3.575 \(\int x^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2} \, dx\)

Optimal. Leaf size=167 \[ \frac{3 a b^2 x^7 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 \left (a+b x^2\right )}+\frac{3 a^2 b x^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 \left (a+b x^2\right )}+\frac{b^3 x^9 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 \left (a+b x^2\right )}+\frac{a^3 x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.131793, antiderivative size = 167, 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 \[ \frac{3 a b^2 x^7 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 \left (a+b x^2\right )}+\frac{3 a^2 b x^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 \left (a+b x^2\right )}+\frac{b^3 x^9 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 \left (a+b x^2\right )}+\frac{a^3 x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 17.6938, size = 136, normalized size = 0.81 \[ \frac{16 a^{3} x^{3} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{315 \left (a + b x^{2}\right )} + \frac{8 a^{2} x^{3} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{105} + \frac{2 a x^{3} \left (a + b x^{2}\right ) \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}}{21} + \frac{x^{3} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)

[Out]

16*a**3*x**3*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/(315*(a + b*x**2)) + 8*a**2*x**
3*sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)/105 + 2*a*x**3*(a + b*x**2)*sqrt(a**2 + 2*
a*b*x**2 + b**2*x**4)/21 + x**3*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/9

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Mathematica [A]  time = 0.026133, size = 61, normalized size = 0.37 \[ \frac{\sqrt{\left (a+b x^2\right )^2} \left (105 a^3 x^3+189 a^2 b x^5+135 a b^2 x^7+35 b^3 x^9\right )}{315 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]

[Out]

(Sqrt[(a + b*x^2)^2]*(105*a^3*x^3 + 189*a^2*b*x^5 + 135*a*b^2*x^7 + 35*b^3*x^9))
/(315*(a + b*x^2))

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Maple [A]  time = 0.009, size = 58, normalized size = 0.4 \[{\frac{{x}^{3} \left ( 35\,{b}^{3}{x}^{6}+135\,a{x}^{4}{b}^{2}+189\,{a}^{2}b{x}^{2}+105\,{a}^{3} \right ) }{315\, \left ( b{x}^{2}+a \right ) ^{3}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x)

[Out]

1/315*x^3*(35*b^3*x^6+135*a*b^2*x^4+189*a^2*b*x^2+105*a^3)*((b*x^2+a)^2)^(3/2)/(
b*x^2+a)^3

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Maxima [A]  time = 0.695685, size = 47, normalized size = 0.28 \[ \frac{1}{9} \, b^{3} x^{9} + \frac{3}{7} \, a b^{2} x^{7} + \frac{3}{5} \, a^{2} b x^{5} + \frac{1}{3} \, a^{3} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)*x^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.26612, size = 47, normalized size = 0.28 \[ \frac{1}{9} \, b^{3} x^{9} + \frac{3}{7} \, a b^{2} x^{7} + \frac{3}{5} \, a^{2} b x^{5} + \frac{1}{3} \, a^{3} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)*x^2,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{2} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)

[Out]

Integral(x**2*((a + b*x**2)**2)**(3/2), x)

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GIAC/XCAS [A]  time = 0.27157, size = 90, normalized size = 0.54 \[ \frac{1}{9} \, b^{3} x^{9}{\rm sign}\left (b x^{2} + a\right ) + \frac{3}{7} \, a b^{2} x^{7}{\rm sign}\left (b x^{2} + a\right ) + \frac{3}{5} \, a^{2} b x^{5}{\rm sign}\left (b x^{2} + a\right ) + \frac{1}{3} \, a^{3} x^{3}{\rm sign}\left (b x^{2} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)*x^2,x, algorithm="giac")

[Out]

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