3.2145 \(\int \frac{\left (a+b \sqrt{x}\right )^5}{x^2} \, dx\)

Optimal. Leaf size=62 \[ -\frac{a^5}{x}-\frac{10 a^4 b}{\sqrt{x}}+10 a^3 b^2 \log (x)+20 a^2 b^3 \sqrt{x}+5 a b^4 x+\frac{2}{3} b^5 x^{3/2} \]

[Out]

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

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Rubi [A]  time = 0.0878741, antiderivative size = 62, 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 \[ -\frac{a^5}{x}-\frac{10 a^4 b}{\sqrt{x}}+10 a^3 b^2 \log (x)+20 a^2 b^3 \sqrt{x}+5 a b^4 x+\frac{2}{3} b^5 x^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^5/x^2,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0386639, size = 62, normalized size = 1. \[ -\frac{a^5}{x}-\frac{10 a^4 b}{\sqrt{x}}+10 a^3 b^2 \log (x)+20 a^2 b^3 \sqrt{x}+5 a b^4 x+\frac{2}{3} b^5 x^{3/2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])^5/x^2,x]

[Out]

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

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Maple [A]  time = 0.003, size = 55, normalized size = 0.9 \[ -{\frac{{a}^{5}}{x}}+5\,a{b}^{4}x+{\frac{2\,{b}^{5}}{3}{x}^{{\frac{3}{2}}}}+10\,{a}^{3}{b}^{2}\ln \left ( x \right ) -10\,{\frac{{a}^{4}b}{\sqrt{x}}}+20\,{a}^{2}{b}^{3}\sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/2))^5/x^2,x)

[Out]

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

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Maxima [A]  time = 1.44531, size = 74, normalized size = 1.19 \[ \frac{2}{3} \, b^{5} x^{\frac{3}{2}} + 5 \, a b^{4} x + 10 \, a^{3} b^{2} \log \left (x\right ) + 20 \, a^{2} b^{3} \sqrt{x} - \frac{10 \, a^{4} b \sqrt{x} + a^{5}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^5/x^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.235766, size = 82, normalized size = 1.32 \[ \frac{15 \, a b^{4} x^{2} + 60 \, a^{3} b^{2} x \log \left (\sqrt{x}\right ) - 3 \, a^{5} + 2 \,{\left (b^{5} x^{2} + 30 \, a^{2} b^{3} x - 15 \, a^{4} b\right )} \sqrt{x}}{3 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^5/x^2,x, algorithm="fricas")

[Out]

1/3*(15*a*b^4*x^2 + 60*a^3*b^2*x*log(sqrt(x)) - 3*a^5 + 2*(b^5*x^2 + 30*a^2*b^3*
x - 15*a^4*b)*sqrt(x))/x

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Sympy [A]  time = 2.0691, size = 61, normalized size = 0.98 \[ - \frac{a^{5}}{x} - \frac{10 a^{4} b}{\sqrt{x}} + 10 a^{3} b^{2} \log{\left (x \right )} + 20 a^{2} b^{3} \sqrt{x} + 5 a b^{4} x + \frac{2 b^{5} x^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.219055, size = 76, normalized size = 1.23 \[ \frac{2}{3} \, b^{5} x^{\frac{3}{2}} + 5 \, a b^{4} x + 10 \, a^{3} b^{2}{\rm ln}\left ({\left | x \right |}\right ) + 20 \, a^{2} b^{3} \sqrt{x} - \frac{10 \, a^{4} b \sqrt{x} + a^{5}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^5/x^2,x, algorithm="giac")

[Out]

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