3}} x e^{\left (-\frac {1}{3} \, {\left (n + 3\right )} \log \left (c\right ) - \frac {1}{3} \, {\left (n + 3\right )} \log \left (x\right )\right )} + \left (\frac {b c^{-n - 3}}{a}\right )^{\frac {2}{3}}}{x^{2}}\right )}{2 \, a n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 2.46, size = 216, normalized size = 0.98 \begin {gather*} \frac {c^{- \frac {n}{3}} x^{- \frac {n}{3}} \Gamma \left (- \frac {1}{3}\right )}{a c n \Gamma \left (\frac {2}{3}\right )} - \frac {\sqrt [3]{b} c^{- \frac {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 {1}{3}\right )}{3 a^{\frac {4}{3}} c n \Gamma \left (\frac {2}{3}\right )} - \frac {\sqrt [3]{b} c^{- \frac {n}{3}} \log {\left (1 - \frac {\sqrt [3]{b} x^{\frac {n}{3}} e^{i \pi }}{\sqrt [3]{a}} \right )} \Gamma \left (- \frac {1}{3}\right )}{3 a^{\frac {4}{3}} c n \Gamma \left (\frac {2}{3}\right )} - \frac {\sqrt [3]{b} c^{- \frac {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 {1}{3}\right )}{3 a^{\frac {4}{3}} c n \Gamma \left (\frac {2}{3}\right )} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{{\left (c\,x\right )}^{\frac {n}{3}+1}\,\left (a+b\,x^n\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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