3.785 \(\int \frac{a+b x}{x^2 \sqrt{c x^2}} \, dx\)

Optimal. Leaf size=26 \[ -\frac{(a+b x)^2}{2 a x \sqrt{c x^2}} \]

[Out]

-(a + b*x)^2/(2*a*x*Sqrt[c*x^2])

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Rubi [A]  time = 0.0138498, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{(a+b x)^2}{2 a x \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)/(x^2*Sqrt[c*x^2]),x]

[Out]

-(a + b*x)^2/(2*a*x*Sqrt[c*x^2])

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Rubi in Sympy [A]  time = 9.06509, size = 32, normalized size = 1.23 \[ - \frac{a \sqrt{c x^{2}}}{2 c x^{3}} - \frac{b \sqrt{c x^{2}}}{c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*sqrt(c*x**2)/(2*c*x**3) - b*sqrt(c*x**2)/(c*x**2)

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Mathematica [A]  time = 0.00900688, size = 23, normalized size = 0.88 \[ \frac{c x (-a-2 b x)}{2 \left (c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(c*x*(-a - 2*b*x))/(2*(c*x^2)^(3/2))

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Maple [A]  time = 0.004, size = 19, normalized size = 0.7 \[ -{\frac{2\,bx+a}{2\,x}{\frac{1}{\sqrt{c{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/x^2/(c*x^2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.34348, size = 26, normalized size = 1. \[ -\frac{b}{\sqrt{c} x} - \frac{a}{2 \, \sqrt{c} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b/(sqrt(c)*x) - 1/2*a/(sqrt(c)*x^2)

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Fricas [A]  time = 0.208279, size = 28, normalized size = 1.08 \[ -\frac{\sqrt{c x^{2}}{\left (2 \, b x + a\right )}}{2 \, c x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*sqrt(c*x^2)*(2*b*x + a)/(c*x^3)

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Sympy [A]  time = 2.0305, size = 31, normalized size = 1.19 \[ - \frac{a}{2 \sqrt{c} x \sqrt{x^{2}}} - \frac{b}{\sqrt{c} \sqrt{x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a/(2*sqrt(c)*x*sqrt(x**2)) - b/(sqrt(c)*sqrt(x**2))

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GIAC/XCAS [A]  time = 0.574227, size = 4, normalized size = 0.15 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x