Optimal. Leaf size=184 \[ \frac {\left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} \sqrt {c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} \sqrt {c} \sqrt {b+\sqrt {b^2-4 a c}}} \]
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Rubi [A]
time = 0.15, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1504, 1180,
211} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (\frac {2 c d-b e}{\sqrt {b^2-4 a c}}+e\right )}{2 \sqrt {2} \sqrt {c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right )}{2 \sqrt {2} \sqrt {c} \sqrt {\sqrt {b^2-4 a c}+b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 1180
Rule 1504
Rubi steps
\begin {align*} \int \frac {x \left (d+e x^4\right )}{a+b x^4+c x^8} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {d+e x^2}{a+b x^2+c x^4} \, dx,x,x^2\right )\\ &=\frac {1}{4} \left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx,x,x^2\right )+\frac {1}{4} \left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx,x,x^2\right )\\ &=\frac {\left (e+\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} \sqrt {c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} \sqrt {c} \sqrt {b+\sqrt {b^2-4 a c}}}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 179, normalized size = 0.97 \begin {gather*} \frac {\frac {\left (2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b+\sqrt {b^2-4 a c}}}}{2 \sqrt {2} \sqrt {c} \sqrt {b^2-4 a c}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 168, normalized size = 0.91
method | result | size |
default | \(2 c \left (-\frac {\left (e \sqrt {-4 a c +b^{2}}-e b +2 c d \right ) \sqrt {2}\, \arctanh \left (\frac {c \,x^{2} \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (e \sqrt {-4 a c +b^{2}}+e b -2 c d \right ) \sqrt {2}\, \arctan \left (\frac {c \,x^{2} \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )\) | \(168\) |
risch | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (16 a^{3} c^{3}-8 a^{2} b^{2} c^{2}+a \,b^{4} c \right ) \textit {\_Z}^{4}+\left (-4 a^{2} b c \,e^{2}+16 a^{2} c^{2} d e +b^{3} e^{2} a -4 a \,b^{2} c d e -4 a b \,c^{2} d^{2}+b^{3} c \,d^{2}\right ) \textit {\_Z}^{2}+a^{2} e^{4}-2 a b d \,e^{3}+2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}-2 b c \,d^{3} e +c^{2} d^{4}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-4 a^{2} c^{2} e +a \,b^{2} c e +4 a b \,c^{2} d -b^{3} c d \right ) \textit {\_R}^{2}+e^{3} a b -a c d \,e^{2}-b^{2} d \,e^{2}+2 d^{2} e b c -c^{2} d^{3}\right ) x^{2}+\left (4 a^{2} b \,c^{2}-a \,b^{3} c \right ) \textit {\_R}^{3}+\left (2 a^{2} c \,e^{2}-a \,b^{2} e^{2}+2 a b c d e -2 a \,c^{2} d^{2}\right ) \textit {\_R} \right )\right )}{4}\) | \(290\) |
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. 1531 vs.
\(2 (149) = 298\).
time = 0.54, size = 1531, normalized size = 8.32 \begin {gather*} \frac {1}{4} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {b c d^{2} - 4 \, a c d e + a b e^{2} + {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} \sqrt {\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3}}}}{a b^{2} c - 4 \, a^{2} c^{2}}} \log \left (-c^{2} d^{4} x^{2} + b c d^{3} x^{2} e - a b d x^{2} e^{3} + a^{2} x^{2} e^{4} + \frac {1}{2} \, \sqrt {\frac {1}{2}} {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d^{3} - {\left (a b^{2} - 4 \, a^{2} c\right )} d e^{2} - {\left ({\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} d - 2 \, {\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} e\right )} \sqrt {\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3}}}\right )} \sqrt {-\frac {b c d^{2} - 4 \, a c d e + a b e^{2} + {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} \sqrt {\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3}}}}{a b^{2} c - 4 \, a^{2} c^{2}}}\right ) - \frac {1}{4} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {b c d^{2} - 4 \, a c d e + a b e^{2} + {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} \sqrt {\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3}}}}{a b^{2} c - 4 \, a^{2} c^{2}}} \log \left (-c^{2} d^{4} x^{2} + b c d^{3} x^{2} e - a b d x^{2} e^{3} + a^{2} x^{2} e^{4} - \frac {1}{2} \, \sqrt {\frac {1}{2}} {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d^{3} - {\left (a b^{2} - 4 \, a^{2} c\right )} d e^{2} - {\left ({\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} d - 2 \, {\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} e\right )} \sqrt {\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3}}}\right )} \sqrt {-\frac {b c d^{2} - 4 \, a c d e + a b e^{2} + {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} \sqrt {\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3}}}}{a b^{2} c - 4 \, a^{2} c^{2}}}\right ) + \frac {1}{4} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {b c d^{2} - 4 \, a c d e + a b e^{2} - {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} \sqrt {\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3}}}}{a b^{2} c - 4 \, a^{2} c^{2}}} \log \left (-c^{2} d^{4} x^{2} + b c d^{3} x^{2} e - a b d x^{2} e^{3} + a^{2} x^{2} e^{4} + \frac {1}{2} \, \sqrt {\frac {1}{2}} {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d^{3} - {\left (a b^{2} - 4 \, a^{2} c\right )} d e^{2} + {\left ({\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} d - 2 \, {\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} e\right )} \sqrt {\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3}}}\right )} \sqrt {-\frac {b c d^{2} - 4 \, a c d e + a b e^{2} - {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} \sqrt {\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3}}}}{a b^{2} c - 4 \, a^{2} c^{2}}}\right ) - \frac {1}{4} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {b c d^{2} - 4 \, a c d e + a b e^{2} - {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} \sqrt {\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3}}}}{a b^{2} c - 4 \, a^{2} c^{2}}} \log \left (-c^{2} d^{4} x^{2} + b c d^{3} x^{2} e - a b d x^{2} e^{3} + a^{2} x^{2} e^{4} - \frac {1}{2} \, \sqrt {\frac {1}{2}} {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d^{3} - {\left (a b^{2} - 4 \, a^{2} c\right )} d e^{2} + {\left ({\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} d - 2 \, {\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} e\right )} \sqrt {\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3}}}\right )} \sqrt {-\frac {b c d^{2} - 4 \, a c d e + a b e^{2} - {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} \sqrt {\frac {c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3}}}}{a b^{2} c - 4 \, a^{2} c^{2}}}\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. 1404 vs.
\(2 (149) = 298\).
time = 7.24, size = 1404, normalized size = 7.63 \begin {gather*} \frac {{\left ({\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 )} d + 2 \, {\left (2 \, a b^{2} c^{2} - 8 \, a^{2} c^{3} - \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} a b^{2} + 4 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} a^{2} c + 2 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} a b c - \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c + \sqrt {b^{2} - 4 \, a c} c} a c^{2} - 2 \, {\left (b^{2} - 4 \, a c\right )} a c^{2}\right )} e\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} x^{2}}{\sqrt {\frac {b + \sqrt {b^{2} - 4 \, a c}}{c}}}\right )}{8 \, {\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 ({\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 )} d + 2 \, {\left (2 \, a b^{2} c^{2} - 8 \, a^{2} c^{3} - \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} a b^{2} + 4 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} a^{2} c + 2 \, \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} a b c - \sqrt {2} \sqrt {b^{2} - 4 \, a c} \sqrt {b c - \sqrt {b^{2} - 4 \, a c} c} a c^{2} - 2 \, {\left (b^{2} - 4 \, a c\right )} a c^{2}\right )} e\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} x^{2}}{\sqrt {\frac {b - \sqrt {b^{2} - 4 \, a c}}{c}}}\right )}{8 \, {\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 |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.05, size = 2500, normalized size = 13.59 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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