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

Optimal. Leaf size=291 \[ \frac{11}{48 a^2 x^3 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{8 a x^3 \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1155 b^{3/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{128 a^{13/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1155 b \left (a+b x^2\right )}{128 a^6 x \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{385 \left (a+b x^2\right )}{128 a^5 x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{231}{128 a^4 x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{33}{64 a^3 x^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

231/(128*a^4*x^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + 1/(8*a*x^3*(a + b*x^2)^3*Sqr
t[a^2 + 2*a*b*x^2 + b^2*x^4]) + 11/(48*a^2*x^3*(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^
2 + b^2*x^4]) + 33/(64*a^3*x^3*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (3
85*(a + b*x^2))/(128*a^5*x^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (1155*b*(a + b*x
^2))/(128*a^6*x*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (1155*b^(3/2)*(a + b*x^2)*Arc
Tan[(Sqrt[b]*x)/Sqrt[a]])/(128*a^(13/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

_______________________________________________________________________________________

Rubi [A]  time = 0.339539, antiderivative size = 291, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{11}{48 a^2 x^3 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{8 a x^3 \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1155 b^{3/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{128 a^{13/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1155 b \left (a+b x^2\right )}{128 a^6 x \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{385 \left (a+b x^2\right )}{128 a^5 x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{231}{128 a^4 x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{33}{64 a^3 x^3 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

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

[Out]

231/(128*a^4*x^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + 1/(8*a*x^3*(a + b*x^2)^3*Sqr
t[a^2 + 2*a*b*x^2 + b^2*x^4]) + 11/(48*a^2*x^3*(a + b*x^2)^2*Sqrt[a^2 + 2*a*b*x^
2 + b^2*x^4]) + 33/(64*a^3*x^3*(a + b*x^2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (3
85*(a + b*x^2))/(128*a^5*x^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (1155*b*(a + b*x
^2))/(128*a^6*x*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (1155*b^(3/2)*(a + b*x^2)*Arc
Tan[(Sqrt[b]*x)/Sqrt[a]])/(128*a^(13/2)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

_______________________________________________________________________________________

Rubi in Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RecursionError

_______________________________________________________________________________________

Mathematica [A]  time = 0.116031, size = 127, normalized size = 0.44 \[ \frac{\sqrt{a} \left (-128 a^5+1408 a^4 b x^2+9207 a^3 b^2 x^4+16863 a^2 b^3 x^6+12705 a b^4 x^8+3465 b^5 x^{10}\right )+3465 b^{3/2} x^3 \left (a+b x^2\right )^4 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{384 a^{13/2} x^3 \left (a+b x^2\right )^3 \sqrt{\left (a+b x^2\right )^2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a]*(-128*a^5 + 1408*a^4*b*x^2 + 9207*a^3*b^2*x^4 + 16863*a^2*b^3*x^6 + 127
05*a*b^4*x^8 + 3465*b^5*x^10) + 3465*b^(3/2)*x^3*(a + b*x^2)^4*ArcTan[(Sqrt[b]*x
)/Sqrt[a]])/(384*a^(13/2)*x^3*(a + b*x^2)^3*Sqrt[(a + b*x^2)^2])

_______________________________________________________________________________________

Maple [A]  time = 0.031, size = 211, normalized size = 0.7 \[{\frac{b{x}^{2}+a}{384\,{a}^{6}{x}^{3}} \left ( 3465\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{11}{b}^{6}+3465\,\sqrt{ab}{x}^{10}{b}^{5}+13860\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{9}a{b}^{5}+12705\,\sqrt{ab}{x}^{8}a{b}^{4}+20790\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{7}{a}^{2}{b}^{4}+16863\,\sqrt{ab}{x}^{6}{a}^{2}{b}^{3}+13860\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{5}{a}^{3}{b}^{3}+9207\,\sqrt{ab}{x}^{4}{a}^{3}{b}^{2}+3465\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{3}{a}^{4}{b}^{2}+1408\,\sqrt{ab}{x}^{2}{a}^{4}b-128\,\sqrt{ab}{a}^{5} \right ){\frac{1}{\sqrt{ab}}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/384*(3465*arctan(x*b/(a*b)^(1/2))*x^11*b^6+3465*(a*b)^(1/2)*x^10*b^5+13860*arc
tan(x*b/(a*b)^(1/2))*x^9*a*b^5+12705*(a*b)^(1/2)*x^8*a*b^4+20790*arctan(x*b/(a*b
)^(1/2))*x^7*a^2*b^4+16863*(a*b)^(1/2)*x^6*a^2*b^3+13860*arctan(x*b/(a*b)^(1/2))
*x^5*a^3*b^3+9207*(a*b)^(1/2)*x^4*a^3*b^2+3465*arctan(x*b/(a*b)^(1/2))*x^3*a^4*b
^2+1408*(a*b)^(1/2)*x^2*a^4*b-128*(a*b)^(1/2)*a^5)*(b*x^2+a)/x^3/(a*b)^(1/2)/a^6
/((b*x^2+a)^2)^(5/2)

_______________________________________________________________________________________

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^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.276518, size = 1, normalized size = 0. \[ \left [\frac{6930 \, b^{5} x^{10} + 25410 \, a b^{4} x^{8} + 33726 \, a^{2} b^{3} x^{6} + 18414 \, a^{3} b^{2} x^{4} + 2816 \, a^{4} b x^{2} - 256 \, a^{5} + 3465 \,{\left (b^{5} x^{11} + 4 \, a b^{4} x^{9} + 6 \, a^{2} b^{3} x^{7} + 4 \, a^{3} b^{2} x^{5} + a^{4} b x^{3}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right )}{768 \,{\left (a^{6} b^{4} x^{11} + 4 \, a^{7} b^{3} x^{9} + 6 \, a^{8} b^{2} x^{7} + 4 \, a^{9} b x^{5} + a^{10} x^{3}\right )}}, \frac{3465 \, b^{5} x^{10} + 12705 \, a b^{4} x^{8} + 16863 \, a^{2} b^{3} x^{6} + 9207 \, a^{3} b^{2} x^{4} + 1408 \, a^{4} b x^{2} - 128 \, a^{5} + 3465 \,{\left (b^{5} x^{11} + 4 \, a b^{4} x^{9} + 6 \, a^{2} b^{3} x^{7} + 4 \, a^{3} b^{2} x^{5} + a^{4} b x^{3}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{\frac{b}{a}}}\right )}{384 \,{\left (a^{6} b^{4} x^{11} + 4 \, a^{7} b^{3} x^{9} + 6 \, a^{8} b^{2} x^{7} + 4 \, a^{9} b x^{5} + a^{10} x^{3}\right )}}\right ] \]

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^4),x, algorithm="fricas")

[Out]

[1/768*(6930*b^5*x^10 + 25410*a*b^4*x^8 + 33726*a^2*b^3*x^6 + 18414*a^3*b^2*x^4
+ 2816*a^4*b*x^2 - 256*a^5 + 3465*(b^5*x^11 + 4*a*b^4*x^9 + 6*a^2*b^3*x^7 + 4*a^
3*b^2*x^5 + a^4*b*x^3)*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)
))/(a^6*b^4*x^11 + 4*a^7*b^3*x^9 + 6*a^8*b^2*x^7 + 4*a^9*b*x^5 + a^10*x^3), 1/38
4*(3465*b^5*x^10 + 12705*a*b^4*x^8 + 16863*a^2*b^3*x^6 + 9207*a^3*b^2*x^4 + 1408
*a^4*b*x^2 - 128*a^5 + 3465*(b^5*x^11 + 4*a*b^4*x^9 + 6*a^2*b^3*x^7 + 4*a^3*b^2*
x^5 + a^4*b*x^3)*sqrt(b/a)*arctan(b*x/(a*sqrt(b/a))))/(a^6*b^4*x^11 + 4*a^7*b^3*
x^9 + 6*a^8*b^2*x^7 + 4*a^9*b*x^5 + a^10*x^3)]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{4} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**4*((a + b*x**2)**2)**(5/2)), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.619341, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

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^4),x, algorithm="giac")

[Out]

sage0*x