3.787 \(\int \frac{a+b x}{x^4 \sqrt{c x^2}} \, dx\)

Optimal. Leaf size=35 \[ -\frac{a}{4 x^3 \sqrt{c x^2}}-\frac{b}{3 x^2 \sqrt{c x^2}} \]

[Out]

-a/(4*x^3*Sqrt[c*x^2]) - b/(3*x^2*Sqrt[c*x^2])

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Rubi [A]  time = 0.0200505, antiderivative size = 35, 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 \[ -\frac{a}{4 x^3 \sqrt{c x^2}}-\frac{b}{3 x^2 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)/(x^4*Sqrt[c*x^2]),x]

[Out]

-a/(4*x^3*Sqrt[c*x^2]) - b/(3*x^2*Sqrt[c*x^2])

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Rubi in Sympy [A]  time = 9.19481, size = 34, normalized size = 0.97 \[ - \frac{a \sqrt{c x^{2}}}{4 c x^{5}} - \frac{b \sqrt{c x^{2}}}{3 c x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)/x**4/(c*x**2)**(1/2),x)

[Out]

-a*sqrt(c*x**2)/(4*c*x**5) - b*sqrt(c*x**2)/(3*c*x**4)

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Mathematica [A]  time = 0.00840179, size = 24, normalized size = 0.69 \[ \frac{-3 a-4 b x}{12 x^3 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)/(x^4*Sqrt[c*x^2]),x]

[Out]

(-3*a - 4*b*x)/(12*x^3*Sqrt[c*x^2])

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Maple [A]  time = 0.004, size = 21, normalized size = 0.6 \[ -{\frac{4\,bx+3\,a}{12\,{x}^{3}}{\frac{1}{\sqrt{c{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/x^4/(c*x^2)^(1/2),x)

[Out]

-1/12*(4*b*x+3*a)/x^3/(c*x^2)^(1/2)

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Maxima [A]  time = 1.34434, size = 26, normalized size = 0.74 \[ -\frac{b}{3 \, \sqrt{c} x^{3}} - \frac{a}{4 \, \sqrt{c} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/(sqrt(c*x^2)*x^4),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.210393, size = 31, normalized size = 0.89 \[ -\frac{\sqrt{c x^{2}}{\left (4 \, b x + 3 \, a\right )}}{12 \, c x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/(sqrt(c*x^2)*x^4),x, algorithm="fricas")

[Out]

-1/12*sqrt(c*x^2)*(4*b*x + 3*a)/(c*x^5)

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Sympy [A]  time = 2.75897, size = 37, normalized size = 1.06 \[ - \frac{a}{4 \sqrt{c} x^{3} \sqrt{x^{2}}} - \frac{b}{3 \sqrt{c} x^{2} \sqrt{x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)/x**4/(c*x**2)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.519624, size = 4, normalized size = 0.11 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)/(sqrt(c*x^2)*x^4),x, algorithm="giac")

[Out]

sage0*x