3.2156 \(\int \frac{\left (a+b \sqrt{x}\right )^{10}}{x} \, dx\)

Optimal. Leaf size=128 \[ a^{10} \log (x)+20 a^9 b \sqrt{x}+45 a^8 b^2 x+80 a^7 b^3 x^{3/2}+105 a^6 b^4 x^2+\frac{504}{5} a^5 b^5 x^{5/2}+70 a^4 b^6 x^3+\frac{240}{7} a^3 b^7 x^{7/2}+\frac{45}{4} a^2 b^8 x^4+\frac{20}{9} a b^9 x^{9/2}+\frac{b^{10} x^5}{5} \]

[Out]

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 + (504*a^
5*b^5*x^(5/2))/5 + 70*a^4*b^6*x^3 + (240*a^3*b^7*x^(7/2))/7 + (45*a^2*b^8*x^4)/4
+ (20*a*b^9*x^(9/2))/9 + (b^10*x^5)/5 + a^10*Log[x]

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Rubi [A]  time = 0.162366, 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 \[ a^{10} \log (x)+20 a^9 b \sqrt{x}+45 a^8 b^2 x+80 a^7 b^3 x^{3/2}+105 a^6 b^4 x^2+\frac{504}{5} a^5 b^5 x^{5/2}+70 a^4 b^6 x^3+\frac{240}{7} a^3 b^7 x^{7/2}+\frac{45}{4} a^2 b^8 x^4+\frac{20}{9} a b^9 x^{9/2}+\frac{b^{10} x^5}{5} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^10/x,x]

[Out]

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 + (504*a^
5*b^5*x^(5/2))/5 + 70*a^4*b^6*x^3 + (240*a^3*b^7*x^(7/2))/7 + (45*a^2*b^8*x^4)/4
+ (20*a*b^9*x^(9/2))/9 + (b^10*x^5)/5 + a^10*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ 2 a^{10} \log{\left (\sqrt{x} \right )} + 20 a^{9} b \sqrt{x} + 90 a^{8} b^{2} \int ^{\sqrt{x}} x\, dx + 80 a^{7} b^{3} x^{\frac{3}{2}} + 105 a^{6} b^{4} x^{2} + \frac{504 a^{5} b^{5} x^{\frac{5}{2}}}{5} + 70 a^{4} b^{6} x^{3} + \frac{240 a^{3} b^{7} x^{\frac{7}{2}}}{7} + \frac{45 a^{2} b^{8} x^{4}}{4} + \frac{20 a b^{9} x^{\frac{9}{2}}}{9} + \frac{b^{10} x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**(1/2))**10/x,x)

[Out]

2*a**10*log(sqrt(x)) + 20*a**9*b*sqrt(x) + 90*a**8*b**2*Integral(x, (x, sqrt(x))
) + 80*a**7*b**3*x**(3/2) + 105*a**6*b**4*x**2 + 504*a**5*b**5*x**(5/2)/5 + 70*a
**4*b**6*x**3 + 240*a**3*b**7*x**(7/2)/7 + 45*a**2*b**8*x**4/4 + 20*a*b**9*x**(9
/2)/9 + b**10*x**5/5

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Mathematica [A]  time = 0.0301907, size = 128, normalized size = 1. \[ a^{10} \log (x)+20 a^9 b \sqrt{x}+45 a^8 b^2 x+80 a^7 b^3 x^{3/2}+105 a^6 b^4 x^2+\frac{504}{5} a^5 b^5 x^{5/2}+70 a^4 b^6 x^3+\frac{240}{7} a^3 b^7 x^{7/2}+\frac{45}{4} a^2 b^8 x^4+\frac{20}{9} a b^9 x^{9/2}+\frac{b^{10} x^5}{5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])^10/x,x]

[Out]

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 + (504*a^
5*b^5*x^(5/2))/5 + 70*a^4*b^6*x^3 + (240*a^3*b^7*x^(7/2))/7 + (45*a^2*b^8*x^4)/4
+ (20*a*b^9*x^(9/2))/9 + (b^10*x^5)/5 + a^10*Log[x]

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Maple [A]  time = 0.005, size = 109, normalized size = 0.9 \[ 45\,{a}^{8}{b}^{2}x+80\,{a}^{7}{b}^{3}{x}^{3/2}+105\,{a}^{6}{b}^{4}{x}^{2}+{\frac{504\,{a}^{5}{b}^{5}}{5}{x}^{{\frac{5}{2}}}}+70\,{a}^{4}{b}^{6}{x}^{3}+{\frac{240\,{a}^{3}{b}^{7}}{7}{x}^{{\frac{7}{2}}}}+{\frac{45\,{a}^{2}{b}^{8}{x}^{4}}{4}}+{\frac{20\,a{b}^{9}}{9}{x}^{{\frac{9}{2}}}}+{\frac{{b}^{10}{x}^{5}}{5}}+{a}^{10}\ln \left ( x \right ) +20\,{a}^{9}b\sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/2))^10/x,x)

[Out]

45*a^8*b^2*x+80*a^7*b^3*x^(3/2)+105*a^6*b^4*x^2+504/5*a^5*b^5*x^(5/2)+70*a^4*b^6
*x^3+240/7*a^3*b^7*x^(7/2)+45/4*a^2*b^8*x^4+20/9*a*b^9*x^(9/2)+1/5*b^10*x^5+a^10
*ln(x)+20*a^9*b*x^(1/2)

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Maxima [A]  time = 1.44199, size = 146, normalized size = 1.14 \[ \frac{1}{5} \, b^{10} x^{5} + \frac{20}{9} \, a b^{9} x^{\frac{9}{2}} + \frac{45}{4} \, a^{2} b^{8} x^{4} + \frac{240}{7} \, a^{3} b^{7} x^{\frac{7}{2}} + 70 \, a^{4} b^{6} x^{3} + \frac{504}{5} \, a^{5} b^{5} x^{\frac{5}{2}} + 105 \, a^{6} b^{4} x^{2} + 80 \, a^{7} b^{3} x^{\frac{3}{2}} + 45 \, a^{8} b^{2} x + a^{10} \log \left (x\right ) + 20 \, a^{9} b \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^10/x,x, algorithm="maxima")

[Out]

1/5*b^10*x^5 + 20/9*a*b^9*x^(9/2) + 45/4*a^2*b^8*x^4 + 240/7*a^3*b^7*x^(7/2) + 7
0*a^4*b^6*x^3 + 504/5*a^5*b^5*x^(5/2) + 105*a^6*b^4*x^2 + 80*a^7*b^3*x^(3/2) + 4
5*a^8*b^2*x + a^10*log(x) + 20*a^9*b*sqrt(x)

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Fricas [A]  time = 0.237965, size = 151, normalized size = 1.18 \[ \frac{1}{5} \, b^{10} x^{5} + \frac{45}{4} \, a^{2} b^{8} x^{4} + 70 \, a^{4} b^{6} x^{3} + 105 \, a^{6} b^{4} x^{2} + 45 \, a^{8} b^{2} x + 2 \, a^{10} \log \left (\sqrt{x}\right ) + \frac{4}{315} \,{\left (175 \, a b^{9} x^{4} + 2700 \, a^{3} b^{7} x^{3} + 7938 \, a^{5} b^{5} x^{2} + 6300 \, a^{7} b^{3} x + 1575 \, a^{9} b\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^10/x,x, algorithm="fricas")

[Out]

1/5*b^10*x^5 + 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(sqrt(x)) + 4/315*(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)*sqrt(x)

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Sympy [A]  time = 6.17491, size = 131, normalized size = 1.02 \[ a^{10} \log{\left (x \right )} + 20 a^{9} b \sqrt{x} + 45 a^{8} b^{2} x + 80 a^{7} b^{3} x^{\frac{3}{2}} + 105 a^{6} b^{4} x^{2} + \frac{504 a^{5} b^{5} x^{\frac{5}{2}}}{5} + 70 a^{4} b^{6} x^{3} + \frac{240 a^{3} b^{7} x^{\frac{7}{2}}}{7} + \frac{45 a^{2} b^{8} x^{4}}{4} + \frac{20 a b^{9} x^{\frac{9}{2}}}{9} + \frac{b^{10} x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/2))**10/x,x)

[Out]

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 + 504*a**5*b**5*x**(5/2)/5 + 70*a**4*b**6*x**3 + 240*a**3*b**7*x*
*(7/2)/7 + 45*a**2*b**8*x**4/4 + 20*a*b**9*x**(9/2)/9 + b**10*x**5/5

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GIAC/XCAS [A]  time = 0.218431, size = 147, normalized size = 1.15 \[ \frac{1}{5} \, b^{10} x^{5} + \frac{20}{9} \, a b^{9} x^{\frac{9}{2}} + \frac{45}{4} \, a^{2} b^{8} x^{4} + \frac{240}{7} \, a^{3} b^{7} x^{\frac{7}{2}} + 70 \, a^{4} b^{6} x^{3} + \frac{504}{5} \, a^{5} b^{5} x^{\frac{5}{2}} + 105 \, a^{6} b^{4} x^{2} + 80 \, a^{7} b^{3} x^{\frac{3}{2}} + 45 \, a^{8} b^{2} x + a^{10}{\rm ln}\left ({\left | x \right |}\right ) + 20 \, a^{9} b \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^10/x,x, algorithm="giac")

[Out]

1/5*b^10*x^5 + 20/9*a*b^9*x^(9/2) + 45/4*a^2*b^8*x^4 + 240/7*a^3*b^7*x^(7/2) + 7
0*a^4*b^6*x^3 + 504/5*a^5*b^5*x^(5/2) + 105*a^6*b^4*x^2 + 80*a^7*b^3*x^(3/2) + 4
5*a^8*b^2*x + a^10*ln(abs(x)) + 20*a^9*b*sqrt(x)