3.2127 \(\int \frac{\left (a+b \sqrt{x}\right )^2}{x^5} \, dx\)

Optimal. Leaf size=32 \[ -\frac{a^2}{4 x^4}-\frac{4 a b}{7 x^{7/2}}-\frac{b^2}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.0432035, antiderivative size = 32, 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 \[ -\frac{a^2}{4 x^4}-\frac{4 a b}{7 x^{7/2}}-\frac{b^2}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^2/x^5,x]

[Out]

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

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Rubi in Sympy [A]  time = 6.49635, size = 29, normalized size = 0.91 \[ - \frac{a^{2}}{4 x^{4}} - \frac{4 a b}{7 x^{\frac{7}{2}}} - \frac{b^{2}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0110346, size = 28, normalized size = 0.88 \[ -\frac{21 a^2+48 a b \sqrt{x}+28 b^2 x}{84 x^4} \]

Antiderivative was successfully verified.

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

[Out]

-(21*a^2 + 48*a*b*Sqrt[x] + 28*b^2*x)/(84*x^4)

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Maple [A]  time = 0.002, size = 25, normalized size = 0.8 \[ -{\frac{{a}^{2}}{4\,{x}^{4}}}-{\frac{4\,ab}{7}{x}^{-{\frac{7}{2}}}}-{\frac{{b}^{2}}{3\,{x}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/2))^2/x^5,x)

[Out]

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

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Maxima [A]  time = 1.41836, size = 32, normalized size = 1. \[ -\frac{28 \, b^{2} x + 48 \, a b \sqrt{x} + 21 \, a^{2}}{84 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/84*(28*b^2*x + 48*a*b*sqrt(x) + 21*a^2)/x^4

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Fricas [A]  time = 0.237111, size = 32, normalized size = 1. \[ -\frac{28 \, b^{2} x + 48 \, a b \sqrt{x} + 21 \, a^{2}}{84 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/84*(28*b^2*x + 48*a*b*sqrt(x) + 21*a^2)/x^4

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Sympy [A]  time = 4.88277, size = 29, normalized size = 0.91 \[ - \frac{a^{2}}{4 x^{4}} - \frac{4 a b}{7 x^{\frac{7}{2}}} - \frac{b^{2}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.214902, size = 32, normalized size = 1. \[ -\frac{28 \, b^{2} x + 48 \, a b \sqrt{x} + 21 \, a^{2}}{84 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/84*(28*b^2*x + 48*a*b*sqrt(x) + 21*a^2)/x^4