Optimal. Leaf size=113 \[ -\frac {b+2 c x^2}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {3 c \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {6 c^2 \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps
used = 5, 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 {6 c^2 \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac {3 c \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {b+2 c x^2}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \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 )^3} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{\left (a+b x+c x^2\right )^3} \, dx,x,x^2\right )\\ &=-\frac {b+2 c x^2}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac {(3 c) \text {Subst}\left (\int \frac {1}{\left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )}{2 \left (b^2-4 a c\right )}\\ &=-\frac {b+2 c x^2}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {3 c \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\left (3 c^2\right ) \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{\left (b^2-4 a c\right )^2}\\ &=-\frac {b+2 c x^2}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {3 c \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {\left (6 c^2\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{\left (b^2-4 a c\right )^2}\\ &=-\frac {b+2 c x^2}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {3 c \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac {6 c^2 \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 106, normalized size = 0.94 \begin {gather*} \frac {-\frac {\left (b+2 c x^2\right ) \left (b^2-6 b c x^2-2 c \left (5 a+3 c x^4\right )\right )}{\left (a+b x^2+c x^4\right )^2}+\frac {24 c^2 \tan ^{-1}\left (\frac {b+2 c x^2}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}}{4 \left (b^2-4 a c\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 126, normalized size = 1.12
method | result | size |
default | \(\frac {2 c \,x^{2}+b}{4 \left (4 a c -b^{2}\right ) \left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {3 c \left (\frac {2 c \,x^{2}+b}{\left (4 a c -b^{2}\right ) \left (c \,x^{4}+b \,x^{2}+a \right )}+\frac {4 c \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}\right )}{2 \left (4 a c -b^{2}\right )}\) | \(126\) |
risch | \(\frac {\frac {3 c^{3} x^{6}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {9 b \,c^{2} x^{4}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {\left (5 a c +b^{2}\right ) c \,x^{2}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {b \left (10 a c -b^{2}\right )}{64 a^{2} c^{2}-32 a \,b^{2} c +4 b^{4}}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}-\frac {3 c^{2} \ln \left (\left (\left (-4 a c +b^{2}\right )^{\frac {5}{2}}-16 a^{2} b \,c^{2}+8 a \,b^{3} c -b^{5}\right ) x^{2}-32 a^{3} c^{2}+16 a^{2} b^{2} c -2 b^{4} a \right )}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}}}+\frac {3 c^{2} \ln \left (\left (\left (-4 a c +b^{2}\right )^{\frac {5}{2}}+16 a^{2} b \,c^{2}-8 a \,b^{3} c +b^{5}\right ) x^{2}+32 a^{3} c^{2}-16 a^{2} b^{2} c +2 b^{4} a \right )}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}}}\) | \(295\) |
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. 392 vs.
\(2 (105) = 210\).
time = 0.40, size = 809, normalized size = 7.16 \begin {gather*} \left [\frac {12 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{6} - b^{5} + 14 \, a b^{3} c - 40 \, a^{2} b c^{2} + 18 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{4} + 4 \, {\left (b^{4} c + a b^{2} c^{2} - 20 \, a^{2} c^{3}\right )} x^{2} + 12 \, {\left (c^{4} x^{8} + 2 \, b c^{3} x^{6} + 2 \, a b c^{2} x^{2} + {\left (b^{2} c^{2} + 2 \, a c^{3}\right )} x^{4} + a^{2} c^{2}\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 )}{4 \, {\left ({\left (b^{6} c^{2} - 12 \, a b^{4} c^{3} + 48 \, a^{2} b^{2} c^{4} - 64 \, a^{3} c^{5}\right )} x^{8} + a^{2} b^{6} - 12 \, a^{3} b^{4} c + 48 \, a^{4} b^{2} c^{2} - 64 \, a^{5} c^{3} + 2 \, {\left (b^{7} c - 12 \, a b^{5} c^{2} + 48 \, a^{2} b^{3} c^{3} - 64 \, a^{3} b c^{4}\right )} x^{6} + {\left (b^{8} - 10 \, a b^{6} c + 24 \, a^{2} b^{4} c^{2} + 32 \, a^{3} b^{2} c^{3} - 128 \, a^{4} c^{4}\right )} x^{4} + 2 \, {\left (a b^{7} - 12 \, a^{2} b^{5} c + 48 \, a^{3} b^{3} c^{2} - 64 \, a^{4} b c^{3}\right )} x^{2}\right )}}, \frac {12 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{6} - b^{5} + 14 \, a b^{3} c - 40 \, a^{2} b c^{2} + 18 \, {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{4} + 4 \, {\left (b^{4} c + a b^{2} c^{2} - 20 \, a^{2} c^{3}\right )} x^{2} - 24 \, {\left (c^{4} x^{8} + 2 \, b c^{3} x^{6} + 2 \, a b c^{2} x^{2} + {\left (b^{2} c^{2} + 2 \, a c^{3}\right )} x^{4} + a^{2} c^{2}\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 )}{4 \, {\left ({\left (b^{6} c^{2} - 12 \, a b^{4} c^{3} + 48 \, a^{2} b^{2} c^{4} - 64 \, a^{3} c^{5}\right )} x^{8} + a^{2} b^{6} - 12 \, a^{3} b^{4} c + 48 \, a^{4} b^{2} c^{2} - 64 \, a^{5} c^{3} + 2 \, {\left (b^{7} c - 12 \, a b^{5} c^{2} + 48 \, a^{2} b^{3} c^{3} - 64 \, a^{3} b c^{4}\right )} x^{6} + {\left (b^{8} - 10 \, a b^{6} c + 24 \, a^{2} b^{4} c^{2} + 32 \, a^{3} b^{2} c^{3} - 128 \, a^{4} c^{4}\right )} x^{4} + 2 \, {\left (a b^{7} - 12 \, a^{2} b^{5} c + 48 \, a^{3} b^{3} c^{2} - 64 \, a^{4} b c^{3}\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. 481 vs.
\(2 (107) = 214\).
time = 1.37, size = 481, normalized size = 4.26 \begin {gather*} - 3 c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \log {\left (x^{2} + \frac {- 192 a^{3} c^{5} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 144 a^{2} b^{2} c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} - 36 a b^{4} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 3 b^{6} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 3 b c^{2}}{6 c^{3}} \right )} + 3 c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \log {\left (x^{2} + \frac {192 a^{3} c^{5} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} - 144 a^{2} b^{2} c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 36 a b^{4} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} - 3 b^{6} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 3 b c^{2}}{6 c^{3}} \right )} + \frac {10 a b c - b^{3} + 18 b c^{2} x^{4} + 12 c^{3} x^{6} + x^{2} \cdot \left (20 a c^{2} + 4 b^{2} c\right )}{64 a^{4} c^{2} - 32 a^{3} b^{2} c + 4 a^{2} b^{4} + x^{8} \cdot \left (64 a^{2} c^{4} - 32 a b^{2} c^{3} + 4 b^{4} c^{2}\right ) + x^{6} \cdot \left (128 a^{2} b c^{3} - 64 a b^{3} c^{2} + 8 b^{5} c\right ) + x^{4} \cdot \left (128 a^{3} c^{3} - 24 a b^{4} c + 4 b^{6}\right ) + x^{2} \cdot \left (128 a^{3} b c^{2} - 64 a^{2} b^{3} c + 8 a b^{5}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.28, size = 144, normalized size = 1.27 \begin {gather*} \frac {6 \, c^{2} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {12 \, c^{3} x^{6} + 18 \, b c^{2} x^{4} + 4 \, b^{2} c x^{2} + 20 \, a c^{2} x^{2} - b^{3} + 10 \, a b c}{4 \, {\left (c x^{4} + b x^{2} + a\right )}^{2} {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.34, size = 386, normalized size = 3.42 \begin {gather*} \frac {\frac {3\,c^3\,x^6}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}-\frac {b^3-10\,a\,b\,c}{4\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {x^2\,\left (b^2\,c+5\,a\,c^2\right )}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}+\frac {9\,b\,c^2\,x^4}{2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}}{x^4\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^8+2\,a\,b\,x^2+2\,b\,c\,x^6}+\frac {6\,c^2\,\mathrm {atan}\left (\frac {\left (x^2\,\left (\frac {36\,c^6}{a\,{\left (4\,a\,c-b^2\right )}^{9/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {36\,b\,c^4\,\left (16\,a^2\,b\,c^4-8\,a\,b^3\,c^3+b^5\,c^2\right )}{a\,{\left (4\,a\,c-b^2\right )}^{15/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )+\frac {72\,b\,c^6}{{\left (4\,a\,c-b^2\right )}^{15/2}}\right )\,\left (b^4\,{\left (4\,a\,c-b^2\right )}^5+16\,a^2\,c^2\,{\left (4\,a\,c-b^2\right )}^5-8\,a\,b^2\,c\,{\left (4\,a\,c-b^2\right )}^5\right )}{72\,c^6}\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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