3.2134 \(\int \frac{\left (a+b \sqrt{x}\right )^3}{x^2} \, dx\)

Optimal. Leaf size=38 \[ -\frac{a^3}{x}-\frac{6 a^2 b}{\sqrt{x}}+3 a b^2 \log (x)+2 b^3 \sqrt{x} \]

[Out]

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

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Rubi [A]  time = 0.0527943, antiderivative size = 38, 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 \[ -\frac{a^3}{x}-\frac{6 a^2 b}{\sqrt{x}}+3 a b^2 \log (x)+2 b^3 \sqrt{x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^3/x^2,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0296743, size = 39, normalized size = 1.03 \[ 3 a b^2 \log (x)-\frac{a^3+6 a^2 b \sqrt{x}-2 b^3 x^{3/2}}{x} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])^3/x^2,x]

[Out]

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

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Maple [A]  time = 0.003, size = 35, normalized size = 0.9 \[ -{\frac{{a}^{3}}{x}}+3\,a{b}^{2}\ln \left ( x \right ) -6\,{\frac{{a}^{2}b}{\sqrt{x}}}+2\,{b}^{3}\sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/2))^3/x^2,x)

[Out]

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

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Maxima [A]  time = 1.43208, size = 47, normalized size = 1.24 \[ 3 \, a b^{2} \log \left (x\right ) + 2 \, b^{3} \sqrt{x} - \frac{6 \, a^{2} b \sqrt{x} + a^{3}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^3/x^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.234188, size = 51, normalized size = 1.34 \[ \frac{6 \, a b^{2} x \log \left (\sqrt{x}\right ) - a^{3} + 2 \,{\left (b^{3} x - 3 \, a^{2} b\right )} \sqrt{x}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^3/x^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.68588, size = 36, normalized size = 0.95 \[ - \frac{a^{3}}{x} - \frac{6 a^{2} b}{\sqrt{x}} + 3 a b^{2} \log{\left (x \right )} + 2 b^{3} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.221688, size = 49, normalized size = 1.29 \[ 3 \, a b^{2}{\rm ln}\left ({\left | x \right |}\right ) + 2 \, b^{3} \sqrt{x} - \frac{6 \, a^{2} b \sqrt{x} + a^{3}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^3/x^2,x, algorithm="giac")

[Out]

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