3.206 \(\int \frac{1}{x^2 \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=235 \[ -\frac{b}{4 a^2 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 b \log (x) (a+b x)}{a^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 b (a+b x) \log (a+b x)}{a^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{4 b}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a+b x}{a^5 x \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 b}{2 a^4 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 b}{3 a^3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.266236, antiderivative size = 235, 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 \[ -\frac{b}{4 a^2 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{5 b \log (x) (a+b x)}{a^6 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{5 b (a+b x) \log (a+b x)}{a^6 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{4 b}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a+b x}{a^5 x \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{3 b}{2 a^4 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{2 b}{3 a^3 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 35.5686, size = 228, normalized size = 0.97 \[ \frac{2 a + 2 b x}{8 a x \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}} + \frac{5}{12 a^{2} x \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{5 \left (2 a + 2 b x\right )}{12 a^{3} x \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{5}{2 a^{4} x \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} - \frac{5 b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a^{6} \left (a + b x\right )} + \frac{5 b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (a + b x \right )}}{a^{6} \left (a + b x\right )} - \frac{5 \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{a^{6} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

(2*a + 2*b*x)/(8*a*x*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)) + 5/(12*a**2*x*(a**2 +
2*a*b*x + b**2*x**2)**(3/2)) + 5*(2*a + 2*b*x)/(12*a**3*x*(a**2 + 2*a*b*x + b**
2*x**2)**(3/2)) + 5/(2*a**4*x*sqrt(a**2 + 2*a*b*x + b**2*x**2)) - 5*b*sqrt(a**2
+ 2*a*b*x + b**2*x**2)*log(x)/(a**6*(a + b*x)) + 5*b*sqrt(a**2 + 2*a*b*x + b**2*
x**2)*log(a + b*x)/(a**6*(a + b*x)) - 5*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(a**6*x
)

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Mathematica [A]  time = 0.07597, size = 103, normalized size = 0.44 \[ \frac{-a \left (12 a^4+125 a^3 b x+260 a^2 b^2 x^2+210 a b^3 x^3+60 b^4 x^4\right )-60 b x \log (x) (a+b x)^4+60 b x (a+b x)^4 \log (a+b x)}{12 a^6 x (a+b x)^3 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

(-(a*(12*a^4 + 125*a^3*b*x + 260*a^2*b^2*x^2 + 210*a*b^3*x^3 + 60*b^4*x^4)) - 60
*b*x*(a + b*x)^4*Log[x] + 60*b*x*(a + b*x)^4*Log[a + b*x])/(12*a^6*x*(a + b*x)^3
*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.023, size = 199, normalized size = 0.9 \[ -{\frac{ \left ( 60\,{b}^{5}\ln \left ( x \right ){x}^{5}-60\,\ln \left ( bx+a \right ){x}^{5}{b}^{5}+240\,a{b}^{4}\ln \left ( x \right ){x}^{4}-240\,\ln \left ( bx+a \right ){x}^{4}a{b}^{4}+360\,{a}^{2}{b}^{3}\ln \left ( x \right ){x}^{3}-360\,\ln \left ( bx+a \right ){x}^{3}{a}^{2}{b}^{3}+60\,a{b}^{4}{x}^{4}+240\,{a}^{3}{b}^{2}\ln \left ( x \right ){x}^{2}-240\,\ln \left ( bx+a \right ){x}^{2}{a}^{3}{b}^{2}+210\,{a}^{2}{b}^{3}{x}^{3}+60\,{a}^{4}b\ln \left ( x \right ) x-60\,\ln \left ( bx+a \right ) x{a}^{4}b+260\,{a}^{3}{b}^{2}{x}^{2}+125\,{a}^{4}bx+12\,{a}^{5} \right ) \left ( bx+a \right ) }{12\,{a}^{6}x} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

-1/12*(60*b^5*ln(x)*x^5-60*ln(b*x+a)*x^5*b^5+240*a*b^4*ln(x)*x^4-240*ln(b*x+a)*x
^4*a*b^4+360*a^2*b^3*ln(x)*x^3-360*ln(b*x+a)*x^3*a^2*b^3+60*a*b^4*x^4+240*a^3*b^
2*ln(x)*x^2-240*ln(b*x+a)*x^2*a^3*b^2+210*a^2*b^3*x^3+60*a^4*b*ln(x)*x-60*ln(b*x
+a)*x*a^4*b+260*a^3*b^2*x^2+125*a^4*b*x+12*a^5)*(b*x+a)/a^6/x/((b*x+a)^2)^(5/2)

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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)^(5/2)*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.231245, size = 266, normalized size = 1.13 \[ -\frac{60 \, a b^{4} x^{4} + 210 \, a^{2} b^{3} x^{3} + 260 \, a^{3} b^{2} x^{2} + 125 \, a^{4} b x + 12 \, a^{5} - 60 \,{\left (b^{5} x^{5} + 4 \, a b^{4} x^{4} + 6 \, a^{2} b^{3} x^{3} + 4 \, a^{3} b^{2} x^{2} + a^{4} b x\right )} \log \left (b x + a\right ) + 60 \,{\left (b^{5} x^{5} + 4 \, a b^{4} x^{4} + 6 \, a^{2} b^{3} x^{3} + 4 \, a^{3} b^{2} x^{2} + a^{4} b x\right )} \log \left (x\right )}{12 \,{\left (a^{6} b^{4} x^{5} + 4 \, a^{7} b^{3} x^{4} + 6 \, a^{8} b^{2} x^{3} + 4 \, a^{9} b x^{2} + a^{10} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*x^2),x, algorithm="fricas")

[Out]

-1/12*(60*a*b^4*x^4 + 210*a^2*b^3*x^3 + 260*a^3*b^2*x^2 + 125*a^4*b*x + 12*a^5 -
60*(b^5*x^5 + 4*a*b^4*x^4 + 6*a^2*b^3*x^3 + 4*a^3*b^2*x^2 + a^4*b*x)*log(b*x +
a) + 60*(b^5*x^5 + 4*a*b^4*x^4 + 6*a^2*b^3*x^3 + 4*a^3*b^2*x^2 + a^4*b*x)*log(x)
)/(a^6*b^4*x^5 + 4*a^7*b^3*x^4 + 6*a^8*b^2*x^3 + 4*a^9*b*x^2 + a^10*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Integral(1/(x**2*((a + b*x)**2)**(5/2)), x)

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GIAC/XCAS [A]  time = 0.55069, 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)^(5/2)*x^2),x, algorithm="giac")

[Out]

sage0*x