Optimal. Leaf size=67 \[ \frac{2 b^2}{a^3 \left (a+b \sqrt{x}\right )}-\frac{6 b^2 \log \left (a+b \sqrt{x}\right )}{a^4}+\frac{3 b^2 \log (x)}{a^4}+\frac{4 b}{a^3 \sqrt{x}}-\frac{1}{a^2 x} \]
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Rubi [A] time = 0.0416603, antiderivative size = 67, 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, 44} \[ \frac{2 b^2}{a^3 \left (a+b \sqrt{x}\right )}-\frac{6 b^2 \log \left (a+b \sqrt{x}\right )}{a^4}+\frac{3 b^2 \log (x)}{a^4}+\frac{4 b}{a^3 \sqrt{x}}-\frac{1}{a^2 x} \]
Antiderivative was successfully verified.
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Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b \sqrt{x}\right )^2 x^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{x^3 (a+b x)^2} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{1}{a^2 x^3}-\frac{2 b}{a^3 x^2}+\frac{3 b^2}{a^4 x}-\frac{b^3}{a^3 (a+b x)^2}-\frac{3 b^3}{a^4 (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{2 b^2}{a^3 \left (a+b \sqrt{x}\right )}-\frac{1}{a^2 x}+\frac{4 b}{a^3 \sqrt{x}}-\frac{6 b^2 \log \left (a+b \sqrt{x}\right )}{a^4}+\frac{3 b^2 \log (x)}{a^4}\\ \end{align*}
Mathematica [A] time = 0.0732362, size = 60, normalized size = 0.9 \[ \frac{a \left (\frac{2 b^2}{a+b \sqrt{x}}-\frac{a}{x}+\frac{4 b}{\sqrt{x}}\right )-6 b^2 \log \left (a+b \sqrt{x}\right )+3 b^2 \log (x)}{a^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 62, normalized size = 0.9 \begin{align*} -{\frac{1}{{a}^{2}x}}+3\,{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{4}}}-6\,{\frac{{b}^{2}\ln \left ( a+b\sqrt{x} \right ) }{{a}^{4}}}+4\,{\frac{b}{{a}^{3}\sqrt{x}}}+2\,{\frac{{b}^{2}}{{a}^{3} \left ( a+b\sqrt{x} \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.97184, size = 85, normalized size = 1.27 \begin{align*} \frac{6 \, b^{2} x + 3 \, a b \sqrt{x} - a^{2}}{a^{3} b x^{\frac{3}{2}} + a^{4} x} - \frac{6 \, b^{2} \log \left (b \sqrt{x} + a\right )}{a^{4}} + \frac{3 \, b^{2} \log \left (x\right )}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.34294, size = 219, normalized size = 3.27 \begin{align*} -\frac{3 \, a^{2} b^{2} x - a^{4} + 6 \,{\left (b^{4} x^{2} - a^{2} b^{2} x\right )} \log \left (b \sqrt{x} + a\right ) - 6 \,{\left (b^{4} x^{2} - a^{2} b^{2} x\right )} \log \left (\sqrt{x}\right ) - 2 \,{\left (3 \, a b^{3} x - 2 \, a^{3} b\right )} \sqrt{x}}{a^{4} b^{2} x^{2} - a^{6} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.05155, size = 235, normalized size = 3.51 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{2}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{1}{2 b^{2} x^{2}} & \text{for}\: a = 0 \\- \frac{1}{a^{2} x} & \text{for}\: b = 0 \\- \frac{a^{3} \sqrt{x}}{a^{5} x^{\frac{3}{2}} + a^{4} b x^{2}} + \frac{3 a^{2} b x}{a^{5} x^{\frac{3}{2}} + a^{4} b x^{2}} + \frac{3 a b^{2} x^{\frac{3}{2}} \log{\left (x \right )}}{a^{5} x^{\frac{3}{2}} + a^{4} b x^{2}} - \frac{6 a b^{2} x^{\frac{3}{2}} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{5} x^{\frac{3}{2}} + a^{4} b x^{2}} + \frac{3 b^{3} x^{2} \log{\left (x \right )}}{a^{5} x^{\frac{3}{2}} + a^{4} b x^{2}} - \frac{6 b^{3} x^{2} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{5} x^{\frac{3}{2}} + a^{4} b x^{2}} - \frac{6 b^{3} x^{2}}{a^{5} x^{\frac{3}{2}} + a^{4} b x^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08536, size = 90, normalized size = 1.34 \begin{align*} -\frac{6 \, b^{2} \log \left ({\left | b \sqrt{x} + a \right |}\right )}{a^{4}} + \frac{3 \, b^{2} \log \left ({\left | x \right |}\right )}{a^{4}} + \frac{6 \, a b^{2} x + 3 \, a^{2} b \sqrt{x} - a^{3}}{{\left (b \sqrt{x} + a\right )} a^{4} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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