Optimal. Leaf size=169 \[ -\frac{15 \left (a+b x^2\right )}{8 a^3 x \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5}{8 a^2 x \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{4 a x \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )}-\frac{15 \sqrt{b} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.0653457, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {1112, 290, 325, 205} \[ -\frac{15 \left (a+b x^2\right )}{8 a^3 x \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5}{8 a^2 x \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{4 a x \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )}-\frac{15 \sqrt{b} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 1112
Rule 290
Rule 325
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx &=\frac{\left (b^2 \left (a b+b^2 x^2\right )\right ) \int \frac{1}{x^2 \left (a b+b^2 x^2\right )^3} \, dx}{\sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{1}{4 a x \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (5 b \left (a b+b^2 x^2\right )\right ) \int \frac{1}{x^2 \left (a b+b^2 x^2\right )^2} \, dx}{4 a \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{5}{8 a^2 x \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{4 a x \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (15 \left (a b+b^2 x^2\right )\right ) \int \frac{1}{x^2 \left (a b+b^2 x^2\right )} \, dx}{8 a^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{5}{8 a^2 x \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{4 a x \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{15 \left (a+b x^2\right )}{8 a^3 x \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (15 b \left (a b+b^2 x^2\right )\right ) \int \frac{1}{a b+b^2 x^2} \, dx}{8 a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{5}{8 a^2 x \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{4 a x \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{15 \left (a+b x^2\right )}{8 a^3 x \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{15 \sqrt{b} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ \end{align*}
Mathematica [A] time = 0.0322544, size = 93, normalized size = 0.55 \[ \frac{-\sqrt{a} \left (8 a^2+25 a b x^2+15 b^2 x^4\right )-15 \sqrt{b} x \left (a+b x^2\right )^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{7/2} x \left (a+b x^2\right ) \sqrt{\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.239, size = 119, normalized size = 0.7 \begin{align*} -{\frac{b{x}^{2}+a}{8\,x{a}^{3}} \left ( 15\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{5}{b}^{3}+15\,\sqrt{ab}{x}^{4}{b}^{2}+30\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{3}a{b}^{2}+25\,\sqrt{ab}{x}^{2}ab+15\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) x{a}^{2}b+8\,\sqrt{ab}{a}^{2} \right ){\frac{1}{\sqrt{ab}}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \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 = 1.32713, size = 428, normalized size = 2.53 \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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