Optimal. Leaf size=107 \[ \frac{a^5}{2 b^6 \left (a+b \sqrt{x}\right )^4}-\frac{10 a^4}{3 b^6 \left (a+b \sqrt{x}\right )^3}+\frac{10 a^3}{b^6 \left (a+b \sqrt{x}\right )^2}-\frac{20 a^2}{b^6 \left (a+b \sqrt{x}\right )}-\frac{10 a \log \left (a+b \sqrt{x}\right )}{b^6}+\frac{2 \sqrt{x}}{b^5} \]
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Rubi [A] time = 0.06793, antiderivative size = 107, 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} \[ \frac{a^5}{2 b^6 \left (a+b \sqrt{x}\right )^4}-\frac{10 a^4}{3 b^6 \left (a+b \sqrt{x}\right )^3}+\frac{10 a^3}{b^6 \left (a+b \sqrt{x}\right )^2}-\frac{20 a^2}{b^6 \left (a+b \sqrt{x}\right )}-\frac{10 a \log \left (a+b \sqrt{x}\right )}{b^6}+\frac{2 \sqrt{x}}{b^5} \]
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 )^5} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^5}{(a+b x)^5} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{1}{b^5}-\frac{a^5}{b^5 (a+b x)^5}+\frac{5 a^4}{b^5 (a+b x)^4}-\frac{10 a^3}{b^5 (a+b x)^3}+\frac{10 a^2}{b^5 (a+b x)^2}-\frac{5 a}{b^5 (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{a^5}{2 b^6 \left (a+b \sqrt{x}\right )^4}-\frac{10 a^4}{3 b^6 \left (a+b \sqrt{x}\right )^3}+\frac{10 a^3}{b^6 \left (a+b \sqrt{x}\right )^2}-\frac{20 a^2}{b^6 \left (a+b \sqrt{x}\right )}+\frac{2 \sqrt{x}}{b^5}-\frac{10 a \log \left (a+b \sqrt{x}\right )}{b^6}\\ \end{align*}
Mathematica [A] time = 0.069917, size = 100, normalized size = 0.93 \[ -\frac{48 a^2 b^3 x^{3/2}+252 a^3 b^2 x+248 a^4 b \sqrt{x}+77 a^5-48 a b^4 x^2+60 a \left (a+b \sqrt{x}\right )^4 \log \left (a+b \sqrt{x}\right )-12 b^5 x^{5/2}}{6 b^6 \left (a+b \sqrt{x}\right )^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 92, normalized size = 0.9 \begin{align*} -10\,{\frac{a\ln \left ( a+b\sqrt{x} \right ) }{{b}^{6}}}+2\,{\frac{\sqrt{x}}{{b}^{5}}}+{\frac{{a}^{5}}{2\,{b}^{6}} \left ( a+b\sqrt{x} \right ) ^{-4}}-{\frac{10\,{a}^{4}}{3\,{b}^{6}} \left ( a+b\sqrt{x} \right ) ^{-3}}+10\,{\frac{{a}^{3}}{{b}^{6} \left ( a+b\sqrt{x} \right ) ^{2}}}-20\,{\frac{{a}^{2}}{{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 = 0.988579, size = 128, normalized size = 1.2 \begin{align*} -\frac{10 \, a \log \left (b \sqrt{x} + a\right )}{b^{6}} + \frac{2 \,{\left (b \sqrt{x} + a\right )}}{b^{6}} - \frac{20 \, a^{2}}{{\left (b \sqrt{x} + a\right )} b^{6}} + \frac{10 \, a^{3}}{{\left (b \sqrt{x} + a\right )}^{2} b^{6}} - \frac{10 \, a^{4}}{3 \,{\left (b \sqrt{x} + a\right )}^{3} b^{6}} + \frac{a^{5}}{2 \,{\left (b \sqrt{x} + a\right )}^{4} b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.31901, size = 414, normalized size = 3.87 \begin{align*} \frac{180 \, a^{3} b^{6} x^{3} - 357 \, a^{5} b^{4} x^{2} + 278 \, a^{7} b^{2} x - 77 \, a^{9} - 60 \,{\left (a b^{8} x^{4} - 4 \, a^{3} b^{6} x^{3} + 6 \, a^{5} b^{4} x^{2} - 4 \, a^{7} b^{2} x + a^{9}\right )} \log \left (b \sqrt{x} + a\right ) + 4 \,{\left (3 \, b^{9} x^{4} - 42 \, a^{2} b^{7} x^{3} + 73 \, a^{4} b^{5} x^{2} - 55 \, a^{6} b^{3} x + 15 \, a^{8} b\right )} \sqrt{x}}{6 \,{\left (b^{14} x^{4} - 4 \, a^{2} b^{12} x^{3} + 6 \, a^{4} b^{10} x^{2} - 4 \, a^{6} b^{8} x + a^{8} b^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.77458, size = 685, normalized size = 6.4 \begin{align*} \begin{cases} - \frac{60 a^{5} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt{x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac{3}{2}} + 6 b^{10} x^{2}} - \frac{15 a^{5}}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt{x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac{3}{2}} + 6 b^{10} x^{2}} - \frac{240 a^{4} b \sqrt{x} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt{x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac{3}{2}} + 6 b^{10} x^{2}} - \frac{360 a^{3} b^{2} x \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt{x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac{3}{2}} + 6 b^{10} x^{2}} + \frac{120 a^{3} b^{2} x}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt{x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac{3}{2}} + 6 b^{10} x^{2}} - \frac{240 a^{2} b^{3} x^{\frac{3}{2}} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt{x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac{3}{2}} + 6 b^{10} x^{2}} + \frac{200 a^{2} b^{3} x^{\frac{3}{2}}}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt{x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac{3}{2}} + 6 b^{10} x^{2}} - \frac{60 a b^{4} x^{2} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt{x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac{3}{2}} + 6 b^{10} x^{2}} + \frac{110 a b^{4} x^{2}}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt{x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac{3}{2}} + 6 b^{10} x^{2}} + \frac{12 b^{5} x^{\frac{5}{2}}}{6 a^{4} b^{6} + 24 a^{3} b^{7} \sqrt{x} + 36 a^{2} b^{8} x + 24 a b^{9} x^{\frac{3}{2}} + 6 b^{10} x^{2}} & \text{for}\: b \neq 0 \\\frac{x^{3}}{3 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 = 1.10234, size = 99, normalized size = 0.93 \begin{align*} -\frac{10 \, a \log \left ({\left | b \sqrt{x} + a \right |}\right )}{b^{6}} + \frac{2 \, \sqrt{x}}{b^{5}} - \frac{120 \, a^{2} b^{3} x^{\frac{3}{2}} + 300 \, a^{3} b^{2} x + 260 \, a^{4} b \sqrt{x} + 77 \, a^{5}}{6 \,{\left (b \sqrt{x} + a\right )}^{4} b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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