3.915 \(\int \frac{1}{x \sqrt{c x^2} (a+b x)^2} \, dx\)

Optimal. Leaf size=78 \[ -\frac{2 b x \log (x)}{a^3 \sqrt{c x^2}}+\frac{2 b x \log (a+b x)}{a^3 \sqrt{c x^2}}-\frac{b x}{a^2 \sqrt{c x^2} (a+b x)}-\frac{1}{a^2 \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0614447, antiderivative size = 78, 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 \[ -\frac{2 b x \log (x)}{a^3 \sqrt{c x^2}}+\frac{2 b x \log (a+b x)}{a^3 \sqrt{c x^2}}-\frac{b x}{a^2 \sqrt{c x^2} (a+b x)}-\frac{1}{a^2 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 23.1525, size = 85, normalized size = 1.09 \[ - \frac{b \sqrt{c x^{2}}}{a^{2} c x \left (a + b x\right )} - \frac{\sqrt{c x^{2}}}{a^{2} c x^{2}} - \frac{2 b \sqrt{c x^{2}} \log{\left (x \right )}}{a^{3} c x} + \frac{2 b \sqrt{c x^{2}} \log{\left (a + b x \right )}}{a^{3} c x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0415879, size = 60, normalized size = 0.77 \[ \frac{c x^2 (-a (a+2 b x)-2 b x \log (x) (a+b x)+2 b x (a+b x) \log (a+b x))}{a^3 \left (c x^2\right )^{3/2} (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 71, normalized size = 0.9 \[ -{\frac{2\,{b}^{2}\ln \left ( x \right ){x}^{2}-2\,{b}^{2}\ln \left ( bx+a \right ){x}^{2}+2\,ab\ln \left ( x \right ) x-2\,\ln \left ( bx+a \right ) xab+2\,abx+{a}^{2}}{{a}^{3} \left ( bx+a \right ) }{\frac{1}{\sqrt{c{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*b^2*ln(x)*x^2-2*b^2*ln(b*x+a)*x^2+2*a*b*ln(x)*x-2*ln(b*x+a)*x*a*b+2*a*b*x+a^
2)/(c*x^2)^(1/2)/a^3/(b*x+a)

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Maxima [A]  time = 1.35887, size = 77, normalized size = 0.99 \[ -\frac{2 \, b x + a}{a^{2} b \sqrt{c} x^{2} + a^{3} \sqrt{c} x} + \frac{2 \, b \log \left (b x + a\right )}{a^{3} \sqrt{c}} - \frac{2 \, b \log \left (x\right )}{a^{3} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^2)*(b*x + a)^2*x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.224598, size = 84, normalized size = 1.08 \[ -\frac{{\left (2 \, a b x + a^{2} - 2 \,{\left (b^{2} x^{2} + a b x\right )} \log \left (\frac{b x + a}{x}\right )\right )} \sqrt{c x^{2}}}{a^{3} b c x^{3} + a^{4} c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^2)*(b*x + a)^2*x),x, algorithm="fricas")

[Out]

-(2*a*b*x + a^2 - 2*(b^2*x^2 + a*b*x)*log((b*x + a)/x))*sqrt(c*x^2)/(a^3*b*c*x^3
+ a^4*c*x^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \sqrt{c x^{2}} \left (a + b x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x*sqrt(c*x**2)*(a + b*x)**2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{2}}{\left (b x + a\right )}^{2} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^2)*(b*x + a)^2*x),x, algorithm="giac")

[Out]

integrate(1/(sqrt(c*x^2)*(b*x + a)^2*x), x)