Optimal. Leaf size=100 \[ \frac {b \left (b^2-3 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^3 \sqrt {b^2-4 a c}}+\frac {\left (b^2-a c\right ) \log \left (a+b x^2+c x^4\right )}{4 c^3}-\frac {b x^2}{2 c^2}+\frac {x^4}{4 c} \]
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Rubi [A] time = 0.12, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1585, 1114, 701, 634, 618, 206, 628} \begin {gather*} \frac {\left (b^2-a c\right ) \log \left (a+b x^2+c x^4\right )}{4 c^3}+\frac {b \left (b^2-3 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^3 \sqrt {b^2-4 a c}}-\frac {b x^2}{2 c^2}+\frac {x^4}{4 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 628
Rule 634
Rule 701
Rule 1114
Rule 1585
Rubi steps
\begin {align*} \int \frac {x^8}{a x+b x^3+c x^5} \, dx &=\int \frac {x^7}{a+b x^2+c x^4} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^3}{a+b x+c x^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {b}{c^2}+\frac {x}{c}+\frac {a b+\left (b^2-a c\right ) x}{c^2 \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac {b x^2}{2 c^2}+\frac {x^4}{4 c}+\frac {\operatorname {Subst}\left (\int \frac {a b+\left (b^2-a c\right ) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 c^2}\\ &=-\frac {b x^2}{2 c^2}+\frac {x^4}{4 c}-\frac {\left (b \left (b^2-3 a c\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^3}+\frac {\left (b^2-a c\right ) \operatorname {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^3}\\ &=-\frac {b x^2}{2 c^2}+\frac {x^4}{4 c}+\frac {\left (b^2-a c\right ) \log \left (a+b x^2+c x^4\right )}{4 c^3}+\frac {\left (b \left (b^2-3 a c\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 c^3}\\ &=-\frac {b x^2}{2 c^2}+\frac {x^4}{4 c}+\frac {b \left (b^2-3 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^3 \sqrt {b^2-4 a c}}+\frac {\left (b^2-a c\right ) \log \left (a+b x^2+c x^4\right )}{4 c^3}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 93, normalized size = 0.93 \begin {gather*} \frac {-\frac {2 b \left (b^2-3 a c\right ) \tan ^{-1}\left (\frac {b+2 c x^2}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}+\left (b^2-a c\right ) \log \left (a+b x^2+c x^4\right )+c x^2 \left (c x^2-2 b\right )}{4 c^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^8}{a x+b x^3+c x^5} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.15, size = 313, normalized size = 3.13 \begin {gather*} \left [\frac {{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{4} - 2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} x^{2} - {\left (b^{3} - 3 \, a b 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^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )}}, \frac {{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{4} - 2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} x^{2} + 2 \, {\left (b^{3} - 3 \, a b 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^{4} - 5 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, {\left (b^{2} c^{3} - 4 \, a c^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.47, size = 92, normalized size = 0.92 \begin {gather*} \frac {c x^{4} - 2 \, b x^{2}}{4 \, c^{2}} + \frac {{\left (b^{2} - a c\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, c^{3}} - \frac {{\left (b^{3} - 3 \, a b 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^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 142, normalized size = 1.42 \begin {gather*} \frac {x^{4}}{4 c}+\frac {3 a b \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 {b^{3} \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \sqrt {4 a c -b^{2}}\, c^{3}}-\frac {b \,x^{2}}{2 c^{2}}-\frac {a \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 c^{2}}+\frac {b^{2} \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {c x^{4} - 2 \, b x^{2}}{4 \, c^{2}} - \frac {-\int \frac {{\left (b^{2} - a c\right )} x^{3} + a b x}{c x^{4} + b x^{2} + a}\,{d x}}{c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.20, size = 842, normalized size = 8.42 \begin {gather*} \frac {x^4}{4\,c}-\frac {\ln \left (c\,x^4+b\,x^2+a\right )\,\left (8\,a^2\,c^2-10\,a\,b^2\,c+2\,b^4\right )}{2\,\left (16\,a\,c^4-4\,b^2\,c^3\right )}-\frac {b\,x^2}{2\,c^2}+\frac {b\,\mathrm {atan}\left (\frac {2\,c^4\,\left (4\,a\,c-b^2\right )\,\left (\frac {\frac {b\,\left (3\,a\,c-b^2\right )\,\left (\frac {8\,a^2\,c^4-8\,a\,b^2\,c^3}{c^4}-\frac {8\,a\,c^2\,\left (8\,a^2\,c^2-10\,a\,b^2\,c+2\,b^4\right )}{16\,a\,c^4-4\,b^2\,c^3}\right )}{8\,c^3\,\sqrt {4\,a\,c-b^2}}-\frac {a\,b\,\left (3\,a\,c-b^2\right )\,\left (8\,a^2\,c^2-10\,a\,b^2\,c+2\,b^4\right )}{c\,\sqrt {4\,a\,c-b^2}\,\left (16\,a\,c^4-4\,b^2\,c^3\right )}}{a}-x^2\,\left (\frac {\frac {b\,\left (\frac {6\,b^3\,c^3-10\,a\,b\,c^4}{c^4}+\frac {4\,b\,c^2\,\left (8\,a^2\,c^2-10\,a\,b^2\,c+2\,b^4\right )}{16\,a\,c^4-4\,b^2\,c^3}\right )\,\left (3\,a\,c-b^2\right )}{8\,c^3\,\sqrt {4\,a\,c-b^2}}+\frac {b^2\,\left (3\,a\,c-b^2\right )\,\left (8\,a^2\,c^2-10\,a\,b^2\,c+2\,b^4\right )}{2\,c\,\sqrt {4\,a\,c-b^2}\,\left (16\,a\,c^4-4\,b^2\,c^3\right )}}{a}+\frac {b\,\left (\frac {2\,a^2\,b\,c^2-3\,a\,b^3\,c+b^5}{c^4}+\frac {\left (\frac {6\,b^3\,c^3-10\,a\,b\,c^4}{c^4}+\frac {4\,b\,c^2\,\left (8\,a^2\,c^2-10\,a\,b^2\,c+2\,b^4\right )}{16\,a\,c^4-4\,b^2\,c^3}\right )\,\left (8\,a^2\,c^2-10\,a\,b^2\,c+2\,b^4\right )}{2\,\left (16\,a\,c^4-4\,b^2\,c^3\right )}-\frac {b^3\,{\left (3\,a\,c-b^2\right )}^2}{2\,c^4\,\left (4\,a\,c-b^2\right )}\right )}{2\,a\,\sqrt {4\,a\,c-b^2}}\right )+\frac {b\,\left (\frac {\left (\frac {8\,a^2\,c^4-8\,a\,b^2\,c^3}{c^4}-\frac {8\,a\,c^2\,\left (8\,a^2\,c^2-10\,a\,b^2\,c+2\,b^4\right )}{16\,a\,c^4-4\,b^2\,c^3}\right )\,\left (8\,a^2\,c^2-10\,a\,b^2\,c+2\,b^4\right )}{2\,\left (16\,a\,c^4-4\,b^2\,c^3\right )}-\frac {a^3\,c^2-2\,a^2\,b^2\,c+a\,b^4}{c^4}+\frac {a\,b^2\,{\left (3\,a\,c-b^2\right )}^2}{c^4\,\left (4\,a\,c-b^2\right )}\right )}{2\,a\,\sqrt {4\,a\,c-b^2}}\right )}{9\,a^2\,b^2\,c^2-6\,a\,b^4\,c+b^6}\right )\,\left (3\,a\,c-b^2\right )}{2\,c^3\,\sqrt {4\,a\,c-b^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.95, size = 391, normalized size = 3.91 \begin {gather*} - \frac {b x^{2}}{2 c^{2}} + \left (- \frac {b \sqrt {- 4 a c + b^{2}} \left (3 a c - b^{2}\right )}{4 c^{3} \left (4 a c - b^{2}\right )} - \frac {a c - b^{2}}{4 c^{3}}\right ) \log {\left (x^{2} + \frac {2 a^{2} c - a b^{2} + 8 a c^{3} \left (- \frac {b \sqrt {- 4 a c + b^{2}} \left (3 a c - b^{2}\right )}{4 c^{3} \left (4 a c - b^{2}\right )} - \frac {a c - b^{2}}{4 c^{3}}\right ) - 2 b^{2} c^{2} \left (- \frac {b \sqrt {- 4 a c + b^{2}} \left (3 a c - b^{2}\right )}{4 c^{3} \left (4 a c - b^{2}\right )} - \frac {a c - b^{2}}{4 c^{3}}\right )}{3 a b c - b^{3}} \right )} + \left (\frac {b \sqrt {- 4 a c + b^{2}} \left (3 a c - b^{2}\right )}{4 c^{3} \left (4 a c - b^{2}\right )} - \frac {a c - b^{2}}{4 c^{3}}\right ) \log {\left (x^{2} + \frac {2 a^{2} c - a b^{2} + 8 a c^{3} \left (\frac {b \sqrt {- 4 a c + b^{2}} \left (3 a c - b^{2}\right )}{4 c^{3} \left (4 a c - b^{2}\right )} - \frac {a c - b^{2}}{4 c^{3}}\right ) - 2 b^{2} c^{2} \left (\frac {b \sqrt {- 4 a c + b^{2}} \left (3 a c - b^{2}\right )}{4 c^{3} \left (4 a c - b^{2}\right )} - \frac {a c - b^{2}}{4 c^{3}}\right )}{3 a b c - b^{3}} \right )} + \frac {x^{4}}{4 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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