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

Optimal. Leaf size=65 \[ -\frac{\sqrt{c x^2} \log (a+b x)}{a^2 x}+\frac{\sqrt{c x^2} \log (x)}{a^2 x}+\frac{\sqrt{c x^2}}{a x (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.0462673, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{\sqrt{c x^2} \log (a+b x)}{a^2 x}+\frac{\sqrt{c x^2} \log (x)}{a^2 x}+\frac{\sqrt{c x^2}}{a x (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.7456, size = 53, normalized size = 0.82 \[ \frac{\sqrt{c x^{2}}}{a x \left (a + b x\right )} + \frac{\sqrt{c x^{2}} \log{\left (x \right )}}{a^{2} x} - \frac{\sqrt{c x^{2}} \log{\left (a + b x \right )}}{a^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0244368, size = 45, normalized size = 0.69 \[ \frac{c x (\log (x) (a+b x)-(a+b x) \log (a+b x)+a)}{a^2 \sqrt{c x^2} (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 52, normalized size = 0.8 \[{\frac{b\ln \left ( x \right ) x-b\ln \left ( bx+a \right ) x+a\ln \left ( x \right ) -a\ln \left ( bx+a \right ) +a}{x{a}^{2} \left ( bx+a \right ) }\sqrt{c{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.36099, size = 51, normalized size = 0.78 \[ \frac{\sqrt{c}}{a b x + a^{2}} - \frac{\sqrt{c} \log \left (b x + a\right )}{a^{2}} + \frac{\sqrt{c} \log \left (x\right )}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.220959, size = 57, normalized size = 0.88 \[ \frac{\sqrt{c x^{2}}{\left ({\left (b x + a\right )} \log \left (\frac{x}{b x + a}\right ) + a\right )}}{a^{2} b x^{2} + a^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(c*x^2)*((b*x + a)*log(x/(b*x + a)) + a)/(a^2*b*x^2 + a^3*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(c*x**2)/(x**2*(a + b*x)**2), 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)^2*x^2),x, algorithm="giac")

[Out]

Exception raised: TypeError