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

Optimal. Leaf size=103 \[ -\frac{a+b x}{a x \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b \log (x) (a+b x)}{a^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b (a+b x) \log (a+b x)}{a^2 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.103942, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{a+b x}{a x \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b \log (x) (a+b x)}{a^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b (a+b x) \log (a+b x)}{a^2 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 19.5447, size = 95, normalized size = 0.92 \[ - \frac{b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a^{2} \left (a + b x\right )} + \frac{b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (a + b x \right )}}{a^{2} \left (a + b x\right )} - \frac{\sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{a^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0327378, size = 41, normalized size = 0.4 \[ -\frac{(a+b x) (-b x \log (a+b x)+a+b x \log (x))}{a^2 x \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.015, size = 40, normalized size = 0.4 \[ -{\frac{ \left ( bx+a \right ) \left ( b\ln \left ( x \right ) x-b\ln \left ( bx+a \right ) x+a \right ) }{{a}^{2}x}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/((b*x+a)^2)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228003, size = 35, normalized size = 0.34 \[ \frac{b x \log \left (b x + a\right ) - b x \log \left (x\right ) - a}{a^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.34171, size = 19, normalized size = 0.18 \[ - \frac{1}{a x} + \frac{b \left (- \log{\left (x \right )} + \log{\left (\frac{a}{b} + x \right )}\right )}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(a*x) + b*(-log(x) + log(a/b + x))/a**2

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GIAC/XCAS [A]  time = 0.21042, size = 50, normalized size = 0.49 \[{\left (\frac{b{\rm ln}\left ({\left | b x + a \right |}\right )}{a^{2}} - \frac{b{\rm ln}\left ({\left | x \right |}\right )}{a^{2}} - \frac{1}{a x}\right )}{\rm sign}\left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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