3.189 \(\int \frac{1}{x^4 \sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=181 \[ \frac{b (a+b x)}{2 a^2 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a+b x}{3 a x^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b^3 \log (x) (a+b x)}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^3 (a+b x) \log (a+b x)}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b^2 (a+b x)}{a^3 x \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

-(a + b*x)/(3*a*x^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (b*(a + b*x))/(2*a^2*x^2*Sq
rt[a^2 + 2*a*b*x + b^2*x^2]) - (b^2*(a + b*x))/(a^3*x*Sqrt[a^2 + 2*a*b*x + b^2*x
^2]) - (b^3*(a + b*x)*Log[x])/(a^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (b^3*(a + b*
x)*Log[a + b*x])/(a^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.156348, antiderivative size = 181, 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 \[ \frac{b (a+b x)}{2 a^2 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a+b x}{3 a x^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b^3 \log (x) (a+b x)}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^3 (a+b x) \log (a+b x)}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b^2 (a+b x)}{a^3 x \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

-(a + b*x)/(3*a*x^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (b*(a + b*x))/(2*a^2*x^2*Sq
rt[a^2 + 2*a*b*x + b^2*x^2]) - (b^2*(a + b*x))/(a^3*x*Sqrt[a^2 + 2*a*b*x + b^2*x
^2]) - (b^3*(a + b*x)*Log[x])/(a^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (b^3*(a + b*
x)*Log[a + b*x])/(a^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi in Sympy [A]  time = 33.1054, size = 177, normalized size = 0.98 \[ - \frac{2 a + 2 b x}{6 a x^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{b \left (2 a + 2 b x\right )}{4 a^{2} x^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} - \frac{b^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a^{4} \left (a + b x\right )} + \frac{b^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (a + b x \right )}}{a^{4} \left (a + b x\right )} - \frac{b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{a^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/((b*x+a)**2)**(1/2),x)

[Out]

-(2*a + 2*b*x)/(6*a*x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2)) + b*(2*a + 2*b*x)/(4*
a**2*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)) - b**3*sqrt(a**2 + 2*a*b*x + b**2*x*
*2)*log(x)/(a**4*(a + b*x)) + b**3*sqrt(a**2 + 2*a*b*x + b**2*x**2)*log(a + b*x)
/(a**4*(a + b*x)) - b**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(a**4*x)

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Mathematica [A]  time = 0.042839, size = 72, normalized size = 0.4 \[ -\frac{(a+b x) \left (a \left (2 a^2-3 a b x+6 b^2 x^2\right )-6 b^3 x^3 \log (a+b x)+6 b^3 x^3 \log (x)\right )}{6 a^4 x^3 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]

[Out]

-((a + b*x)*(a*(2*a^2 - 3*a*b*x + 6*b^2*x^2) + 6*b^3*x^3*Log[x] - 6*b^3*x^3*Log[
a + b*x]))/(6*a^4*x^3*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.008, size = 69, normalized size = 0.4 \[ -{\frac{ \left ( bx+a \right ) \left ( 6\,{b}^{3}\ln \left ( x \right ){x}^{3}-6\,{b}^{3}\ln \left ( bx+a \right ){x}^{3}+6\,a{b}^{2}{x}^{2}-3\,{a}^{2}bx+2\,{a}^{3} \right ) }{6\,{x}^{3}{a}^{4}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/((b*x+a)^2)^(1/2),x)

[Out]

-1/6*(b*x+a)*(6*b^3*ln(x)*x^3-6*b^3*ln(b*x+a)*x^3+6*a*b^2*x^2-3*a^2*b*x+2*a^3)/(
(b*x+a)^2)^(1/2)/x^3/a^4

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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^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227492, size = 73, normalized size = 0.4 \[ \frac{6 \, b^{3} x^{3} \log \left (b x + a\right ) - 6 \, b^{3} x^{3} \log \left (x\right ) - 6 \, a b^{2} x^{2} + 3 \, a^{2} b x - 2 \, a^{3}}{6 \, a^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt((b*x + a)^2)*x^4),x, algorithm="fricas")

[Out]

1/6*(6*b^3*x^3*log(b*x + a) - 6*b^3*x^3*log(x) - 6*a*b^2*x^2 + 3*a^2*b*x - 2*a^3
)/(a^4*x^3)

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Sympy [A]  time = 1.58118, size = 44, normalized size = 0.24 \[ - \frac{2 a^{2} - 3 a b x + 6 b^{2} x^{2}}{6 a^{3} x^{3}} + \frac{b^{3} \left (- \log{\left (x \right )} + \log{\left (\frac{a}{b} + x \right )}\right )}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/((b*x+a)**2)**(1/2),x)

[Out]

-(2*a**2 - 3*a*b*x + 6*b**2*x**2)/(6*a**3*x**3) + b**3*(-log(x) + log(a/b + x))/
a**4

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GIAC/XCAS [A]  time = 0.208094, size = 88, normalized size = 0.49 \[ \frac{1}{6} \,{\left (\frac{6 \, b^{3}{\rm ln}\left ({\left | b x + a \right |}\right )}{a^{4}} - \frac{6 \, b^{3}{\rm ln}\left ({\left | x \right |}\right )}{a^{4}} - \frac{6 \, a b^{2} x^{2} - 3 \, a^{2} b x + 2 \, a^{3}}{a^{4} x^{3}}\right )}{\rm sign}\left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt((b*x + a)^2)*x^4),x, algorithm="giac")

[Out]

1/6*(6*b^3*ln(abs(b*x + a))/a^4 - 6*b^3*ln(abs(x))/a^4 - (6*a*b^2*x^2 - 3*a^2*b*
x + 2*a^3)/(a^4*x^3))*sign(b*x + a)