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

Optimal. Leaf size=26 \[ -\frac{\sqrt{c x^2} (a+b x)^2}{2 a x^3} \]

[Out]

-(Sqrt[c*x^2]*(a + b*x)^2)/(2*a*x^3)

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Rubi [A]  time = 0.0146623, 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 \[ -\frac{\sqrt{c x^2} (a+b x)^2}{2 a x^3} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[c*x^2]*(a + b*x)^2)/(2*a*x^3)

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Rubi in Sympy [A]  time = 8.29241, size = 29, normalized size = 1.12 \[ - \frac{a \sqrt{c x^{2}}}{2 x^{3}} - \frac{b \sqrt{c x^{2}}}{x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00663325, size = 22, normalized size = 0.85 \[ -\frac{\sqrt{c x^2} (a+2 b x)}{2 x^3} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[c*x^2]*(a + 2*b*x))/(2*x^3)

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Maple [A]  time = 0.006, size = 19, normalized size = 0.7 \[ -{\frac{2\,bx+a}{2\,{x}^{3}}\sqrt{c{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 0.207393, size = 24, normalized size = 0.92 \[ -\frac{\sqrt{c x^{2}}{\left (2 \, b x + a\right )}}{2 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*sqrt(c*x^2)*(2*b*x + a)/x^3

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Sympy [A]  time = 1.8043, size = 36, normalized size = 1.38 \[ - \frac{a \sqrt{c} \sqrt{x^{2}}}{2 x^{3}} - \frac{b \sqrt{c} \sqrt{x^{2}}}{x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.211441, size = 26, normalized size = 1. \[ -\frac{{\left (2 \, b x{\rm sign}\left (x\right ) + a{\rm sign}\left (x\right )\right )} \sqrt{c}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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