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

Optimal. Leaf size=27 \[ \frac{b x \log (x)}{\sqrt{c x^2}}-\frac{a}{\sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0178087, antiderivative size = 27, 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 \[ \frac{b x \log (x)}{\sqrt{c x^2}}-\frac{a}{\sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.06173, size = 31, normalized size = 1.15 \[ - \frac{a \sqrt{c x^{2}}}{c x^{2}} + \frac{b \sqrt{c x^{2}} \log{\left (x \right )}}{c x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0102027, size = 23, normalized size = 0.85 \[ \frac{c x^2 (b x \log (x)-a)}{\left (c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 18, normalized size = 0.7 \[{(b\ln \left ( x \right ) x-a){\frac{1}{\sqrt{c{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/x/(c*x^2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.35178, size = 23, normalized size = 0.85 \[ \frac{b \log \left (x\right )}{\sqrt{c}} - \frac{a}{\sqrt{c} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*log(x)/sqrt(c) - a/(sqrt(c)*x)

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Fricas [A]  time = 0.221174, size = 31, normalized size = 1.15 \[ \frac{\sqrt{c x^{2}}{\left (b x \log \left (x\right ) - a\right )}}{c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(c*x^2)*(b*x*log(x) - a)/(c*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.219873, size = 63, normalized size = 2.33 \[ -\frac{b{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2}} \right |}\right ) - \frac{2 \, a \sqrt{c}}{\sqrt{c} x - \sqrt{c x^{2}}}}{\sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(b*ln(abs(-sqrt(c)*x + sqrt(c*x^2))) - 2*a*sqrt(c)/(sqrt(c)*x - sqrt(c*x^2)))/s
qrt(c)