Optimal. Leaf size=164 \[ \frac {b \left (b^2-2 a c\right ) \log \left (a+b x^n+c x^{2 n}\right )}{2 a^4 n}-\frac {b \log (x) \left (b^2-2 a c\right )}{a^4}-\frac {x^{-n} \left (b^2-a c\right )}{a^3 n}+\frac {b x^{-2 n}}{2 a^2 n}-\frac {\left (2 a^2 c^2-4 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac {b+2 c x^n}{\sqrt {b^2-4 a c}}\right )}{a^4 n \sqrt {b^2-4 a c}}-\frac {x^{-3 n}}{3 a n} \]
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Rubi [A] time = 0.23, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1357, 709, 800, 634, 618, 206, 628} \[ -\frac {\left (2 a^2 c^2-4 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac {b+2 c x^n}{\sqrt {b^2-4 a c}}\right )}{a^4 n \sqrt {b^2-4 a c}}-\frac {x^{-n} \left (b^2-a c\right )}{a^3 n}+\frac {b \left (b^2-2 a c\right ) \log \left (a+b x^n+c x^{2 n}\right )}{2 a^4 n}-\frac {b \log (x) \left (b^2-2 a c\right )}{a^4}+\frac {b x^{-2 n}}{2 a^2 n}-\frac {x^{-3 n}}{3 a n} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 628
Rule 634
Rule 709
Rule 800
Rule 1357
Rubi steps
\begin {align*} \int \frac {x^{-1-3 n}}{a+b x^n+c x^{2 n}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^4 \left (a+b x+c x^2\right )} \, dx,x,x^n\right )}{n}\\ &=-\frac {x^{-3 n}}{3 a n}+\frac {\operatorname {Subst}\left (\int \frac {-b-c x}{x^3 \left (a+b x+c x^2\right )} \, dx,x,x^n\right )}{a n}\\ &=-\frac {x^{-3 n}}{3 a n}+\frac {\operatorname {Subst}\left (\int \left (-\frac {b}{a x^3}+\frac {b^2-a c}{a^2 x^2}+\frac {-b^3+2 a b c}{a^3 x}+\frac {b^4-3 a b^2 c+a^2 c^2+b c \left (b^2-2 a c\right ) x}{a^3 \left (a+b x+c x^2\right )}\right ) \, dx,x,x^n\right )}{a n}\\ &=-\frac {x^{-3 n}}{3 a n}+\frac {b x^{-2 n}}{2 a^2 n}-\frac {\left (b^2-a c\right ) x^{-n}}{a^3 n}-\frac {b \left (b^2-2 a c\right ) \log (x)}{a^4}+\frac {\operatorname {Subst}\left (\int \frac {b^4-3 a b^2 c+a^2 c^2+b c \left (b^2-2 a c\right ) x}{a+b x+c x^2} \, dx,x,x^n\right )}{a^4 n}\\ &=-\frac {x^{-3 n}}{3 a n}+\frac {b x^{-2 n}}{2 a^2 n}-\frac {\left (b^2-a c\right ) x^{-n}}{a^3 n}-\frac {b \left (b^2-2 a c\right ) \log (x)}{a^4}+\frac {\left (b \left (b^2-2 a c\right )\right ) \operatorname {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^n\right )}{2 a^4 n}+\frac {\left (b^4-4 a b^2 c+2 a^2 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^n\right )}{2 a^4 n}\\ &=-\frac {x^{-3 n}}{3 a n}+\frac {b x^{-2 n}}{2 a^2 n}-\frac {\left (b^2-a c\right ) x^{-n}}{a^3 n}-\frac {b \left (b^2-2 a c\right ) \log (x)}{a^4}+\frac {b \left (b^2-2 a c\right ) \log \left (a+b x^n+c x^{2 n}\right )}{2 a^4 n}-\frac {\left (b^4-4 a b^2 c+2 a^2 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^n\right )}{a^4 n}\\ &=-\frac {x^{-3 n}}{3 a n}+\frac {b x^{-2 n}}{2 a^2 n}-\frac {\left (b^2-a c\right ) x^{-n}}{a^3 n}-\frac {\left (b^4-4 a b^2 c+2 a^2 c^2\right ) \tanh ^{-1}\left (\frac {b+2 c x^n}{\sqrt {b^2-4 a c}}\right )}{a^4 \sqrt {b^2-4 a c} n}-\frac {b \left (b^2-2 a c\right ) \log (x)}{a^4}+\frac {b \left (b^2-2 a c\right ) \log \left (a+b x^n+c x^{2 n}\right )}{2 a^4 n}\\ \end {align*}
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Mathematica [A] time = 0.44, size = 143, normalized size = 0.87 \[ \frac {-2 a^3 x^{-3 n}-\frac {6 \left (2 a^2 c^2-4 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac {b+2 c x^n}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}}+3 a^2 b x^{-2 n}+6 a x^{-n} \left (a c-b^2\right )+3 b \left (b^2-2 a c\right ) \log \left (a+x^n \left (b+c x^n\right )\right )-6 b n \log (x) \left (b^2-2 a c\right )}{6 a^4 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 522, normalized size = 3.18 \[ \left [-\frac {2 \, a^{3} b^{2} - 8 \, a^{4} c + 6 \, {\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} n x^{3 \, n} \log \relax (x) - 3 \, {\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} \sqrt {b^{2} - 4 \, a c} x^{3 \, n} \log \left (\frac {2 \, c^{2} x^{2 \, n} + b^{2} - 2 \, a c + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} x^{n} - \sqrt {b^{2} - 4 \, a c} b}{c x^{2 \, n} + b x^{n} + a}\right ) - 3 \, {\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} x^{3 \, n} \log \left (c x^{2 \, n} + b x^{n} + a\right ) + 6 \, {\left (a b^{4} - 5 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} x^{2 \, n} - 3 \, {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x^{n}}{6 \, {\left (a^{4} b^{2} - 4 \, a^{5} c\right )} n x^{3 \, n}}, -\frac {2 \, a^{3} b^{2} - 8 \, a^{4} c + 6 \, {\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} n x^{3 \, n} \log \relax (x) + 6 \, {\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c} x^{3 \, n} \arctan \left (-\frac {2 \, \sqrt {-b^{2} + 4 \, a c} c x^{n} + \sqrt {-b^{2} + 4 \, a c} b}{b^{2} - 4 \, a c}\right ) - 3 \, {\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} x^{3 \, n} \log \left (c x^{2 \, n} + b x^{n} + a\right ) + 6 \, {\left (a b^{4} - 5 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} x^{2 \, n} - 3 \, {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x^{n}}{6 \, {\left (a^{4} b^{2} - 4 \, a^{5} c\right )} n x^{3 \, n}}\right ] \]
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 \[ \int \frac {x^{-3 \, n - 1}}{c x^{2 \, n} + b x^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.20, size = 1300, normalized size = 7.93 \[ \frac {8 a^{2} b \,c^{2} n^{2} \ln \relax (x )}{4 a^{5} c \,n^{2}-a^{4} b^{2} n^{2}}-\frac {6 a \,b^{3} c \,n^{2} \ln \relax (x )}{4 a^{5} c \,n^{2}-a^{4} b^{2} n^{2}}+\frac {b^{5} n^{2} \ln \relax (x )}{4 a^{5} c \,n^{2}-a^{4} b^{2} n^{2}}-\frac {4 b \,c^{2} \ln \left (x^{n}-\frac {-2 a^{2} b \,c^{2}+4 a \,b^{3} c -b^{5}+\sqrt {-16 a^{5} c^{5}+68 a^{4} b^{2} c^{4}-96 a^{3} b^{4} c^{3}+52 a^{2} b^{6} c^{2}-12 a \,b^{8} c +b^{10}}}{2 \left (2 a^{2} c^{2}-4 a \,b^{2} c +b^{4}\right ) c}\right )}{\left (4 a c -b^{2}\right ) a^{2} n}-\frac {4 b \,c^{2} \ln \left (x^{n}+\frac {2 a^{2} b \,c^{2}-4 a \,b^{3} c +b^{5}+\sqrt {-16 a^{5} c^{5}+68 a^{4} b^{2} c^{4}-96 a^{3} b^{4} c^{3}+52 a^{2} b^{6} c^{2}-12 a \,b^{8} c +b^{10}}}{2 \left (2 a^{2} c^{2}-4 a \,b^{2} c +b^{4}\right ) c}\right )}{\left (4 a c -b^{2}\right ) a^{2} n}+\frac {3 b^{3} c \ln \left (x^{n}-\frac {-2 a^{2} b \,c^{2}+4 a \,b^{3} c -b^{5}+\sqrt {-16 a^{5} c^{5}+68 a^{4} b^{2} c^{4}-96 a^{3} b^{4} c^{3}+52 a^{2} b^{6} c^{2}-12 a \,b^{8} c +b^{10}}}{2 \left (2 a^{2} c^{2}-4 a \,b^{2} c +b^{4}\right ) c}\right )}{\left (4 a c -b^{2}\right ) a^{3} n}+\frac {3 b^{3} c \ln \left (x^{n}+\frac {2 a^{2} b \,c^{2}-4 a \,b^{3} c +b^{5}+\sqrt {-16 a^{5} c^{5}+68 a^{4} b^{2} c^{4}-96 a^{3} b^{4} c^{3}+52 a^{2} b^{6} c^{2}-12 a \,b^{8} c +b^{10}}}{2 \left (2 a^{2} c^{2}-4 a \,b^{2} c +b^{4}\right ) c}\right )}{\left (4 a c -b^{2}\right ) a^{3} n}-\frac {b^{5} \ln \left (x^{n}-\frac {-2 a^{2} b \,c^{2}+4 a \,b^{3} c -b^{5}+\sqrt {-16 a^{5} c^{5}+68 a^{4} b^{2} c^{4}-96 a^{3} b^{4} c^{3}+52 a^{2} b^{6} c^{2}-12 a \,b^{8} c +b^{10}}}{2 \left (2 a^{2} c^{2}-4 a \,b^{2} c +b^{4}\right ) c}\right )}{2 \left (4 a c -b^{2}\right ) a^{4} n}-\frac {b^{5} \ln \left (x^{n}+\frac {2 a^{2} b \,c^{2}-4 a \,b^{3} c +b^{5}+\sqrt {-16 a^{5} c^{5}+68 a^{4} b^{2} c^{4}-96 a^{3} b^{4} c^{3}+52 a^{2} b^{6} c^{2}-12 a \,b^{8} c +b^{10}}}{2 \left (2 a^{2} c^{2}-4 a \,b^{2} c +b^{4}\right ) c}\right )}{2 \left (4 a c -b^{2}\right ) a^{4} n}-\frac {x^{-3 n}}{3 a n}+\frac {b \,x^{-2 n}}{2 a^{2} n}+\frac {c \,x^{-n}}{a^{2} n}-\frac {b^{2} x^{-n}}{a^{3} n}-\frac {\sqrt {-16 a^{5} c^{5}+68 a^{4} b^{2} c^{4}-96 a^{3} b^{4} c^{3}+52 a^{2} b^{6} c^{2}-12 a \,b^{8} c +b^{10}}\, \ln \left (x^{n}-\frac {-2 a^{2} b \,c^{2}+4 a \,b^{3} c -b^{5}+\sqrt {-16 a^{5} c^{5}+68 a^{4} b^{2} c^{4}-96 a^{3} b^{4} c^{3}+52 a^{2} b^{6} c^{2}-12 a \,b^{8} c +b^{10}}}{2 \left (2 a^{2} c^{2}-4 a \,b^{2} c +b^{4}\right ) c}\right )}{2 \left (4 a c -b^{2}\right ) a^{4} n}+\frac {\sqrt {-16 a^{5} c^{5}+68 a^{4} b^{2} c^{4}-96 a^{3} b^{4} c^{3}+52 a^{2} b^{6} c^{2}-12 a \,b^{8} c +b^{10}}\, \ln \left (x^{n}+\frac {2 a^{2} b \,c^{2}-4 a \,b^{3} c +b^{5}+\sqrt {-16 a^{5} c^{5}+68 a^{4} b^{2} c^{4}-96 a^{3} b^{4} c^{3}+52 a^{2} b^{6} c^{2}-12 a \,b^{8} c +b^{10}}}{2 \left (2 a^{2} c^{2}-4 a \,b^{2} c +b^{4}\right ) c}\right )}{2 \left (4 a c -b^{2}\right ) a^{4} 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 \[ \frac {3 \, a b x^{n} - 2 \, a^{2} - 6 \, {\left (b^{2} - a c\right )} x^{2 \, n}}{6 \, a^{3} n x^{3 \, n}} + \int -\frac {b^{3} - 2 \, a b c + {\left (b^{2} c - a c^{2}\right )} x^{n}}{a^{3} c x x^{2 \, n} + a^{3} b x x^{n} + a^{4} x}\,{d x} \]
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 \[ \int \frac {1}{x^{3\,n+1}\,\left (a+b\,x^n+c\,x^{2\,n}\right )} \,d x \]
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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