3.2133 \(\int \frac{\left (a+b \sqrt{x}\right )^3}{x} \, dx\)

Optimal. Leaf size=37 \[ a^3 \log (x)+6 a^2 b \sqrt{x}+3 a b^2 x+\frac{2}{3} b^3 x^{3/2} \]

[Out]

6*a^2*b*Sqrt[x] + 3*a*b^2*x + (2*b^3*x^(3/2))/3 + a^3*Log[x]

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Rubi [A]  time = 0.049088, antiderivative size = 37, 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 \[ a^3 \log (x)+6 a^2 b \sqrt{x}+3 a b^2 x+\frac{2}{3} b^3 x^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^3/x,x]

[Out]

6*a^2*b*Sqrt[x] + 3*a*b^2*x + (2*b^3*x^(3/2))/3 + a^3*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ 2 a^{3} \log{\left (\sqrt{x} \right )} + 6 a^{2} b \sqrt{x} + 6 a b^{2} \int ^{\sqrt{x}} x\, dx + \frac{2 b^{3} x^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**(1/2))**3/x,x)

[Out]

2*a**3*log(sqrt(x)) + 6*a**2*b*sqrt(x) + 6*a*b**2*Integral(x, (x, sqrt(x))) + 2*
b**3*x**(3/2)/3

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Mathematica [A]  time = 0.0152542, size = 37, normalized size = 1. \[ a^3 \log (x)+6 a^2 b \sqrt{x}+3 a b^2 x+\frac{2}{3} b^3 x^{3/2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])^3/x,x]

[Out]

6*a^2*b*Sqrt[x] + 3*a*b^2*x + (2*b^3*x^(3/2))/3 + a^3*Log[x]

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Maple [A]  time = 0.003, size = 32, normalized size = 0.9 \[ 3\,a{b}^{2}x+{\frac{2\,{b}^{3}}{3}{x}^{{\frac{3}{2}}}}+{a}^{3}\ln \left ( x \right ) +6\,{a}^{2}b\sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/2))^3/x,x)

[Out]

3*a*b^2*x+2/3*b^3*x^(3/2)+a^3*ln(x)+6*a^2*b*x^(1/2)

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Maxima [A]  time = 1.42985, size = 42, normalized size = 1.14 \[ \frac{2}{3} \, b^{3} x^{\frac{3}{2}} + 3 \, a b^{2} x + a^{3} \log \left (x\right ) + 6 \, a^{2} b \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^3/x,x, algorithm="maxima")

[Out]

2/3*b^3*x^(3/2) + 3*a*b^2*x + a^3*log(x) + 6*a^2*b*sqrt(x)

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Fricas [A]  time = 0.235659, size = 46, normalized size = 1.24 \[ 3 \, a b^{2} x + 2 \, a^{3} \log \left (\sqrt{x}\right ) + \frac{2}{3} \,{\left (b^{3} x + 9 \, a^{2} b\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^3/x,x, algorithm="fricas")

[Out]

3*a*b^2*x + 2*a^3*log(sqrt(x)) + 2/3*(b^3*x + 9*a^2*b)*sqrt(x)

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Sympy [A]  time = 0.665255, size = 37, normalized size = 1. \[ a^{3} \log{\left (x \right )} + 6 a^{2} b \sqrt{x} + 3 a b^{2} x + \frac{2 b^{3} x^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/2))**3/x,x)

[Out]

a**3*log(x) + 6*a**2*b*sqrt(x) + 3*a*b**2*x + 2*b**3*x**(3/2)/3

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GIAC/XCAS [A]  time = 0.217069, size = 43, normalized size = 1.16 \[ \frac{2}{3} \, b^{3} x^{\frac{3}{2}} + 3 \, a b^{2} x + a^{3}{\rm ln}\left ({\left | x \right |}\right ) + 6 \, a^{2} b \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^3/x,x, algorithm="giac")

[Out]

2/3*b^3*x^(3/2) + 3*a*b^2*x + a^3*ln(abs(x)) + 6*a^2*b*sqrt(x)