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

Optimal. Leaf size=67 \[ \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{10 b^2}-\frac{a \left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{8 b^2} \]

[Out]

-(a*(a + b*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2))/(8*b^2) + (a^2 + 2*a*b*x^2 +
b^2*x^4)^(5/2)/(10*b^2)

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Rubi [A]  time = 0.128327, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}{10 b^2}-\frac{a \left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{8 b^2} \]

Antiderivative was successfully verified.

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

[Out]

-(a*(a + b*x^2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2))/(8*b^2) + (a^2 + 2*a*b*x^2 +
b^2*x^4)^(5/2)/(10*b^2)

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Rubi in Sympy [A]  time = 13.6864, size = 65, normalized size = 0.97 \[ - \frac{a \left (2 a + 2 b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}}{16 b^{2}} + \frac{\left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{10 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*(2*a + 2*b*x**2)*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)/(16*b**2) + (a**2 + 2
*a*b*x**2 + b**2*x**4)**(5/2)/(10*b**2)

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Mathematica [A]  time = 0.0278878, size = 61, normalized size = 0.91 \[ \frac{x^4 \sqrt{\left (a+b x^2\right )^2} \left (10 a^3+20 a^2 b x^2+15 a b^2 x^4+4 b^3 x^6\right )}{40 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(x^4*Sqrt[(a + b*x^2)^2]*(10*a^3 + 20*a^2*b*x^2 + 15*a*b^2*x^4 + 4*b^3*x^6))/(40
*(a + b*x^2))

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Maple [A]  time = 0.009, size = 58, normalized size = 0.9 \[{\frac{{x}^{4} \left ( 4\,{b}^{3}{x}^{6}+15\,a{b}^{2}{x}^{4}+20\,{a}^{2}b{x}^{2}+10\,{a}^{3} \right ) }{40\, \left ( b{x}^{2}+a \right ) ^{3}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/40*x^4*(4*b^3*x^6+15*a*b^2*x^4+20*a^2*b*x^2+10*a^3)*((b*x^2+a)^2)^(3/2)/(b*x^2
+a)^3

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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)^(3/2)*x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.264811, size = 47, normalized size = 0.7 \[ \frac{1}{10} \, b^{3} x^{10} + \frac{3}{8} \, a b^{2} x^{8} + \frac{1}{2} \, a^{2} b x^{6} + \frac{1}{4} \, a^{3} x^{4} \]

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^3,x, algorithm="fricas")

[Out]

1/10*b^3*x^10 + 3/8*a*b^2*x^8 + 1/2*a^2*b*x^6 + 1/4*a^3*x^4

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.26998, size = 61, normalized size = 0.91 \[ \frac{1}{40} \,{\left (4 \, b^{3} x^{10} + 15 \, a b^{2} x^{8} + 20 \, a^{2} b x^{6} + 10 \, a^{3} x^{4}\right )}{\rm sign}\left (b x^{2} + a\right ) \]

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^3,x, algorithm="giac")

[Out]

1/40*(4*b^3*x^10 + 15*a*b^2*x^8 + 20*a^2*b*x^6 + 10*a^3*x^4)*sign(b*x^2 + a)