3.2135 \(\int \frac{\left (a+b \sqrt{x}\right )^3}{x^3} \, dx\)

Optimal. Leaf size=21 \[ -\frac{\left (a+b \sqrt{x}\right )^4}{2 a x^2} \]

[Out]

-(a + b*Sqrt[x])^4/(2*a*x^2)

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Rubi [A]  time = 0.0156728, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{\left (a+b \sqrt{x}\right )^4}{2 a x^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^3/x^3,x]

[Out]

-(a + b*Sqrt[x])^4/(2*a*x^2)

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Rubi in Sympy [A]  time = 2.70663, size = 17, normalized size = 0.81 \[ - \frac{\left (a + b \sqrt{x}\right )^{4}}{2 a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a + b*sqrt(x))**4/(2*a*x**2)

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Mathematica [A]  time = 0.015595, size = 41, normalized size = 1.95 \[ -\frac{a^3}{2 x^2}-\frac{2 a^2 b}{x^{3/2}}-\frac{3 a b^2}{x}-\frac{2 b^3}{\sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

-a^3/(2*x^2) - (2*a^2*b)/x^(3/2) - (3*a*b^2)/x - (2*b^3)/Sqrt[x]

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Maple [B]  time = 0.003, size = 36, normalized size = 1.7 \[ -2\,{\frac{{b}^{3}}{\sqrt{x}}}-3\,{\frac{a{b}^{2}}{x}}-2\,{\frac{{a}^{2}b}{{x}^{3/2}}}-{\frac{{a}^{3}}{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/2))^3/x^3,x)

[Out]

-2*b^3/x^(1/2)-3*a*b^2/x-2*a^2*b/x^(3/2)-1/2*a^3/x^2

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Maxima [A]  time = 1.48042, size = 45, normalized size = 2.14 \[ -\frac{4 \, b^{3} x^{\frac{3}{2}} + 6 \, a b^{2} x + 4 \, a^{2} b \sqrt{x} + a^{3}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(4*b^3*x^(3/2) + 6*a*b^2*x + 4*a^2*b*sqrt(x) + a^3)/x^2

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Fricas [A]  time = 0.238723, size = 43, normalized size = 2.05 \[ -\frac{6 \, a b^{2} x + a^{3} + 4 \,{\left (b^{3} x + a^{2} b\right )} \sqrt{x}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(6*a*b^2*x + a^3 + 4*(b^3*x + a^2*b)*sqrt(x))/x^2

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Sympy [A]  time = 2.2877, size = 39, normalized size = 1.86 \[ - \frac{a^{3}}{2 x^{2}} - \frac{2 a^{2} b}{x^{\frac{3}{2}}} - \frac{3 a b^{2}}{x} - \frac{2 b^{3}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**3/(2*x**2) - 2*a**2*b/x**(3/2) - 3*a*b**2/x - 2*b**3/sqrt(x)

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GIAC/XCAS [A]  time = 0.21961, size = 45, normalized size = 2.14 \[ -\frac{4 \, b^{3} x^{\frac{3}{2}} + 6 \, a b^{2} x + 4 \, a^{2} b \sqrt{x} + a^{3}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(4*b^3*x^(3/2) + 6*a*b^2*x + 4*a^2*b*sqrt(x) + a^3)/x^2