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

Optimal. Leaf size=255 \[ \frac{b^5 x^{15} \sqrt{a^2+2 a b x^2+b^2 x^4}}{15 \left (a+b x^2\right )}+\frac{5 a b^4 x^{13} \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 \left (a+b x^2\right )}+\frac{10 a^2 b^3 x^{11} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 \left (a+b x^2\right )}+\frac{a^5 x^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 \left (a+b x^2\right )}+\frac{5 a^4 b x^7 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 \left (a+b x^2\right )}+\frac{10 a^3 b^2 x^9 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 \left (a+b x^2\right )} \]

[Out]

(a^5*x^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(5*(a + b*x^2)) + (5*a^4*b*x^7*Sqrt[a^
2 + 2*a*b*x^2 + b^2*x^4])/(7*(a + b*x^2)) + (10*a^3*b^2*x^9*Sqrt[a^2 + 2*a*b*x^2
+ b^2*x^4])/(9*(a + b*x^2)) + (10*a^2*b^3*x^11*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])
/(11*(a + b*x^2)) + (5*a*b^4*x^13*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(13*(a + b*x^
2)) + (b^5*x^15*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(15*(a + b*x^2))

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Rubi [A]  time = 0.185073, 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 \[ \frac{b^5 x^{15} \sqrt{a^2+2 a b x^2+b^2 x^4}}{15 \left (a+b x^2\right )}+\frac{5 a b^4 x^{13} \sqrt{a^2+2 a b x^2+b^2 x^4}}{13 \left (a+b x^2\right )}+\frac{10 a^2 b^3 x^{11} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 \left (a+b x^2\right )}+\frac{a^5 x^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 \left (a+b x^2\right )}+\frac{5 a^4 b x^7 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 \left (a+b x^2\right )}+\frac{10 a^3 b^2 x^9 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(a^5*x^5*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(5*(a + b*x^2)) + (5*a^4*b*x^7*Sqrt[a^
2 + 2*a*b*x^2 + b^2*x^4])/(7*(a + b*x^2)) + (10*a^3*b^2*x^9*Sqrt[a^2 + 2*a*b*x^2
+ b^2*x^4])/(9*(a + b*x^2)) + (10*a^2*b^3*x^11*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])
/(11*(a + b*x^2)) + (5*a*b^4*x^13*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(13*(a + b*x^
2)) + (b^5*x^15*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])/(15*(a + b*x^2))

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Rubi in Sympy [A]  time = 27.4097, size = 207, normalized size = 0.81 \[ \frac{256 a^{5} x^{5} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{45045 \left (a + b x^{2}\right )} + \frac{128 a^{4} x^{5} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{9009} + \frac{32 a^{3} x^{5} \left (a + b x^{2}\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{1287} + \frac{16 a^{2} x^{5} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{429} + \frac{2 a x^{5} \left (a + b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{39} + \frac{x^{5} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

256*a**5*x**5*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(45045*(a + b*x**2)) + 128*a**
4*x**5*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/9009 + 32*a**3*x**5*(a + b*x**2)*sqrt
(a**2 + 2*a*b*x**2 + b**2*x**4)/1287 + 16*a**2*x**5*(a**2 + 2*a*b*x**2 + b**2*x*
*4)**(3/2)/429 + 2*a*x**5*(a + b*x**2)*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/39
+ x**5*(a**2 + 2*a*b*x**2 + b**2*x**4)**(5/2)/15

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Mathematica [A]  time = 0.0349085, size = 83, normalized size = 0.33 \[ \frac{x^5 \sqrt{\left (a+b x^2\right )^2} \left (9009 a^5+32175 a^4 b x^2+50050 a^3 b^2 x^4+40950 a^2 b^3 x^6+17325 a b^4 x^8+3003 b^5 x^{10}\right )}{45045 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(x^5*Sqrt[(a + b*x^2)^2]*(9009*a^5 + 32175*a^4*b*x^2 + 50050*a^3*b^2*x^4 + 40950
*a^2*b^3*x^6 + 17325*a*b^4*x^8 + 3003*b^5*x^10))/(45045*(a + b*x^2))

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Maple [A]  time = 0.01, size = 80, normalized size = 0.3 \[{\frac{{x}^{5} \left ( 3003\,{b}^{5}{x}^{10}+17325\,a{b}^{4}{x}^{8}+40950\,{a}^{2}{b}^{3}{x}^{6}+50050\,{a}^{3}{b}^{2}{x}^{4}+32175\,{a}^{4}b{x}^{2}+9009\,{a}^{5} \right ) }{45045\, \left ( b{x}^{2}+a \right ) ^{5}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/45045*x^5*(3003*b^5*x^10+17325*a*b^4*x^8+40950*a^2*b^3*x^6+50050*a^3*b^2*x^4+3
2175*a^4*b*x^2+9009*a^5)*((b*x^2+a)^2)^(5/2)/(b*x^2+a)^5

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Maxima [A]  time = 0.701013, size = 77, normalized size = 0.3 \[ \frac{1}{15} \, b^{5} x^{15} + \frac{5}{13} \, a b^{4} x^{13} + \frac{10}{11} \, a^{2} b^{3} x^{11} + \frac{10}{9} \, a^{3} b^{2} x^{9} + \frac{5}{7} \, a^{4} b x^{7} + \frac{1}{5} \, a^{5} x^{5} \]

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^4,x, algorithm="maxima")

[Out]

1/15*b^5*x^15 + 5/13*a*b^4*x^13 + 10/11*a^2*b^3*x^11 + 10/9*a^3*b^2*x^9 + 5/7*a^
4*b*x^7 + 1/5*a^5*x^5

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Fricas [A]  time = 0.259928, size = 77, normalized size = 0.3 \[ \frac{1}{15} \, b^{5} x^{15} + \frac{5}{13} \, a b^{4} x^{13} + \frac{10}{11} \, a^{2} b^{3} x^{11} + \frac{10}{9} \, a^{3} b^{2} x^{9} + \frac{5}{7} \, a^{4} b x^{7} + \frac{1}{5} \, a^{5} x^{5} \]

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^4,x, algorithm="fricas")

[Out]

1/15*b^5*x^15 + 5/13*a*b^4*x^13 + 10/11*a^2*b^3*x^11 + 10/9*a^3*b^2*x^9 + 5/7*a^
4*b*x^7 + 1/5*a^5*x^5

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.270671, size = 142, normalized size = 0.56 \[ \frac{1}{15} \, b^{5} x^{15}{\rm sign}\left (b x^{2} + a\right ) + \frac{5}{13} \, a b^{4} x^{13}{\rm sign}\left (b x^{2} + a\right ) + \frac{10}{11} \, a^{2} b^{3} x^{11}{\rm sign}\left (b x^{2} + a\right ) + \frac{10}{9} \, a^{3} b^{2} x^{9}{\rm sign}\left (b x^{2} + a\right ) + \frac{5}{7} \, a^{4} b x^{7}{\rm sign}\left (b x^{2} + a\right ) + \frac{1}{5} \, a^{5} x^{5}{\rm sign}\left (b x^{2} + a\right ) \]

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^4,x, algorithm="giac")

[Out]

1/15*b^5*x^15*sign(b*x^2 + a) + 5/13*a*b^4*x^13*sign(b*x^2 + a) + 10/11*a^2*b^3*
x^11*sign(b*x^2 + a) + 10/9*a^3*b^2*x^9*sign(b*x^2 + a) + 5/7*a^4*b*x^7*sign(b*x
^2 + a) + 1/5*a^5*x^5*sign(b*x^2 + a)