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

Optimal. Leaf size=194 \[ \frac{1}{3 a^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{1}{4 a (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\log (x) (a+b x)}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(a+b x) \log (a+b x)}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{1}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{1}{2 a^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

1/(a^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + 1/(4*a*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x +
b^2*x^2]) + 1/(3*a^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + 1/(2*a^3*(a +
b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((a + b*x)*Log[x])/(a^5*Sqrt[a^2 + 2*a*b*x
+ b^2*x^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 = 0.223142, antiderivative size = 194, 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 \[ \frac{1}{3 a^2 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{1}{4 a (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{\log (x) (a+b x)}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{(a+b x) \log (a+b x)}{a^5 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{1}{a^4 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{1}{2 a^3 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

1/(a^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + 1/(4*a*(a + b*x)^3*Sqrt[a^2 + 2*a*b*x +
b^2*x^2]) + 1/(3*a^2*(a + b*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + 1/(2*a^3*(a +
b*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + ((a + b*x)*Log[x])/(a^5*Sqrt[a^2 + 2*a*b*x
+ b^2*x^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 = 29.6095, size = 185, normalized size = 0.95 \[ \frac{2 a + 2 b x}{8 a \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}} + \frac{1}{3 a^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{2 a + 2 b x}{4 a^{3} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{1}{a^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} + \frac{\sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a^{5} \left (a + b x\right )} - \frac{\sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (a + b x \right )}}{a^{5} \left (a + b x\right )} \]

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)**(5/2),x)

[Out]

(2*a + 2*b*x)/(8*a*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)) + 1/(3*a**2*(a**2 + 2*a*
b*x + b**2*x**2)**(3/2)) + (2*a + 2*b*x)/(4*a**3*(a**2 + 2*a*b*x + b**2*x**2)**(
3/2)) + 1/(a**4*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**5*(a + b*x)) - sqrt(a**2 + 2*a*b*x + b**2*x**2)*log(a + b*x)/(a*
*5*(a + b*x))

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Mathematica [A]  time = 0.0656461, size = 84, normalized size = 0.43 \[ \frac{a \left (25 a^3+52 a^2 b x+42 a b^2 x^2+12 b^3 x^3\right )+12 \log (x) (a+b x)^4-12 (a+b x)^4 \log (a+b x)}{12 a^5 (a+b x)^3 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

(a*(25*a^3 + 52*a^2*b*x + 42*a*b^2*x^2 + 12*b^3*x^3) + 12*(a + b*x)^4*Log[x] - 1
2*(a + b*x)^4*Log[a + b*x])/(12*a^5*(a + b*x)^3*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.01, size = 173, normalized size = 0.9 \[{\frac{ \left ( 12\,\ln \left ( x \right ){x}^{4}{b}^{4}-12\,\ln \left ( bx+a \right ){x}^{4}{b}^{4}+48\,\ln \left ( x \right ){x}^{3}a{b}^{3}-48\,\ln \left ( bx+a \right ){x}^{3}a{b}^{3}+72\,\ln \left ( x \right ){x}^{2}{a}^{2}{b}^{2}-72\,\ln \left ( bx+a \right ){x}^{2}{a}^{2}{b}^{2}+12\,{x}^{3}a{b}^{3}+48\,\ln \left ( x \right ) x{a}^{3}b-48\,\ln \left ( bx+a \right ) x{a}^{3}b+42\,{x}^{2}{a}^{2}{b}^{2}+12\,\ln \left ( x \right ){a}^{4}-12\,{a}^{4}\ln \left ( bx+a \right ) +52\,x{a}^{3}b+25\,{a}^{4} \right ) \left ( bx+a \right ) }{12\,{a}^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(12*ln(x)*x^4*b^4-12*ln(b*x+a)*x^4*b^4+48*ln(x)*x^3*a*b^3-48*ln(b*x+a)*x^3*
a*b^3+72*ln(x)*x^2*a^2*b^2-72*ln(b*x+a)*x^2*a^2*b^2+12*x^3*a*b^3+48*ln(x)*x*a^3*
b-48*ln(b*x+a)*x*a^3*b+42*x^2*a^2*b^2+12*ln(x)*a^4-12*a^4*ln(b*x+a)+52*x*a^3*b+2
5*a^4)*(b*x+a)/a^5/((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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.231948, size = 227, normalized size = 1.17 \[ \frac{12 \, a b^{3} x^{3} + 42 \, a^{2} b^{2} x^{2} + 52 \, a^{3} b x + 25 \, a^{4} - 12 \,{\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} \log \left (b x + a\right ) + 12 \,{\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}\right )} \log \left (x\right )}{12 \,{\left (a^{5} b^{4} x^{4} + 4 \, a^{6} b^{3} x^{3} + 6 \, a^{7} b^{2} x^{2} + 4 \, a^{8} b x + a^{9}\right )}} \]

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),x, algorithm="fricas")

[Out]

1/12*(12*a*b^3*x^3 + 42*a^2*b^2*x^2 + 52*a^3*b*x + 25*a^4 - 12*(b^4*x^4 + 4*a*b^
3*x^3 + 6*a^2*b^2*x^2 + 4*a^3*b*x + a^4)*log(b*x + a) + 12*(b^4*x^4 + 4*a*b^3*x^
3 + 6*a^2*b^2*x^2 + 4*a^3*b*x + a^4)*log(x))/(a^5*b^4*x^4 + 4*a^6*b^3*x^3 + 6*a^
7*b^2*x^2 + 4*a^8*b*x + a^9)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.564207, 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),x, algorithm="giac")

[Out]

sage0*x