Integrand size = 26, antiderivative size = 610 \[ \int \frac {x^{-1+\frac {n}{3}}}{a+b x^n+c x^{2 n}} \, dx=-\frac {2^{2/3} \sqrt {3} c^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3} n}+\frac {2^{2/3} \sqrt {3} c^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right )^{2/3} n}+\frac {2^{2/3} c^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3} n}-\frac {2^{2/3} c^{2/3} \log \left (\sqrt [3]{b+\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right )^{2/3} n}-\frac {c^{2/3} \log \left (\left (b-\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x^{n/3}+2^{2/3} c^{2/3} x^{2 n/3}\right )}{\sqrt [3]{2} \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3} n}+\frac {c^{2/3} \log \left (\left (b+\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x^{n/3}+2^{2/3} c^{2/3} x^{2 n/3}\right )}{\sqrt [3]{2} \sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right )^{2/3} n} \]
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Time = 0.74 (sec) , antiderivative size = 610, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {1395, 1361, 206, 31, 648, 631, 210, 642} \[ \int \frac {x^{-1+\frac {n}{3}}}{a+b x^n+c x^{2 n}} \, dx=-\frac {2^{2/3} \sqrt {3} c^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{n \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {2^{2/3} \sqrt {3} c^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{\sqrt {b^2-4 a c}+b}}}{\sqrt {3}}\right )}{n \sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}+\frac {2^{2/3} c^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{n \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}-\frac {2^{2/3} c^{2/3} \log \left (\sqrt [3]{\sqrt {b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{n \sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}}-\frac {c^{2/3} \log \left (-\sqrt [3]{2} \sqrt [3]{c} x^{n/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^{2 n/3}\right )}{\sqrt [3]{2} n \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3}}+\frac {c^{2/3} \log \left (-\sqrt [3]{2} \sqrt [3]{c} x^{n/3} \sqrt [3]{\sqrt {b^2-4 a c}+b}+\left (\sqrt {b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^{2 n/3}\right )}{\sqrt [3]{2} n \sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}+b\right )^{2/3}} \]
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Rule 31
Rule 206
Rule 210
Rule 631
Rule 642
Rule 648
Rule 1361
Rule 1395
Rubi steps \begin{align*} \text {integral}& = \frac {3 \text {Subst}\left (\int \frac {1}{a+b x^3+c x^6} \, dx,x,x^{n/3}\right )}{n} \\ & = \frac {(3 c) \text {Subst}\left (\int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^3} \, dx,x,x^{n/3}\right )}{\sqrt {b^2-4 a c} n}-\frac {(3 c) \text {Subst}\left (\int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^3} \, dx,x,x^{n/3}\right )}{\sqrt {b^2-4 a c} n} \\ & = \frac {\left (2^{2/3} c\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt [3]{b-\sqrt {b^2-4 a c}}}{\sqrt [3]{2}}+\sqrt [3]{c} x} \, dx,x,x^{n/3}\right )}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3} n}+\frac {\left (2^{2/3} c\right ) \text {Subst}\left (\int \frac {2^{2/3} \sqrt [3]{b-\sqrt {b^2-4 a c}}-\sqrt [3]{c} x}{\frac {\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac {\sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx,x,x^{n/3}\right )}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3} n}-\frac {\left (2^{2/3} c\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt [3]{b+\sqrt {b^2-4 a c}}}{\sqrt [3]{2}}+\sqrt [3]{c} x} \, dx,x,x^{n/3}\right )}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right )^{2/3} n}-\frac {\left (2^{2/3} c\right ) \text {Subst}\left (\int \frac {2^{2/3} \sqrt [3]{b+\sqrt {b^2-4 a c}}-\sqrt [3]{c} x}{\frac {\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac {\sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx,x,x^{n/3}\right )}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right )^{2/3} n} \\ & = \frac {2^{2/3} c^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3} n}-\frac {2^{2/3} c^{2/3} \log \left (\sqrt [3]{b+\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right )^{2/3} n}-\frac {c^{2/3} \text {Subst}\left (\int \frac {-\frac {\sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}}}{\sqrt [3]{2}}+2 c^{2/3} x}{\frac {\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac {\sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx,x,x^{n/3}\right )}{\sqrt [3]{2} \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3} n}+\frac {(3 c) \text {Subst}\left (\int \frac {1}{\frac {\left (b-\sqrt {b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac {\sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx,x,x^{n/3}\right )}{2^{2/3} \sqrt {b^2-4 a c} \sqrt [3]{b-\sqrt {b^2-4 a c}} n}+\frac {c^{2/3} \text {Subst}\left (\int \frac {-\frac {\sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}}}{\sqrt [3]{2}}+2 c^{2/3} x}{\frac {\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac {\sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx,x,x^{n/3}\right )}{\sqrt [3]{2} \sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right )^{2/3} n}-\frac {(3 c) \text {Subst}\left (\int \frac {1}{\frac {\left (b+\sqrt {b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac {\sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx,x,x^{n/3}\right )}{2^{2/3} \sqrt {b^2-4 a c} \sqrt [3]{b+\sqrt {b^2-4 a c}} n} \\ & = \frac {2^{2/3} c^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3} n}-\frac {2^{2/3} c^{2/3} \log \left (\sqrt [3]{b+\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right )^{2/3} n}-\frac {c^{2/3} \log \left (\left (b-\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x^{n/3}+2^{2/3} c^{2/3} x^{2 n/3}\right )}{\sqrt [3]{2} \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3} n}+\frac {c^{2/3} \log \left (\left (b+\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x^{n/3}+2^{2/3} c^{2/3} x^{2 n/3}\right )}{\sqrt [3]{2} \sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right )^{2/3} n}+\frac {\left (3\ 2^{2/3} c^{2/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3} n}-\frac {\left (3\ 2^{2/3} c^{2/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right )^{2/3} n} \\ & = -\frac {2^{2/3} \sqrt {3} c^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3} n}+\frac {2^{2/3} \sqrt {3} c^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right )^{2/3} n}+\frac {2^{2/3} c^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3} n}-\frac {2^{2/3} c^{2/3} \log \left (\sqrt [3]{b+\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right )^{2/3} n}-\frac {c^{2/3} \log \left (\left (b-\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x^{n/3}+2^{2/3} c^{2/3} x^{2 n/3}\right )}{\sqrt [3]{2} \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3} n}+\frac {c^{2/3} \log \left (\left (b+\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x^{n/3}+2^{2/3} c^{2/3} x^{2 n/3}\right )}{\sqrt [3]{2} \sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right )^{2/3} n} \\ \end{align*}
Time = 0.64 (sec) , antiderivative size = 526, normalized size of antiderivative = 0.86 \[ \int \frac {x^{-1+\frac {n}{3}}}{a+b x^n+c x^{2 n}} \, dx=\frac {c^{2/3} \left (-2 \sqrt {3} \left (b+\sqrt {b^2-4 a c}\right )^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )+2 \sqrt {3} \left (b-\sqrt {b^2-4 a c}\right )^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x^{n/3}}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )+2 \left (b+\sqrt {b^2-4 a c}\right )^{2/3} \log \left (\sqrt [3]{b-\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )-2 \left (b-\sqrt {b^2-4 a c}\right )^{2/3} \log \left (\sqrt [3]{b+\sqrt {b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x^{n/3}\right )-\left (b+\sqrt {b^2-4 a c}\right )^{2/3} \log \left (\left (b-\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x^{n/3}+2^{2/3} c^{2/3} x^{2 n/3}\right )+\left (b-\sqrt {b^2-4 a c}\right )^{2/3} \log \left (\left (b+\sqrt {b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x^{n/3}+2^{2/3} c^{2/3} x^{2 n/3}\right )\right )}{\sqrt [3]{2} \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )^{2/3} \left (b+\sqrt {b^2-4 a c}\right )^{2/3} n} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.70 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.43
method | result | size |
risch | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (64 a^{5} c^{3} n^{6}-48 a^{4} b^{2} c^{2} n^{6}+12 a^{3} b^{4} c \,n^{6}-a^{2} b^{6} n^{6}\right ) \textit {\_Z}^{6}+\left (16 a^{2} b \,c^{2} n^{3}-8 a \,b^{3} c \,n^{3}+b^{5} n^{3}\right ) \textit {\_Z}^{3}+c^{2}\right )}{\sum }\textit {\_R} \ln \left (x^{\frac {n}{3}}+\left (-\frac {16 n^{4} b \,a^{4} c^{2}}{2 a \,c^{2}-b^{2} c}+\frac {8 n^{4} b^{3} a^{3} c}{2 a \,c^{2}-b^{2} c}-\frac {n^{4} b^{5} a^{2}}{2 a \,c^{2}-b^{2} c}\right ) \textit {\_R}^{4}+\left (\frac {4 n \,a^{2} c^{2}}{2 a \,c^{2}-b^{2} c}-\frac {5 n \,b^{2} a c}{2 a \,c^{2}-b^{2} c}+\frac {n \,b^{4}}{2 a \,c^{2}-b^{2} c}\right ) \textit {\_R} \right )\) | \(260\) |
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Leaf count of result is larger than twice the leaf count of optimal. 2461 vs. \(2 (465) = 930\).
Time = 0.35 (sec) , antiderivative size = 2461, normalized size of antiderivative = 4.03 \[ \int \frac {x^{-1+\frac {n}{3}}}{a+b x^n+c x^{2 n}} \, dx=\text {Too large to display} \]
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\[ \int \frac {x^{-1+\frac {n}{3}}}{a+b x^n+c x^{2 n}} \, dx=\int \frac {x^{\frac {n}{3} - 1}}{a + b x^{n} + c x^{2 n}}\, dx \]
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\[ \int \frac {x^{-1+\frac {n}{3}}}{a+b x^n+c x^{2 n}} \, dx=\int { \frac {x^{\frac {1}{3} \, n - 1}}{c x^{2 \, n} + b x^{n} + a} \,d x } \]
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\[ \int \frac {x^{-1+\frac {n}{3}}}{a+b x^n+c x^{2 n}} \, dx=\int { \frac {x^{\frac {1}{3} \, n - 1}}{c x^{2 \, n} + b x^{n} + a} \,d x } \]
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Timed out. \[ \int \frac {x^{-1+\frac {n}{3}}}{a+b x^n+c x^{2 n}} \, dx=\int \frac {x^{\frac {n}{3}-1}}{a+b\,x^n+c\,x^{2\,n}} \,d x \]
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