Optimal. Leaf size=128 \[ a^{10} \log (x)+20 a^9 b \sqrt {x}+45 a^8 b^2 x+80 a^7 b^3 x^{3/2}+105 a^6 b^4 x^2+\frac {504}{5} a^5 b^5 x^{5/2}+70 a^4 b^6 x^3+\frac {240}{7} a^3 b^7 x^{7/2}+\frac {45}{4} a^2 b^8 x^4+\frac {20}{9} a b^9 x^{9/2}+\frac {b^{10} x^5}{5} \]
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Rubi [A] time = 0.07, antiderivative size = 128, 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} \[ 80 a^7 b^3 x^{3/2}+105 a^6 b^4 x^2+\frac {504}{5} a^5 b^5 x^{5/2}+70 a^4 b^6 x^3+\frac {240}{7} a^3 b^7 x^{7/2}+\frac {45}{4} a^2 b^8 x^4+45 a^8 b^2 x+20 a^9 b \sqrt {x}+a^{10} \log (x)+\frac {20}{9} a b^9 x^{9/2}+\frac {b^{10} x^5}{5} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
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*}
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Mathematica [A] time = 0.04, size = 128, normalized size = 1.00 \[ a^{10} \log (x)+20 a^9 b \sqrt {x}+45 a^8 b^2 x+80 a^7 b^3 x^{3/2}+105 a^6 b^4 x^2+\frac {504}{5} a^5 b^5 x^{5/2}+70 a^4 b^6 x^3+\frac {240}{7} a^3 b^7 x^{7/2}+\frac {45}{4} a^2 b^8 x^4+\frac {20}{9} a b^9 x^{9/2}+\frac {b^{10} x^5}{5} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.09, size = 112, normalized size = 0.88 \[ \frac {1}{5} \, b^{10} x^{5} + \frac {45}{4} \, a^{2} b^{8} x^{4} + 70 \, a^{4} b^{6} x^{3} + 105 \, a^{6} b^{4} x^{2} + 45 \, a^{8} b^{2} x + 2 \, a^{10} \log \left (\sqrt {x}\right ) + \frac {4}{315} \, {\left (175 \, a b^{9} x^{4} + 2700 \, a^{3} b^{7} x^{3} + 7938 \, a^{5} b^{5} x^{2} + 6300 \, a^{7} b^{3} x + 1575 \, a^{9} b\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 109, normalized size = 0.85 \[ \frac {1}{5} \, b^{10} x^{5} + \frac {20}{9} \, a b^{9} x^{\frac {9}{2}} + \frac {45}{4} \, a^{2} b^{8} x^{4} + \frac {240}{7} \, a^{3} b^{7} x^{\frac {7}{2}} + 70 \, a^{4} b^{6} x^{3} + \frac {504}{5} \, a^{5} b^{5} x^{\frac {5}{2}} + 105 \, a^{6} b^{4} x^{2} + 80 \, a^{7} b^{3} x^{\frac {3}{2}} + 45 \, a^{8} b^{2} x + a^{10} \log \left ({\left | x \right |}\right ) + 20 \, a^{9} b \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 109, normalized size = 0.85 \[ \frac {b^{10} x^{5}}{5}+\frac {20 a \,b^{9} x^{\frac {9}{2}}}{9}+\frac {45 a^{2} b^{8} x^{4}}{4}+\frac {240 a^{3} b^{7} x^{\frac {7}{2}}}{7}+70 a^{4} b^{6} x^{3}+\frac {504 a^{5} b^{5} x^{\frac {5}{2}}}{5}+105 a^{6} b^{4} x^{2}+80 a^{7} b^{3} x^{\frac {3}{2}}+a^{10} \ln \relax (x )+45 a^{8} b^{2} x +20 a^{9} b \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.85, size = 108, normalized size = 0.84 \[ \frac {1}{5} \, b^{10} x^{5} + \frac {20}{9} \, a b^{9} x^{\frac {9}{2}} + \frac {45}{4} \, a^{2} b^{8} x^{4} + \frac {240}{7} \, a^{3} b^{7} x^{\frac {7}{2}} + 70 \, a^{4} b^{6} x^{3} + \frac {504}{5} \, a^{5} b^{5} x^{\frac {5}{2}} + 105 \, a^{6} b^{4} x^{2} + 80 \, a^{7} b^{3} x^{\frac {3}{2}} + 45 \, a^{8} b^{2} x + a^{10} \log \relax (x) + 20 \, a^{9} b \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 111, normalized size = 0.87 \[ 2\,a^{10}\,\ln \left (\sqrt {x}\right )+\frac {b^{10}\,x^5}{5}+45\,a^8\,b^2\,x+20\,a^9\,b\,\sqrt {x}+\frac {20\,a\,b^9\,x^{9/2}}{9}+105\,a^6\,b^4\,x^2+70\,a^4\,b^6\,x^3+\frac {45\,a^2\,b^8\,x^4}{4}+80\,a^7\,b^3\,x^{3/2}+\frac {504\,a^5\,b^5\,x^{5/2}}{5}+\frac {240\,a^3\,b^7\,x^{7/2}}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.12, size = 131, normalized size = 1.02 \[ a^{10} \log {\relax (x )} + 20 a^{9} b \sqrt {x} + 45 a^{8} b^{2} x + 80 a^{7} b^{3} x^{\frac {3}{2}} + 105 a^{6} b^{4} x^{2} + \frac {504 a^{5} b^{5} x^{\frac {5}{2}}}{5} + 70 a^{4} b^{6} x^{3} + \frac {240 a^{3} b^{7} x^{\frac {7}{2}}}{7} + \frac {45 a^{2} b^{8} x^{4}}{4} + \frac {20 a b^{9} x^{\frac {9}{2}}}{9} + \frac {b^{10} x^{5}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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