Optimal. Leaf size=174 \[ -\frac {1}{a x}-\frac {\sqrt {c} \left (1+\frac {b}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a \sqrt {b+\sqrt {b^2-4 a c}}} \]
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Rubi [A]
time = 0.15, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1599, 1137,
1180, 211} \begin {gather*} -\frac {\sqrt {c} \left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} a \sqrt {\sqrt {b^2-4 a c}+b}}-\frac {1}{a x} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 1137
Rule 1180
Rule 1599
Rubi steps
\begin {align*} \int \frac {1}{x \left (a x+b x^3+c x^5\right )} \, dx &=\int \frac {1}{x^2 \left (a+b x^2+c x^4\right )} \, dx\\ &=-\frac {1}{a x}+\frac {\int \frac {-b-c x^2}{a+b x^2+c x^4} \, dx}{a}\\ &=-\frac {1}{a x}-\frac {\left (c \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx}{2 a}-\frac {\left (c \left (1+\frac {b}{\sqrt {b^2-4 a c}}\right )\right ) \int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx}{2 a}\\ &=-\frac {1}{a x}-\frac {\sqrt {c} \left (1+\frac {b}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a \sqrt {b+\sqrt {b^2-4 a c}}}\\ \end {align*}
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Mathematica [A]
time = 0.26, size = 191, normalized size = 1.10 \begin {gather*} -\frac {\frac {2}{x}+\frac {\sqrt {2} \sqrt {c} \left (b+\sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \sqrt {c} \left (-b+\sqrt {b^2-4 a c}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 159, normalized size = 0.91
method | result | size |
default | \(\frac {4 c \left (-\frac {\left (-b -\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \arctanh \left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (b -\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{a}-\frac {1}{a x}\) | \(159\) |
risch | \(-\frac {1}{a x}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (16 a^{5} c^{2}-8 a^{4} b^{2} c +a^{3} b^{4}\right ) \textit {\_Z}^{4}+\left (12 a^{2} b \,c^{2}-7 a \,b^{3} c +b^{5}\right ) \textit {\_Z}^{2}+c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (40 a^{5} c^{2}-22 a^{4} b^{2} c +3 a^{3} b^{4}\right ) \textit {\_R}^{4}+\left (25 a^{2} b \,c^{2}-14 a \,b^{3} c +2 b^{5}\right ) \textit {\_R}^{2}+2 c^{3}\right ) x +\left (4 a^{4} c^{2}-5 a^{3} b^{2} c +a^{2} b^{4}\right ) \textit {\_R}^{3}\right )\right )}{2}\) | \(170\) |
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. 1116 vs.
\(2 (137) = 274\).
time = 0.36, size = 1116, normalized size = 6.41 \begin {gather*} -\frac {\sqrt {\frac {1}{2}} a x \sqrt {-\frac {b^{3} - 3 \, a b c + {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{a^{6} b^{2} - 4 \, a^{7} c}}}{a^{3} b^{2} - 4 \, a^{4} c}} \log \left (-2 \, {\left (b^{2} c^{2} - a c^{3}\right )} x + \sqrt {\frac {1}{2}} {\left (b^{5} - 5 \, a b^{3} c + 4 \, a^{2} b c^{2} - {\left (a^{3} b^{4} - 6 \, a^{4} b^{2} c + 8 \, a^{5} c^{2}\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{a^{6} b^{2} - 4 \, a^{7} c}}\right )} \sqrt {-\frac {b^{3} - 3 \, a b c + {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{a^{6} b^{2} - 4 \, a^{7} c}}}{a^{3} b^{2} - 4 \, a^{4} c}}\right ) - \sqrt {\frac {1}{2}} a x \sqrt {-\frac {b^{3} - 3 \, a b c + {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{a^{6} b^{2} - 4 \, a^{7} c}}}{a^{3} b^{2} - 4 \, a^{4} c}} \log \left (-2 \, {\left (b^{2} c^{2} - a c^{3}\right )} x - \sqrt {\frac {1}{2}} {\left (b^{5} - 5 \, a b^{3} c + 4 \, a^{2} b c^{2} - {\left (a^{3} b^{4} - 6 \, a^{4} b^{2} c + 8 \, a^{5} c^{2}\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{a^{6} b^{2} - 4 \, a^{7} c}}\right )} \sqrt {-\frac {b^{3} - 3 \, a b c + {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{a^{6} b^{2} - 4 \, a^{7} c}}}{a^{3} b^{2} - 4 \, a^{4} c}}\right ) + \sqrt {\frac {1}{2}} a x \sqrt {-\frac {b^{3} - 3 \, a b c - {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{a^{6} b^{2} - 4 \, a^{7} c}}}{a^{3} b^{2} - 4 \, a^{4} c}} \log \left (-2 \, {\left (b^{2} c^{2} - a c^{3}\right )} x + \sqrt {\frac {1}{2}} {\left (b^{5} - 5 \, a b^{3} c + 4 \, a^{2} b c^{2} + {\left (a^{3} b^{4} - 6 \, a^{4} b^{2} c + 8 \, a^{5} c^{2}\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{a^{6} b^{2} - 4 \, a^{7} c}}\right )} \sqrt {-\frac {b^{3} - 3 \, a b c - {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{a^{6} b^{2} - 4 \, a^{7} c}}}{a^{3} b^{2} - 4 \, a^{4} c}}\right ) - \sqrt {\frac {1}{2}} a x \sqrt {-\frac {b^{3} - 3 \, a b c - {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{a^{6} b^{2} - 4 \, a^{7} c}}}{a^{3} b^{2} - 4 \, a^{4} c}} \log \left (-2 \, {\left (b^{2} c^{2} - a c^{3}\right )} x - \sqrt {\frac {1}{2}} {\left (b^{5} - 5 \, a b^{3} c + 4 \, a^{2} b c^{2} + {\left (a^{3} b^{4} - 6 \, a^{4} b^{2} c + 8 \, a^{5} c^{2}\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{a^{6} b^{2} - 4 \, a^{7} c}}\right )} \sqrt {-\frac {b^{3} - 3 \, a b c - {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} \sqrt {\frac {b^{4} - 2 \, a b^{2} c + a^{2} c^{2}}{a^{6} b^{2} - 4 \, a^{7} c}}}{a^{3} b^{2} - 4 \, a^{4} c}}\right ) + 2}{2 \, a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.99, size = 148, normalized size = 0.85 \begin {gather*} \operatorname {RootSum} {\left (t^{4} \cdot \left (256 a^{5} c^{2} - 128 a^{4} b^{2} c + 16 a^{3} b^{4}\right ) + t^{2} \cdot \left (48 a^{2} b c^{2} - 28 a b^{3} c + 4 b^{5}\right ) + c^{3}, \left ( t \mapsto t \log {\left (x + \frac {- 64 t^{3} a^{5} c^{2} + 48 t^{3} a^{4} b^{2} c - 8 t^{3} a^{3} b^{4} - 10 t a^{2} b c^{2} + 10 t a b^{3} c - 2 t b^{5}}{a c^{3} - b^{2} c^{2}} \right )} \right )\right )} - \frac {1}{a x} \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. 1839 vs.
\(2 (137) = 274\).
time = 4.90, size = 1839, normalized size = 10.57 \begin {gather*} -\frac {{\left (2 \, a^{2} b^{4} c^{2} - 8 \, a^{3} b^{2} c^{3} - \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} a^{2} b^{4} + 4 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} a^{3} b^{2} c + 2 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} a^{2} b^{3} c - \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} a^{2} b^{2} c^{2} - 2 \, {\left (b^{2} - 4 \, a c\right )} a^{2} b^{2} c^{2} + {\left (2 \, b^{4} c^{2} - 16 \, a b^{2} c^{3} + 32 \, a^{2} c^{4} - \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} b^{4} + 8 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} a b^{2} c + 2 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} b^{3} c - 16 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} a^{2} c^{2} - 8 \, \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^{2} + 4 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} a c^{3} - 2 \, {\left (b^{2} - 4 \, a c\right )} b^{2} c^{2} + 8 \, {\left (b^{2} - 4 \, a c\right )} a c^{3}\right )} a^{2} + 2 \, {\left (\sqrt {2} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} a b^{5} - 8 \, \sqrt {2} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} a^{2} b^{3} c - 2 \, \sqrt {2} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} a b^{4} c - 2 \, a b^{5} c + 16 \, \sqrt {2} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} a^{3} b c^{2} + 8 \, \sqrt {2} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} a^{2} b^{2} c^{2} + \sqrt {2} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} a b^{3} c^{2} + 16 \, a^{2} b^{3} c^{2} - 4 \, \sqrt {2} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} a^{2} b c^{3} - 32 \, a^{3} b c^{3} + 2 \, {\left (b^{2} - 4 \, a c\right )} a b^{3} c - 8 \, {\left (b^{2} - 4 \, a c\right )} a^{2} b c^{2}\right )} {\left | a \right |}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} x}{\sqrt {\frac {a b + \sqrt {a^{2} b^{2} - 4 \, a^{3} c}}{a c}}}\right )}{8 \, {\left (a^{3} b^{4} - 8 \, a^{4} b^{2} c - 2 \, a^{3} b^{3} c + 16 \, a^{5} c^{2} + 8 \, a^{4} b c^{2} + a^{3} b^{2} c^{2} - 4 \, a^{4} c^{3}\right )} {\left | a \right |} {\left | c \right |}} - \frac {{\left (2 \, a^{2} b^{4} c^{2} - 8 \, a^{3} b^{2} c^{3} - \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} a^{2} b^{4} + 4 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} a^{3} b^{2} c + 2 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} a^{2} b^{3} c - \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} a^{2} b^{2} c^{2} - 2 \, {\left (b^{2} - 4 \, a c\right )} a^{2} b^{2} c^{2} + {\left (2 \, b^{4} c^{2} - 16 \, a b^{2} c^{3} + 32 \, a^{2} c^{4} - \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} b^{4} + 8 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} a b^{2} c + 2 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} b^{3} c - 16 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} a^{2} c^{2} - 8 \, \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^{2} + 4 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} a c^{3} - 2 \, {\left (b^{2} - 4 \, a c\right )} b^{2} c^{2} + 8 \, {\left (b^{2} - 4 \, a c\right )} a c^{3}\right )} a^{2} + 2 \, {\left (\sqrt {2} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} a b^{5} - 8 \, \sqrt {2} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} a^{2} b^{3} c - 2 \, \sqrt {2} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} a b^{4} c + 2 \, a b^{5} c + 16 \, \sqrt {2} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} a^{3} b c^{2} + 8 \, \sqrt {2} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} a^{2} b^{2} c^{2} + \sqrt {2} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} a b^{3} c^{2} - 16 \, a^{2} b^{3} c^{2} - 4 \, \sqrt {2} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} a^{2} b c^{3} + 32 \, a^{3} b c^{3} - 2 \, {\left (b^{2} - 4 \, a c\right )} a b^{3} c + 8 \, {\left (b^{2} - 4 \, a c\right )} a^{2} b c^{2}\right )} {\left | a \right |}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} x}{\sqrt {\frac {a b - \sqrt {a^{2} b^{2} - 4 \, a^{3} c}}{a c}}}\right )}{8 \, {\left (a^{3} b^{4} - 8 \, a^{4} b^{2} c - 2 \, a^{3} b^{3} c + 16 \, a^{5} c^{2} + 8 \, a^{4} b c^{2} + a^{3} b^{2} c^{2} - 4 \, a^{4} c^{3}\right )} {\left | a \right |} {\left | c \right |}} - \frac {1}{a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.86, size = 2997, normalized size = 17.22 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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