Optimal. Leaf size=54 \[ \frac{\log \left (a x^4+2 a x^2+a+b\right )}{4 a}-\frac{\tan ^{-1}\left (\frac{\sqrt{a} \left (x^2+1\right )}{\sqrt{b}}\right )}{2 \sqrt{a} \sqrt{b}} \]
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Rubi [A] time = 0.0449957, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1114, 634, 618, 204, 628} \[ \frac{\log \left (a x^4+2 a x^2+a+b\right )}{4 a}-\frac{\tan ^{-1}\left (\frac{\sqrt{a} \left (x^2+1\right )}{\sqrt{b}}\right )}{2 \sqrt{a} \sqrt{b}} \]
Antiderivative was successfully verified.
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Rule 1114
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x^3}{a+b+2 a x^2+a x^4} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{a+b+2 a x+a x^2} \, dx,x,x^2\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{a+b+2 a x+a x^2} \, dx,x,x^2\right )\right )+\frac{\operatorname{Subst}\left (\int \frac{2 a+2 a x}{a+b+2 a x+a x^2} \, dx,x,x^2\right )}{4 a}\\ &=\frac{\log \left (a+b+2 a x^2+a x^4\right )}{4 a}+\operatorname{Subst}\left (\int \frac{1}{-4 a b-x^2} \, dx,x,2 a \left (1+x^2\right )\right )\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{a} \left (1+x^2\right )}{\sqrt{b}}\right )}{2 \sqrt{a} \sqrt{b}}+\frac{\log \left (a+b+2 a x^2+a x^4\right )}{4 a}\\ \end{align*}
Mathematica [A] time = 0.016579, size = 49, normalized size = 0.91 \[ \frac{\log \left (a \left (x^2+1\right )^2+b\right )-\frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} \left (x^2+1\right )}{\sqrt{b}}\right )}{\sqrt{b}}}{4 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 47, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( a{x}^{4}+2\,a{x}^{2}+a+b \right ) }{4\,a}}-{\frac{1}{2}\arctan \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 = 1.56662, size = 312, normalized size = 5.78 \begin{align*} \left [\frac{b \log \left (a x^{4} + 2 \, a x^{2} + a + b\right ) - \sqrt{-a b} \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 b}, \frac{b \log \left (a x^{4} + 2 \, a x^{2} + a + b\right ) + 2 \, \sqrt{a b} \arctan \left (\frac{\sqrt{a b}}{a x^{2} + a}\right )}{4 \, a b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.508829, size = 117, normalized size = 2.17 \begin{align*} \left (\frac{1}{4 a} - \frac{\sqrt{- a^{3} b}}{4 a^{2} b}\right ) \log{\left (x^{2} + \frac{- 4 a b \left (\frac{1}{4 a} - \frac{\sqrt{- a^{3} b}}{4 a^{2} b}\right ) + a + b}{a} \right )} + \left (\frac{1}{4 a} + \frac{\sqrt{- a^{3} b}}{4 a^{2} b}\right ) \log{\left (x^{2} + \frac{- 4 a b \left (\frac{1}{4 a} + \frac{\sqrt{- a^{3} b}}{4 a^{2} b}\right ) + a + b}{a} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 3.41815, size = 57, normalized size = 1.06 \begin{align*} -\frac{\arctan \left (\frac{a x^{2} + a}{\sqrt{a b}}\right )}{2 \, \sqrt{a b}} + \frac{\log \left (a x^{4} + 2 \, a x^{2} + a + b\right )}{4 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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