Optimal. Leaf size=122 \[ -\frac{15 a^8 b^2}{x^3}-\frac{48 a^7 b^3}{x^{5/2}}-\frac{105 a^6 b^4}{x^2}-\frac{168 a^5 b^5}{x^{3/2}}-\frac{210 a^4 b^6}{x}-\frac{240 a^3 b^7}{\sqrt{x}}+45 a^2 b^8 \log (x)-\frac{20 a^9 b}{7 x^{7/2}}-\frac{a^{10}}{4 x^4}+20 a b^9 \sqrt{x}+b^{10} x \]
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Rubi [A] time = 0.0674332, antiderivative size = 122, 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} \[ -\frac{15 a^8 b^2}{x^3}-\frac{48 a^7 b^3}{x^{5/2}}-\frac{105 a^6 b^4}{x^2}-\frac{168 a^5 b^5}{x^{3/2}}-\frac{210 a^4 b^6}{x}-\frac{240 a^3 b^7}{\sqrt{x}}+45 a^2 b^8 \log (x)-\frac{20 a^9 b}{7 x^{7/2}}-\frac{a^{10}}{4 x^4}+20 a b^9 \sqrt{x}+b^{10} x \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+b \sqrt{x}\right )^{10}}{x^5} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^{10}}{x^9} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (10 a b^9+\frac{a^{10}}{x^9}+\frac{10 a^9 b}{x^8}+\frac{45 a^8 b^2}{x^7}+\frac{120 a^7 b^3}{x^6}+\frac{210 a^6 b^4}{x^5}+\frac{252 a^5 b^5}{x^4}+\frac{210 a^4 b^6}{x^3}+\frac{120 a^3 b^7}{x^2}+\frac{45 a^2 b^8}{x}+b^{10} x\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{a^{10}}{4 x^4}-\frac{20 a^9 b}{7 x^{7/2}}-\frac{15 a^8 b^2}{x^3}-\frac{48 a^7 b^3}{x^{5/2}}-\frac{105 a^6 b^4}{x^2}-\frac{168 a^5 b^5}{x^{3/2}}-\frac{210 a^4 b^6}{x}-\frac{240 a^3 b^7}{\sqrt{x}}+20 a b^9 \sqrt{x}+b^{10} x+45 a^2 b^8 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0761258, size = 122, normalized size = 1. \[ -\frac{15 a^8 b^2}{x^3}-\frac{48 a^7 b^3}{x^{5/2}}-\frac{105 a^6 b^4}{x^2}-\frac{168 a^5 b^5}{x^{3/2}}-\frac{210 a^4 b^6}{x}-\frac{240 a^3 b^7}{\sqrt{x}}+45 a^2 b^8 \log (x)-\frac{20 a^9 b}{7 x^{7/2}}-\frac{a^{10}}{4 x^4}+20 a b^9 \sqrt{x}+b^{10} x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 109, normalized size = 0.9 \begin{align*} -{\frac{{a}^{10}}{4\,{x}^{4}}}-{\frac{20\,{a}^{9}b}{7}{x}^{-{\frac{7}{2}}}}-15\,{\frac{{a}^{8}{b}^{2}}{{x}^{3}}}-48\,{\frac{{a}^{7}{b}^{3}}{{x}^{5/2}}}-105\,{\frac{{a}^{6}{b}^{4}}{{x}^{2}}}-168\,{\frac{{a}^{5}{b}^{5}}{{x}^{3/2}}}-210\,{\frac{{a}^{4}{b}^{6}}{x}}+{b}^{10}x+45\,{a}^{2}{b}^{8}\ln \left ( x \right ) -240\,{\frac{{a}^{3}{b}^{7}}{\sqrt{x}}}+20\,a{b}^{9}\sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.988544, size = 147, normalized size = 1.2 \begin{align*} b^{10} x + 45 \, a^{2} b^{8} \log \left (x\right ) + 20 \, a b^{9} \sqrt{x} - \frac{6720 \, a^{3} b^{7} x^{\frac{7}{2}} + 5880 \, a^{4} b^{6} x^{3} + 4704 \, a^{5} b^{5} x^{\frac{5}{2}} + 2940 \, a^{6} b^{4} x^{2} + 1344 \, a^{7} b^{3} x^{\frac{3}{2}} + 420 \, a^{8} b^{2} x + 80 \, a^{9} b \sqrt{x} + 7 \, a^{10}}{28 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.41334, size = 281, normalized size = 2.3 \begin{align*} \frac{28 \, b^{10} x^{5} + 2520 \, a^{2} b^{8} x^{4} \log \left (\sqrt{x}\right ) - 5880 \, a^{4} b^{6} x^{3} - 2940 \, a^{6} b^{4} x^{2} - 420 \, a^{8} b^{2} x - 7 \, a^{10} + 16 \,{\left (35 \, a b^{9} x^{4} - 420 \, a^{3} b^{7} x^{3} - 294 \, a^{5} b^{5} x^{2} - 84 \, a^{7} b^{3} x - 5 \, a^{9} b\right )} \sqrt{x}}{28 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.69892, size = 124, normalized size = 1.02 \begin{align*} - \frac{a^{10}}{4 x^{4}} - \frac{20 a^{9} b}{7 x^{\frac{7}{2}}} - \frac{15 a^{8} b^{2}}{x^{3}} - \frac{48 a^{7} b^{3}}{x^{\frac{5}{2}}} - \frac{105 a^{6} b^{4}}{x^{2}} - \frac{168 a^{5} b^{5}}{x^{\frac{3}{2}}} - \frac{210 a^{4} b^{6}}{x} - \frac{240 a^{3} b^{7}}{\sqrt{x}} + 45 a^{2} b^{8} \log{\left (x \right )} + 20 a b^{9} \sqrt{x} + b^{10} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12026, size = 149, normalized size = 1.22 \begin{align*} b^{10} x + 45 \, a^{2} b^{8} \log \left ({\left | x \right |}\right ) + 20 \, a b^{9} \sqrt{x} - \frac{6720 \, a^{3} b^{7} x^{\frac{7}{2}} + 5880 \, a^{4} b^{6} x^{3} + 4704 \, a^{5} b^{5} x^{\frac{5}{2}} + 2940 \, a^{6} b^{4} x^{2} + 1344 \, a^{7} b^{3} x^{\frac{3}{2}} + 420 \, a^{8} b^{2} x + 80 \, a^{9} b \sqrt{x} + 7 \, a^{10}}{28 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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