Optimal. Leaf size=169 \[ \frac {2 \sqrt {2} \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x^{n/2}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}} n}-\frac {2 \sqrt {2} \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x^{n/2}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}} n} \]
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Rubi [A]
time = 0.13, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1395, 1107,
211} \begin {gather*} \frac {2 \sqrt {2} \sqrt {c} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} x^{n/2}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{n \sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {2 \sqrt {2} \sqrt {c} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} x^{n/2}}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{n \sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 1107
Rule 1395
Rubi steps
\begin {align*} \int \frac {x^{-1+\frac {n}{2}}}{a+b x^n+c x^{2 n}} \, dx &=\frac {2 \text {Subst}\left (\int \frac {1}{a+b x^2+c x^4} \, dx,x,x^{n/2}\right )}{n}\\ &=\frac {(2 c) \text {Subst}\left (\int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx,x,x^{n/2}\right )}{\sqrt {b^2-4 a c} n}-\frac {(2 c) \text {Subst}\left (\int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx,x,x^{n/2}\right )}{\sqrt {b^2-4 a c} n}\\ &=\frac {2 \sqrt {2} \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x^{n/2}}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}} n}-\frac {2 \sqrt {2} \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x^{n/2}}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}} n}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.11, size = 60, normalized size = 0.36 \begin {gather*} \frac {\text {RootSum}\left [a+b \text {$\#$1}^2+c \text {$\#$1}^4\&,\frac {-n \log (x)+2 \log \left (x^{n/2}-\text {$\#$1}\right )}{b \text {$\#$1}+2 c \text {$\#$1}^3}\&\right ]}{2 n} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.15, size = 114, normalized size = 0.67
method | result | size |
risch | \(\munderset {\textit {\_R} =\RootOf \left (\left (16 a^{3} c^{2} n^{4}-8 a^{2} b^{2} c \,n^{4}+a \,b^{4} n^{4}\right ) \textit {\_Z}^{4}+\left (-4 a b c \,n^{2}+b^{3} n^{2}\right ) \textit {\_Z}^{2}+c \right )}{\sum }\textit {\_R} \ln \left (x^{\frac {n}{2}}+\left (4 n^{3} b \,a^{2}-\frac {n^{3} b^{3} a}{c}\right ) \textit {\_R}^{3}+\left (2 a n -\frac {n \,b^{2}}{c}\right ) \textit {\_R} \right )\) | \(114\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 801 vs.
\(2 (129) = 258\).
time = 0.36, size = 801, normalized size = 4.74 \begin {gather*} \frac {1}{2} \, \sqrt {2} \sqrt {-\frac {{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2} \sqrt {\frac {1}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} n^{4}}} + b}{{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2}}} \log \left (\frac {4 \, c x x^{\frac {1}{2} \, n - 1} + \sqrt {2} {\left ({\left (a b^{3} - 4 \, a^{2} b c\right )} n^{3} \sqrt {\frac {1}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} n^{4}}} - {\left (b^{2} - 4 \, a c\right )} n\right )} \sqrt {-\frac {{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2} \sqrt {\frac {1}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} n^{4}}} + b}{{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2}}}}{x}\right ) - \frac {1}{2} \, \sqrt {2} \sqrt {-\frac {{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2} \sqrt {\frac {1}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} n^{4}}} + b}{{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2}}} \log \left (\frac {4 \, c x x^{\frac {1}{2} \, n - 1} - \sqrt {2} {\left ({\left (a b^{3} - 4 \, a^{2} b c\right )} n^{3} \sqrt {\frac {1}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} n^{4}}} - {\left (b^{2} - 4 \, a c\right )} n\right )} \sqrt {-\frac {{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2} \sqrt {\frac {1}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} n^{4}}} + b}{{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2}}}}{x}\right ) - \frac {1}{2} \, \sqrt {2} \sqrt {\frac {{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2} \sqrt {\frac {1}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} n^{4}}} - b}{{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2}}} \log \left (\frac {4 \, c x x^{\frac {1}{2} \, n - 1} + \sqrt {2} {\left ({\left (a b^{3} - 4 \, a^{2} b c\right )} n^{3} \sqrt {\frac {1}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} n^{4}}} + {\left (b^{2} - 4 \, a c\right )} n\right )} \sqrt {\frac {{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2} \sqrt {\frac {1}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} n^{4}}} - b}{{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2}}}}{x}\right ) + \frac {1}{2} \, \sqrt {2} \sqrt {\frac {{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2} \sqrt {\frac {1}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} n^{4}}} - b}{{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2}}} \log \left (\frac {4 \, c x x^{\frac {1}{2} \, n - 1} - \sqrt {2} {\left ({\left (a b^{3} - 4 \, a^{2} b c\right )} n^{3} \sqrt {\frac {1}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} n^{4}}} + {\left (b^{2} - 4 \, a c\right )} n\right )} \sqrt {\frac {{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2} \sqrt {\frac {1}{{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} n^{4}}} - b}{{\left (a b^{2} - 4 \, a^{2} c\right )} n^{2}}}}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1035 vs.
\(2 (129) = 258\).
time = 6.50, size = 1035, normalized size = 6.12 \begin {gather*} \frac {\frac {{\left (\sqrt {2} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} b^{4} - 8 \, \sqrt {2} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} a b^{2} c - 2 \, \sqrt {2} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} b^{3} c - 2 \, b^{4} c + 16 \, \sqrt {2} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} a^{2} c^{2} + 8 \, \sqrt {2} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} a b c^{2} + \sqrt {2} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} b^{2} c^{2} + 16 \, a b^{2} c^{2} - 2 \, b^{3} c^{2} - 4 \, \sqrt {2} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} a c^{3} - 32 \, a^{2} c^{3} + 8 \, a b c^{3} + \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} b^{3} - 4 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} a b c - 2 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} b^{2} c + \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} b c^{2} + 2 \, {\left (b^{2} - 4 \, a c\right )} b^{2} c - 8 \, {\left (b^{2} - 4 \, a c\right )} a c^{2} + 2 \, {\left (b^{2} - 4 \, a c\right )} b c^{2}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x^{n}}}{\sqrt {\frac {b + \sqrt {b^{2} - 4 \, a c}}{c}}}\right )}{{\left (a b^{4} - 8 \, a^{2} b^{2} c - 2 \, a b^{3} c + 16 \, a^{3} c^{2} + 8 \, a^{2} b c^{2} + a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} {\left | c \right |}} + \frac {{\left (\sqrt {2} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} b^{4} - 8 \, \sqrt {2} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} a b^{2} c - 2 \, \sqrt {2} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} b^{3} c + 2 \, b^{4} c + 16 \, \sqrt {2} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} a^{2} c^{2} + 8 \, \sqrt {2} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} a b c^{2} + \sqrt {2} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} b^{2} c^{2} - 16 \, a b^{2} c^{2} - 2 \, b^{3} c^{2} - 4 \, \sqrt {2} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} a c^{3} + 32 \, a^{2} c^{3} + 8 \, a b c^{3} + \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} b^{3} - 4 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} a b c - 2 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} b^{2} c + \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} b c^{2} - 2 \, {\left (b^{2} - 4 \, a c\right )} b^{2} c + 8 \, {\left (b^{2} - 4 \, a c\right )} a c^{2} + 2 \, {\left (b^{2} - 4 \, a c\right )} b c^{2}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {x^{n}}}{\sqrt {\frac {b - \sqrt {b^{2} - 4 \, a c}}{c}}}\right )}{{\left (a b^{4} - 8 \, a^{2} b^{2} c - 2 \, a b^{3} c + 16 \, a^{3} c^{2} + 8 \, a^{2} b c^{2} + a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} {\left | c \right |}}}{2 \, n} \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 = -0.01 \begin {gather*} \int \frac {x^{\frac {n}{2}-1}}{a+b\,x^n+c\,x^{2\,n}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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