Optimal. Leaf size=82 \[ \frac {\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac {b-2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^2 \sqrt {b^2-4 a c}}+\frac {b \log \left (a-b x^2+c x^4\right )}{4 c^2}+\frac {x^2}{2 c} \]
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Rubi [A] time = 0.09, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {1114, 703, 634, 618, 206, 628} \[ \frac {\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac {b-2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^2 \sqrt {b^2-4 a c}}+\frac {b \log \left (a-b x^2+c x^4\right )}{4 c^2}+\frac {x^2}{2 c} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 628
Rule 634
Rule 703
Rule 1114
Rubi steps
\begin {align*} \int \frac {x^5}{a-b x^2+c x^4} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2}{a-b x+c x^2} \, dx,x,x^2\right )\\ &=\frac {x^2}{2 c}+\frac {\operatorname {Subst}\left (\int \frac {-a+b x}{a-b x+c x^2} \, dx,x,x^2\right )}{2 c}\\ &=\frac {x^2}{2 c}+\frac {b \operatorname {Subst}\left (\int \frac {-b+2 c x}{a-b x+c x^2} \, dx,x,x^2\right )}{4 c^2}+\frac {\left (b^2-2 a c\right ) \operatorname {Subst}\left (\int \frac {1}{a-b x+c x^2} \, dx,x,x^2\right )}{4 c^2}\\ &=\frac {x^2}{2 c}+\frac {b \log \left (a-b x^2+c x^4\right )}{4 c^2}-\frac {\left (b^2-2 a c\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,-b+2 c x^2\right )}{2 c^2}\\ &=\frac {x^2}{2 c}+\frac {\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac {b-2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^2 \sqrt {b^2-4 a c}}+\frac {b \log \left (a-b x^2+c x^4\right )}{4 c^2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 80, normalized size = 0.98 \[ \frac {\frac {2 \left (b^2-2 a c\right ) \tan ^{-1}\left (\frac {2 c x^2-b}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}+b \log \left (a-b x^2+c x^4\right )+2 c x^2}{4 c^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 259, normalized size = 3.16 \[ \left [\frac {2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} - {\left (b^{2} - 2 \, a c\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{4} - 2 \, b c x^{2} + b^{2} - 2 \, a c + {\left (2 \, c x^{2} - b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{4} - b x^{2} + a}\right ) + {\left (b^{3} - 4 \, a b c\right )} \log \left (c x^{4} - b x^{2} + a\right )}{4 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}, \frac {2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} - 2 \, {\left (b^{2} - 2 \, a c\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {{\left (2 \, c x^{2} - b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) + {\left (b^{3} - 4 \, a b c\right )} \log \left (c x^{4} - b x^{2} + a\right )}{4 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.53, size = 78, normalized size = 0.95 \[ \frac {x^{2}}{2 \, c} + \frac {b \log \left (c x^{4} - b x^{2} + a\right )}{4 \, c^{2}} + \frac {{\left (b^{2} - 2 \, a c\right )} \arctan \left (\frac {2 \, c x^{2} - b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt {-b^{2} + 4 \, a c} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 116, normalized size = 1.41 \[ -\frac {a \arctan \left (\frac {2 c \,x^{2}-b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c}+\frac {b^{2} \arctan \left (\frac {2 c \,x^{2}-b}{\sqrt {4 a c -b^{2}}}\right )}{2 \sqrt {4 a c -b^{2}}\, c^{2}}+\frac {x^{2}}{2 c}+\frac {b \ln \left (c \,x^{4}-b \,x^{2}+a \right )}{4 c^{2}} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.74, size = 656, normalized size = 8.00 \[ \frac {x^2}{2\,c}-\frac {\ln \left (c\,x^4-b\,x^2+a\right )\,\left (2\,b^3-8\,a\,b\,c\right )}{2\,\left (16\,a\,c^3-4\,b^2\,c^2\right )}-\frac {\mathrm {atan}\left (\frac {2\,c^2\,\left (4\,a\,c-b^2\right )\,\left (\frac {\frac {\left (8\,a\,b+\frac {8\,a\,c^2\,\left (2\,b^3-8\,a\,b\,c\right )}{16\,a\,c^3-4\,b^2\,c^2}\right )\,\left (2\,a\,c-b^2\right )}{8\,c^2\,\sqrt {4\,a\,c-b^2}}+\frac {a\,\left (2\,b^3-8\,a\,b\,c\right )\,\left (2\,a\,c-b^2\right )}{\sqrt {4\,a\,c-b^2}\,\left (16\,a\,c^3-4\,b^2\,c^2\right )}}{a}+x^2\,\left (\frac {\frac {\left (2\,a\,c-b^2\right )\,\left (\frac {4\,a\,c^3-6\,b^2\,c^2}{c^2}-\frac {4\,b\,c^2\,\left (2\,b^3-8\,a\,b\,c\right )}{16\,a\,c^3-4\,b^2\,c^2}\right )}{8\,c^2\,\sqrt {4\,a\,c-b^2}}-\frac {b\,\left (2\,b^3-8\,a\,b\,c\right )\,\left (2\,a\,c-b^2\right )}{2\,\sqrt {4\,a\,c-b^2}\,\left (16\,a\,c^3-4\,b^2\,c^2\right )}}{a}+\frac {b\,\left (\frac {\left (2\,b^3-8\,a\,b\,c\right )\,\left (\frac {4\,a\,c^3-6\,b^2\,c^2}{c^2}-\frac {4\,b\,c^2\,\left (2\,b^3-8\,a\,b\,c\right )}{16\,a\,c^3-4\,b^2\,c^2}\right )}{2\,\left (16\,a\,c^3-4\,b^2\,c^2\right )}-\frac {b^3-a\,b\,c}{c^2}+\frac {b\,{\left (2\,a\,c-b^2\right )}^2}{2\,c^2\,\left (4\,a\,c-b^2\right )}\right )}{2\,a\,\sqrt {4\,a\,c-b^2}}\right )+\frac {b\,\left (\frac {a\,b^2}{c^2}+\frac {\left (2\,b^3-8\,a\,b\,c\right )\,\left (8\,a\,b+\frac {8\,a\,c^2\,\left (2\,b^3-8\,a\,b\,c\right )}{16\,a\,c^3-4\,b^2\,c^2}\right )}{2\,\left (16\,a\,c^3-4\,b^2\,c^2\right )}-\frac {a\,{\left (2\,a\,c-b^2\right )}^2}{c^2\,\left (4\,a\,c-b^2\right )}\right )}{2\,a\,\sqrt {4\,a\,c-b^2}}\right )}{4\,a^2\,c^2-4\,a\,b^2\,c+b^4}\right )\,\left (2\,a\,c-b^2\right )}{2\,c^2\,\sqrt {4\,a\,c-b^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.76, size = 311, normalized size = 3.79 \[ \left (\frac {b}{4 c^{2}} - \frac {\sqrt {- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 c^{2} \left (4 a c - b^{2}\right )}\right ) \log {\left (x^{2} + \frac {a b - 8 a c^{2} \left (\frac {b}{4 c^{2}} - \frac {\sqrt {- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 c^{2} \left (4 a c - b^{2}\right )}\right ) + 2 b^{2} c \left (\frac {b}{4 c^{2}} - \frac {\sqrt {- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 c^{2} \left (4 a c - b^{2}\right )}\right )}{2 a c - b^{2}} \right )} + \left (\frac {b}{4 c^{2}} + \frac {\sqrt {- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 c^{2} \left (4 a c - b^{2}\right )}\right ) \log {\left (x^{2} + \frac {a b - 8 a c^{2} \left (\frac {b}{4 c^{2}} + \frac {\sqrt {- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 c^{2} \left (4 a c - b^{2}\right )}\right ) + 2 b^{2} c \left (\frac {b}{4 c^{2}} + \frac {\sqrt {- 4 a c + b^{2}} \left (2 a c - b^{2}\right )}{4 c^{2} \left (4 a c - b^{2}\right )}\right )}{2 a c - b^{2}} \right )} + \frac {x^{2}}{2 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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