Optimal. Leaf size=205 \[ -\frac {1}{2 a d x^2}-\frac {\left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^2 \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )}-\frac {(b d+a e) \log (x)}{a^2 d^2}+\frac {e^3 \log \left (d+e x^2\right )}{2 d^2 \left (c d^2-b d e+a e^2\right )}+\frac {\left (b c d-b^2 e+a c e\right ) \log \left (a+b x^2+c x^4\right )}{4 a^2 \left (c d^2-b d e+a e^2\right )} \]
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Rubi [A]
time = 0.27, antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1265, 907, 648,
632, 212, 642} \begin {gather*} \frac {\left (a c e+b^2 (-e)+b c d\right ) \log \left (a+b x^2+c x^4\right )}{4 a^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^2 \sqrt {b^2-4 a c} \left (a e^2-b d e+c d^2\right )}-\frac {\log (x) (a e+b d)}{a^2 d^2}+\frac {e^3 \log \left (d+e x^2\right )}{2 d^2 \left (a e^2-b d e+c d^2\right )}-\frac {1}{2 a d x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 907
Rule 1265
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 (d+e x) \left (a+b x+c x^2\right )} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{a d x^2}+\frac {-b d-a e}{a^2 d^2 x}+\frac {e^4}{d^2 \left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac {b^2 c d-a c^2 d-b^3 e+2 a b c e+c \left (b c d-b^2 e+a c e\right ) x}{a^2 \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{2 a d x^2}-\frac {(b d+a e) \log (x)}{a^2 d^2}+\frac {e^3 \log \left (d+e x^2\right )}{2 d^2 \left (c d^2-b d e+a e^2\right )}+\frac {\text {Subst}\left (\int \frac {b^2 c d-a c^2 d-b^3 e+2 a b c e+c \left (b c d-b^2 e+a c e\right ) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 a^2 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {1}{2 a d x^2}-\frac {(b d+a e) \log (x)}{a^2 d^2}+\frac {e^3 \log \left (d+e x^2\right )}{2 d^2 \left (c d^2-b d e+a e^2\right )}+\frac {\left (b c d-b^2 e+a c e\right ) \text {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^2 \left (c d^2-b d e+a e^2\right )}+\frac {\left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e\right ) \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^2 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {1}{2 a d x^2}-\frac {(b d+a e) \log (x)}{a^2 d^2}+\frac {e^3 \log \left (d+e x^2\right )}{2 d^2 \left (c d^2-b d e+a e^2\right )}+\frac {\left (b c d-b^2 e+a c e\right ) \log \left (a+b x^2+c x^4\right )}{4 a^2 \left (c d^2-b d e+a e^2\right )}-\frac {\left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 a^2 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac {1}{2 a d x^2}-\frac {\left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^2 \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )}-\frac {(b d+a e) \log (x)}{a^2 d^2}+\frac {e^3 \log \left (d+e x^2\right )}{2 d^2 \left (c d^2-b d e+a e^2\right )}+\frac {\left (b c d-b^2 e+a c e\right ) \log \left (a+b x^2+c x^4\right )}{4 a^2 \left (c d^2-b d e+a e^2\right )}\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 331, normalized size = 1.61 \begin {gather*} \frac {1}{4} \left (-\frac {2}{a d x^2}-\frac {4 (b d+a e) \log (x)}{a^2 d^2}+\frac {\left (b^3 e-b c \left (\sqrt {b^2-4 a c} d+3 a e\right )+a c \left (2 c d-\sqrt {b^2-4 a c} e\right )+b^2 \left (-c d+\sqrt {b^2-4 a c} e\right )\right ) \log \left (b-\sqrt {b^2-4 a c}+2 c x^2\right )}{a^2 \sqrt {b^2-4 a c} \left (-c d^2+e (b d-a e)\right )}+\frac {\left (-b^3 e+b c \left (-\sqrt {b^2-4 a c} d+3 a e\right )+b^2 \left (c d+\sqrt {b^2-4 a c} e\right )-a c \left (2 c d+\sqrt {b^2-4 a c} e\right )\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )}{a^2 \sqrt {b^2-4 a c} \left (-c d^2+e (b d-a e)\right )}+\frac {2 e^3 \log \left (d+e x^2\right )}{c d^4+d^2 e (-b d+a e)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 215, normalized size = 1.05
method | result | size |
default | \(\frac {\frac {\left (a \,c^{2} e -b^{2} c e +b \,c^{2} d \right ) \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2 c}+\frac {2 \left (2 a b c e -a \,c^{2} d -b^{3} e +b^{2} c d -\frac {\left (a \,c^{2} e -b^{2} c e +b \,c^{2} d \right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{2 \left (a \,e^{2}-d e b +c \,d^{2}\right ) a^{2}}+\frac {e^{3} \ln \left (e \,x^{2}+d \right )}{2 d^{2} \left (a \,e^{2}-d e b +c \,d^{2}\right )}-\frac {1}{2 a d \,x^{2}}+\frac {\left (-a e -b d \right ) \ln \left (x \right )}{a^{2} d^{2}}\) | \(215\) |
risch | \(-\frac {1}{2 a d \,x^{2}}-\frac {e \ln \left (x \right )}{a \,d^{2}}-\frac {\ln \left (x \right ) b}{a^{2} d}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (4 a^{4} c \,e^{2}-a^{3} b^{2} e^{2}-4 a^{3} b c d e +4 a^{3} c^{2} d^{2}+a^{2} b^{3} d e -a^{2} b^{2} c \,d^{2}\right ) \textit {\_Z}^{2}+\left (-4 a^{2} c^{2} e +5 a \,b^{2} c e -4 a b \,c^{2} d -b^{4} e +b^{3} c d \right ) \textit {\_Z} +c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-12 d^{2} a^{5} e^{4} c +3 d^{2} b^{2} a^{4} e^{4}+16 d^{3} b \,a^{4} e^{3} c -14 a^{4} c^{2} d^{4} e^{2}-4 d^{3} b^{3} a^{3} e^{3}-8 d^{4} b^{2} a^{3} e^{2} c +22 d^{5} b \,a^{3} e \,c^{2}-10 d^{6} a^{3} c^{3}+3 d^{4} b^{4} a^{2} e^{2}-6 d^{5} b^{3} a^{2} e c +3 d^{6} a^{2} b^{2} c^{2}\right ) \textit {\_R}^{3}+\left (-16 d b \,a^{3} e^{4} c +22 d^{2} a^{3} e^{3} c^{2}+4 d \,b^{3} a^{2} e^{4}-21 d^{2} b^{2} a^{2} e^{3} c +16 a^{2} b \,c^{2} d^{3} e^{2}+3 d^{4} a^{2} e \,c^{3}+4 d^{2} b^{4} a \,e^{3}-4 d^{3} b^{3} a \,e^{2} c -4 d^{4} b^{2} a e \,c^{2}+4 d^{5} b a \,c^{3}\right ) \textit {\_R}^{2}+\left (-4 a^{2} c^{2} e^{4}+8 a \,b^{2} e^{4} c -4 a b \,c^{2} d \,e^{3}-4 a \,c^{3} d^{2} e^{2}-2 b^{4} e^{4}-2 c^{4} d^{4}\right ) \textit {\_R} +2 c^{3} e^{3}\right ) x^{2}+\left (-4 a^{5} c \,d^{3} e^{3}+a^{4} b^{2} d^{3} e^{3}-3 a^{4} b c \,d^{4} e^{2}+4 a^{4} c^{2} d^{5} e +a^{3} b^{3} d^{4} e^{2}-2 a^{3} b^{2} c \,d^{5} e +a^{3} b \,c^{2} d^{6}\right ) \textit {\_R}^{3}+\left (-8 a^{4} c d \,e^{4}+2 a^{3} b^{2} d \,e^{4}-11 a^{3} b c \,d^{2} e^{3}+8 a^{3} c^{2} d^{3} e^{2}+3 a^{2} b^{3} d^{2} e^{3}-10 a^{2} b^{2} c \,d^{3} e^{2}+10 a^{2} b \,c^{2} d^{4} e -a^{2} c^{3} d^{5}+2 a \,b^{4} d^{3} e^{2}-4 a \,b^{3} c \,d^{4} e +2 a \,b^{2} c^{2} d^{5}\right ) \textit {\_R}^{2}+\left (6 a^{2} b c \,e^{4}-2 a \,b^{3} e^{4}+6 b^{2} c d \,e^{3} a -2 c^{3} d^{3} e a -2 b^{4} d \,e^{3}-2 b \,c^{3} d^{4}\right ) \textit {\_R} \right )\right )}{2}+\frac {e^{3} \ln \left (-e \,x^{2}-d \right )}{2 d^{2} \left (a \,e^{2}-d e b +c \,d^{2}\right )}\) | \(847\) |
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 [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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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 = 4.37, size = 237, normalized size = 1.16 \begin {gather*} \frac {{\left (b c d - b^{2} e + a c e\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, {\left (a^{2} c d^{2} - a^{2} b d e + a^{3} e^{2}\right )}} + \frac {e^{4} \log \left ({\left | x^{2} e + d \right |}\right )}{2 \, {\left (c d^{4} e - b d^{3} e^{2} + a d^{2} e^{3}\right )}} + \frac {{\left (b^{2} c d - 2 \, a c^{2} d - b^{3} e + 3 \, a b c e\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, {\left (a^{2} c d^{2} - a^{2} b d e + a^{3} e^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {{\left (b d + a e\right )} \log \left (x^{2}\right )}{2 \, a^{2} d^{2}} + \frac {b d x^{2} + a x^{2} e - a d}{2 \, a^{2} d^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 62.95, size = 2500, normalized size = 12.20 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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