3.2144 \(\int \frac{\left (a+b \sqrt{x}\right )^5}{x} \, dx\)

Optimal. Leaf size=65 \[ a^5 \log (x)+10 a^4 b \sqrt{x}+10 a^3 b^2 x+\frac{20}{3} a^2 b^3 x^{3/2}+\frac{5}{2} a b^4 x^2+\frac{2}{5} b^5 x^{5/2} \]

[Out]

10*a^4*b*Sqrt[x] + 10*a^3*b^2*x + (20*a^2*b^3*x^(3/2))/3 + (5*a*b^4*x^2)/2 + (2*
b^5*x^(5/2))/5 + a^5*Log[x]

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Rubi [A]  time = 0.0780266, 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 \[ a^5 \log (x)+10 a^4 b \sqrt{x}+10 a^3 b^2 x+\frac{20}{3} a^2 b^3 x^{3/2}+\frac{5}{2} a b^4 x^2+\frac{2}{5} b^5 x^{5/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^5/x,x]

[Out]

10*a^4*b*Sqrt[x] + 10*a^3*b^2*x + (20*a^2*b^3*x^(3/2))/3 + (5*a*b^4*x^2)/2 + (2*
b^5*x^(5/2))/5 + a^5*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ 2 a^{5} \log{\left (\sqrt{x} \right )} + 10 a^{4} b \sqrt{x} + 20 a^{3} b^{2} \int ^{\sqrt{x}} x\, dx + \frac{20 a^{2} b^{3} x^{\frac{3}{2}}}{3} + \frac{5 a b^{4} x^{2}}{2} + \frac{2 b^{5} x^{\frac{5}{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**(1/2))**5/x,x)

[Out]

2*a**5*log(sqrt(x)) + 10*a**4*b*sqrt(x) + 20*a**3*b**2*Integral(x, (x, sqrt(x)))
+ 20*a**2*b**3*x**(3/2)/3 + 5*a*b**4*x**2/2 + 2*b**5*x**(5/2)/5

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Mathematica [A]  time = 0.0191907, size = 65, normalized size = 1. \[ a^5 \log (x)+10 a^4 b \sqrt{x}+10 a^3 b^2 x+\frac{20}{3} a^2 b^3 x^{3/2}+\frac{5}{2} a b^4 x^2+\frac{2}{5} b^5 x^{5/2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])^5/x,x]

[Out]

10*a^4*b*Sqrt[x] + 10*a^3*b^2*x + (20*a^2*b^3*x^(3/2))/3 + (5*a*b^4*x^2)/2 + (2*
b^5*x^(5/2))/5 + a^5*Log[x]

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Maple [A]  time = 0.003, size = 54, normalized size = 0.8 \[ 10\,{a}^{3}{b}^{2}x+{\frac{20\,{a}^{2}{b}^{3}}{3}{x}^{{\frac{3}{2}}}}+{\frac{5\,a{b}^{4}{x}^{2}}{2}}+{\frac{2\,{b}^{5}}{5}{x}^{{\frac{5}{2}}}}+{a}^{5}\ln \left ( x \right ) +10\,{a}^{4}b\sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/2))^5/x,x)

[Out]

10*a^3*b^2*x+20/3*a^2*b^3*x^(3/2)+5/2*a*b^4*x^2+2/5*b^5*x^(5/2)+a^5*ln(x)+10*a^4
*b*x^(1/2)

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Maxima [A]  time = 1.4508, size = 72, normalized size = 1.11 \[ \frac{2}{5} \, b^{5} x^{\frac{5}{2}} + \frac{5}{2} \, a b^{4} x^{2} + \frac{20}{3} \, a^{2} b^{3} x^{\frac{3}{2}} + 10 \, a^{3} b^{2} x + a^{5} \log \left (x\right ) + 10 \, a^{4} b \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^5/x,x, algorithm="maxima")

[Out]

2/5*b^5*x^(5/2) + 5/2*a*b^4*x^2 + 20/3*a^2*b^3*x^(3/2) + 10*a^3*b^2*x + a^5*log(
x) + 10*a^4*b*sqrt(x)

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Fricas [A]  time = 0.236069, size = 77, normalized size = 1.18 \[ \frac{5}{2} \, a b^{4} x^{2} + 10 \, a^{3} b^{2} x + 2 \, a^{5} \log \left (\sqrt{x}\right ) + \frac{2}{15} \,{\left (3 \, b^{5} x^{2} + 50 \, a^{2} b^{3} x + 75 \, a^{4} b\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^5/x,x, algorithm="fricas")

[Out]

5/2*a*b^4*x^2 + 10*a^3*b^2*x + 2*a^5*log(sqrt(x)) + 2/15*(3*b^5*x^2 + 50*a^2*b^3
*x + 75*a^4*b)*sqrt(x)

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Sympy [A]  time = 1.31507, size = 66, normalized size = 1.02 \[ a^{5} \log{\left (x \right )} + 10 a^{4} b \sqrt{x} + 10 a^{3} b^{2} x + \frac{20 a^{2} b^{3} x^{\frac{3}{2}}}{3} + \frac{5 a b^{4} x^{2}}{2} + \frac{2 b^{5} x^{\frac{5}{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/2))**5/x,x)

[Out]

a**5*log(x) + 10*a**4*b*sqrt(x) + 10*a**3*b**2*x + 20*a**2*b**3*x**(3/2)/3 + 5*a
*b**4*x**2/2 + 2*b**5*x**(5/2)/5

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GIAC/XCAS [A]  time = 0.215347, size = 73, normalized size = 1.12 \[ \frac{2}{5} \, b^{5} x^{\frac{5}{2}} + \frac{5}{2} \, a b^{4} x^{2} + \frac{20}{3} \, a^{2} b^{3} x^{\frac{3}{2}} + 10 \, a^{3} b^{2} x + a^{5}{\rm ln}\left ({\left | x \right |}\right ) + 10 \, a^{4} b \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^5/x,x, algorithm="giac")

[Out]

2/5*b^5*x^(5/2) + 5/2*a*b^4*x^2 + 20/3*a^2*b^3*x^(3/2) + 10*a^3*b^2*x + a^5*ln(a
bs(x)) + 10*a^4*b*sqrt(x)