3.622 \(\int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{x^{22}} \, dx\)

Optimal. Leaf size=255 \[ -\frac{b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 x^{11} \left (a+b x^2\right )}-\frac{5 a b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 x^{13} \left (a+b x^2\right )}-\frac{2 a^2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^{15} \left (a+b x^2\right )}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{21 x^{21} \left (a+b x^2\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{19 x^{19} \left (a+b x^2\right )}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{17 x^{17} \left (a+b x^2\right )} \]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(21*x^21*(a + b*x^2)) - (5*a^4*b*Sqrt[a^2
+ 2*a*b*x^2 + b^2*x^4])/(19*x^19*(a + b*x^2)) - (10*a^3*b^2*Sqrt[a^2 + 2*a*b*x^
2 + b^2*x^4])/(17*x^17*(a + b*x^2)) - (2*a^2*b^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]
)/(3*x^15*(a + b*x^2)) - (5*a*b^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(13*x^13*(a +
b*x^2)) - (b^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(11*x^11*(a + b*x^2))

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Rubi [A]  time = 0.179276, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 x^{11} \left (a+b x^2\right )}-\frac{5 a b^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 x^{13} \left (a+b x^2\right )}-\frac{2 a^2 b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^{15} \left (a+b x^2\right )}-\frac{a^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{21 x^{21} \left (a+b x^2\right )}-\frac{5 a^4 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{19 x^{19} \left (a+b x^2\right )}-\frac{10 a^3 b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{17 x^{17} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^22,x]

[Out]

-(a^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(21*x^21*(a + b*x^2)) - (5*a^4*b*Sqrt[a^2
+ 2*a*b*x^2 + b^2*x^4])/(19*x^19*(a + b*x^2)) - (10*a^3*b^2*Sqrt[a^2 + 2*a*b*x^
2 + b^2*x^4])/(17*x^17*(a + b*x^2)) - (2*a^2*b^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]
)/(3*x^15*(a + b*x^2)) - (5*a*b^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(13*x^13*(a +
b*x^2)) - (b^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(11*x^11*(a + b*x^2))

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Rubi in Sympy [A]  time = 26.3625, size = 211, normalized size = 0.83 \[ \frac{256 a b^{4} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{969969 x^{13} \left (a + b x^{2}\right )} + \frac{32 a b^{2} \left (a + b x^{2}\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{6783 x^{17}} + \frac{10 a \left (a + b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{399 x^{21}} - \frac{128 b^{4} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{74613 x^{13}} - \frac{16 b^{2} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{969 x^{17}} - \frac{29 \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{399 x^{21}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**4+2*a*b*x**2+a**2)**(5/2)/x**22,x)

[Out]

256*a*b**4*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(969969*x**13*(a + b*x**2)) + 32*
a*b**2*(a + b*x**2)*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(6783*x**17) + 10*a*(a +
b*x**2)*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/(399*x**21) - 128*b**4*sqrt(a**2
+ 2*a*b*x**2 + b**2*x**4)/(74613*x**13) - 16*b**2*(a**2 + 2*a*b*x**2 + b**2*x**
4)**(3/2)/(969*x**17) - 29*(a**2 + 2*a*b*x**2 + b**2*x**4)**(5/2)/(399*x**21)

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Mathematica [A]  time = 0.0309283, size = 83, normalized size = 0.33 \[ -\frac{\sqrt{\left (a+b x^2\right )^2} \left (46189 a^5+255255 a^4 b x^2+570570 a^3 b^2 x^4+646646 a^2 b^3 x^6+373065 a b^4 x^8+88179 b^5 x^{10}\right )}{969969 x^{21} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2)/x^22,x]

[Out]

-(Sqrt[(a + b*x^2)^2]*(46189*a^5 + 255255*a^4*b*x^2 + 570570*a^3*b^2*x^4 + 64664
6*a^2*b^3*x^6 + 373065*a*b^4*x^8 + 88179*b^5*x^10))/(969969*x^21*(a + b*x^2))

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Maple [A]  time = 0.012, size = 80, normalized size = 0.3 \[ -{\frac{88179\,{b}^{5}{x}^{10}+373065\,a{b}^{4}{x}^{8}+646646\,{a}^{2}{b}^{3}{x}^{6}+570570\,{a}^{3}{b}^{2}{x}^{4}+255255\,{a}^{4}b{x}^{2}+46189\,{a}^{5}}{969969\,{x}^{21} \left ( b{x}^{2}+a \right ) ^{5}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^4+2*a*b*x^2+a^2)^(5/2)/x^22,x)

[Out]

-1/969969*(88179*b^5*x^10+373065*a*b^4*x^8+646646*a^2*b^3*x^6+570570*a^3*b^2*x^4
+255255*a^4*b*x^2+46189*a^5)*((b*x^2+a)^2)^(5/2)/x^21/(b*x^2+a)^5

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Maxima [A]  time = 0.698523, size = 80, normalized size = 0.31 \[ -\frac{88179 \, b^{5} x^{10} + 373065 \, a b^{4} x^{8} + 646646 \, a^{2} b^{3} x^{6} + 570570 \, a^{3} b^{2} x^{4} + 255255 \, a^{4} b x^{2} + 46189 \, a^{5}}{969969 \, x^{21}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)/x^22,x, algorithm="maxima")

[Out]

-1/969969*(88179*b^5*x^10 + 373065*a*b^4*x^8 + 646646*a^2*b^3*x^6 + 570570*a^3*b
^2*x^4 + 255255*a^4*b*x^2 + 46189*a^5)/x^21

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Fricas [A]  time = 0.259814, size = 80, normalized size = 0.31 \[ -\frac{88179 \, b^{5} x^{10} + 373065 \, a b^{4} x^{8} + 646646 \, a^{2} b^{3} x^{6} + 570570 \, a^{3} b^{2} x^{4} + 255255 \, a^{4} b x^{2} + 46189 \, a^{5}}{969969 \, x^{21}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)/x^22,x, algorithm="fricas")

[Out]

-1/969969*(88179*b^5*x^10 + 373065*a*b^4*x^8 + 646646*a^2*b^3*x^6 + 570570*a^3*b
^2*x^4 + 255255*a^4*b*x^2 + 46189*a^5)/x^21

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**4+2*a*b*x**2+a**2)**(5/2)/x**22,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.273466, size = 144, normalized size = 0.56 \[ -\frac{88179 \, b^{5} x^{10}{\rm sign}\left (b x^{2} + a\right ) + 373065 \, a b^{4} x^{8}{\rm sign}\left (b x^{2} + a\right ) + 646646 \, a^{2} b^{3} x^{6}{\rm sign}\left (b x^{2} + a\right ) + 570570 \, a^{3} b^{2} x^{4}{\rm sign}\left (b x^{2} + a\right ) + 255255 \, a^{4} b x^{2}{\rm sign}\left (b x^{2} + a\right ) + 46189 \, a^{5}{\rm sign}\left (b x^{2} + a\right )}{969969 \, x^{21}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)/x^22,x, algorithm="giac")

[Out]

-1/969969*(88179*b^5*x^10*sign(b*x^2 + a) + 373065*a*b^4*x^8*sign(b*x^2 + a) + 6
46646*a^2*b^3*x^6*sign(b*x^2 + a) + 570570*a^3*b^2*x^4*sign(b*x^2 + a) + 255255*
a^4*b*x^2*sign(b*x^2 + a) + 46189*a^5*sign(b*x^2 + a))/x^21