Optimal. Leaf size=65 \[ a^5 \log (x)+10 a^4 b \sqrt {x}+10 a^3 b^2 x+\frac {20}{3} a^2 b^3 x^{3/2}+\frac {5}{2} a b^4 x^2+\frac {2}{5} b^5 x^{5/2} \]
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Rubi [A] time = 0.03, antiderivative size = 65, normalized size of antiderivative = 1.00, 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 43
Rule 266
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*}
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Mathematica [A] time = 0.02, size = 65, normalized size = 1.00 \[ a^5 \log (x)+10 a^4 b \sqrt {x}+10 a^3 b^2 x+\frac {20}{3} a^2 b^3 x^{3/2}+\frac {5}{2} a b^4 x^2+\frac {2}{5} b^5 x^{5/2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.19, size = 57, normalized size = 0.88 \[ \frac {5}{2} \, a b^{4} x^{2} + 10 \, a^{3} b^{2} x + 2 \, a^{5} \log \left (\sqrt {x}\right ) + \frac {2}{15} \, {\left (3 \, b^{5} x^{2} + 50 \, a^{2} b^{3} x + 75 \, a^{4} b\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 54, normalized size = 0.83 \[ \frac {2}{5} \, b^{5} x^{\frac {5}{2}} + \frac {5}{2} \, a b^{4} x^{2} + \frac {20}{3} \, a^{2} b^{3} x^{\frac {3}{2}} + 10 \, a^{3} b^{2} x + a^{5} \log \left ({\left | x \right |}\right ) + 10 \, a^{4} b \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 54, normalized size = 0.83 \[ \frac {2 b^{5} x^{\frac {5}{2}}}{5}+\frac {5 a \,b^{4} x^{2}}{2}+\frac {20 a^{2} b^{3} x^{\frac {3}{2}}}{3}+a^{5} \ln \relax (x )+10 a^{3} b^{2} x +10 a^{4} b \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.87, size = 53, normalized size = 0.82 \[ \frac {2}{5} \, b^{5} x^{\frac {5}{2}} + \frac {5}{2} \, a b^{4} x^{2} + \frac {20}{3} \, a^{2} b^{3} x^{\frac {3}{2}} + 10 \, a^{3} b^{2} x + a^{5} \log \relax (x) + 10 \, a^{4} b \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 56, normalized size = 0.86 \[ 2\,a^5\,\ln \left (\sqrt {x}\right )+\frac {2\,b^5\,x^{5/2}}{5}+10\,a^3\,b^2\,x+\frac {5\,a\,b^4\,x^2}{2}+10\,a^4\,b\,\sqrt {x}+\frac {20\,a^2\,b^3\,x^{3/2}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.46, size = 66, normalized size = 1.02 \[ a^{5} \log {\relax (x )} + 10 a^{4} b \sqrt {x} + 10 a^{3} b^{2} x + \frac {20 a^{2} b^{3} x^{\frac {3}{2}}}{3} + \frac {5 a b^{4} x^{2}}{2} + \frac {2 b^{5} x^{\frac {5}{2}}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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