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

Optimal. Leaf size=28 \[ \frac{a \sqrt{c x^2} \log (x)}{x}+b \sqrt{c x^2} \]

[Out]

b*Sqrt[c*x^2] + (a*Sqrt[c*x^2]*Log[x])/x

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Rubi [A]  time = 0.0156603, antiderivative size = 28, 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 \[ \frac{a \sqrt{c x^2} \log (x)}{x}+b \sqrt{c x^2} \]

Antiderivative was successfully verified.

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

[Out]

b*Sqrt[c*x^2] + (a*Sqrt[c*x^2]*Log[x])/x

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a \sqrt{c x^{2}} \log{\left (x \right )}}{x} + \frac{\sqrt{c x^{2}} \int b\, dx}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*sqrt(c*x**2)*log(x)/x + sqrt(c*x**2)*Integral(b, x)/x

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Mathematica [A]  time = 0.00687355, size = 20, normalized size = 0.71 \[ \frac{c x (a \log (x)+b x)}{\sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(c*x*(b*x + a*Log[x]))/Sqrt[c*x^2]

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Maple [A]  time = 0.02, size = 20, normalized size = 0.7 \[{\frac{bx+a\ln \left ( x \right ) }{x}\sqrt{c{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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^2,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 0.212534, size = 26, normalized size = 0.93 \[ \frac{\sqrt{c x^{2}}{\left (b x + a \log \left (x\right )\right )}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(c*x^2)*(b*x + a*log(x))/x

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2}} \left (a + b x\right )}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(c*x**2)*(a + b*x)/x**2, x)

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GIAC/XCAS [A]  time = 0.20913, size = 23, normalized size = 0.82 \[{\left (b x{\rm sign}\left (x\right ) + a{\rm ln}\left ({\left | x \right |}\right ){\rm sign}\left (x\right )\right )} \sqrt{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*x*sign(x) + a*ln(abs(x))*sign(x))*sqrt(c)