Optimal. Leaf size=123 \[ \frac{3 x}{256 a^2 b^3 \left (a+b x^2\right )}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 a^{5/2} b^{7/2}}-\frac{x^3}{16 b^2 \left (a+b x^2\right )^4}+\frac{x}{128 a b^3 \left (a+b x^2\right )^2}-\frac{x}{32 b^3 \left (a+b x^2\right )^3}-\frac{x^5}{10 b \left (a+b x^2\right )^5} \]
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Rubi [A] time = 0.0721446, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {28, 288, 199, 205} \[ \frac{3 x}{256 a^2 b^3 \left (a+b x^2\right )}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 a^{5/2} b^{7/2}}-\frac{x^3}{16 b^2 \left (a+b x^2\right )^4}+\frac{x}{128 a b^3 \left (a+b x^2\right )^2}-\frac{x}{32 b^3 \left (a+b x^2\right )^3}-\frac{x^5}{10 b \left (a+b x^2\right )^5} \]
Antiderivative was successfully verified.
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Rule 28
Rule 288
Rule 199
Rule 205
Rubi steps
\begin{align*} \int \frac{x^6}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac{x^6}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=-\frac{x^5}{10 b \left (a+b x^2\right )^5}+\frac{1}{2} b^4 \int \frac{x^4}{\left (a b+b^2 x^2\right )^5} \, dx\\ &=-\frac{x^5}{10 b \left (a+b x^2\right )^5}-\frac{x^3}{16 b^2 \left (a+b x^2\right )^4}+\frac{1}{16} \left (3 b^2\right ) \int \frac{x^2}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac{x^5}{10 b \left (a+b x^2\right )^5}-\frac{x^3}{16 b^2 \left (a+b x^2\right )^4}-\frac{x}{32 b^3 \left (a+b x^2\right )^3}+\frac{1}{32} \int \frac{1}{\left (a b+b^2 x^2\right )^3} \, dx\\ &=-\frac{x^5}{10 b \left (a+b x^2\right )^5}-\frac{x^3}{16 b^2 \left (a+b x^2\right )^4}-\frac{x}{32 b^3 \left (a+b x^2\right )^3}+\frac{x}{128 a b^3 \left (a+b x^2\right )^2}+\frac{3 \int \frac{1}{\left (a b+b^2 x^2\right )^2} \, dx}{128 a b}\\ &=-\frac{x^5}{10 b \left (a+b x^2\right )^5}-\frac{x^3}{16 b^2 \left (a+b x^2\right )^4}-\frac{x}{32 b^3 \left (a+b x^2\right )^3}+\frac{x}{128 a b^3 \left (a+b x^2\right )^2}+\frac{3 x}{256 a^2 b^3 \left (a+b x^2\right )}+\frac{3 \int \frac{1}{a b+b^2 x^2} \, dx}{256 a^2 b^2}\\ &=-\frac{x^5}{10 b \left (a+b x^2\right )^5}-\frac{x^3}{16 b^2 \left (a+b x^2\right )^4}-\frac{x}{32 b^3 \left (a+b x^2\right )^3}+\frac{x}{128 a b^3 \left (a+b x^2\right )^2}+\frac{3 x}{256 a^2 b^3 \left (a+b x^2\right )}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 a^{5/2} b^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0585359, size = 91, normalized size = 0.74 \[ \frac{-128 a^2 b^2 x^5-70 a^3 b x^3-15 a^4 x+70 a b^3 x^7+15 b^4 x^9}{1280 a^2 b^3 \left (a+b x^2\right )^5}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 a^{5/2} b^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 78, normalized size = 0.6 \begin{align*}{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{5}} \left ({\frac{3\,b{x}^{9}}{256\,{a}^{2}}}+{\frac{7\,{x}^{7}}{128\,a}}-{\frac{{x}^{5}}{10\,b}}-{\frac{7\,a{x}^{3}}{128\,{b}^{2}}}-{\frac{3\,{a}^{2}x}{256\,{b}^{3}}} \right ) }+{\frac{3}{256\,{a}^{2}{b}^{3}}\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 = 1.49462, size = 838, normalized size = 6.81 \begin{align*} \left [\frac{30 \, a b^{5} x^{9} + 140 \, a^{2} b^{4} x^{7} - 256 \, a^{3} b^{3} x^{5} - 140 \, a^{4} b^{2} x^{3} - 30 \, a^{5} b x - 15 \,{\left (b^{5} x^{10} + 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} + 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} + a^{5}\right )} \sqrt{-a b} \log \left (\frac{b x^{2} - 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right )}{2560 \,{\left (a^{3} b^{9} x^{10} + 5 \, a^{4} b^{8} x^{8} + 10 \, a^{5} b^{7} x^{6} + 10 \, a^{6} b^{6} x^{4} + 5 \, a^{7} b^{5} x^{2} + a^{8} b^{4}\right )}}, \frac{15 \, a b^{5} x^{9} + 70 \, a^{2} b^{4} x^{7} - 128 \, a^{3} b^{3} x^{5} - 70 \, a^{4} b^{2} x^{3} - 15 \, a^{5} b x + 15 \,{\left (b^{5} x^{10} + 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} + 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} + a^{5}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right )}{1280 \,{\left (a^{3} b^{9} x^{10} + 5 \, a^{4} b^{8} x^{8} + 10 \, a^{5} b^{7} x^{6} + 10 \, a^{6} b^{6} x^{4} + 5 \, a^{7} b^{5} x^{2} + a^{8} b^{4}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.14973, size = 196, normalized size = 1.59 \begin{align*} - \frac{3 \sqrt{- \frac{1}{a^{5} b^{7}}} \log{\left (- a^{3} b^{3} \sqrt{- \frac{1}{a^{5} b^{7}}} + x \right )}}{512} + \frac{3 \sqrt{- \frac{1}{a^{5} b^{7}}} \log{\left (a^{3} b^{3} \sqrt{- \frac{1}{a^{5} b^{7}}} + x \right )}}{512} + \frac{- 15 a^{4} x - 70 a^{3} b x^{3} - 128 a^{2} b^{2} x^{5} + 70 a b^{3} x^{7} + 15 b^{4} x^{9}}{1280 a^{7} b^{3} + 6400 a^{6} b^{4} x^{2} + 12800 a^{5} b^{5} x^{4} + 12800 a^{4} b^{6} x^{6} + 6400 a^{3} b^{7} x^{8} + 1280 a^{2} b^{8} x^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15764, size = 113, normalized size = 0.92 \begin{align*} \frac{3 \, \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{256 \, \sqrt{a b} a^{2} b^{3}} + \frac{15 \, b^{4} x^{9} + 70 \, a b^{3} x^{7} - 128 \, a^{2} b^{2} x^{5} - 70 \, a^{3} b x^{3} - 15 \, a^{4} x}{1280 \,{\left (b x^{2} + a\right )}^{5} a^{2} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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