3.606 \(\int x^{10} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.184808, 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 \[ \frac{b^5 x^{21} \sqrt{a^2+2 a b x^2+b^2 x^4}}{21 \left (a+b x^2\right )}+\frac{5 a b^4 x^{19} \sqrt{a^2+2 a b x^2+b^2 x^4}}{19 \left (a+b x^2\right )}+\frac{10 a^2 b^3 x^{17} \sqrt{a^2+2 a b x^2+b^2 x^4}}{17 \left (a+b x^2\right )}+\frac{a^5 x^{11} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 \left (a+b x^2\right )}+\frac{5 a^4 b x^{13} \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 \left (a+b x^2\right )}+\frac{2 a^3 b^2 x^{15} \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^10*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 26.508, size = 207, normalized size = 0.81 \[ \frac{256 a^{5} x^{11} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{969969 \left (a + b x^{2}\right )} + \frac{128 a^{4} x^{11} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{88179} + \frac{32 a^{3} x^{11} \left (a + b x^{2}\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{6783} + \frac{80 a^{2} x^{11} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{6783} + \frac{10 a x^{11} \left (a + b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{399} + \frac{x^{11} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**10*(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

256*a**5*x**11*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(969969*(a + b*x**2)) + 128*a
**4*x**11*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/88179 + 32*a**3*x**11*(a + b*x**2)
*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/6783 + 80*a**2*x**11*(a**2 + 2*a*b*x**2 + b
**2*x**4)**(3/2)/6783 + 10*a*x**11*(a + b*x**2)*(a**2 + 2*a*b*x**2 + b**2*x**4)*
*(3/2)/399 + x**11*(a**2 + 2*a*b*x**2 + b**2*x**4)**(5/2)/21

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Mathematica [A]  time = 0.035721, size = 83, normalized size = 0.33 \[ \frac{x^{11} \sqrt{\left (a+b x^2\right )^2} \left (88179 a^5+373065 a^4 b x^2+646646 a^3 b^2 x^4+570570 a^2 b^3 x^6+255255 a b^4 x^8+46189 b^5 x^{10}\right )}{969969 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[x^10*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(x^11*Sqrt[(a + b*x^2)^2]*(88179*a^5 + 373065*a^4*b*x^2 + 646646*a^3*b^2*x^4 + 5
70570*a^2*b^3*x^6 + 255255*a*b^4*x^8 + 46189*b^5*x^10))/(969969*(a + b*x^2))

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Maple [A]  time = 0.011, size = 80, normalized size = 0.3 \[{\frac{{x}^{11} \left ( 46189\,{b}^{5}{x}^{10}+255255\,a{b}^{4}{x}^{8}+570570\,{a}^{2}{b}^{3}{x}^{6}+646646\,{a}^{3}{b}^{2}{x}^{4}+373065\,{a}^{4}b{x}^{2}+88179\,{a}^{5} \right ) }{969969\, \left ( b{x}^{2}+a \right ) ^{5}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^10*(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)

[Out]

1/969969*x^11*(46189*b^5*x^10+255255*a*b^4*x^8+570570*a^2*b^3*x^6+646646*a^3*b^2
*x^4+373065*a^4*b*x^2+88179*a^5)*((b*x^2+a)^2)^(5/2)/(b*x^2+a)^5

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Maxima [A]  time = 0.68635, size = 77, normalized size = 0.3 \[ \frac{1}{21} \, b^{5} x^{21} + \frac{5}{19} \, a b^{4} x^{19} + \frac{10}{17} \, a^{2} b^{3} x^{17} + \frac{2}{3} \, a^{3} b^{2} x^{15} + \frac{5}{13} \, a^{4} b x^{13} + \frac{1}{11} \, a^{5} x^{11} \]

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^10,x, algorithm="maxima")

[Out]

1/21*b^5*x^21 + 5/19*a*b^4*x^19 + 10/17*a^2*b^3*x^17 + 2/3*a^3*b^2*x^15 + 5/13*a
^4*b*x^13 + 1/11*a^5*x^11

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Fricas [A]  time = 0.258596, size = 77, normalized size = 0.3 \[ \frac{1}{21} \, b^{5} x^{21} + \frac{5}{19} \, a b^{4} x^{19} + \frac{10}{17} \, a^{2} b^{3} x^{17} + \frac{2}{3} \, a^{3} b^{2} x^{15} + \frac{5}{13} \, a^{4} b x^{13} + \frac{1}{11} \, a^{5} x^{11} \]

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^10,x, algorithm="fricas")

[Out]

1/21*b^5*x^21 + 5/19*a*b^4*x^19 + 10/17*a^2*b^3*x^17 + 2/3*a^3*b^2*x^15 + 5/13*a
^4*b*x^13 + 1/11*a^5*x^11

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{10} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac{5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**10*(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

Integral(x**10*((a + b*x**2)**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.273178, size = 142, normalized size = 0.56 \[ \frac{1}{21} \, b^{5} x^{21}{\rm sign}\left (b x^{2} + a\right ) + \frac{5}{19} \, a b^{4} x^{19}{\rm sign}\left (b x^{2} + a\right ) + \frac{10}{17} \, a^{2} b^{3} x^{17}{\rm sign}\left (b x^{2} + a\right ) + \frac{2}{3} \, a^{3} b^{2} x^{15}{\rm sign}\left (b x^{2} + a\right ) + \frac{5}{13} \, a^{4} b x^{13}{\rm sign}\left (b x^{2} + a\right ) + \frac{1}{11} \, a^{5} x^{11}{\rm sign}\left (b x^{2} + a\right ) \]

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^10,x, algorithm="giac")

[Out]

1/21*b^5*x^21*sign(b*x^2 + a) + 5/19*a*b^4*x^19*sign(b*x^2 + a) + 10/17*a^2*b^3*
x^17*sign(b*x^2 + a) + 2/3*a^3*b^2*x^15*sign(b*x^2 + a) + 5/13*a^4*b*x^13*sign(b
*x^2 + a) + 1/11*a^5*x^11*sign(b*x^2 + a)