Optimal. Leaf size=66 \[ -\frac{10 a^3 b^2}{x}-\frac{20 a^2 b^3}{\sqrt{x}}-\frac{10 a^4 b}{3 x^{3/2}}-\frac{a^5}{2 x^2}+5 a b^4 \log (x)+2 b^5 \sqrt{x} \]
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Rubi [A] time = 0.0306667, antiderivative size = 66, 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} \[ -\frac{10 a^3 b^2}{x}-\frac{20 a^2 b^3}{\sqrt{x}}-\frac{10 a^4 b}{3 x^{3/2}}-\frac{a^5}{2 x^2}+5 a b^4 \log (x)+2 b^5 \sqrt{x} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+b \sqrt{x}\right )^5}{x^3} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^5} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (b^5+\frac{a^5}{x^5}+\frac{5 a^4 b}{x^4}+\frac{10 a^3 b^2}{x^3}+\frac{10 a^2 b^3}{x^2}+\frac{5 a b^4}{x}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{a^5}{2 x^2}-\frac{10 a^4 b}{3 x^{3/2}}-\frac{10 a^3 b^2}{x}-\frac{20 a^2 b^3}{\sqrt{x}}+2 b^5 \sqrt{x}+5 a b^4 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0317971, size = 66, normalized size = 1. \[ -\frac{10 a^3 b^2}{x}-\frac{20 a^2 b^3}{\sqrt{x}}-\frac{10 a^4 b}{3 x^{3/2}}-\frac{a^5}{2 x^2}+5 a b^4 \log (x)+2 b^5 \sqrt{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 57, normalized size = 0.9 \begin{align*} -{\frac{{a}^{5}}{2\,{x}^{2}}}-{\frac{10\,{a}^{4}b}{3}{x}^{-{\frac{3}{2}}}}-10\,{\frac{{a}^{3}{b}^{2}}{x}}+5\,a{b}^{4}\ln \left ( x \right ) -20\,{\frac{{a}^{2}{b}^{3}}{\sqrt{x}}}+2\,{b}^{5}\sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.966082, size = 77, normalized size = 1.17 \begin{align*} 5 \, a b^{4} \log \left (x\right ) + 2 \, b^{5} \sqrt{x} - \frac{120 \, a^{2} b^{3} x^{\frac{3}{2}} + 60 \, a^{3} b^{2} x + 20 \, a^{4} b \sqrt{x} + 3 \, a^{5}}{6 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51722, size = 147, normalized size = 2.23 \begin{align*} \frac{60 \, a b^{4} x^{2} \log \left (\sqrt{x}\right ) - 60 \, a^{3} b^{2} x - 3 \, a^{5} + 4 \,{\left (3 \, b^{5} x^{2} - 30 \, a^{2} b^{3} x - 5 \, a^{4} b\right )} \sqrt{x}}{6 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.906333, size = 65, normalized size = 0.98 \begin{align*} - \frac{a^{5}}{2 x^{2}} - \frac{10 a^{4} b}{3 x^{\frac{3}{2}}} - \frac{10 a^{3} b^{2}}{x} - \frac{20 a^{2} b^{3}}{\sqrt{x}} + 5 a b^{4} \log{\left (x \right )} + 2 b^{5} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11602, size = 78, normalized size = 1.18 \begin{align*} 5 \, a b^{4} \log \left ({\left | x \right |}\right ) + 2 \, b^{5} \sqrt{x} - \frac{120 \, a^{2} b^{3} x^{\frac{3}{2}} + 60 \, a^{3} b^{2} x + 20 \, a^{4} b \sqrt{x} + 3 \, a^{5}}{6 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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