Optimal. Leaf size=38 \[ -\frac {\tanh ^{-1}\left (\frac {b+2 c x^4}{\sqrt {b^2-4 a c}}\right )}{2 \sqrt {b^2-4 a c}} \]
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Rubi [A] time = 0.03, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1352, 618, 206} \[ -\frac {\tanh ^{-1}\left (\frac {b+2 c x^4}{\sqrt {b^2-4 a c}}\right )}{2 \sqrt {b^2-4 a c}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 1352
Rubi steps
\begin {align*} \int \frac {x^3}{a+b x^4+c x^8} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^4\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^4\right )\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {b+2 c x^4}{\sqrt {b^2-4 a c}}\right )}{2 \sqrt {b^2-4 a c}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 42, normalized size = 1.11 \[ \frac {\tan ^{-1}\left (\frac {b+2 c x^4}{\sqrt {4 a c-b^2}}\right )}{2 \sqrt {4 a c-b^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 129, normalized size = 3.39 \[ \left [\frac {\log \left (\frac {2 \, c^{2} x^{8} + 2 \, b c x^{4} + b^{2} - 2 \, a c - {\left (2 \, c x^{4} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{8} + b x^{4} + a}\right )}{4 \, \sqrt {b^{2} - 4 \, a c}}, -\frac {\sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {{\left (2 \, c x^{4} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right )}{2 \, {\left (b^{2} - 4 \, a c\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 17.34, size = 36, normalized size = 0.95 \[ \frac {\arctan \left (\frac {2 \, c x^{4} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt {-b^{2} + 4 \, a c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 37, normalized size = 0.97 \[ \frac {\arctan \left (\frac {2 c \,x^{4}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \sqrt {4 a c -b^{2}}} \]
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 = 1.37, size = 260, normalized size = 6.84 \[ -\frac {\mathrm {atan}\left (\frac {{\left (4\,a\,c-b^2\right )}^2\,\left (\frac {\left (\frac {4\,a\,c^4}{4\,a\,c-b^2}-\frac {4\,a\,b^2\,c^4}{{\left (4\,a\,c-b^2\right )}^2}\right )\,\left (b^3-3\,a\,b\,c\right )}{8\,a^3\,c^2\,\sqrt {4\,a\,c-b^2}}-x^4\,\left (\frac {\left (\frac {2\,c^4}{\sqrt {4\,a\,c-b^2}}-\frac {6\,b^2\,c^4}{{\left (4\,a\,c-b^2\right )}^{3/2}}\right )\,\left (a\,c-b^2\right )}{8\,a^3\,c^2}-\frac {\left (b^3-3\,a\,b\,c\right )\,\left (\frac {6\,b\,c^4}{4\,a\,c-b^2}-\frac {2\,b^3\,c^4}{{\left (4\,a\,c-b^2\right )}^2}\right )}{8\,a^3\,c^2\,\sqrt {4\,a\,c-b^2}}\right )+\frac {b\,c^2\,\left (a\,c-b^2\right )}{a^2\,{\left (4\,a\,c-b^2\right )}^{3/2}}\right )}{2\,c^4}\right )}{2\,\sqrt {4\,a\,c-b^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.77, size = 131, normalized size = 3.45 \[ - \frac {\sqrt {- \frac {1}{4 a c - b^{2}}} \log {\left (x^{4} + \frac {- 4 a c \sqrt {- \frac {1}{4 a c - b^{2}}} + b^{2} \sqrt {- \frac {1}{4 a c - b^{2}}} + b}{2 c} \right )}}{4} + \frac {\sqrt {- \frac {1}{4 a c - b^{2}}} \log {\left (x^{4} + \frac {4 a c \sqrt {- \frac {1}{4 a c - b^{2}}} - b^{2} \sqrt {- \frac {1}{4 a c - b^{2}}} + b}{2 c} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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