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

Optimal. Leaf size=248 \[ \frac{b^5 x^{11} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{11 \left (a+b x^2\right )^5}+\frac{5 a b^4 x^9 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{9 \left (a+b x^2\right )^5}+\frac{10 a^2 b^3 x^7 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{7 \left (a+b x^2\right )^5}+\frac{a^5 x \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{\left (a+b x^2\right )^5}+\frac{5 a^4 b x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{3 \left (a+b x^2\right )^5}+\frac{2 a^3 b^2 x^5 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{\left (a+b x^2\right )^5} \]

[Out]

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

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Rubi [A]  time = 0.141133, antiderivative size = 248, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{b^5 x^{11} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{11 \left (a+b x^2\right )^5}+\frac{5 a b^4 x^9 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{9 \left (a+b x^2\right )^5}+\frac{10 a^2 b^3 x^7 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{7 \left (a+b x^2\right )^5}+\frac{a^5 x \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{\left (a+b x^2\right )^5}+\frac{5 a^4 b x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{3 \left (a+b x^2\right )^5}+\frac{2 a^3 b^2 x^5 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{\left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 46.4279, size = 197, normalized size = 0.79 \[ \frac{256 a^{5} x \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{693 \left (a + b x^{2}\right )} + \frac{128 a^{4} x \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{693} + \frac{32 a^{3} x \left (a + b x^{2}\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{231} + \frac{80 a^{2} x \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{693} + \frac{10 a x \left (a + b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{99} + \frac{x \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{11} \]

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)

[Out]

256*a**5*x*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(693*(a + b*x**2)) + 128*a**4*x*s
qrt(a**2 + 2*a*b*x**2 + b**2*x**4)/693 + 32*a**3*x*(a + b*x**2)*sqrt(a**2 + 2*a*
b*x**2 + b**2*x**4)/231 + 80*a**2*x*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/693 +
10*a*x*(a + b*x**2)*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/99 + x*(a**2 + 2*a*b
*x**2 + b**2*x**4)**(5/2)/11

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Mathematica [A]  time = 0.0320447, size = 81, normalized size = 0.33 \[ \frac{\sqrt{\left (a+b x^2\right )^2} \left (693 a^5 x+1155 a^4 b x^3+1386 a^3 b^2 x^5+990 a^2 b^3 x^7+385 a b^4 x^9+63 b^5 x^{11}\right )}{693 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(a + b*x^2)^2]*(693*a^5*x + 1155*a^4*b*x^3 + 1386*a^3*b^2*x^5 + 990*a^2*b^
3*x^7 + 385*a*b^4*x^9 + 63*b^5*x^11))/(693*(a + b*x^2))

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Maple [A]  time = 0.006, size = 78, normalized size = 0.3 \[{\frac{x \left ( 63\,{b}^{5}{x}^{10}+385\,a{b}^{4}{x}^{8}+990\,{a}^{2}{b}^{3}{x}^{6}+1386\,{a}^{3}{b}^{2}{x}^{4}+1155\,{a}^{4}b{x}^{2}+693\,{a}^{5} \right ) }{693\, \left ( b{x}^{2}+a \right ) ^{5}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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)

[Out]

1/693*x*(63*b^5*x^10+385*a*b^4*x^8+990*a^2*b^3*x^6+1386*a^3*b^2*x^4+1155*a^4*b*x
^2+693*a^5)*((b*x^2+a)^2)^(5/2)/(b*x^2+a)^5

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Maxima [A]  time = 0.693496, size = 73, normalized size = 0.29 \[ \frac{1}{11} \, b^{5} x^{11} + \frac{5}{9} \, a b^{4} x^{9} + \frac{10}{7} \, a^{2} b^{3} x^{7} + 2 \, a^{3} b^{2} x^{5} + \frac{5}{3} \, a^{4} b x^{3} + a^{5} x \]

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, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.255879, size = 73, normalized size = 0.29 \[ \frac{1}{11} \, b^{5} x^{11} + \frac{5}{9} \, a b^{4} x^{9} + \frac{10}{7} \, a^{2} b^{3} x^{7} + 2 \, a^{3} b^{2} x^{5} + \frac{5}{3} \, a^{4} b x^{3} + a^{5} x \]

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, algorithm="fricas")

[Out]

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

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

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)

[Out]

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

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GIAC/XCAS [A]  time = 0.271644, size = 138, normalized size = 0.56 \[ \frac{1}{11} \, b^{5} x^{11}{\rm sign}\left (b x^{2} + a\right ) + \frac{5}{9} \, a b^{4} x^{9}{\rm sign}\left (b x^{2} + a\right ) + \frac{10}{7} \, a^{2} b^{3} x^{7}{\rm sign}\left (b x^{2} + a\right ) + 2 \, a^{3} b^{2} x^{5}{\rm sign}\left (b x^{2} + a\right ) + \frac{5}{3} \, a^{4} b x^{3}{\rm sign}\left (b x^{2} + a\right ) + a^{5} x{\rm sign}\left (b x^{2} + a\right ) \]

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, algorithm="giac")

[Out]

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