Optimal. Leaf size=95 \[ \frac{2 b^4}{a^5 \left (a+b \sqrt{x}\right )}+\frac{8 b^3}{a^5 \sqrt{x}}-\frac{3 b^2}{a^4 x}-\frac{10 b^4 \log \left (a+b \sqrt{x}\right )}{a^6}+\frac{5 b^4 \log (x)}{a^6}+\frac{4 b}{3 a^3 x^{3/2}}-\frac{1}{2 a^2 x^2} \]
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Rubi [A] time = 0.0611979, antiderivative size = 95, 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} \[ \frac{2 b^4}{a^5 \left (a+b \sqrt{x}\right )}+\frac{8 b^3}{a^5 \sqrt{x}}-\frac{3 b^2}{a^4 x}-\frac{10 b^4 \log \left (a+b \sqrt{x}\right )}{a^6}+\frac{5 b^4 \log (x)}{a^6}+\frac{4 b}{3 a^3 x^{3/2}}-\frac{1}{2 a^2 x^2} \]
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 )^2 x^3} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{x^5 (a+b x)^2} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{1}{a^2 x^5}-\frac{2 b}{a^3 x^4}+\frac{3 b^2}{a^4 x^3}-\frac{4 b^3}{a^5 x^2}+\frac{5 b^4}{a^6 x}-\frac{b^5}{a^5 (a+b x)^2}-\frac{5 b^5}{a^6 (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{2 b^4}{a^5 \left (a+b \sqrt{x}\right )}-\frac{1}{2 a^2 x^2}+\frac{4 b}{3 a^3 x^{3/2}}-\frac{3 b^2}{a^4 x}+\frac{8 b^3}{a^5 \sqrt{x}}-\frac{10 b^4 \log \left (a+b \sqrt{x}\right )}{a^6}+\frac{5 b^4 \log (x)}{a^6}\\ \end{align*}
Mathematica [A] time = 0.0925992, size = 91, normalized size = 0.96 \[ \frac{\frac{a \left (-10 a^2 b^2 x+5 a^3 b \sqrt{x}-3 a^4+30 a b^3 x^{3/2}+60 b^4 x^2\right )}{x^2 \left (a+b \sqrt{x}\right )}-60 b^4 \log \left (a+b \sqrt{x}\right )+30 b^4 \log (x)}{6 a^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 84, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,{a}^{2}{x}^{2}}}+{\frac{4\,b}{3\,{a}^{3}}{x}^{-{\frac{3}{2}}}}-3\,{\frac{{b}^{2}}{{a}^{4}x}}+5\,{\frac{{b}^{4}\ln \left ( x \right ) }{{a}^{6}}}-10\,{\frac{{b}^{4}\ln \left ( a+b\sqrt{x} \right ) }{{a}^{6}}}+8\,{\frac{{b}^{3}}{{a}^{5}\sqrt{x}}}+2\,{\frac{{b}^{4}}{{a}^{5} \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 = 0.970946, size = 119, normalized size = 1.25 \begin{align*} \frac{60 \, b^{4} x^{2} + 30 \, a b^{3} x^{\frac{3}{2}} - 10 \, a^{2} b^{2} x + 5 \, a^{3} b \sqrt{x} - 3 \, a^{4}}{6 \,{\left (a^{5} b x^{\frac{5}{2}} + a^{6} x^{2}\right )}} - \frac{10 \, b^{4} \log \left (b \sqrt{x} + a\right )}{a^{6}} + \frac{5 \, b^{4} \log \left (x\right )}{a^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.30237, size = 286, normalized size = 3.01 \begin{align*} -\frac{30 \, a^{2} b^{4} x^{2} - 15 \, a^{4} b^{2} x - 3 \, a^{6} + 60 \,{\left (b^{6} x^{3} - a^{2} b^{4} x^{2}\right )} \log \left (b \sqrt{x} + a\right ) - 60 \,{\left (b^{6} x^{3} - a^{2} b^{4} x^{2}\right )} \log \left (\sqrt{x}\right ) - 4 \,{\left (15 \, a b^{5} x^{2} - 10 \, a^{3} b^{3} x - 2 \, a^{5} b\right )} \sqrt{x}}{6 \,{\left (a^{6} b^{2} x^{3} - a^{8} x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.78722, size = 333, normalized size = 3.51 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{3}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{1}{3 b^{2} x^{3}} & \text{for}\: a = 0 \\- \frac{1}{2 a^{2} x^{2}} & \text{for}\: b = 0 \\- \frac{3 a^{5} \sqrt{x}}{6 a^{7} x^{\frac{5}{2}} + 6 a^{6} b x^{3}} + \frac{5 a^{4} b x}{6 a^{7} x^{\frac{5}{2}} + 6 a^{6} b x^{3}} - \frac{10 a^{3} b^{2} x^{\frac{3}{2}}}{6 a^{7} x^{\frac{5}{2}} + 6 a^{6} b x^{3}} + \frac{30 a^{2} b^{3} x^{2}}{6 a^{7} x^{\frac{5}{2}} + 6 a^{6} b x^{3}} + \frac{30 a b^{4} x^{\frac{5}{2}} \log{\left (x \right )}}{6 a^{7} x^{\frac{5}{2}} + 6 a^{6} b x^{3}} - \frac{60 a b^{4} x^{\frac{5}{2}} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{6 a^{7} x^{\frac{5}{2}} + 6 a^{6} b x^{3}} + \frac{60 a b^{4} x^{\frac{5}{2}}}{6 a^{7} x^{\frac{5}{2}} + 6 a^{6} b x^{3}} + \frac{30 b^{5} x^{3} \log{\left (x \right )}}{6 a^{7} x^{\frac{5}{2}} + 6 a^{6} b x^{3}} - \frac{60 b^{5} x^{3} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{6 a^{7} x^{\frac{5}{2}} + 6 a^{6} b x^{3}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11077, size = 122, normalized size = 1.28 \begin{align*} -\frac{10 \, b^{4} \log \left ({\left | b \sqrt{x} + a \right |}\right )}{a^{6}} + \frac{5 \, b^{4} \log \left ({\left | x \right |}\right )}{a^{6}} + \frac{60 \, a b^{4} x^{2} + 30 \, a^{2} b^{3} x^{\frac{3}{2}} - 10 \, a^{3} b^{2} x + 5 \, a^{4} b \sqrt{x} - 3 \, a^{5}}{6 \,{\left (b \sqrt{x} + a\right )} a^{6} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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