Optimal. Leaf size=74 \[ -\frac {b+2 c x^2}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {2 c \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1121, 628, 632,
212} \begin {gather*} \frac {2 c \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {b+2 c x^2}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 628
Rule 632
Rule 1121
Rubi steps
\begin {align*} \int \frac {x}{\left (a+b x^2+c x^4\right )^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{\left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )\\ &=-\frac {b+2 c x^2}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {c \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{b^2-4 a c}\\ &=-\frac {b+2 c x^2}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {(2 c) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{b^2-4 a c}\\ &=-\frac {b+2 c x^2}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {2 c \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 79, normalized size = 1.07 \begin {gather*} -\frac {\frac {b+2 c x^2}{a+b x^2+c x^4}+\frac {4 c \tan ^{-1}\left (\frac {b+2 c x^2}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}}{2 \left (b^2-4 a c\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 75, normalized size = 1.01
method | result | size |
default | \(\frac {2 c \,x^{2}+b}{2 \left (4 a c -b^{2}\right ) \left (c \,x^{4}+b \,x^{2}+a \right )}+\frac {2 c \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}\) | \(75\) |
risch | \(\frac {\frac {c \,x^{2}}{4 a c -b^{2}}+\frac {b}{8 a c -2 b^{2}}}{c \,x^{4}+b \,x^{2}+a}+\frac {c \ln \left (\left (\left (-4 a c +b^{2}\right )^{\frac {3}{2}}+4 a b c -b^{3}\right ) x^{2}+8 a^{2} c -2 a \,b^{2}\right )}{\left (-4 a c +b^{2}\right )^{\frac {3}{2}}}-\frac {c \ln \left (\left (\left (-4 a c +b^{2}\right )^{\frac {3}{2}}-4 a b c +b^{3}\right ) x^{2}-8 a^{2} c +2 a \,b^{2}\right )}{\left (-4 a c +b^{2}\right )^{\frac {3}{2}}}\) | \(151\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 168 vs.
\(2 (68) = 136\).
time = 0.36, size = 361, normalized size = 4.88 \begin {gather*} \left [-\frac {b^{3} - 4 \, a b c + 2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} + 2 \, {\left (c^{2} x^{4} + b c x^{2} + a c\right )} \sqrt {b^{2} - 4 \, a c} \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 )}{2 \, {\left (a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{4} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{2}\right )}}, -\frac {b^{3} - 4 \, a b c + 2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} - 4 \, {\left (c^{2} x^{4} + b c x^{2} + a c\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {{\left (2 \, c x^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right )}{2 \, {\left (a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} + {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{4} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 267 vs.
\(2 (66) = 132\).
time = 0.69, size = 267, normalized size = 3.61 \begin {gather*} - c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \log {\left (x^{2} + \frac {- 16 a^{2} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} + 8 a b^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} - b^{4} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} + b c}{2 c^{2}} \right )} + c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \log {\left (x^{2} + \frac {16 a^{2} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} - 8 a b^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} + b^{4} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} + b c}{2 c^{2}} \right )} + \frac {b + 2 c x^{2}}{8 a^{2} c - 2 a b^{2} + x^{4} \cdot \left (8 a c^{2} - 2 b^{2} c\right ) + x^{2} \cdot \left (8 a b c - 2 b^{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.93, size = 82, normalized size = 1.11 \begin {gather*} -\frac {2 \, c \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} - 4 \, a c\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {2 \, c x^{2} + b}{2 \, {\left (c x^{4} + b x^{2} + a\right )} {\left (b^{2} - 4 \, a c\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.31, size = 172, normalized size = 2.32 \begin {gather*} \frac {\frac {b}{2\,\left (4\,a\,c-b^2\right )}+\frac {c\,x^2}{4\,a\,c-b^2}}{c\,x^4+b\,x^2+a}-\frac {2\,c\,\mathrm {atan}\left (\frac {b^3-4\,a\,b\,c}{{\left (4\,a\,c-b^2\right )}^{3/2}}-\frac {x^2\,{\left (4\,a\,c-b^2\right )}^4\,\left (\frac {4\,c^4}{a\,{\left (4\,a\,c-b^2\right )}^{7/2}}+\frac {4\,c^2\,\left (b^3\,c^2-4\,a\,b\,c^3\right )\,\left (b^3-4\,a\,b\,c\right )}{a\,{\left (4\,a\,c-b^2\right )}^{13/2}}\right )}{8\,c^4}\right )}{{\left (4\,a\,c-b^2\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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