Optimal. Leaf size=75 \[ \frac{2 b^3}{a^4 \sqrt{x}}-\frac{b^2}{a^3 x}-\frac{2 b^4 \log \left (a+b \sqrt{x}\right )}{a^5}+\frac{b^4 \log (x)}{a^5}+\frac{2 b}{3 a^2 x^{3/2}}-\frac{1}{2 a x^2} \]
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Rubi [A] time = 0.0383596, antiderivative size = 75, 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} \[ \frac{2 b^3}{a^4 \sqrt{x}}-\frac{b^2}{a^3 x}-\frac{2 b^4 \log \left (a+b \sqrt{x}\right )}{a^5}+\frac{b^4 \log (x)}{a^5}+\frac{2 b}{3 a^2 x^{3/2}}-\frac{1}{2 a x^2} \]
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 ) x^3} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{x^5 (a+b x)} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{1}{a x^5}-\frac{b}{a^2 x^4}+\frac{b^2}{a^3 x^3}-\frac{b^3}{a^4 x^2}+\frac{b^4}{a^5 x}-\frac{b^5}{a^5 (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{1}{2 a x^2}+\frac{2 b}{3 a^2 x^{3/2}}-\frac{b^2}{a^3 x}+\frac{2 b^3}{a^4 \sqrt{x}}-\frac{2 b^4 \log \left (a+b \sqrt{x}\right )}{a^5}+\frac{b^4 \log (x)}{a^5}\\ \end{align*}
Mathematica [A] time = 0.0433211, size = 69, normalized size = 0.92 \[ \frac{\frac{a \left (4 a^2 b \sqrt{x}-3 a^3-6 a b^2 x+12 b^3 x^{3/2}\right )}{x^2}-12 b^4 \log \left (a+b \sqrt{x}\right )+6 b^4 \log (x)}{6 a^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 66, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,a{x}^{2}}}+{\frac{2\,b}{3\,{a}^{2}}{x}^{-{\frac{3}{2}}}}-{\frac{{b}^{2}}{{a}^{3}x}}+{\frac{{b}^{4}\ln \left ( x \right ) }{{a}^{5}}}-2\,{\frac{{b}^{4}\ln \left ( a+b\sqrt{x} \right ) }{{a}^{5}}}+2\,{\frac{{b}^{3}}{{a}^{4}\sqrt{x}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.953942, size = 86, normalized size = 1.15 \begin{align*} -\frac{2 \, b^{4} \log \left (b \sqrt{x} + a\right )}{a^{5}} + \frac{b^{4} \log \left (x\right )}{a^{5}} + \frac{12 \, b^{3} x^{\frac{3}{2}} - 6 \, a b^{2} x + 4 \, a^{2} b \sqrt{x} - 3 \, a^{3}}{6 \, a^{4} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.33795, size = 173, normalized size = 2.31 \begin{align*} -\frac{12 \, b^{4} x^{2} \log \left (b \sqrt{x} + a\right ) - 12 \, b^{4} x^{2} \log \left (\sqrt{x}\right ) + 6 \, a^{2} b^{2} x + 3 \, a^{4} - 4 \,{\left (3 \, a b^{3} x + a^{3} b\right )} \sqrt{x}}{6 \, a^{5} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.0342, size = 99, normalized size = 1.32 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{5}{2}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2}{5 b x^{\frac{5}{2}}} & \text{for}\: a = 0 \\- \frac{1}{2 a x^{2}} & \text{for}\: b = 0 \\- \frac{1}{2 a x^{2}} + \frac{2 b}{3 a^{2} x^{\frac{3}{2}}} - \frac{b^{2}}{a^{3} x} + \frac{2 b^{3}}{a^{4} \sqrt{x}} + \frac{b^{4} \log{\left (x \right )}}{a^{5}} - \frac{2 b^{4} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{5}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11152, size = 93, normalized size = 1.24 \begin{align*} -\frac{2 \, b^{4} \log \left ({\left | b \sqrt{x} + a \right |}\right )}{a^{5}} + \frac{b^{4} \log \left ({\left | x \right |}\right )}{a^{5}} + \frac{12 \, a b^{3} x^{\frac{3}{2}} - 6 \, a^{2} b^{2} x + 4 \, a^{3} b \sqrt{x} - 3 \, a^{4}}{6 \, a^{5} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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