3.195 \(\int \frac{1}{x \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=126 \[ \frac{1}{2 a (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{1}{a^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\log (x) (a+b x)}{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(a+b x) \log (a+b x)}{a^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.151634, antiderivative size = 126, 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 \[ \frac{1}{2 a (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{1}{a^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\log (x) (a+b x)}{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(a+b x) \log (a+b x)}{a^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

1/(a^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + 1/(2*a*(a + b*x)*Sqrt[a^2 + 2*a*b*x + b^
2*x^2]) + ((a + b*x)*Log[x])/(a^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - ((a + b*x)*Lo
g[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 = 20.3059, size = 124, normalized size = 0.98 \[ \frac{2 a + 2 b x}{4 a \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{1}{a^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{\sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a^{3} \left (a + b x\right )} - \frac{\sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (a + b x \right )}}{a^{3} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0474618, size = 62, normalized size = 0.49 \[ \frac{a (3 a+2 b x)+2 \log (x) (a+b x)^2-2 (a+b x)^2 \log (a+b x)}{2 a^3 (a+b x) \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 91, normalized size = 0.7 \[{\frac{ \left ( 2\,{b}^{2}\ln \left ( x \right ){x}^{2}-2\,{b}^{2}\ln \left ( bx+a \right ){x}^{2}+4\,\ln \left ( x \right ) xab-4\,\ln \left ( bx+a \right ) xab+2\,{a}^{2}\ln \left ( x \right ) -2\,{a}^{2}\ln \left ( bx+a \right ) +2\,abx+3\,{a}^{2} \right ) \left ( bx+a \right ) }{2\,{a}^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227501, size = 108, normalized size = 0.86 \[ \frac{2 \, a b x + 3 \, a^{2} - 2 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \log \left (b x + a\right ) + 2 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{3} b^{2} x^{2} + 2 \, a^{4} b x + a^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*a*b*x + 3*a^2 - 2*(b^2*x^2 + 2*a*b*x + a^2)*log(b*x + a) + 2*(b^2*x^2 + 2
*a*b*x + a^2)*log(x))/(a^3*b^2*x^2 + 2*a^4*b*x + a^5)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x*((a + b*x)**2)**(3/2)), x)

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GIAC/XCAS [A]  time = 0.563499, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x