Optimal. Leaf size=84 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{16 a^{3/2} b^{5/2}}+\frac{x}{16 a b^2 \left (a+b x^2\right )}-\frac{x}{8 b^2 \left (a+b x^2\right )^2}-\frac{x^3}{6 b \left (a+b x^2\right )^3} \]
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Rubi [A] time = 0.0411935, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {28, 288, 199, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{16 a^{3/2} b^{5/2}}+\frac{x}{16 a b^2 \left (a+b x^2\right )}-\frac{x}{8 b^2 \left (a+b x^2\right )^2}-\frac{x^3}{6 b \left (a+b x^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 28
Rule 288
Rule 199
Rule 205
Rubi steps
\begin{align*} \int \frac{x^4}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac{x^4}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac{x^3}{6 b \left (a+b x^2\right )^3}+\frac{1}{2} b^2 \int \frac{x^2}{\left (a b+b^2 x^2\right )^3} \, dx\\ &=-\frac{x^3}{6 b \left (a+b x^2\right )^3}-\frac{x}{8 b^2 \left (a+b x^2\right )^2}+\frac{1}{8} \int \frac{1}{\left (a b+b^2 x^2\right )^2} \, dx\\ &=-\frac{x^3}{6 b \left (a+b x^2\right )^3}-\frac{x}{8 b^2 \left (a+b x^2\right )^2}+\frac{x}{16 a b^2 \left (a+b x^2\right )}+\frac{\int \frac{1}{a b+b^2 x^2} \, dx}{16 a b}\\ &=-\frac{x^3}{6 b \left (a+b x^2\right )^3}-\frac{x}{8 b^2 \left (a+b x^2\right )^2}+\frac{x}{16 a b^2 \left (a+b x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{16 a^{3/2} b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0448912, size = 69, normalized size = 0.82 \[ \frac{-3 a^2 x-8 a b x^3+3 b^2 x^5}{48 a b^2 \left (a+b x^2\right )^3}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{16 a^{3/2} b^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 58, normalized size = 0.7 \begin{align*}{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{3}} \left ({\frac{{x}^{5}}{16\,a}}-{\frac{{x}^{3}}{6\,b}}-{\frac{ax}{16\,{b}^{2}}} \right ) }+{\frac{1}{16\,{b}^{2}a}\arctan \left ({bx{\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 = 1.80431, size = 537, normalized size = 6.39 \begin{align*} \left [\frac{6 \, a b^{3} x^{5} - 16 \, a^{2} b^{2} x^{3} - 6 \, a^{3} b x - 3 \,{\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \sqrt{-a b} \log \left (\frac{b x^{2} - 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right )}{96 \,{\left (a^{2} b^{6} x^{6} + 3 \, a^{3} b^{5} x^{4} + 3 \, a^{4} b^{4} x^{2} + a^{5} b^{3}\right )}}, \frac{3 \, a b^{3} x^{5} - 8 \, a^{2} b^{2} x^{3} - 3 \, a^{3} b x + 3 \,{\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right )}{48 \,{\left (a^{2} b^{6} x^{6} + 3 \, a^{3} b^{5} x^{4} + 3 \, a^{4} b^{4} x^{2} + a^{5} b^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.618193, size = 143, normalized size = 1.7 \begin{align*} - \frac{\sqrt{- \frac{1}{a^{3} b^{5}}} \log{\left (- a^{2} b^{2} \sqrt{- \frac{1}{a^{3} b^{5}}} + x \right )}}{32} + \frac{\sqrt{- \frac{1}{a^{3} b^{5}}} \log{\left (a^{2} b^{2} \sqrt{- \frac{1}{a^{3} b^{5}}} + x \right )}}{32} + \frac{- 3 a^{2} x - 8 a b x^{3} + 3 b^{2} x^{5}}{48 a^{4} b^{2} + 144 a^{3} b^{3} x^{2} + 144 a^{2} b^{4} x^{4} + 48 a b^{5} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15908, size = 84, normalized size = 1. \begin{align*} \frac{\arctan \left (\frac{b x}{\sqrt{a b}}\right )}{16 \, \sqrt{a b} a b^{2}} + \frac{3 \, b^{2} x^{5} - 8 \, a b x^{3} - 3 \, a^{2} x}{48 \,{\left (b x^{2} + a\right )}^{3} a b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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