3.791 \(\int \frac{a+b x}{\left (c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=29 \[ -\frac{(a+b x)^2}{2 a c x \sqrt{c x^2}} \]

[Out]

-(a + b*x)^2/(2*a*c*x*Sqrt[c*x^2])

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Rubi [A]  time = 0.0157361, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{(a+b x)^2}{2 a c x \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)/(c*x^2)^(3/2),x]

[Out]

-(a + b*x)^2/(2*a*c*x*Sqrt[c*x^2])

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Rubi in Sympy [A]  time = 6.38163, size = 36, normalized size = 1.24 \[ - \frac{a \sqrt{c x^{2}}}{2 c^{2} x^{3}} - \frac{b \sqrt{c x^{2}}}{c^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00393579, size = 22, normalized size = 0.76 \[ \frac{x (-a-2 b x)}{2 \left (c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)/(c*x^2)^(3/2),x]

[Out]

(x*(-a - 2*b*x))/(2*(c*x^2)^(3/2))

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Maple [A]  time = 0.004, size = 17, normalized size = 0.6 \[ -{\frac{x \left ( 2\,bx+a \right ) }{2} \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/(c*x^2)^(3/2),x)

[Out]

-1/2*x*(2*b*x+a)/(c*x^2)^(3/2)

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Maxima [A]  time = 1.34866, size = 31, normalized size = 1.07 \[ -\frac{b}{\sqrt{c x^{2}} c} - \frac{a}{2 \, c^{\frac{3}{2}} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b/(sqrt(c*x^2)*c) - 1/2*a/(c^(3/2)*x^2)

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Fricas [A]  time = 0.210717, size = 28, normalized size = 0.97 \[ -\frac{\sqrt{c x^{2}}{\left (2 \, b x + a\right )}}{2 \, c^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*sqrt(c*x^2)*(2*b*x + a)/(c^2*x^3)

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Sympy [A]  time = 2.03309, size = 34, normalized size = 1.17 \[ - \frac{a x}{2 c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} - \frac{b x^{2}}{c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.538674, size = 4, normalized size = 0.14 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x