3.651 \(\int \frac{1}{x \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx\)

Optimal. Leaf size=223 \[ \frac{1}{6 a^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{8 a \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\log (x) \left (a+b x^2\right )}{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{2 a^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{4 a^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

1/(2*a^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + 1/(8*a*(a + b*x^2)^3*Sqrt[a^2 + 2*a*
b*x^2 + b^2*x^4]) + 1/(6*a^2*(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + 1/
(4*a^3*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + ((a + b*x^2)*Log[x])/(a^5*
Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - ((a + b*x^2)*Log[a + b*x^2])/(2*a^5*Sqrt[a^2
+ 2*a*b*x^2 + b^2*x^4])

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Rubi [A]  time = 0.293692, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{1}{6 a^2 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{8 a \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\log (x) \left (a+b x^2\right )}{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{2 a^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{4 a^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)),x]

[Out]

1/(2*a^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + 1/(8*a*(a + b*x^2)^3*Sqrt[a^2 + 2*a*
b*x^2 + b^2*x^4]) + 1/(6*a^2*(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + 1/
(4*a^3*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + ((a + b*x^2)*Log[x])/(a^5*
Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - ((a + b*x^2)*Log[a + b*x^2])/(2*a^5*Sqrt[a^2
+ 2*a*b*x^2 + b^2*x^4])

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Rubi in Sympy [A]  time = 36.7566, size = 211, normalized size = 0.95 \[ \frac{2 a + 2 b x^{2}}{16 a \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}} + \frac{1}{6 a^{2} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}} + \frac{2 a + 2 b x^{2}}{8 a^{3} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}} + \frac{1}{2 a^{4} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}} + \frac{\sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}} \log{\left (x^{2} \right )}}{2 a^{5} \left (a + b x^{2}\right )} - \frac{\sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}} \log{\left (a + b x^{2} \right )}}{2 a^{5} \left (a + b x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

(2*a + 2*b*x**2)/(16*a*(a**2 + 2*a*b*x**2 + b**2*x**4)**(5/2)) + 1/(6*a**2*(a**2
+ 2*a*b*x**2 + b**2*x**4)**(3/2)) + (2*a + 2*b*x**2)/(8*a**3*(a**2 + 2*a*b*x**2
+ b**2*x**4)**(3/2)) + 1/(2*a**4*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)) + sqrt(a*
*2 + 2*a*b*x**2 + b**2*x**4)*log(x**2)/(2*a**5*(a + b*x**2)) - sqrt(a**2 + 2*a*b
*x**2 + b**2*x**4)*log(a + b*x**2)/(2*a**5*(a + b*x**2))

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Mathematica [A]  time = 0.0741545, size = 96, normalized size = 0.43 \[ \frac{a \left (25 a^3+52 a^2 b x^2+42 a b^2 x^4+12 b^3 x^6\right )+24 \log (x) \left (a+b x^2\right )^4-12 \left (a+b x^2\right )^4 \log \left (a+b x^2\right )}{24 a^5 \left (a+b x^2\right )^3 \sqrt{\left (a+b x^2\right )^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)),x]

[Out]

(a*(25*a^3 + 52*a^2*b*x^2 + 42*a*b^2*x^4 + 12*b^3*x^6) + 24*(a + b*x^2)^4*Log[x]
- 12*(a + b*x^2)^4*Log[a + b*x^2])/(24*a^5*(a + b*x^2)^3*Sqrt[(a + b*x^2)^2])

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Maple [A]  time = 0.026, size = 193, normalized size = 0.9 \[ -{\frac{ \left ( 12\,\ln \left ( b{x}^{2}+a \right ){x}^{8}{b}^{4}-24\,\ln \left ( x \right ){x}^{8}{b}^{4}+48\,\ln \left ( b{x}^{2}+a \right ){x}^{6}a{b}^{3}-96\,\ln \left ( x \right ){x}^{6}a{b}^{3}-12\,a{b}^{3}{x}^{6}+72\,\ln \left ( b{x}^{2}+a \right ){x}^{4}{a}^{2}{b}^{2}-144\,\ln \left ( x \right ){x}^{4}{a}^{2}{b}^{2}-42\,{a}^{2}{b}^{2}{x}^{4}+48\,\ln \left ( b{x}^{2}+a \right ){x}^{2}{a}^{3}b-96\,\ln \left ( x \right ){x}^{2}{a}^{3}b-52\,{a}^{3}b{x}^{2}+12\,\ln \left ( b{x}^{2}+a \right ){a}^{4}-24\,{a}^{4}\ln \left ( x \right ) -25\,{a}^{4} \right ) \left ( b{x}^{2}+a \right ) }{24\,{a}^{5}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)

[Out]

-1/24*(12*ln(b*x^2+a)*x^8*b^4-24*ln(x)*x^8*b^4+48*ln(b*x^2+a)*x^6*a*b^3-96*ln(x)
*x^6*a*b^3-12*a*b^3*x^6+72*ln(b*x^2+a)*x^4*a^2*b^2-144*ln(x)*x^4*a^2*b^2-42*a^2*
b^2*x^4+48*ln(b*x^2+a)*x^2*a^3*b-96*ln(x)*x^2*a^3*b-52*a^3*b*x^2+12*ln(b*x^2+a)*
a^4-24*a^4*ln(x)-25*a^4)*(b*x^2+a)/a^5/((b*x^2+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^4 + 2*a*b*x^2 + a^2)^(5/2)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.270221, size = 240, normalized size = 1.08 \[ \frac{12 \, a b^{3} x^{6} + 42 \, a^{2} b^{2} x^{4} + 52 \, a^{3} b x^{2} + 25 \, a^{4} - 12 \,{\left (b^{4} x^{8} + 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} + 4 \, a^{3} b x^{2} + a^{4}\right )} \log \left (b x^{2} + a\right ) + 24 \,{\left (b^{4} x^{8} + 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} + 4 \, a^{3} b x^{2} + a^{4}\right )} \log \left (x\right )}{24 \,{\left (a^{5} b^{4} x^{8} + 4 \, a^{6} b^{3} x^{6} + 6 \, a^{7} b^{2} x^{4} + 4 \, a^{8} b x^{2} + a^{9}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)*x),x, algorithm="fricas")

[Out]

1/24*(12*a*b^3*x^6 + 42*a^2*b^2*x^4 + 52*a^3*b*x^2 + 25*a^4 - 12*(b^4*x^8 + 4*a*
b^3*x^6 + 6*a^2*b^2*x^4 + 4*a^3*b*x^2 + a^4)*log(b*x^2 + a) + 24*(b^4*x^8 + 4*a*
b^3*x^6 + 6*a^2*b^2*x^4 + 4*a^3*b*x^2 + a^4)*log(x))/(a^5*b^4*x^8 + 4*a^6*b^3*x^
6 + 6*a^7*b^2*x^4 + 4*a^8*b*x^2 + a^9)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.620811, 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^4 + 2*a*b*x^2 + a^2)^(5/2)*x),x, algorithm="giac")

[Out]

sage0*x