Optimal. Leaf size=54 \[ \frac{a^2}{b^3 \sqrt{a+\frac{b}{x^2}}}+\frac{2 a \sqrt{a+\frac{b}{x^2}}}{b^3}-\frac{\left (a+\frac{b}{x^2}\right )^{3/2}}{3 b^3} \]
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Rubi [A] time = 0.0301785, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{a^2}{b^3 \sqrt{a+\frac{b}{x^2}}}+\frac{2 a \sqrt{a+\frac{b}{x^2}}}{b^3}-\frac{\left (a+\frac{b}{x^2}\right )^{3/2}}{3 b^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x^2}\right )^{3/2} x^7} \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{(a+b x)^{3/2}} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^2}{b^2 (a+b x)^{3/2}}-\frac{2 a}{b^2 \sqrt{a+b x}}+\frac{\sqrt{a+b x}}{b^2}\right ) \, dx,x,\frac{1}{x^2}\right )\right )\\ &=\frac{a^2}{b^3 \sqrt{a+\frac{b}{x^2}}}+\frac{2 a \sqrt{a+\frac{b}{x^2}}}{b^3}-\frac{\left (a+\frac{b}{x^2}\right )^{3/2}}{3 b^3}\\ \end{align*}
Mathematica [A] time = 0.0095832, size = 42, normalized size = 0.78 \[ \frac{8 a^2 x^4+4 a b x^2-b^2}{3 b^3 x^4 \sqrt{a+\frac{b}{x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 50, normalized size = 0.9 \begin{align*}{\frac{ \left ( a{x}^{2}+b \right ) \left ( 8\,{a}^{2}{x}^{4}+4\,ab{x}^{2}-{b}^{2} \right ) }{3\,{b}^{3}{x}^{6}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05125, size = 62, normalized size = 1.15 \begin{align*} -\frac{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}}}{3 \, b^{3}} + \frac{2 \, \sqrt{a + \frac{b}{x^{2}}} a}{b^{3}} + \frac{a^{2}}{\sqrt{a + \frac{b}{x^{2}}} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54787, size = 107, normalized size = 1.98 \begin{align*} \frac{{\left (8 \, a^{2} x^{4} + 4 \, a b x^{2} - b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{3 \,{\left (a b^{3} x^{4} + b^{4} x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.84307, size = 423, normalized size = 7.83 \begin{align*} \frac{8 a^{\frac{9}{2}} b^{\frac{7}{2}} x^{6} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a^{\frac{7}{2}} b^{6} x^{7} + 6 a^{\frac{5}{2}} b^{7} x^{5} + 3 a^{\frac{3}{2}} b^{8} x^{3}} + \frac{12 a^{\frac{7}{2}} b^{\frac{9}{2}} x^{4} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a^{\frac{7}{2}} b^{6} x^{7} + 6 a^{\frac{5}{2}} b^{7} x^{5} + 3 a^{\frac{3}{2}} b^{8} x^{3}} + \frac{3 a^{\frac{5}{2}} b^{\frac{11}{2}} x^{2} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a^{\frac{7}{2}} b^{6} x^{7} + 6 a^{\frac{5}{2}} b^{7} x^{5} + 3 a^{\frac{3}{2}} b^{8} x^{3}} - \frac{a^{\frac{3}{2}} b^{\frac{13}{2}} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a^{\frac{7}{2}} b^{6} x^{7} + 6 a^{\frac{5}{2}} b^{7} x^{5} + 3 a^{\frac{3}{2}} b^{8} x^{3}} - \frac{8 a^{5} b^{3} x^{7}}{3 a^{\frac{7}{2}} b^{6} x^{7} + 6 a^{\frac{5}{2}} b^{7} x^{5} + 3 a^{\frac{3}{2}} b^{8} x^{3}} - \frac{16 a^{4} b^{4} x^{5}}{3 a^{\frac{7}{2}} b^{6} x^{7} + 6 a^{\frac{5}{2}} b^{7} x^{5} + 3 a^{\frac{3}{2}} b^{8} x^{3}} - \frac{8 a^{3} b^{5} x^{3}}{3 a^{\frac{7}{2}} b^{6} x^{7} + 6 a^{\frac{5}{2}} b^{7} x^{5} + 3 a^{\frac{3}{2}} b^{8} x^{3}} \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*} \int \frac{1}{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} x^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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