3.586 \(\int x^{11} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx\)

Optimal. Leaf size=255 \[ \frac{b^5 x^{22} \sqrt{a^2+2 a b x^2+b^2 x^4}}{22 \left (a+b x^2\right )}+\frac{a b^4 x^{20} \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 \left (a+b x^2\right )}+\frac{5 a^2 b^3 x^{18} \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 \left (a+b x^2\right )}+\frac{a^5 x^{12} \sqrt{a^2+2 a b x^2+b^2 x^4}}{12 \left (a+b x^2\right )}+\frac{5 a^4 b x^{14} \sqrt{a^2+2 a b x^2+b^2 x^4}}{14 \left (a+b x^2\right )}+\frac{5 a^3 b^2 x^{16} \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 \left (a+b x^2\right )} \]

[Out]

(a^5*x^12*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(12*(a + b*x^2)) + (5*a^4*b*x^14*Sqrt
[a^2 + 2*a*b*x^2 + b^2*x^4])/(14*(a + b*x^2)) + (5*a^3*b^2*x^16*Sqrt[a^2 + 2*a*b
*x^2 + b^2*x^4])/(8*(a + b*x^2)) + (5*a^2*b^3*x^18*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^
4])/(9*(a + b*x^2)) + (a*b^4*x^20*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(4*(a + b*x^2
)) + (b^5*x^22*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(22*(a + b*x^2))

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Rubi [A]  time = 0.380891, antiderivative size = 255, 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 \[ \frac{b^5 x^{22} \sqrt{a^2+2 a b x^2+b^2 x^4}}{22 \left (a+b x^2\right )}+\frac{a b^4 x^{20} \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 \left (a+b x^2\right )}+\frac{5 a^2 b^3 x^{18} \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 \left (a+b x^2\right )}+\frac{a^5 x^{12} \sqrt{a^2+2 a b x^2+b^2 x^4}}{12 \left (a+b x^2\right )}+\frac{5 a^4 b x^{14} \sqrt{a^2+2 a b x^2+b^2 x^4}}{14 \left (a+b x^2\right )}+\frac{5 a^3 b^2 x^{16} \sqrt{a^2+2 a b x^2+b^2 x^4}}{8 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^11*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(a^5*x^12*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(12*(a + b*x^2)) + (5*a^4*b*x^14*Sqrt
[a^2 + 2*a*b*x^2 + b^2*x^4])/(14*(a + b*x^2)) + (5*a^3*b^2*x^16*Sqrt[a^2 + 2*a*b
*x^2 + b^2*x^4])/(8*(a + b*x^2)) + (5*a^2*b^3*x^18*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^
4])/(9*(a + b*x^2)) + (a*b^4*x^20*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(4*(a + b*x^2
)) + (b^5*x^22*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(22*(a + b*x^2))

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Rubi in Sympy [A]  time = 26.9294, size = 199, normalized size = 0.78 \[ \frac{a^{5} x^{12} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{5544 \left (a + b x^{2}\right )} + \frac{a^{4} x^{12} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{924} + \frac{a^{3} x^{12} \left (a + b x^{2}\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{264} + \frac{a^{2} x^{12} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{99} + \frac{a x^{12} \left (a + b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{44} + \frac{x^{12} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{22} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11*(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

a**5*x**12*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(5544*(a + b*x**2)) + a**4*x**12*
sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/924 + a**3*x**12*(a + b*x**2)*sqrt(a**2 + 2*
a*b*x**2 + b**2*x**4)/264 + a**2*x**12*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/99
+ a*x**12*(a + b*x**2)*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/44 + x**12*(a**2
+ 2*a*b*x**2 + b**2*x**4)**(5/2)/22

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Mathematica [A]  time = 0.0354557, size = 83, normalized size = 0.33 \[ \frac{x^{12} \sqrt{\left (a+b x^2\right )^2} \left (462 a^5+1980 a^4 b x^2+3465 a^3 b^2 x^4+3080 a^2 b^3 x^6+1386 a b^4 x^8+252 b^5 x^{10}\right )}{5544 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[x^11*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]

[Out]

(x^12*Sqrt[(a + b*x^2)^2]*(462*a^5 + 1980*a^4*b*x^2 + 3465*a^3*b^2*x^4 + 3080*a^
2*b^3*x^6 + 1386*a*b^4*x^8 + 252*b^5*x^10))/(5544*(a + b*x^2))

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Maple [A]  time = 0.01, size = 80, normalized size = 0.3 \[{\frac{{x}^{12} \left ( 252\,{b}^{5}{x}^{10}+1386\,a{b}^{4}{x}^{8}+3080\,{a}^{2}{b}^{3}{x}^{6}+3465\,{a}^{3}{b}^{2}{x}^{4}+1980\,{a}^{4}b{x}^{2}+462\,{a}^{5} \right ) }{5544\, \left ( b{x}^{2}+a \right ) ^{5}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11*(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)

[Out]

1/5544*x^12*(252*b^5*x^10+1386*a*b^4*x^8+3080*a^2*b^3*x^6+3465*a^3*b^2*x^4+1980*
a^4*b*x^2+462*a^5)*((b*x^2+a)^2)^(5/2)/(b*x^2+a)^5

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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((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)*x^11,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.25975, size = 77, normalized size = 0.3 \[ \frac{1}{22} \, b^{5} x^{22} + \frac{1}{4} \, a b^{4} x^{20} + \frac{5}{9} \, a^{2} b^{3} x^{18} + \frac{5}{8} \, a^{3} b^{2} x^{16} + \frac{5}{14} \, a^{4} b x^{14} + \frac{1}{12} \, a^{5} x^{12} \]

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^11,x, algorithm="fricas")

[Out]

1/22*b^5*x^22 + 1/4*a*b^4*x^20 + 5/9*a^2*b^3*x^18 + 5/8*a^3*b^2*x^16 + 5/14*a^4*
b*x^14 + 1/12*a^5*x^12

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{11} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac{5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**11*(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)

[Out]

Integral(x**11*((a + b*x**2)**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.270208, size = 142, normalized size = 0.56 \[ \frac{1}{22} \, b^{5} x^{22}{\rm sign}\left (b x^{2} + a\right ) + \frac{1}{4} \, a b^{4} x^{20}{\rm sign}\left (b x^{2} + a\right ) + \frac{5}{9} \, a^{2} b^{3} x^{18}{\rm sign}\left (b x^{2} + a\right ) + \frac{5}{8} \, a^{3} b^{2} x^{16}{\rm sign}\left (b x^{2} + a\right ) + \frac{5}{14} \, a^{4} b x^{14}{\rm sign}\left (b x^{2} + a\right ) + \frac{1}{12} \, a^{5} x^{12}{\rm sign}\left (b x^{2} + a\right ) \]

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^11,x, algorithm="giac")

[Out]

1/22*b^5*x^22*sign(b*x^2 + a) + 1/4*a*b^4*x^20*sign(b*x^2 + a) + 5/9*a^2*b^3*x^1
8*sign(b*x^2 + a) + 5/8*a^3*b^2*x^16*sign(b*x^2 + a) + 5/14*a^4*b*x^14*sign(b*x^
2 + a) + 1/12*a^5*x^12*sign(b*x^2 + a)