Optimal. Leaf size=128 \[ 80 a^7 b^3 x^{3/2}+105 a^6 b^4 x^2+\frac{504}{5} a^5 b^5 x^{5/2}+70 a^4 b^6 x^3+\frac{240}{7} a^3 b^7 x^{7/2}+\frac{45}{4} a^2 b^8 x^4+45 a^8 b^2 x+20 a^9 b \sqrt{x}+a^{10} \log (x)+\frac{20}{9} a b^9 x^{9/2}+\frac{b^{10} x^5}{5} \]
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Rubi [A] time = 0.0656418, antiderivative size = 128, 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} \[ 80 a^7 b^3 x^{3/2}+105 a^6 b^4 x^2+\frac{504}{5} a^5 b^5 x^{5/2}+70 a^4 b^6 x^3+\frac{240}{7} a^3 b^7 x^{7/2}+\frac{45}{4} a^2 b^8 x^4+45 a^8 b^2 x+20 a^9 b \sqrt{x}+a^{10} \log (x)+\frac{20}{9} a b^9 x^{9/2}+\frac{b^{10} x^5}{5} \]
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} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^{10}}{x} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (10 a^9 b+\frac{a^{10}}{x}+45 a^8 b^2 x+120 a^7 b^3 x^2+210 a^6 b^4 x^3+252 a^5 b^5 x^4+210 a^4 b^6 x^5+120 a^3 b^7 x^6+45 a^2 b^8 x^7+10 a b^9 x^8+b^{10} x^9\right ) \, dx,x,\sqrt{x}\right )\\ &=20 a^9 b \sqrt{x}+45 a^8 b^2 x+80 a^7 b^3 x^{3/2}+105 a^6 b^4 x^2+\frac{504}{5} a^5 b^5 x^{5/2}+70 a^4 b^6 x^3+\frac{240}{7} a^3 b^7 x^{7/2}+\frac{45}{4} a^2 b^8 x^4+\frac{20}{9} a b^9 x^{9/2}+\frac{b^{10} x^5}{5}+a^{10} \log (x)\\ \end{align*}
Mathematica [A] time = 0.0404464, size = 128, normalized size = 1. \[ 80 a^7 b^3 x^{3/2}+105 a^6 b^4 x^2+\frac{504}{5} a^5 b^5 x^{5/2}+70 a^4 b^6 x^3+\frac{240}{7} a^3 b^7 x^{7/2}+\frac{45}{4} a^2 b^8 x^4+45 a^8 b^2 x+20 a^9 b \sqrt{x}+a^{10} \log (x)+\frac{20}{9} a b^9 x^{9/2}+\frac{b^{10} x^5}{5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 109, normalized size = 0.9 \begin{align*} 45\,{a}^{8}{b}^{2}x+80\,{a}^{7}{b}^{3}{x}^{3/2}+105\,{a}^{6}{b}^{4}{x}^{2}+{\frac{504\,{a}^{5}{b}^{5}}{5}{x}^{{\frac{5}{2}}}}+70\,{a}^{4}{b}^{6}{x}^{3}+{\frac{240\,{a}^{3}{b}^{7}}{7}{x}^{{\frac{7}{2}}}}+{\frac{45\,{a}^{2}{b}^{8}{x}^{4}}{4}}+{\frac{20\,a{b}^{9}}{9}{x}^{{\frac{9}{2}}}}+{\frac{{b}^{10}{x}^{5}}{5}}+{a}^{10}\ln \left ( x \right ) +20\,{a}^{9}b\sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.946887, size = 146, normalized size = 1.14 \begin{align*} \frac{1}{5} \, b^{10} x^{5} + \frac{20}{9} \, a b^{9} x^{\frac{9}{2}} + \frac{45}{4} \, a^{2} b^{8} x^{4} + \frac{240}{7} \, a^{3} b^{7} x^{\frac{7}{2}} + 70 \, a^{4} b^{6} x^{3} + \frac{504}{5} \, a^{5} b^{5} x^{\frac{5}{2}} + 105 \, a^{6} b^{4} x^{2} + 80 \, a^{7} b^{3} x^{\frac{3}{2}} + 45 \, a^{8} b^{2} x + a^{10} \log \left (x\right ) + 20 \, a^{9} b \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53265, size = 277, normalized size = 2.16 \begin{align*} \frac{1}{5} \, b^{10} x^{5} + \frac{45}{4} \, a^{2} b^{8} x^{4} + 70 \, a^{4} b^{6} x^{3} + 105 \, a^{6} b^{4} x^{2} + 45 \, a^{8} b^{2} x + 2 \, a^{10} \log \left (\sqrt{x}\right ) + \frac{4}{315} \,{\left (175 \, a b^{9} x^{4} + 2700 \, a^{3} b^{7} x^{3} + 7938 \, a^{5} b^{5} x^{2} + 6300 \, a^{7} b^{3} x + 1575 \, a^{9} b\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.79492, size = 131, normalized size = 1.02 \begin{align*} a^{10} \log{\left (x \right )} + 20 a^{9} b \sqrt{x} + 45 a^{8} b^{2} x + 80 a^{7} b^{3} x^{\frac{3}{2}} + 105 a^{6} b^{4} x^{2} + \frac{504 a^{5} b^{5} x^{\frac{5}{2}}}{5} + 70 a^{4} b^{6} x^{3} + \frac{240 a^{3} b^{7} x^{\frac{7}{2}}}{7} + \frac{45 a^{2} b^{8} x^{4}}{4} + \frac{20 a b^{9} x^{\frac{9}{2}}}{9} + \frac{b^{10} x^{5}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08016, size = 147, normalized size = 1.15 \begin{align*} \frac{1}{5} \, b^{10} x^{5} + \frac{20}{9} \, a b^{9} x^{\frac{9}{2}} + \frac{45}{4} \, a^{2} b^{8} x^{4} + \frac{240}{7} \, a^{3} b^{7} x^{\frac{7}{2}} + 70 \, a^{4} b^{6} x^{3} + \frac{504}{5} \, a^{5} b^{5} x^{\frac{5}{2}} + 105 \, a^{6} b^{4} x^{2} + 80 \, a^{7} b^{3} x^{\frac{3}{2}} + 45 \, a^{8} b^{2} x + a^{10} \log \left ({\left | x \right |}\right ) + 20 \, a^{9} b \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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