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

Optimal. Leaf size=29 \[ \frac{a x \log (x)}{\sqrt{c x^2}}+\frac{b x^2}{\sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0149554, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{a x \log (x)}{\sqrt{c x^2}}+\frac{b x^2}{\sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)/Sqrt[c*x^2],x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00582401, size = 19, normalized size = 0.66 \[ \frac{x (a \log (x)+b x)}{\sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(b*x + a*Log[x]))/Sqrt[c*x^2]

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Maple [A]  time = 0.005, size = 18, normalized size = 0.6 \[{x \left ( bx+a\ln \left ( x \right ) \right ){\frac{1}{\sqrt{c{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/(c*x^2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.33998, size = 27, normalized size = 0.93 \[ \frac{a \log \left (x\right )}{\sqrt{c}} + \frac{\sqrt{c x^{2}} b}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*log(x)/sqrt(c) + sqrt(c*x^2)*b/c

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Fricas [A]  time = 0.214175, size = 30, normalized size = 1.03 \[ \frac{\sqrt{c x^{2}}{\left (b x + a \log \left (x\right )\right )}}{c x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(c*x^2)*(b*x + a*log(x))/(c*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a + b*x)/sqrt(c*x**2), x)

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GIAC/XCAS [A]  time = 0.212215, size = 47, normalized size = 1.62 \[ -\frac{a{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2}} \right |}\right )}{\sqrt{c}} + \frac{\sqrt{c x^{2}} b}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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