3.2146 \(\int \frac{\left (a+b \sqrt{x}\right )^5}{x^3} \, dx\)

Optimal. Leaf size=66 \[ -\frac{a^5}{2 x^2}-\frac{10 a^4 b}{3 x^{3/2}}-\frac{10 a^3 b^2}{x}-\frac{20 a^2 b^3}{\sqrt{x}}+5 a b^4 \log (x)+2 b^5 \sqrt{x} \]

[Out]

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

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Rubi [A]  time = 0.0843245, antiderivative size = 66, 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 \[ -\frac{a^5}{2 x^2}-\frac{10 a^4 b}{3 x^{3/2}}-\frac{10 a^3 b^2}{x}-\frac{20 a^2 b^3}{\sqrt{x}}+5 a b^4 \log (x)+2 b^5 \sqrt{x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^5/x^3,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**(1/2))**5/x**3,x)

[Out]

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

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Mathematica [A]  time = 0.0386744, size = 67, normalized size = 1.02 \[ -\frac{3 a^5+20 a^4 b \sqrt{x}+60 a^3 b^2 x+120 a^2 b^3 x^{3/2}-30 a b^4 x^2 \log (x)-12 b^5 x^{5/2}}{6 x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])^5/x^3,x]

[Out]

-(3*a^5 + 20*a^4*b*Sqrt[x] + 60*a^3*b^2*x + 120*a^2*b^3*x^(3/2) - 12*b^5*x^(5/2)
- 30*a*b^4*x^2*Log[x])/(6*x^2)

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Maple [A]  time = 0.005, size = 57, normalized size = 0.9 \[ -{\frac{{a}^{5}}{2\,{x}^{2}}}-{\frac{10\,{a}^{4}b}{3}{x}^{-{\frac{3}{2}}}}-10\,{\frac{{a}^{3}{b}^{2}}{x}}+5\,a{b}^{4}\ln \left ( x \right ) -20\,{\frac{{a}^{2}{b}^{3}}{\sqrt{x}}}+2\,{b}^{5}\sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/2))^5/x^3,x)

[Out]

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

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Maxima [A]  time = 1.43993, size = 77, normalized size = 1.17 \[ 5 \, a b^{4} \log \left (x\right ) + 2 \, b^{5} \sqrt{x} - \frac{120 \, a^{2} b^{3} x^{\frac{3}{2}} + 60 \, a^{3} b^{2} x + 20 \, a^{4} b \sqrt{x} + 3 \, a^{5}}{6 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^5/x^3,x, algorithm="maxima")

[Out]

5*a*b^4*log(x) + 2*b^5*sqrt(x) - 1/6*(120*a^2*b^3*x^(3/2) + 60*a^3*b^2*x + 20*a^
4*b*sqrt(x) + 3*a^5)/x^2

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Fricas [A]  time = 0.231575, size = 84, normalized size = 1.27 \[ \frac{60 \, a b^{4} x^{2} \log \left (\sqrt{x}\right ) - 60 \, a^{3} b^{2} x - 3 \, a^{5} + 4 \,{\left (3 \, b^{5} x^{2} - 30 \, a^{2} b^{3} x - 5 \, a^{4} b\right )} \sqrt{x}}{6 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^5/x^3,x, algorithm="fricas")

[Out]

1/6*(60*a*b^4*x^2*log(sqrt(x)) - 60*a^3*b^2*x - 3*a^5 + 4*(3*b^5*x^2 - 30*a^2*b^
3*x - 5*a^4*b)*sqrt(x))/x^2

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Sympy [A]  time = 2.43716, size = 65, normalized size = 0.98 \[ - \frac{a^{5}}{2 x^{2}} - \frac{10 a^{4} b}{3 x^{\frac{3}{2}}} - \frac{10 a^{3} b^{2}}{x} - \frac{20 a^{2} b^{3}}{\sqrt{x}} + 5 a b^{4} \log{\left (x \right )} + 2 b^{5} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/2))**5/x**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.218907, size = 78, normalized size = 1.18 \[ 5 \, a b^{4}{\rm ln}\left ({\left | x \right |}\right ) + 2 \, b^{5} \sqrt{x} - \frac{120 \, a^{2} b^{3} x^{\frac{3}{2}} + 60 \, a^{3} b^{2} x + 20 \, a^{4} b \sqrt{x} + 3 \, a^{5}}{6 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^5/x^3,x, algorithm="giac")

[Out]

5*a*b^4*ln(abs(x)) + 2*b^5*sqrt(x) - 1/6*(120*a^2*b^3*x^(3/2) + 60*a^3*b^2*x + 2
0*a^4*b*sqrt(x) + 3*a^5)/x^2