Optimal. Leaf size=83 \[ \frac{2 a^5}{b^6 \left (a+b \sqrt{x}\right )}-\frac{8 a^3 \sqrt{x}}{b^5}+\frac{3 a^2 x}{b^4}+\frac{10 a^4 \log \left (a+b \sqrt{x}\right )}{b^6}-\frac{4 a x^{3/2}}{3 b^3}+\frac{x^2}{2 b^2} \]
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Rubi [A] time = 0.0596308, antiderivative size = 83, 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} \[ \frac{2 a^5}{b^6 \left (a+b \sqrt{x}\right )}-\frac{8 a^3 \sqrt{x}}{b^5}+\frac{3 a^2 x}{b^4}+\frac{10 a^4 \log \left (a+b \sqrt{x}\right )}{b^6}-\frac{4 a x^{3/2}}{3 b^3}+\frac{x^2}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^2}{\left (a+b \sqrt{x}\right )^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^5}{(a+b x)^2} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{4 a^3}{b^5}+\frac{3 a^2 x}{b^4}-\frac{2 a x^2}{b^3}+\frac{x^3}{b^2}-\frac{a^5}{b^5 (a+b x)^2}+\frac{5 a^4}{b^5 (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{2 a^5}{b^6 \left (a+b \sqrt{x}\right )}-\frac{8 a^3 \sqrt{x}}{b^5}+\frac{3 a^2 x}{b^4}-\frac{4 a x^{3/2}}{3 b^3}+\frac{x^2}{2 b^2}+\frac{10 a^4 \log \left (a+b \sqrt{x}\right )}{b^6}\\ \end{align*}
Mathematica [A] time = 0.0569048, size = 78, normalized size = 0.94 \[ \frac{18 a^2 b^2 x+\frac{12 a^5}{a+b \sqrt{x}}-48 a^3 b \sqrt{x}+60 a^4 \log \left (a+b \sqrt{x}\right )-8 a b^3 x^{3/2}+3 b^4 x^2}{6 b^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 72, normalized size = 0.9 \begin{align*} 3\,{\frac{{a}^{2}x}{{b}^{4}}}-{\frac{4\,a}{3\,{b}^{3}}{x}^{{\frac{3}{2}}}}+{\frac{{x}^{2}}{2\,{b}^{2}}}+10\,{\frac{{a}^{4}\ln \left ( a+b\sqrt{x} \right ) }{{b}^{6}}}-8\,{\frac{{a}^{3}\sqrt{x}}{{b}^{5}}}+2\,{\frac{{a}^{5}}{{b}^{6} \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 = 0.971959, size = 128, normalized size = 1.54 \begin{align*} \frac{10 \, a^{4} \log \left (b \sqrt{x} + a\right )}{b^{6}} + \frac{{\left (b \sqrt{x} + a\right )}^{4}}{2 \, b^{6}} - \frac{10 \,{\left (b \sqrt{x} + a\right )}^{3} a}{3 \, b^{6}} + \frac{10 \,{\left (b \sqrt{x} + a\right )}^{2} a^{2}}{b^{6}} - \frac{20 \,{\left (b \sqrt{x} + a\right )} a^{3}}{b^{6}} + \frac{2 \, a^{5}}{{\left (b \sqrt{x} + a\right )} b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28098, size = 230, normalized size = 2.77 \begin{align*} \frac{3 \, b^{6} x^{3} + 15 \, a^{2} b^{4} x^{2} - 18 \, a^{4} b^{2} x - 12 \, a^{6} + 60 \,{\left (a^{4} b^{2} x - a^{6}\right )} \log \left (b \sqrt{x} + a\right ) - 4 \,{\left (2 \, a b^{5} x^{2} + 10 \, a^{3} b^{3} x - 15 \, a^{5} b\right )} \sqrt{x}}{6 \,{\left (b^{8} x - a^{2} b^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.24107, size = 212, normalized size = 2.55 \begin{align*} \begin{cases} \frac{60 a^{5} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{6 a b^{6} + 6 b^{7} \sqrt{x}} + \frac{60 a^{5}}{6 a b^{6} + 6 b^{7} \sqrt{x}} + \frac{60 a^{4} b \sqrt{x} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{6 a b^{6} + 6 b^{7} \sqrt{x}} - \frac{30 a^{3} b^{2} x}{6 a b^{6} + 6 b^{7} \sqrt{x}} + \frac{10 a^{2} b^{3} x^{\frac{3}{2}}}{6 a b^{6} + 6 b^{7} \sqrt{x}} - \frac{5 a b^{4} x^{2}}{6 a b^{6} + 6 b^{7} \sqrt{x}} + \frac{3 b^{5} x^{\frac{5}{2}}}{6 a b^{6} + 6 b^{7} \sqrt{x}} & \text{for}\: b \neq 0 \\\frac{x^{3}}{3 a^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11074, size = 105, normalized size = 1.27 \begin{align*} \frac{10 \, a^{4} \log \left ({\left | b \sqrt{x} + a \right |}\right )}{b^{6}} + \frac{2 \, a^{5}}{{\left (b \sqrt{x} + a\right )} b^{6}} + \frac{3 \, b^{6} x^{2} - 8 \, a b^{5} x^{\frac{3}{2}} + 18 \, a^{2} b^{4} x - 48 \, a^{3} b^{3} \sqrt{x}}{6 \, b^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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