Optimal. Leaf size=69 \[ -\frac{(a+b) \tanh ^{-1}\left (\frac{\sqrt{a} \left (x^2+1\right )}{\sqrt{b}}\right )}{2 a^{3/2} \sqrt{b}}-\frac{\log \left (a x^4+2 a x^2+a-b\right )}{2 a}+\frac{x^2}{2 a} \]
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Rubi [A] time = 0.0840896, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {1114, 703, 634, 618, 206, 628} \[ -\frac{(a+b) \tanh ^{-1}\left (\frac{\sqrt{a} \left (x^2+1\right )}{\sqrt{b}}\right )}{2 a^{3/2} \sqrt{b}}-\frac{\log \left (a x^4+2 a x^2+a-b\right )}{2 a}+\frac{x^2}{2 a} \]
Antiderivative was successfully verified.
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Rule 1114
Rule 703
Rule 634
Rule 618
Rule 206
Rule 628
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) \tanh ^{-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*}
Mathematica [A] time = 0.0379755, size = 62, normalized size = 0.9 \[ \frac{x^2-\log \left (a \left (x^2+1\right )^2-b\right )}{2 a}-\frac{(a+b) \tanh ^{-1}\left (\frac{\sqrt{a} \left (x^2+1\right )}{\sqrt{b}}\right )}{2 a^{3/2} \sqrt{b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 86, normalized size = 1.3 \begin{align*}{\frac{{x}^{2}}{2\,a}}-{\frac{\ln \left ( a{x}^{4}+2\,a{x}^{2}+a-b \right ) }{2\,a}}-{\frac{1}{2}{\it Artanh} \left ({\frac{2\,a{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{b}{2\,a}{\it Artanh} \left ({\frac{2\,a{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49233, size = 374, normalized size = 5.42 \begin{align*} \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 ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.08395, size = 138, normalized size = 2. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 3.48536, size = 81, normalized size = 1.17 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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