Optimal. Leaf size=174 \[ -\frac {d}{4 a x^4}+\frac {b d-a e}{2 a^2 x^2}+\frac {\left (b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^3 \sqrt {b^2-4 a c}}+\frac {\left (b^2 d-a b e-a (c d-a f)\right ) \log (x)}{a^3}-\frac {\left (b^2 d-a b e-a (c d-a f)\right ) \log \left (a+b x^2+c x^4\right )}{4 a^3} \]
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Rubi [A]
time = 0.27, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1677, 1642,
648, 632, 212, 642} \begin {gather*} -\frac {\log \left (a+b x^2+c x^4\right ) \left (-a b e-a (c d-a f)+b^2 d\right )}{4 a^3}+\frac {\log (x) \left (-a b e-a (c d-a f)+b^2 d\right )}{a^3}+\frac {b d-a e}{2 a^2 x^2}+\frac {\tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (2 a^2 c e-a b^2 e-a b (3 c d-a f)+b^3 d\right )}{2 a^3 \sqrt {b^2-4 a c}}-\frac {d}{4 a x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 1642
Rule 1677
Rubi steps
\begin {align*} \int \frac {d+e x^2+f x^4}{x^5 \left (a+b x^2+c x^4\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {d+e x+f x^2}{x^3 \left (a+b x+c x^2\right )} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {d}{a x^3}+\frac {-b d+a e}{a^2 x^2}+\frac {b^2 d-a b e-a (c d-a f)}{a^3 x}+\frac {-b^3 d+a b^2 e-a^2 c e+a b (2 c d-a f)-c \left (b^2 d-a b e-a (c d-a f)\right ) x}{a^3 \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac {d}{4 a x^4}+\frac {b d-a e}{2 a^2 x^2}+\frac {\left (b^2 d-a b e-a (c d-a f)\right ) \log (x)}{a^3}+\frac {\text {Subst}\left (\int \frac {-b^3 d+a b^2 e-a^2 c e+a b (2 c d-a f)-c \left (b^2 d-a b e-a (c d-a f)\right ) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 a^3}\\ &=-\frac {d}{4 a x^4}+\frac {b d-a e}{2 a^2 x^2}+\frac {\left (b^2 d-a b e-a (c d-a f)\right ) \log (x)}{a^3}-\frac {\left (b^2 d-a b e-a (c d-a f)\right ) \text {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^3}-\frac {\left (b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)\right ) \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^3}\\ &=-\frac {d}{4 a x^4}+\frac {b d-a e}{2 a^2 x^2}+\frac {\left (b^2 d-a b e-a (c d-a f)\right ) \log (x)}{a^3}-\frac {\left (b^2 d-a b e-a (c d-a f)\right ) \log \left (a+b x^2+c x^4\right )}{4 a^3}+\frac {\left (b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 a^3}\\ &=-\frac {d}{4 a x^4}+\frac {b d-a e}{2 a^2 x^2}+\frac {\left (b^3 d-a b^2 e+2 a^2 c e-a b (3 c d-a f)\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^3 \sqrt {b^2-4 a c}}+\frac {\left (b^2 d-a b e-a (c d-a f)\right ) \log (x)}{a^3}-\frac {\left (b^2 d-a b e-a (c d-a f)\right ) \log \left (a+b x^2+c x^4\right )}{4 a^3}\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 314, normalized size = 1.80 \begin {gather*} -\frac {\frac {a^2 d}{x^4}+\frac {2 a (-b d+a e)}{x^2}-4 \left (b^2 d-a b e+a (-c d+a f)\right ) \log (x)+\frac {\left (b^3 d+b^2 \left (\sqrt {b^2-4 a c} d-a e\right )+a b \left (-3 c d-\sqrt {b^2-4 a c} e+a f\right )+a \left (-c \sqrt {b^2-4 a c} d+2 a c e+a \sqrt {b^2-4 a c} f\right )\right ) \log \left (b-\sqrt {b^2-4 a c}+2 c x^2\right )}{\sqrt {b^2-4 a c}}+\frac {\left (-b^3 d+b^2 \left (\sqrt {b^2-4 a c} d+a e\right )-a b \left (-3 c d+\sqrt {b^2-4 a c} e+a f\right )+a \left (-c \left (\sqrt {b^2-4 a c} d+2 a e\right )+a \sqrt {b^2-4 a c} f\right )\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )}{\sqrt {b^2-4 a c}}}{4 a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 203, normalized size = 1.17
method | result | size |
default | \(-\frac {\frac {\left (a^{2} c f -a b c e -a \,c^{2} d +b^{2} c d \right ) \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2 c}+\frac {2 \left (a^{2} b f +a^{2} c e -a \,b^{2} e -2 a b c d +b^{3} d -\frac {\left (a^{2} c f -a b c e -a \,c^{2} d +b^{2} c d \right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{2 a^{3}}-\frac {d}{4 x^{4} a}-\frac {a e -b d}{2 a^{2} x^{2}}+\frac {\left (a^{2} f -a b e -a c d +b^{2} d \right ) \ln \left (x \right )}{a^{3}}\) | \(203\) |
risch | \(\frac {-\frac {\left (a e -b d \right ) x^{2}}{2 a^{2}}-\frac {d}{4 a}}{x^{4}}+\frac {\ln \left (x \right ) f}{a}-\frac {\ln \left (x \right ) b e}{a^{2}}-\frac {\ln \left (x \right ) c d}{a^{2}}+\frac {\ln \left (x \right ) b^{2} d}{a^{3}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (4 c \,a^{4}-b^{2} a^{3}\right ) \textit {\_Z}^{2}+\left (4 a^{3} c f -a^{2} b^{2} f -4 a^{2} b c e -4 a^{2} c^{2} d +a \,b^{3} e +5 a \,b^{2} c d -b^{4} d \right ) \textit {\_Z} +c \,f^{2} a^{2}-a b c e f -2 a \,c^{2} d f +a \,c^{2} e^{2}+b^{2} c d f -e d \,c^{2} b +c^{3} d^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (10 c \,a^{5}-3 a^{4} b^{2}\right ) \textit {\_R}^{2}+\left (5 a^{4} c f -4 e c b \,a^{3}-5 d \,c^{2} a^{3}+4 d c \,b^{2} a^{2}\right ) \textit {\_R} +2 e^{2} c^{2} a^{2}-4 e d \,c^{2} b a +2 d^{2} c^{2} b^{2}\right ) x^{2}-a^{5} b \,\textit {\_R}^{2}+\left (2 a^{4} b f +a^{4} c e -2 a^{3} b^{2} e -3 a^{3} b c d +2 a^{2} b^{3} d \right ) \textit {\_R} -2 a^{3} c e f +2 a^{2} b c d f +2 a^{2} b c \,e^{2}+2 a^{2} c^{2} d e -4 a \,b^{2} c d e -2 a b \,c^{2} d^{2}+2 b^{3} c \,d^{2}\right )\right )}{2}\) | \(411\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.91, size = 609, normalized size = 3.50 \begin {gather*} \left [\frac {{\left (a^{2} b f + {\left (b^{3} - 3 \, a b c\right )} d - {\left (a b^{2} - 2 \, a^{2} c\right )} e\right )} \sqrt {b^{2} - 4 \, a c} x^{4} \log \left (\frac {2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c + {\left (2 \, c x^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) - {\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d - {\left (a b^{3} - 4 \, a^{2} b c\right )} e + {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} f\right )} x^{4} \log \left (c x^{4} + b x^{2} + a\right ) + 4 \, {\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d - {\left (a b^{3} - 4 \, a^{2} b c\right )} e + {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} f\right )} x^{4} \log \left (x\right ) + 2 \, {\left ({\left (a b^{3} - 4 \, a^{2} b c\right )} d - {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} e\right )} x^{2} - {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d}{4 \, {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} x^{4}}, \frac {2 \, {\left (a^{2} b f + {\left (b^{3} - 3 \, a b c\right )} d - {\left (a b^{2} - 2 \, a^{2} c\right )} e\right )} \sqrt {-b^{2} + 4 \, a c} x^{4} \arctan \left (-\frac {{\left (2 \, c x^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) - {\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d - {\left (a b^{3} - 4 \, a^{2} b c\right )} e + {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} f\right )} x^{4} \log \left (c x^{4} + b x^{2} + a\right ) + 4 \, {\left ({\left (b^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d - {\left (a b^{3} - 4 \, a^{2} b c\right )} e + {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} f\right )} x^{4} \log \left (x\right ) + 2 \, {\left ({\left (a b^{3} - 4 \, a^{2} b c\right )} d - {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} e\right )} x^{2} - {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d}{4 \, {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} x^{4}}\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 [A]
time = 4.70, size = 212, normalized size = 1.22 \begin {gather*} -\frac {{\left (b^{2} d - a c d + a^{2} f - a b e\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, a^{3}} + \frac {{\left (b^{2} d - a c d + a^{2} f - a b e\right )} \log \left (x^{2}\right )}{2 \, a^{3}} - \frac {{\left (b^{3} d - 3 \, a b c d + a^{2} b f - a b^{2} e + 2 \, a^{2} c e\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt {-b^{2} + 4 \, a c} a^{3}} - \frac {3 \, b^{2} d x^{4} - 3 \, a c d x^{4} + 3 \, a^{2} f x^{4} - 3 \, a b x^{4} e - 2 \, a b d x^{2} + 2 \, a^{2} x^{2} e + a^{2} d}{4 \, a^{3} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 9.92, size = 2500, normalized size = 14.37 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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