3.188 \(\int \frac{1}{x^3 \sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=142 \[ \frac{b (a+b x)}{a^2 x \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a+b x}{2 a x^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^2 \log (x) (a+b x)}{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b^2 (a+b x) \log (a+b x)}{a^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.130204, antiderivative size = 142, 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 \[ \frac{b (a+b x)}{a^2 x \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a+b x}{2 a x^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^2 \log (x) (a+b x)}{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b^2 (a+b x) \log (a+b x)}{a^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

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

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Rubi in Sympy [A]  time = 25.6482, size = 136, normalized size = 0.96 \[ - \frac{2 a + 2 b x}{4 a x^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a^{3} \left (a + b x\right )} - \frac{b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (a + b x \right )}}{a^{3} \left (a + b x\right )} + \frac{b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{a^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/((b*x+a)**2)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0368905, size = 59, normalized size = 0.42 \[ -\frac{(a+b x) \left (2 b^2 x^2 \log (a+b x)+a (a-2 b x)-2 b^2 x^2 \log (x)\right )}{2 a^3 x^2 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

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

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Maple [A]  time = 0.009, size = 58, normalized size = 0.4 \[{\frac{ \left ( bx+a \right ) \left ( 2\,{b}^{2}\ln \left ( x \right ){x}^{2}-2\,{b}^{2}\ln \left ( bx+a \right ){x}^{2}+2\,abx-{a}^{2} \right ) }{2\,{x}^{2}{a}^{3}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/((b*x+a)^2)^(1/2),x)

[Out]

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

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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^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.225608, size = 55, normalized size = 0.39 \[ -\frac{2 \, b^{2} x^{2} \log \left (b x + a\right ) - 2 \, b^{2} x^{2} \log \left (x\right ) - 2 \, a b x + a^{2}}{2 \, a^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt((b*x + a)^2)*x^3),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.4459, size = 31, normalized size = 0.22 \[ \frac{- a + 2 b x}{2 a^{2} x^{2}} + \frac{b^{2} \left (\log{\left (x \right )} - \log{\left (\frac{a}{b} + x \right )}\right )}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/((b*x+a)**2)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.20813, size = 73, normalized size = 0.51 \[ -\frac{1}{2} \,{\left (\frac{2 \, b^{2}{\rm ln}\left ({\left | b x + a \right |}\right )}{a^{3}} - \frac{2 \, b^{2}{\rm ln}\left ({\left | x \right |}\right )}{a^{3}} - \frac{2 \, a b x - a^{2}}{a^{3} x^{2}}\right )}{\rm sign}\left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt((b*x + a)^2)*x^3),x, algorithm="giac")

[Out]

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