Optimal. Leaf size=244 \[ -\frac {d}{6 a x^6}+\frac {b d-a e}{4 a^2 x^4}-\frac {b^2 d-a b e-a (c d-a f)}{2 a^3 x^2}-\frac {\left (b^4 d-a b^3 e+3 a^2 b c e+2 a^2 c (c d-a f)-a b^2 (4 c d-a f)\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^4 \sqrt {b^2-4 a c}}-\frac {\left (b^3 d-a b^2 e+a^2 c e-a b (2 c d-a f)\right ) \log (x)}{a^4}+\frac {\left (b^3 d-a b^2 e+a^2 c e-a b (2 c d-a f)\right ) \log \left (a+b x^2+c x^4\right )}{4 a^4} \]
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Rubi [A]
time = 0.38, antiderivative size = 244, 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 {-a b e-a (c d-a f)+b^2 d}{2 a^3 x^2}+\frac {b d-a e}{4 a^2 x^4}+\frac {\log \left (a+b x^2+c x^4\right ) \left (a^2 c e-a b^2 e-a b (2 c d-a f)+b^3 d\right )}{4 a^4}-\frac {\log (x) \left (a^2 c e-a b^2 e-a b (2 c d-a f)+b^3 d\right )}{a^4}-\frac {\tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (3 a^2 b c e+2 a^2 c (c d-a f)-a b^3 e-a b^2 (4 c d-a f)+b^4 d\right )}{2 a^4 \sqrt {b^2-4 a c}}-\frac {d}{6 a x^6} \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^7 \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^4 \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^4}+\frac {-b d+a e}{a^2 x^3}+\frac {b^2 d-a b e-a (c d-a f)}{a^3 x^2}+\frac {-b^3 d+a b^2 e-a^2 c e+a b (2 c d-a f)}{a^4 x}+\frac {b^4 d-a b^3 e+2 a^2 b c e+a^2 c (c d-a f)-a b^2 (3 c d-a f)+c \left (b^3 d-a b^2 e+a^2 c e-a b (2 c d-a f)\right ) x}{a^4 \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac {d}{6 a x^6}+\frac {b d-a e}{4 a^2 x^4}-\frac {b^2 d-a b e-a (c d-a f)}{2 a^3 x^2}-\frac {\left (b^3 d-a b^2 e+a^2 c e-a b (2 c d-a f)\right ) \log (x)}{a^4}+\frac {\text {Subst}\left (\int \frac {b^4 d-a b^3 e+2 a^2 b c e+a^2 c (c d-a f)-a b^2 (3 c d-a f)+c \left (b^3 d-a b^2 e+a^2 c e-a b (2 c d-a f)\right ) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 a^4}\\ &=-\frac {d}{6 a x^6}+\frac {b d-a e}{4 a^2 x^4}-\frac {b^2 d-a b e-a (c d-a f)}{2 a^3 x^2}-\frac {\left (b^3 d-a b^2 e+a^2 c e-a b (2 c d-a f)\right ) \log (x)}{a^4}+\frac {\left (b^3 d-a b^2 e+a^2 c e-a b (2 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^4}+\frac {\left (b^4 d-a b^3 e+3 a^2 b c e+2 a^2 c (c d-a f)-a b^2 (4 c d-a f)\right ) \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^4}\\ &=-\frac {d}{6 a x^6}+\frac {b d-a e}{4 a^2 x^4}-\frac {b^2 d-a b e-a (c d-a f)}{2 a^3 x^2}-\frac {\left (b^3 d-a b^2 e+a^2 c e-a b (2 c d-a f)\right ) \log (x)}{a^4}+\frac {\left (b^3 d-a b^2 e+a^2 c e-a b (2 c d-a f)\right ) \log \left (a+b x^2+c x^4\right )}{4 a^4}-\frac {\left (b^4 d-a b^3 e+3 a^2 b c e+2 a^2 c (c d-a f)-a b^2 (4 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^4}\\ &=-\frac {d}{6 a x^6}+\frac {b d-a e}{4 a^2 x^4}-\frac {b^2 d-a b e-a (c d-a f)}{2 a^3 x^2}-\frac {\left (b^4 d-a b^3 e+3 a^2 b c e+2 a^2 c (c d-a f)-a b^2 (4 c d-a f)\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^4 \sqrt {b^2-4 a c}}-\frac {\left (b^3 d-a b^2 e+a^2 c e-a b (2 c d-a f)\right ) \log (x)}{a^4}+\frac {\left (b^3 d-a b^2 e+a^2 c e-a b (2 c d-a f)\right ) \log \left (a+b x^2+c x^4\right )}{4 a^4}\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 416, normalized size = 1.70 \begin {gather*} \frac {-\frac {2 a^3 d}{x^6}+\frac {3 a^2 (b d-a e)}{x^4}+\frac {6 a \left (-b^2 d+a b e+a (c d-a f)\right )}{x^2}-12 \left (b^3 d-a b^2 e+a^2 c e+a b (-2 c d+a f)\right ) \log (x)+\frac {3 \left (b^4 d+b^3 \left (\sqrt {b^2-4 a c} d-a e\right )+a^2 c \left (2 c d+\sqrt {b^2-4 a c} e-2 a f\right )+a b^2 \left (-4 c d-\sqrt {b^2-4 a c} e+a f\right )+a b \left (-2 c \sqrt {b^2-4 a c} d+3 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 {3 \left (-b^4 d+b^3 \left (\sqrt {b^2-4 a c} d+a e\right )-a b^2 \left (-4 c d+\sqrt {b^2-4 a c} e+a f\right )+a^2 c \left (-2 c d+\sqrt {b^2-4 a c} e+2 a f\right )+a b \left (-2 c \sqrt {b^2-4 a c} d-3 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}}}{12 a^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 294, normalized size = 1.20
method | result | size |
default | \(-\frac {\frac {\left (-a^{2} b c f -a^{2} c^{2} e +a \,b^{2} c e +2 a b \,c^{2} d -b^{3} c d \right ) \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2 c}+\frac {2 \left (a^{3} c f -a^{2} b^{2} f -2 a^{2} b c e -a^{2} c^{2} d +a \,b^{3} e +3 a \,b^{2} c d -b^{4} d -\frac {\left (-a^{2} b c f -a^{2} c^{2} e +a \,b^{2} c e +2 a b \,c^{2} d -b^{3} c d \right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{2 a^{4}}-\frac {d}{6 a \,x^{6}}-\frac {a e -b d}{4 a^{2} x^{4}}-\frac {a^{2} f -a b e -a c d +b^{2} d}{2 a^{3} x^{2}}+\frac {\left (-a^{2} b f -a^{2} c e +a \,b^{2} e +2 a b c d -b^{3} d \right ) \ln \left (x \right )}{a^{4}}\) | \(294\) |
risch | \(\frac {-\frac {\left (a^{2} f -a b e -a c d +b^{2} d \right ) x^{4}}{2 a^{3}}-\frac {\left (a e -b d \right ) x^{2}}{4 a^{2}}-\frac {d}{6 a}}{x^{6}}-\frac {\ln \left (x \right ) b f}{a^{2}}-\frac {\ln \left (x \right ) c e}{a^{2}}+\frac {\ln \left (x \right ) b^{2} e}{a^{3}}+\frac {2 \ln \left (x \right ) b c d}{a^{3}}-\frac {\ln \left (x \right ) b^{3} d}{a^{4}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (4 c \,a^{5}-a^{4} b^{2}\right ) \textit {\_Z}^{2}+\left (-4 c \,a^{3} b f -4 e \,a^{3} c^{2}+a^{2} b^{3} f +5 a^{2} b^{2} c e +8 a^{2} b \,c^{2} d -a \,b^{4} e -6 a \,b^{3} c d +b^{5} d \right ) \textit {\_Z} +a^{2} c^{2} f^{2}-a b \,c^{2} e f -2 a \,c^{3} d f +a \,c^{3} e^{2}+b^{2} c^{2} d f -b \,c^{3} d e +c^{4} d^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (10 a^{7} c -3 b^{2} a^{6}\right ) \textit {\_R}^{2}+\left (-4 a^{5} b c f -5 a^{5} c^{2} e +4 a^{4} b^{2} c e +9 a^{4} b \,c^{2} d -4 a^{3} b^{3} c d \right ) \textit {\_R} +2 a^{4} c^{2} f^{2}-4 a^{3} b \,c^{2} e f -4 a^{3} c^{3} d f +4 a^{2} b^{2} c^{2} d f +2 a^{2} b^{2} c^{2} e^{2}+4 a^{2} b \,c^{3} d e +2 a^{2} c^{4} d^{2}-4 a \,b^{3} c^{2} d e -4 a \,b^{2} c^{3} d^{2}+2 b^{4} c^{2} d^{2}\right ) x^{2}-a^{7} b \,\textit {\_R}^{2}+\left (a^{6} c f -2 a^{5} b^{2} f -3 a^{5} b c e -a^{5} c^{2} d +2 a^{4} b^{3} e +5 a^{4} b^{2} c d -2 a^{3} b^{4} d \right ) \textit {\_R} +2 a^{4} b c \,f^{2}+2 a^{4} c^{2} e f -4 a^{3} b^{2} c e f -6 a^{3} b \,c^{2} d f -2 a^{3} b \,c^{2} e^{2}-2 a^{3} c^{3} d e +4 a^{2} b^{3} c d f +2 a^{2} b^{3} c \,e^{2}+8 a^{2} b^{2} c^{2} d e +4 a^{2} b \,c^{3} d^{2}-4 a \,b^{4} c d e -6 a \,b^{3} c^{2} d^{2}+2 b^{5} c \,d^{2}\right )\right )}{2}\) | \(667\) |
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 = 1.64, size = 834, normalized size = 3.42 \begin {gather*} \left [-\frac {3 \, \sqrt {b^{2} - 4 \, a c} {\left ({\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} d - {\left (a b^{3} - 3 \, a^{2} b c\right )} e + {\left (a^{2} b^{2} - 2 \, a^{3} c\right )} f\right )} x^{6} \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 ) - 3 \, {\left ({\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} d - {\left (a b^{4} - 5 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} e + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} f\right )} x^{6} \log \left (c x^{4} + b x^{2} + a\right ) + 12 \, {\left ({\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} d - {\left (a b^{4} - 5 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} e + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} f\right )} x^{6} \log \left (x\right ) + 6 \, {\left ({\left (a b^{4} - 5 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} d - {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} e + {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} f\right )} x^{4} - 3 \, {\left ({\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} d - {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} e\right )} x^{2} + 2 \, {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} d}{12 \, {\left (a^{4} b^{2} - 4 \, a^{5} c\right )} x^{6}}, -\frac {6 \, \sqrt {-b^{2} + 4 \, a c} {\left ({\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} d - {\left (a b^{3} - 3 \, a^{2} b c\right )} e + {\left (a^{2} b^{2} - 2 \, a^{3} c\right )} f\right )} x^{6} \arctan \left (-\frac {{\left (2 \, c x^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) - 3 \, {\left ({\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} d - {\left (a b^{4} - 5 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} e + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} f\right )} x^{6} \log \left (c x^{4} + b x^{2} + a\right ) + 12 \, {\left ({\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} d - {\left (a b^{4} - 5 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} e + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} f\right )} x^{6} \log \left (x\right ) + 6 \, {\left ({\left (a b^{4} - 5 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} d - {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} e + {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} f\right )} x^{4} - 3 \, {\left ({\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} d - {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} e\right )} x^{2} + 2 \, {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} d}{12 \, {\left (a^{4} b^{2} - 4 \, a^{5} c\right )} x^{6}}\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 = 6.81, size = 313, normalized size = 1.28 \begin {gather*} \frac {{\left (b^{3} d - 2 \, a b c d + a^{2} b f - a b^{2} e + a^{2} c e\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, a^{4}} - \frac {{\left (b^{3} d - 2 \, a b c d + a^{2} b f - a b^{2} e + a^{2} c e\right )} \log \left (x^{2}\right )}{2 \, a^{4}} + \frac {{\left (b^{4} d - 4 \, a b^{2} c d + 2 \, a^{2} c^{2} d + a^{2} b^{2} f - 2 \, a^{3} c f - a b^{3} e + 3 \, a^{2} b 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^{4}} + \frac {11 \, b^{3} d x^{6} - 22 \, a b c d x^{6} + 11 \, a^{2} b f x^{6} - 11 \, a b^{2} x^{6} e + 11 \, a^{2} c x^{6} e - 6 \, a b^{2} d x^{4} + 6 \, a^{2} c d x^{4} - 6 \, a^{3} f x^{4} + 6 \, a^{2} b x^{4} e + 3 \, a^{2} b d x^{2} - 3 \, a^{3} x^{2} e - 2 \, a^{3} d}{12 \, a^{4} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 13.83, size = 2500, normalized size = 10.25 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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