3.434 \(\int \frac{a+b x}{x^{5/2}} \, dx\)

Optimal. Leaf size=19 \[ -\frac{2 a}{3 x^{3/2}}-\frac{2 b}{\sqrt{x}} \]

[Out]

(-2*a)/(3*x^(3/2)) - (2*b)/Sqrt[x]

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Rubi [A]  time = 0.0131644, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{2 a}{3 x^{3/2}}-\frac{2 b}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)/x^(5/2),x]

[Out]

(-2*a)/(3*x^(3/2)) - (2*b)/Sqrt[x]

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Rubi in Sympy [A]  time = 2.3363, size = 19, normalized size = 1. \[ - \frac{2 a}{3 x^{\frac{3}{2}}} - \frac{2 b}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a/(3*x**(3/2)) - 2*b/sqrt(x)

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Mathematica [A]  time = 0.00556482, size = 15, normalized size = 0.79 \[ -\frac{2 (a+3 b x)}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)/x^(5/2),x]

[Out]

(-2*(a + 3*b*x))/(3*x^(3/2))

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Maple [A]  time = 0.003, size = 12, normalized size = 0.6 \[ -{\frac{6\,bx+2\,a}{3}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/x^(5/2),x)

[Out]

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

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Maxima [A]  time = 1.34283, size = 15, normalized size = 0.79 \[ -\frac{2 \,{\left (3 \, b x + a\right )}}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(3*b*x + a)/x^(3/2)

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Fricas [A]  time = 0.206976, size = 15, normalized size = 0.79 \[ -\frac{2 \,{\left (3 \, b x + a\right )}}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(3*b*x + a)/x^(3/2)

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Sympy [A]  time = 2.06573, size = 19, normalized size = 1. \[ - \frac{2 a}{3 x^{\frac{3}{2}}} - \frac{2 b}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)/x**(5/2),x)

[Out]

-2*a/(3*x**(3/2)) - 2*b/sqrt(x)

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GIAC/XCAS [A]  time = 0.2063, size = 15, normalized size = 0.79 \[ -\frac{2 \,{\left (3 \, b x + a\right )}}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(3*b*x + a)/x^(3/2)