Optimal. Leaf size=122 \[ \frac{7 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 a^{3/2} b^{9/2}}-\frac{7 x^5}{80 b^2 \left (a+b x^2\right )^4}-\frac{7 x^3}{96 b^3 \left (a+b x^2\right )^3}+\frac{7 x}{256 a b^4 \left (a+b x^2\right )}-\frac{7 x}{128 b^4 \left (a+b x^2\right )^2}-\frac{x^7}{10 b \left (a+b x^2\right )^5} \]
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Rubi [A] time = 0.0721806, antiderivative size = 122, 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} \[ \frac{7 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 a^{3/2} b^{9/2}}-\frac{7 x^5}{80 b^2 \left (a+b x^2\right )^4}-\frac{7 x^3}{96 b^3 \left (a+b x^2\right )^3}+\frac{7 x}{256 a b^4 \left (a+b x^2\right )}-\frac{7 x}{128 b^4 \left (a+b x^2\right )^2}-\frac{x^7}{10 b \left (a+b x^2\right )^5} \]
Antiderivative was successfully verified.
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Rule 28
Rule 288
Rule 199
Rule 205
Rubi steps
\begin{align*} \int \frac{x^8}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac{x^8}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=-\frac{x^7}{10 b \left (a+b x^2\right )^5}+\frac{1}{10} \left (7 b^4\right ) \int \frac{x^6}{\left (a b+b^2 x^2\right )^5} \, dx\\ &=-\frac{x^7}{10 b \left (a+b x^2\right )^5}-\frac{7 x^5}{80 b^2 \left (a+b x^2\right )^4}+\frac{1}{16} \left (7 b^2\right ) \int \frac{x^4}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac{x^7}{10 b \left (a+b x^2\right )^5}-\frac{7 x^5}{80 b^2 \left (a+b x^2\right )^4}-\frac{7 x^3}{96 b^3 \left (a+b x^2\right )^3}+\frac{7}{32} \int \frac{x^2}{\left (a b+b^2 x^2\right )^3} \, dx\\ &=-\frac{x^7}{10 b \left (a+b x^2\right )^5}-\frac{7 x^5}{80 b^2 \left (a+b x^2\right )^4}-\frac{7 x^3}{96 b^3 \left (a+b x^2\right )^3}-\frac{7 x}{128 b^4 \left (a+b x^2\right )^2}+\frac{7 \int \frac{1}{\left (a b+b^2 x^2\right )^2} \, dx}{128 b^2}\\ &=-\frac{x^7}{10 b \left (a+b x^2\right )^5}-\frac{7 x^5}{80 b^2 \left (a+b x^2\right )^4}-\frac{7 x^3}{96 b^3 \left (a+b x^2\right )^3}-\frac{7 x}{128 b^4 \left (a+b x^2\right )^2}+\frac{7 x}{256 a b^4 \left (a+b x^2\right )}+\frac{7 \int \frac{1}{a b+b^2 x^2} \, dx}{256 a b^3}\\ &=-\frac{x^7}{10 b \left (a+b x^2\right )^5}-\frac{7 x^5}{80 b^2 \left (a+b x^2\right )^4}-\frac{7 x^3}{96 b^3 \left (a+b x^2\right )^3}-\frac{7 x}{128 b^4 \left (a+b x^2\right )^2}+\frac{7 x}{256 a b^4 \left (a+b x^2\right )}+\frac{7 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 a^{3/2} b^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.0578995, size = 91, normalized size = 0.75 \[ \frac{7 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 a^{3/2} b^{9/2}}-\frac{x \left (896 a^2 b^2 x^4+490 a^3 b x^2+105 a^4+790 a b^3 x^6-105 b^4 x^8\right )}{3840 a b^4 \left (a+b x^2\right )^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 80, normalized size = 0.7 \begin{align*}{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{5}} \left ({\frac{7\,{x}^{9}}{256\,a}}-{\frac{79\,{x}^{7}}{384\,b}}-{\frac{7\,a{x}^{5}}{30\,{b}^{2}}}-{\frac{49\,{a}^{2}{x}^{3}}{384\,{b}^{3}}}-{\frac{7\,x{a}^{3}}{256\,{b}^{4}}} \right ) }+{\frac{7}{256\,a{b}^{4}}\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 = 1.51243, size = 855, normalized size = 7.01 \begin{align*} \left [\frac{210 \, a b^{5} x^{9} - 1580 \, a^{2} b^{4} x^{7} - 1792 \, a^{3} b^{3} x^{5} - 980 \, a^{4} b^{2} x^{3} - 210 \, a^{5} b x - 105 \,{\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 )}{7680 \,{\left (a^{2} b^{10} x^{10} + 5 \, a^{3} b^{9} x^{8} + 10 \, a^{4} b^{8} x^{6} + 10 \, a^{5} b^{7} x^{4} + 5 \, a^{6} b^{6} x^{2} + a^{7} b^{5}\right )}}, \frac{105 \, a b^{5} x^{9} - 790 \, a^{2} b^{4} x^{7} - 896 \, a^{3} b^{3} x^{5} - 490 \, a^{4} b^{2} x^{3} - 105 \, a^{5} b x + 105 \,{\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 )}{3840 \,{\left (a^{2} b^{10} x^{10} + 5 \, a^{3} b^{9} x^{8} + 10 \, a^{4} b^{8} x^{6} + 10 \, a^{5} b^{7} x^{4} + 5 \, a^{6} b^{6} x^{2} + a^{7} b^{5}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.2115, size = 194, normalized size = 1.59 \begin{align*} - \frac{7 \sqrt{- \frac{1}{a^{3} b^{9}}} \log{\left (- a^{2} b^{4} \sqrt{- \frac{1}{a^{3} b^{9}}} + x \right )}}{512} + \frac{7 \sqrt{- \frac{1}{a^{3} b^{9}}} \log{\left (a^{2} b^{4} \sqrt{- \frac{1}{a^{3} b^{9}}} + x \right )}}{512} + \frac{- 105 a^{4} x - 490 a^{3} b x^{3} - 896 a^{2} b^{2} x^{5} - 790 a b^{3} x^{7} + 105 b^{4} x^{9}}{3840 a^{6} b^{4} + 19200 a^{5} b^{5} x^{2} + 38400 a^{4} b^{6} x^{4} + 38400 a^{3} b^{7} x^{6} + 19200 a^{2} b^{8} x^{8} + 3840 a b^{9} x^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13508, size = 113, normalized size = 0.93 \begin{align*} \frac{7 \, \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{256 \, \sqrt{a b} a b^{4}} + \frac{105 \, b^{4} x^{9} - 790 \, a b^{3} x^{7} - 896 \, a^{2} b^{2} x^{5} - 490 \, a^{3} b x^{3} - 105 \, a^{4} x}{3840 \,{\left (b x^{2} + a\right )}^{5} a b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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