Optimal. Leaf size=64 \[ -\frac{3 x}{8 b^2 \left (a+b x^2\right )}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 \sqrt{a} b^{5/2}}-\frac{x^3}{4 b \left (a+b x^2\right )^2} \]
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Rubi [A] time = 0.0197249, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {288, 205} \[ -\frac{3 x}{8 b^2 \left (a+b x^2\right )}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 \sqrt{a} b^{5/2}}-\frac{x^3}{4 b \left (a+b x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 288
Rule 205
Rubi steps
\begin{align*} \int \frac{x^4}{\left (a+b x^2\right )^3} \, dx &=-\frac{x^3}{4 b \left (a+b x^2\right )^2}+\frac{3 \int \frac{x^2}{\left (a+b x^2\right )^2} \, dx}{4 b}\\ &=-\frac{x^3}{4 b \left (a+b x^2\right )^2}-\frac{3 x}{8 b^2 \left (a+b x^2\right )}+\frac{3 \int \frac{1}{a+b x^2} \, dx}{8 b^2}\\ &=-\frac{x^3}{4 b \left (a+b x^2\right )^2}-\frac{3 x}{8 b^2 \left (a+b x^2\right )}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 \sqrt{a} b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0384839, size = 55, normalized size = 0.86 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 \sqrt{a} b^{5/2}}-\frac{3 a x+5 b x^3}{8 b^2 \left (a+b x^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 47, normalized size = 0.7 \begin{align*}{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{2}} \left ( -{\frac{5\,{x}^{3}}{8\,b}}-{\frac{3\,ax}{8\,{b}^{2}}} \right ) }+{\frac{3}{8\,{b}^{2}}\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 = 1.28876, size = 404, normalized size = 6.31 \begin{align*} \left [-\frac{10 \, a b^{2} x^{3} + 6 \, a^{2} b x + 3 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt{-a b} \log \left (\frac{b x^{2} - 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right )}{16 \,{\left (a b^{5} x^{4} + 2 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )}}, -\frac{5 \, a b^{2} x^{3} + 3 \, a^{2} b x - 3 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right )}{8 \,{\left (a b^{5} x^{4} + 2 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.48288, size = 109, normalized size = 1.7 \begin{align*} - \frac{3 \sqrt{- \frac{1}{a b^{5}}} \log{\left (- a b^{2} \sqrt{- \frac{1}{a b^{5}}} + x \right )}}{16} + \frac{3 \sqrt{- \frac{1}{a b^{5}}} \log{\left (a b^{2} \sqrt{- \frac{1}{a b^{5}}} + x \right )}}{16} - \frac{3 a x + 5 b x^{3}}{8 a^{2} b^{2} + 16 a b^{3} x^{2} + 8 b^{4} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.36908, size = 61, normalized size = 0.95 \begin{align*} \frac{3 \, \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} b^{2}} - \frac{5 \, b x^{3} + 3 \, a x}{8 \,{\left (b x^{2} + a\right )}^{2} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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