Optimal. Leaf size=69 \[ \frac {(a-b) \tan ^{-1}\left (\frac {\sqrt {a} \left (x^2+1\right )}{\sqrt {b}}\right )}{2 a^{3/2} \sqrt {b}}-\frac {\log \left (a x^4+2 a x^2+a+b\right )}{2 a}+\frac {x^2}{2 a} \]
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Rubi [A] time = 0.08, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1114, 703, 634, 618, 204, 628} \[ \frac {(a-b) \tan ^{-1}\left (\frac {\sqrt {a} \left (x^2+1\right )}{\sqrt {b}}\right )}{2 a^{3/2} \sqrt {b}}-\frac {\log \left (a x^4+2 a x^2+a+b\right )}{2 a}+\frac {x^2}{2 a} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 703
Rule 1114
Rubi steps
\begin {align*} \int \frac {x^5}{a+b+2 a x^2+a x^4} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2}{a+b+2 a x+a x^2} \, dx,x,x^2\right )\\ &=\frac {x^2}{2 a}+\frac {\operatorname {Subst}\left (\int \frac {-a-b-2 a x}{a+b+2 a x+a x^2} \, dx,x,x^2\right )}{2 a}\\ &=\frac {x^2}{2 a}-\frac {\operatorname {Subst}\left (\int \frac {2 a+2 a x}{a+b+2 a x+a x^2} \, dx,x,x^2\right )}{2 a}+\frac {(a-b) \operatorname {Subst}\left (\int \frac {1}{a+b+2 a x+a x^2} \, dx,x,x^2\right )}{2 a}\\ &=\frac {x^2}{2 a}-\frac {\log \left (a+b+2 a x^2+a x^4\right )}{2 a}-\frac {(a-b) \operatorname {Subst}\left (\int \frac {1}{-4 a b-x^2} \, dx,x,2 a \left (1+x^2\right )\right )}{a}\\ &=\frac {x^2}{2 a}+\frac {(a-b) \tan ^{-1}\left (\frac {\sqrt {a} \left (1+x^2\right )}{\sqrt {b}}\right )}{2 a^{3/2} \sqrt {b}}-\frac {\log \left (a+b+2 a x^2+a x^4\right )}{2 a}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 62, normalized size = 0.90 \[ \frac {\sqrt {a} \left (x^2-\log \left (a \left (x^2+1\right )^2+b\right )\right )+\frac {(a-b) \tan ^{-1}\left (\frac {\sqrt {a} \left (x^2+1\right )}{\sqrt {b}}\right )}{\sqrt {b}}}{2 a^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 157, normalized size = 2.28 \[ \left [\frac {2 \, a b x^{2} - 2 \, a b \log \left (a x^{4} + 2 \, a x^{2} + a + b\right ) + \sqrt {-a b} {\left (a - b\right )} \log \left (\frac {a x^{4} + 2 \, a x^{2} + 2 \, \sqrt {-a b} {\left (x^{2} + 1\right )} + a - b}{a x^{4} + 2 \, a x^{2} + a + b}\right )}{4 \, a^{2} b}, \frac {a b x^{2} - a b \log \left (a x^{4} + 2 \, a x^{2} + a + b\right ) - \sqrt {a b} {\left (a - b\right )} \arctan \left (\frac {\sqrt {a b}}{a x^{2} + a}\right )}{2 \, a^{2} b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 58, normalized size = 0.84 \[ \frac {x^{2}}{2 \, a} + \frac {{\left (a - b\right )} \arctan \left (\frac {a x^{2} + a}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a} - \frac {\log \left (a x^{4} + 2 \, a x^{2} + a + b\right )}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 84, normalized size = 1.22 \[ -\frac {b \arctan \left (\frac {2 a \,x^{2}+2 a}{2 \sqrt {a b}}\right )}{2 \sqrt {a b}\, a}+\frac {x^{2}}{2 a}+\frac {\arctan \left (\frac {2 a \,x^{2}+2 a}{2 \sqrt {a b}}\right )}{2 \sqrt {a b}}-\frac {\ln \left (a \,x^{4}+2 a \,x^{2}+a +b \right )}{2 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.03, size = 58, normalized size = 0.84 \[ \frac {x^{2}}{2 \, a} + \frac {{\left (a - b\right )} \arctan \left (\frac {a x^{2} + a}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a} - \frac {\log \left (a x^{4} + 2 \, a x^{2} + a + b\right )}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 302, normalized size = 4.38 \[ \frac {x^2}{2\,a}-\frac {\ln \left (a\,x^4+2\,a\,x^2+a+b\right )}{2\,a}-\frac {\mathrm {atan}\left (\frac {a\,b\,\left (x^2\,\left (\frac {\frac {\sqrt {a}\,\left (2\,a-2\,b\right )}{\sqrt {b}}+\frac {\left (a-b\right )\,\left (4\,a\,b-12\,a^2\right )}{4\,a^{3/2}\,\sqrt {b}}}{a+b}+\frac {\sqrt {a}\,\left (6\,a-2\,b-\frac {{\left (a-b\right )}^2}{b}+\frac {2\,a\,b-6\,a^2}{a}\right )}{\sqrt {b}\,\left (a+b\right )}\right )-\frac {\frac {\left (a-b\right )\,\left (16\,a\,b-\frac {8\,a^3+8\,b\,a^2}{a}+16\,a^2\right )}{4\,a^{3/2}\,\sqrt {b}}-\frac {\left (16\,a^3+16\,b\,a^2\right )\,\left (a-b\right )}{8\,a^{5/2}\,\sqrt {b}}}{a+b}+\frac {\sqrt {a}\,\left (4\,a+4\,b-\frac {8\,a\,b-\frac {8\,a^3+8\,b\,a^2}{2\,a}+8\,a^2}{a}-\frac {{\left (a-b\right )}^2\,\left (a^3+b\,a^2\right )}{a^3\,b}\right )}{\sqrt {b}\,\left (a+b\right )}\right )}{a^2-2\,a\,b+b^2}\right )\,\left (a-b\right )}{2\,a^{3/2}\,\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.60, size = 144, normalized size = 2.09 \[ \left (- \frac {1}{2 a} - \frac {\sqrt {- a^{3} b} \left (a - b\right )}{4 a^{3} b}\right ) \log {\left (x^{2} + \frac {4 a b \left (- \frac {1}{2 a} - \frac {\sqrt {- a^{3} b} \left (a - b\right )}{4 a^{3} b}\right ) + a + b}{a - b} \right )} + \left (- \frac {1}{2 a} + \frac {\sqrt {- a^{3} b} \left (a - b\right )}{4 a^{3} b}\right ) \log {\left (x^{2} + \frac {4 a b \left (- \frac {1}{2 a} + \frac {\sqrt {- a^{3} b} \left (a - b\right )}{4 a^{3} b}\right ) + a + b}{a - b} \right )} + \frac {x^{2}}{2 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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