Optimal. Leaf size=76 \[ \frac{5}{8 a^2 x \left (a+b x^2\right )}-\frac{15 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{7/2}}-\frac{15}{8 a^3 x}+\frac{1}{4 a x \left (a+b x^2\right )^2} \]
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Rubi [A] time = 0.0258144, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {290, 325, 205} \[ \frac{5}{8 a^2 x \left (a+b x^2\right )}-\frac{15 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{7/2}}-\frac{15}{8 a^3 x}+\frac{1}{4 a x \left (a+b x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 290
Rule 325
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (a+b x^2\right )^3} \, dx &=\frac{1}{4 a x \left (a+b x^2\right )^2}+\frac{5 \int \frac{1}{x^2 \left (a+b x^2\right )^2} \, dx}{4 a}\\ &=\frac{1}{4 a x \left (a+b x^2\right )^2}+\frac{5}{8 a^2 x \left (a+b x^2\right )}+\frac{15 \int \frac{1}{x^2 \left (a+b x^2\right )} \, dx}{8 a^2}\\ &=-\frac{15}{8 a^3 x}+\frac{1}{4 a x \left (a+b x^2\right )^2}+\frac{5}{8 a^2 x \left (a+b x^2\right )}-\frac{(15 b) \int \frac{1}{a+b x^2} \, dx}{8 a^3}\\ &=-\frac{15}{8 a^3 x}+\frac{1}{4 a x \left (a+b x^2\right )^2}+\frac{5}{8 a^2 x \left (a+b x^2\right )}-\frac{15 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0387667, size = 68, normalized size = 0.89 \[ -\frac{8 a^2+25 a b x^2+15 b^2 x^4}{8 a^3 x \left (a+b x^2\right )^2}-\frac{15 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 66, normalized size = 0.9 \begin{align*} -{\frac{1}{{a}^{3}x}}-{\frac{7\,{b}^{2}{x}^{3}}{8\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{9\,bx}{8\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{15\,b}{8\,{a}^{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 = 1.32791, size = 428, normalized size = 5.63 \begin{align*} \left [-\frac{30 \, b^{2} x^{4} + 50 \, a b x^{2} - 15 \,{\left (b^{2} x^{5} + 2 \, a b x^{3} + a^{2} x\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) + 16 \, a^{2}}{16 \,{\left (a^{3} b^{2} x^{5} + 2 \, a^{4} b x^{3} + a^{5} x\right )}}, -\frac{15 \, b^{2} x^{4} + 25 \, a b x^{2} + 15 \,{\left (b^{2} x^{5} + 2 \, a b x^{3} + a^{2} x\right )} \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right ) + 8 \, a^{2}}{8 \,{\left (a^{3} b^{2} x^{5} + 2 \, a^{4} b x^{3} + a^{5} x\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.636164, size = 114, normalized size = 1.5 \begin{align*} \frac{15 \sqrt{- \frac{b}{a^{7}}} \log{\left (- \frac{a^{4} \sqrt{- \frac{b}{a^{7}}}}{b} + x \right )}}{16} - \frac{15 \sqrt{- \frac{b}{a^{7}}} \log{\left (\frac{a^{4} \sqrt{- \frac{b}{a^{7}}}}{b} + x \right )}}{16} - \frac{8 a^{2} + 25 a b x^{2} + 15 b^{2} x^{4}}{8 a^{5} x + 16 a^{4} b x^{3} + 8 a^{3} b^{2} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.10063, size = 77, normalized size = 1.01 \begin{align*} -\frac{15 \, b \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{3}} - \frac{7 \, b^{2} x^{3} + 9 \, a b x}{8 \,{\left (b x^{2} + a\right )}^{2} a^{3}} - \frac{1}{a^{3} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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