Optimal. Leaf size=100 \[ -\frac{63 b^2}{8 a^5 x}-\frac{63 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{11/2}}+\frac{21 b}{8 a^4 x^3}+\frac{9}{8 a^2 x^5 \left (a+b x^2\right )}-\frac{63}{40 a^3 x^5}+\frac{1}{4 a x^5 \left (a+b x^2\right )^2} \]
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Rubi [A] time = 0.041861, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {290, 325, 205} \[ -\frac{63 b^2}{8 a^5 x}-\frac{63 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{11/2}}+\frac{21 b}{8 a^4 x^3}+\frac{9}{8 a^2 x^5 \left (a+b x^2\right )}-\frac{63}{40 a^3 x^5}+\frac{1}{4 a x^5 \left (a+b x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 290
Rule 325
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{x^6 \left (a+b x^2\right )^3} \, dx &=\frac{1}{4 a x^5 \left (a+b x^2\right )^2}+\frac{9 \int \frac{1}{x^6 \left (a+b x^2\right )^2} \, dx}{4 a}\\ &=\frac{1}{4 a x^5 \left (a+b x^2\right )^2}+\frac{9}{8 a^2 x^5 \left (a+b x^2\right )}+\frac{63 \int \frac{1}{x^6 \left (a+b x^2\right )} \, dx}{8 a^2}\\ &=-\frac{63}{40 a^3 x^5}+\frac{1}{4 a x^5 \left (a+b x^2\right )^2}+\frac{9}{8 a^2 x^5 \left (a+b x^2\right )}-\frac{(63 b) \int \frac{1}{x^4 \left (a+b x^2\right )} \, dx}{8 a^3}\\ &=-\frac{63}{40 a^3 x^5}+\frac{21 b}{8 a^4 x^3}+\frac{1}{4 a x^5 \left (a+b x^2\right )^2}+\frac{9}{8 a^2 x^5 \left (a+b x^2\right )}+\frac{\left (63 b^2\right ) \int \frac{1}{x^2 \left (a+b x^2\right )} \, dx}{8 a^4}\\ &=-\frac{63}{40 a^3 x^5}+\frac{21 b}{8 a^4 x^3}-\frac{63 b^2}{8 a^5 x}+\frac{1}{4 a x^5 \left (a+b x^2\right )^2}+\frac{9}{8 a^2 x^5 \left (a+b x^2\right )}-\frac{\left (63 b^3\right ) \int \frac{1}{a+b x^2} \, dx}{8 a^5}\\ &=-\frac{63}{40 a^3 x^5}+\frac{21 b}{8 a^4 x^3}-\frac{63 b^2}{8 a^5 x}+\frac{1}{4 a x^5 \left (a+b x^2\right )^2}+\frac{9}{8 a^2 x^5 \left (a+b x^2\right )}-\frac{63 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{11/2}}\\ \end{align*}
Mathematica [A] time = 0.050852, size = 90, normalized size = 0.9 \[ -\frac{168 a^2 b^2 x^4-24 a^3 b x^2+8 a^4+525 a b^3 x^6+315 b^4 x^8}{40 a^5 x^5 \left (a+b x^2\right )^2}-\frac{63 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{11/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 89, normalized size = 0.9 \begin{align*} -{\frac{1}{5\,{a}^{3}{x}^{5}}}-6\,{\frac{{b}^{2}}{{a}^{5}x}}+{\frac{b}{{a}^{4}{x}^{3}}}-{\frac{15\,{b}^{4}{x}^{3}}{8\,{a}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{17\,{b}^{3}x}{8\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{63\,{b}^{3}}{8\,{a}^{5}}\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 = 1.25866, size = 560, normalized size = 5.6 \begin{align*} \left [-\frac{630 \, b^{4} x^{8} + 1050 \, a b^{3} x^{6} + 336 \, a^{2} b^{2} x^{4} - 48 \, a^{3} b x^{2} + 16 \, a^{4} - 315 \,{\left (b^{4} x^{9} + 2 \, a b^{3} x^{7} + a^{2} b^{2} x^{5}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right )}{80 \,{\left (a^{5} b^{2} x^{9} + 2 \, a^{6} b x^{7} + a^{7} x^{5}\right )}}, -\frac{315 \, b^{4} x^{8} + 525 \, a b^{3} x^{6} + 168 \, a^{2} b^{2} x^{4} - 24 \, a^{3} b x^{2} + 8 \, a^{4} + 315 \,{\left (b^{4} x^{9} + 2 \, a b^{3} x^{7} + a^{2} b^{2} x^{5}\right )} \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right )}{40 \,{\left (a^{5} b^{2} x^{9} + 2 \, a^{6} b x^{7} + a^{7} x^{5}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.21165, size = 150, normalized size = 1.5 \begin{align*} \frac{63 \sqrt{- \frac{b^{5}}{a^{11}}} \log{\left (- \frac{a^{6} \sqrt{- \frac{b^{5}}{a^{11}}}}{b^{3}} + x \right )}}{16} - \frac{63 \sqrt{- \frac{b^{5}}{a^{11}}} \log{\left (\frac{a^{6} \sqrt{- \frac{b^{5}}{a^{11}}}}{b^{3}} + x \right )}}{16} - \frac{8 a^{4} - 24 a^{3} b x^{2} + 168 a^{2} b^{2} x^{4} + 525 a b^{3} x^{6} + 315 b^{4} x^{8}}{40 a^{7} x^{5} + 80 a^{6} b x^{7} + 40 a^{5} b^{2} x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.51769, size = 108, normalized size = 1.08 \begin{align*} -\frac{63 \, b^{3} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{5}} - \frac{15 \, b^{4} x^{3} + 17 \, a b^{3} x}{8 \,{\left (b x^{2} + a\right )}^{2} a^{5}} - \frac{30 \, b^{2} x^{4} - 5 \, a b x^{2} + a^{2}}{5 \, a^{5} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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