3.61 \(\int x^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx\)

Optimal. Leaf size=36 \[ \frac{\left (a+b x^3\right ) \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{18 b} \]

[Out]

((a + b*x^3)*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2))/(18*b)

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Rubi [A]  time = 0.0688459, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{\left (a+b x^3\right ) \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}{18 b} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]

[Out]

((a + b*x^3)*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2))/(18*b)

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Rubi in Sympy [A]  time = 9.21185, size = 34, normalized size = 0.94 \[ \frac{\left (2 a + 2 b x^{3}\right ) \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{5}{2}}}{36 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(b**2*x**6+2*a*b*x**3+a**2)**(5/2),x)

[Out]

(2*a + 2*b*x**3)*(a**2 + 2*a*b*x**3 + b**2*x**6)**(5/2)/(36*b)

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Mathematica [B]  time = 0.0356727, size = 82, normalized size = 2.28 \[ \frac{x^3 \sqrt{\left (a+b x^3\right )^2} \left (6 a^5+15 a^4 b x^3+20 a^3 b^2 x^6+15 a^2 b^3 x^9+6 a b^4 x^{12}+b^5 x^{15}\right )}{18 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]

[Out]

(x^3*Sqrt[(a + b*x^3)^2]*(6*a^5 + 15*a^4*b*x^3 + 20*a^3*b^2*x^6 + 15*a^2*b^3*x^9
+ 6*a*b^4*x^12 + b^5*x^15))/(18*(a + b*x^3))

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Maple [B]  time = 0.01, size = 79, normalized size = 2.2 \[{\frac{{x}^{3} \left ({b}^{5}{x}^{15}+6\,a{b}^{4}{x}^{12}+15\,{a}^{2}{b}^{3}{x}^{9}+20\,{a}^{3}{b}^{2}{x}^{6}+15\,{a}^{4}b{x}^{3}+6\,{a}^{5} \right ) }{18\, \left ( b{x}^{3}+a \right ) ^{5}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x)

[Out]

1/18*x^3*(b^5*x^15+6*a*b^4*x^12+15*a^2*b^3*x^9+20*a^3*b^2*x^6+15*a^4*b*x^3+6*a^5
)*((b*x^3+a)^2)^(5/2)/(b*x^3+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^6 + 2*a*b*x^3 + a^2)^(5/2)*x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.267522, size = 77, normalized size = 2.14 \[ \frac{1}{18} \, b^{5} x^{18} + \frac{1}{3} \, a b^{4} x^{15} + \frac{5}{6} \, a^{2} b^{3} x^{12} + \frac{10}{9} \, a^{3} b^{2} x^{9} + \frac{5}{6} \, a^{4} b x^{6} + \frac{1}{3} \, a^{5} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^(5/2)*x^2,x, algorithm="fricas")

[Out]

1/18*b^5*x^18 + 1/3*a*b^4*x^15 + 5/6*a^2*b^3*x^12 + 10/9*a^3*b^2*x^9 + 5/6*a^4*b
*x^6 + 1/3*a^5*x^3

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{2} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(b**2*x**6+2*a*b*x**3+a**2)**(5/2),x)

[Out]

Integral(x**2*((a + b*x**3)**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.269938, size = 89, normalized size = 2.47 \[ \frac{1}{18} \,{\left (b^{5} x^{18} + 6 \, a b^{4} x^{15} + 15 \, a^{2} b^{3} x^{12} + 20 \, a^{3} b^{2} x^{9} + 15 \, a^{4} b x^{6} + 6 \, a^{5} x^{3}\right )}{\rm sign}\left (b x^{3} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^(5/2)*x^2,x, algorithm="giac")

[Out]

1/18*(b^5*x^18 + 6*a*b^4*x^15 + 15*a^2*b^3*x^12 + 20*a^3*b^2*x^9 + 15*a^4*b*x^6
+ 6*a^5*x^3)*sign(b*x^3 + a)