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

Optimal. Leaf size=42 \[ \frac{\sqrt{c x^2} \log (x)}{a x}-\frac{\sqrt{c x^2} \log (a+b x)}{a x} \]

[Out]

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

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Rubi [A]  time = 0.0219454, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{\sqrt{c x^2} \log (x)}{a x}-\frac{\sqrt{c x^2} \log (a+b x)}{a x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.0785, size = 32, normalized size = 0.76 \[ \frac{\sqrt{c x^{2}} \log{\left (x \right )}}{a x} - \frac{\sqrt{c x^{2}} \log{\left (a + b x \right )}}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0108631, size = 26, normalized size = 0.62 \[ \frac{c x (\log (x)-\log (a+b x))}{a \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 26, normalized size = 0.6 \[{\frac{\ln \left ( x \right ) -\ln \left ( bx+a \right ) }{ax}\sqrt{c{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.35115, size = 32, normalized size = 0.76 \[ -\frac{\sqrt{c} \log \left (b x + a\right )}{a} + \frac{\sqrt{c} \log \left (x\right )}{a} \]

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]

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

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Fricas [A]  time = 0.211578, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{c x^{2}} \log \left (\frac{x}{b x + a}\right )}{a x}, -\frac{2 \, \sqrt{-c} \arctan \left (\frac{2 \, b c x^{2} + a c x}{\sqrt{c x^{2}} a \sqrt{-c}}\right )}{a}\right ] \]

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)*log(x/(b*x + a))/(a*x), -2*sqrt(-c)*arctan((2*b*c*x^2 + a*c*x)/(sqr
t(c*x^2)*a*sqrt(-c)))/a]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

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]

Exception raised: TypeError