Optimal. Leaf size=53 \[ \frac{2}{a^2 \left (a+b \sqrt{x}\right )}-\frac{2 \log \left (a+b \sqrt{x}\right )}{a^3}+\frac{\log (x)}{a^3}+\frac{1}{a \left (a+b \sqrt{x}\right )^2} \]
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Rubi [A] time = 0.0307791, antiderivative size = 53, 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} \[ \frac{2}{a^2 \left (a+b \sqrt{x}\right )}-\frac{2 \log \left (a+b \sqrt{x}\right )}{a^3}+\frac{\log (x)}{a^3}+\frac{1}{a \left (a+b \sqrt{x}\right )^2} \]
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 )^3 x} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{x (a+b x)^3} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{1}{a^3 x}-\frac{b}{a (a+b x)^3}-\frac{b}{a^2 (a+b x)^2}-\frac{b}{a^3 (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{1}{a \left (a+b \sqrt{x}\right )^2}+\frac{2}{a^2 \left (a+b \sqrt{x}\right )}-\frac{2 \log \left (a+b \sqrt{x}\right )}{a^3}+\frac{\log (x)}{a^3}\\ \end{align*}
Mathematica [A] time = 0.0508998, size = 44, normalized size = 0.83 \[ \frac{\frac{a \left (3 a+2 b \sqrt{x}\right )}{\left (a+b \sqrt{x}\right )^2}-2 \log \left (a+b \sqrt{x}\right )+\log (x)}{a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 48, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( x \right ) }{{a}^{3}}}-2\,{\frac{\ln \left ( a+b\sqrt{x} \right ) }{{a}^{3}}}+{\frac{1}{a} \left ( a+b\sqrt{x} \right ) ^{-2}}+2\,{\frac{1}{{a}^{2} \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 = 0.984636, size = 73, normalized size = 1.38 \begin{align*} \frac{2 \, b \sqrt{x} + 3 \, a}{a^{2} b^{2} x + 2 \, a^{3} b \sqrt{x} + a^{4}} - \frac{2 \, \log \left (b \sqrt{x} + a\right )}{a^{3}} + \frac{\log \left (x\right )}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.35832, size = 254, normalized size = 4.79 \begin{align*} -\frac{a^{2} b^{2} x - 3 \, a^{4} + 2 \,{\left (b^{4} x^{2} - 2 \, a^{2} b^{2} x + a^{4}\right )} \log \left (b \sqrt{x} + a\right ) - 2 \,{\left (b^{4} x^{2} - 2 \, a^{2} b^{2} x + a^{4}\right )} \log \left (\sqrt{x}\right ) - 2 \,{\left (a b^{3} x - 2 \, a^{3} b\right )} \sqrt{x}}{a^{3} b^{4} x^{2} - 2 \, a^{5} b^{2} x + a^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.36452, size = 364, normalized size = 6.87 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{3}{2}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2}{3 b^{3} x^{\frac{3}{2}}} & \text{for}\: a = 0 \\\frac{\log{\left (x \right )}}{a^{3}} & \text{for}\: b = 0 \\\frac{a^{2} \sqrt{x} \log{\left (x \right )}}{a^{5} \sqrt{x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac{3}{2}}} - \frac{2 a^{2} \sqrt{x} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{5} \sqrt{x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac{3}{2}}} + \frac{2 a b x \log{\left (x \right )}}{a^{5} \sqrt{x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac{3}{2}}} - \frac{4 a b x \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{5} \sqrt{x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac{3}{2}}} - \frac{4 a b x}{a^{5} \sqrt{x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac{3}{2}}} + \frac{b^{2} x^{\frac{3}{2}} \log{\left (x \right )}}{a^{5} \sqrt{x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac{3}{2}}} - \frac{2 b^{2} x^{\frac{3}{2}} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{5} \sqrt{x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac{3}{2}}} - \frac{3 b^{2} x^{\frac{3}{2}}}{a^{5} \sqrt{x} + 2 a^{4} b x + a^{3} b^{2} x^{\frac{3}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10212, size = 65, normalized size = 1.23 \begin{align*} -\frac{2 \, \log \left ({\left | b \sqrt{x} + a \right |}\right )}{a^{3}} + \frac{\log \left ({\left | x \right |}\right )}{a^{3}} + \frac{2 \, a b \sqrt{x} + 3 \, a^{2}}{{\left (b \sqrt{x} + a\right )}^{2} a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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