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

Optimal. Leaf size=209 \[ \frac{6 b^2 \log (x) (a+b x)}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{6 b^2 (a+b x) \log (a+b x)}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 b^2}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 b (a+b x)}{a^4 x \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^2}{2 a^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a+b x}{2 a^3 x^2 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

(3*b^2)/(a^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + b^2/(2*a^3*(a + b*x)*Sqrt[a^2 + 2*
a*b*x + b^2*x^2]) - (a + b*x)/(2*a^3*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (3*b*(
a + b*x))/(a^4*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (6*b^2*(a + b*x)*Log[x])/(a^5*
Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (6*b^2*(a + b*x)*Log[a + b*x])/(a^5*Sqrt[a^2 +
2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.212621, antiderivative size = 209, 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 \[ \frac{6 b^2 \log (x) (a+b x)}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{6 b^2 (a+b x) \log (a+b x)}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 b^2}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{3 b (a+b x)}{a^4 x \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^2}{2 a^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a+b x}{2 a^3 x^2 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(3*b^2)/(a^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + b^2/(2*a^3*(a + b*x)*Sqrt[a^2 + 2*
a*b*x + b^2*x^2]) - (a + b*x)/(2*a^3*x^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (3*b*(
a + b*x))/(a^4*x*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (6*b^2*(a + b*x)*Log[x])/(a^5*
Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (6*b^2*(a + b*x)*Log[a + b*x])/(a^5*Sqrt[a^2 +
2*a*b*x + b^2*x^2])

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Rubi in Sympy [A]  time = 33.8407, size = 207, normalized size = 0.99 \[ \frac{2 a + 2 b x}{4 a x^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{2}{a^{2} x^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} - \frac{3 \left (2 a + 2 b x\right )}{2 a^{3} x^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{6 b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a^{5} \left (a + b x\right )} - \frac{6 b^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (a + b x \right )}}{a^{5} \left (a + b x\right )} + \frac{6 b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{a^{5} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(2*a + 2*b*x)/(4*a*x**2*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)) + 2/(a**2*x**2*sqrt
(a**2 + 2*a*b*x + b**2*x**2)) - 3*(2*a + 2*b*x)/(2*a**3*x**2*sqrt(a**2 + 2*a*b*x
+ b**2*x**2)) + 6*b**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)*log(x)/(a**5*(a + b*x))
- 6*b**2*sqrt(a**2 + 2*a*b*x + b**2*x**2)*log(a + b*x)/(a**5*(a + b*x)) + 6*b*s
qrt(a**2 + 2*a*b*x + b**2*x**2)/(a**5*x)

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Mathematica [A]  time = 0.0665776, size = 99, normalized size = 0.47 \[ \frac{a \left (-a^3+4 a^2 b x+18 a b^2 x^2+12 b^3 x^3\right )+12 b^2 x^2 \log (x) (a+b x)^2-12 b^2 x^2 (a+b x)^2 \log (a+b x)}{2 a^5 x^2 (a+b x) \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

(a*(-a^3 + 4*a^2*b*x + 18*a*b^2*x^2 + 12*b^3*x^3) + 12*b^2*x^2*(a + b*x)^2*Log[x
] - 12*b^2*x^2*(a + b*x)^2*Log[a + b*x])/(2*a^5*x^2*(a + b*x)*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.009, size = 136, normalized size = 0.7 \[{\frac{ \left ( 12\,\ln \left ( x \right ){x}^{4}{b}^{4}-12\,\ln \left ( bx+a \right ){x}^{4}{b}^{4}+24\,\ln \left ( x \right ){x}^{3}a{b}^{3}-24\,\ln \left ( bx+a \right ){x}^{3}a{b}^{3}+12\,\ln \left ( x \right ){x}^{2}{a}^{2}{b}^{2}-12\,\ln \left ( bx+a \right ){x}^{2}{a}^{2}{b}^{2}+12\,{x}^{3}a{b}^{3}+18\,{x}^{2}{a}^{2}{b}^{2}+4\,x{a}^{3}b-{a}^{4} \right ) \left ( bx+a \right ) }{2\,{a}^{5}{x}^{2}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(12*ln(x)*x^4*b^4-12*ln(b*x+a)*x^4*b^4+24*ln(x)*x^3*a*b^3-24*ln(b*x+a)*x^3*a
*b^3+12*ln(x)*x^2*a^2*b^2-12*ln(b*x+a)*x^2*a^2*b^2+12*x^3*a*b^3+18*x^2*a^2*b^2+4
*x*a^3*b-a^4)*(b*x+a)/a^5/x^2/((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^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230859, size = 176, normalized size = 0.84 \[ \frac{12 \, a b^{3} x^{3} + 18 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x - a^{4} - 12 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{3} + a^{2} b^{2} x^{2}\right )} \log \left (b x + a\right ) + 12 \,{\left (b^{4} x^{4} + 2 \, a b^{3} x^{3} + a^{2} b^{2} x^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{5} b^{2} x^{4} + 2 \, a^{6} b x^{3} + a^{7} x^{2}\right )}} \]

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^3),x, algorithm="fricas")

[Out]

1/2*(12*a*b^3*x^3 + 18*a^2*b^2*x^2 + 4*a^3*b*x - a^4 - 12*(b^4*x^4 + 2*a*b^3*x^3
+ a^2*b^2*x^2)*log(b*x + a) + 12*(b^4*x^4 + 2*a*b^3*x^3 + a^2*b^2*x^2)*log(x))/
(a^5*b^2*x^4 + 2*a^6*b*x^3 + a^7*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.572573, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

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^3),x, algorithm="giac")

[Out]

sage0*x