Optimal. Leaf size=65 \[ \frac{20}{3} a^2 b^3 x^{3/2}+10 a^3 b^2 x+10 a^4 b \sqrt{x}+a^5 \log (x)+\frac{5}{2} a b^4 x^2+\frac{2}{5} b^5 x^{5/2} \]
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Rubi [A] time = 0.0311541, antiderivative size = 65, 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} \[ \frac{20}{3} a^2 b^3 x^{3/2}+10 a^3 b^2 x+10 a^4 b \sqrt{x}+a^5 \log (x)+\frac{5}{2} a b^4 x^2+\frac{2}{5} b^5 x^{5/2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+b \sqrt{x}\right )^5}{x} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (5 a^4 b+\frac{a^5}{x}+10 a^3 b^2 x+10 a^2 b^3 x^2+5 a b^4 x^3+b^5 x^4\right ) \, dx,x,\sqrt{x}\right )\\ &=10 a^4 b \sqrt{x}+10 a^3 b^2 x+\frac{20}{3} a^2 b^3 x^{3/2}+\frac{5}{2} a b^4 x^2+\frac{2}{5} b^5 x^{5/2}+a^5 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0211607, size = 65, normalized size = 1. \[ \frac{20}{3} a^2 b^3 x^{3/2}+10 a^3 b^2 x+10 a^4 b \sqrt{x}+a^5 \log (x)+\frac{5}{2} a b^4 x^2+\frac{2}{5} b^5 x^{5/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 54, normalized size = 0.8 \begin{align*} 10\,{a}^{3}{b}^{2}x+{\frac{20\,{a}^{2}{b}^{3}}{3}{x}^{{\frac{3}{2}}}}+{\frac{5\,a{b}^{4}{x}^{2}}{2}}+{\frac{2\,{b}^{5}}{5}{x}^{{\frac{5}{2}}}}+{a}^{5}\ln \left ( x \right ) +10\,{a}^{4}b\sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.975073, size = 72, normalized size = 1.11 \begin{align*} \frac{2}{5} \, b^{5} x^{\frac{5}{2}} + \frac{5}{2} \, a b^{4} x^{2} + \frac{20}{3} \, a^{2} b^{3} x^{\frac{3}{2}} + 10 \, a^{3} b^{2} x + a^{5} \log \left (x\right ) + 10 \, a^{4} b \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48419, size = 140, normalized size = 2.15 \begin{align*} \frac{5}{2} \, a b^{4} x^{2} + 10 \, a^{3} b^{2} x + 2 \, a^{5} \log \left (\sqrt{x}\right ) + \frac{2}{15} \,{\left (3 \, b^{5} x^{2} + 50 \, a^{2} b^{3} x + 75 \, a^{4} b\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.566302, size = 66, normalized size = 1.02 \begin{align*} a^{5} \log{\left (x \right )} + 10 a^{4} b \sqrt{x} + 10 a^{3} b^{2} x + \frac{20 a^{2} b^{3} x^{\frac{3}{2}}}{3} + \frac{5 a b^{4} x^{2}}{2} + \frac{2 b^{5} x^{\frac{5}{2}}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11264, size = 73, normalized size = 1.12 \begin{align*} \frac{2}{5} \, b^{5} x^{\frac{5}{2}} + \frac{5}{2} \, a b^{4} x^{2} + \frac{20}{3} \, a^{2} b^{3} x^{\frac{3}{2}} + 10 \, a^{3} b^{2} x + a^{5} \log \left ({\left | x \right |}\right ) + 10 \, a^{4} b \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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