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

Optimal. Leaf size=41 \[ -\frac{a}{3 c x^2 \sqrt{c x^2}}-\frac{b}{2 c x \sqrt{c x^2}} \]

[Out]

-a/(3*c*x^2*Sqrt[c*x^2]) - b/(2*c*x*Sqrt[c*x^2])

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Rubi [A]  time = 0.0221236, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{a}{3 c x^2 \sqrt{c x^2}}-\frac{b}{2 c x \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

-a/(3*c*x^2*Sqrt[c*x^2]) - b/(2*c*x*Sqrt[c*x^2])

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Rubi in Sympy [A]  time = 10.1134, size = 37, normalized size = 0.9 \[ - \frac{a \sqrt{c x^{2}}}{3 c^{2} x^{4}} - \frac{b \sqrt{c x^{2}}}{2 c^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*sqrt(c*x**2)/(3*c**2*x**4) - b*sqrt(c*x**2)/(2*c**2*x**3)

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Mathematica [A]  time = 0.0112656, size = 25, normalized size = 0.61 \[ \frac{c x^2 (-2 a-3 b x)}{6 \left (c x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(c*x^2*(-2*a - 3*b*x))/(6*(c*x^2)^(5/2))

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Maple [A]  time = 0.004, size = 18, normalized size = 0.4 \[ -{\frac{3\,bx+2\,a}{6} \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/x/(c*x^2)^(3/2),x)

[Out]

-1/6*(3*b*x+2*a)/(c*x^2)^(3/2)

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Maxima [A]  time = 1.33946, size = 26, normalized size = 0.63 \[ -\frac{b}{2 \, c^{\frac{3}{2}} x^{2}} - \frac{a}{3 \, c^{\frac{3}{2}} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.21, size = 31, normalized size = 0.76 \[ -\frac{\sqrt{c x^{2}}{\left (3 \, b x + 2 \, a\right )}}{6 \, c^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*sqrt(c*x^2)*(3*b*x + 2*a)/(c^2*x^4)

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Sympy [A]  time = 2.43401, size = 32, normalized size = 0.78 \[ - \frac{a}{3 c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} - \frac{b x}{2 c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{b x + a}{\left (c x^{2}\right )^{\frac{3}{2}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x + a)/((c*x^2)^(3/2)*x), x)