3.433 \(\int \frac{a+b x}{x^{3/2}} \, dx\)

Optimal. Leaf size=17 \[ 2 b \sqrt{x}-\frac{2 a}{\sqrt{x}} \]

[Out]

(-2*a)/Sqrt[x] + 2*b*Sqrt[x]

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Rubi [A]  time = 0.012567, antiderivative size = 17, 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 \[ 2 b \sqrt{x}-\frac{2 a}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)/x^(3/2),x]

[Out]

(-2*a)/Sqrt[x] + 2*b*Sqrt[x]

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Rubi in Sympy [A]  time = 2.31643, size = 15, normalized size = 0.88 \[ - \frac{2 a}{\sqrt{x}} + 2 b \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a/sqrt(x) + 2*b*sqrt(x)

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Mathematica [A]  time = 0.00537955, size = 14, normalized size = 0.82 \[ \frac{2 (b x-a)}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)/x^(3/2),x]

[Out]

(2*(-a + b*x))/Sqrt[x]

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Maple [A]  time = 0.004, size = 12, normalized size = 0.7 \[ -2\,{\frac{-bx+a}{\sqrt{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/x^(3/2),x)

[Out]

-2*(-b*x+a)/x^(1/2)

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Maxima [A]  time = 1.3457, size = 18, normalized size = 1.06 \[ 2 \, b \sqrt{x} - \frac{2 \, a}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/x^(3/2),x, algorithm="maxima")

[Out]

2*b*sqrt(x) - 2*a/sqrt(x)

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Fricas [A]  time = 0.210057, size = 16, normalized size = 0.94 \[ \frac{2 \,{\left (b x - a\right )}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/x^(3/2),x, algorithm="fricas")

[Out]

2*(b*x - a)/sqrt(x)

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Sympy [A]  time = 1.31541, size = 15, normalized size = 0.88 \[ - \frac{2 a}{\sqrt{x}} + 2 b \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)/x**(3/2),x)

[Out]

-2*a/sqrt(x) + 2*b*sqrt(x)

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GIAC/XCAS [A]  time = 0.202568, size = 18, normalized size = 1.06 \[ 2 \, b \sqrt{x} - \frac{2 \, a}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/x^(3/2),x, algorithm="giac")

[Out]

2*b*sqrt(x) - 2*a/sqrt(x)