3.2137 \(\int \frac{\left (a+b \sqrt{x}\right )^3}{x^5} \, dx\)

Optimal. Leaf size=45 \[ -\frac{a^3}{4 x^4}-\frac{6 a^2 b}{7 x^{7/2}}-\frac{a b^2}{x^3}-\frac{2 b^3}{5 x^{5/2}} \]

[Out]

-a^3/(4*x^4) - (6*a^2*b)/(7*x^(7/2)) - (a*b^2)/x^3 - (2*b^3)/(5*x^(5/2))

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Rubi [A]  time = 0.0550816, antiderivative size = 45, 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 \[ -\frac{a^3}{4 x^4}-\frac{6 a^2 b}{7 x^{7/2}}-\frac{a b^2}{x^3}-\frac{2 b^3}{5 x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^3/x^5,x]

[Out]

-a^3/(4*x^4) - (6*a^2*b)/(7*x^(7/2)) - (a*b^2)/x^3 - (2*b^3)/(5*x^(5/2))

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Rubi in Sympy [A]  time = 8.58788, size = 42, normalized size = 0.93 \[ - \frac{a^{3}}{4 x^{4}} - \frac{6 a^{2} b}{7 x^{\frac{7}{2}}} - \frac{a b^{2}}{x^{3}} - \frac{2 b^{3}}{5 x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**3/(4*x**4) - 6*a**2*b/(7*x**(7/2)) - a*b**2/x**3 - 2*b**3/(5*x**(5/2))

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Mathematica [A]  time = 0.0160295, size = 41, normalized size = 0.91 \[ -\frac{35 a^3+120 a^2 b \sqrt{x}+140 a b^2 x+56 b^3 x^{3/2}}{140 x^4} \]

Antiderivative was successfully verified.

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

[Out]

-(35*a^3 + 120*a^2*b*Sqrt[x] + 140*a*b^2*x + 56*b^3*x^(3/2))/(140*x^4)

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Maple [A]  time = 0.003, size = 36, normalized size = 0.8 \[ -{\frac{{a}^{3}}{4\,{x}^{4}}}-{\frac{6\,{a}^{2}b}{7}{x}^{-{\frac{7}{2}}}}-{\frac{a{b}^{2}}{{x}^{3}}}-{\frac{2\,{b}^{3}}{5}{x}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/2))^3/x^5,x)

[Out]

-1/4*a^3/x^4-6/7*a^2*b/x^(7/2)-a*b^2/x^3-2/5*b^3/x^(5/2)

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Maxima [A]  time = 1.43958, size = 47, normalized size = 1.04 \[ -\frac{56 \, b^{3} x^{\frac{3}{2}} + 140 \, a b^{2} x + 120 \, a^{2} b \sqrt{x} + 35 \, a^{3}}{140 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/140*(56*b^3*x^(3/2) + 140*a*b^2*x + 120*a^2*b*sqrt(x) + 35*a^3)/x^4

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Fricas [A]  time = 0.241591, size = 49, normalized size = 1.09 \[ -\frac{140 \, a b^{2} x + 35 \, a^{3} + 8 \,{\left (7 \, b^{3} x + 15 \, a^{2} b\right )} \sqrt{x}}{140 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/140*(140*a*b^2*x + 35*a^3 + 8*(7*b^3*x + 15*a^2*b)*sqrt(x))/x^4

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Sympy [A]  time = 5.04537, size = 42, normalized size = 0.93 \[ - \frac{a^{3}}{4 x^{4}} - \frac{6 a^{2} b}{7 x^{\frac{7}{2}}} - \frac{a b^{2}}{x^{3}} - \frac{2 b^{3}}{5 x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**3/(4*x**4) - 6*a**2*b/(7*x**(7/2)) - a*b**2/x**3 - 2*b**3/(5*x**(5/2))

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GIAC/XCAS [A]  time = 0.215818, size = 47, normalized size = 1.04 \[ -\frac{56 \, b^{3} x^{\frac{3}{2}} + 140 \, a b^{2} x + 120 \, a^{2} b \sqrt{x} + 35 \, a^{3}}{140 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/140*(56*b^3*x^(3/2) + 140*a*b^2*x + 120*a^2*b*sqrt(x) + 35*a^3)/x^4