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

Optimal. Leaf size=78 \[ \frac{\left (a+b x^3\right )^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{15 b^2}-\frac{a \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{12 b^2} \]

[Out]

-(a*(a + b*x^3)^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(12*b^2) + ((a + b*x^3)^4*Sqr
t[a^2 + 2*a*b*x^3 + b^2*x^6])/(15*b^2)

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Rubi [A]  time = 0.127671, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{\left (a+b x^3\right )^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}{15 b^2}-\frac{a \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}{12 b^2} \]

Antiderivative was successfully verified.

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

[Out]

-(a*(a + b*x^3)^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(12*b^2) + ((a + b*x^3)^4*Sqr
t[a^2 + 2*a*b*x^3 + b^2*x^6])/(15*b^2)

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Rubi in Sympy [A]  time = 14.7323, size = 65, normalized size = 0.83 \[ - \frac{a \left (2 a + 2 b x^{3}\right ) \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{24 b^{2}} + \frac{\left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{5}{2}}}{15 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*(2*a + 2*b*x**3)*(a**2 + 2*a*b*x**3 + b**2*x**6)**(3/2)/(24*b**2) + (a**2 + 2
*a*b*x**3 + b**2*x**6)**(5/2)/(15*b**2)

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Mathematica [A]  time = 0.0311487, size = 61, normalized size = 0.78 \[ \frac{x^6 \sqrt{\left (a+b x^3\right )^2} \left (10 a^3+20 a^2 b x^3+15 a b^2 x^6+4 b^3 x^9\right )}{60 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

(x^6*Sqrt[(a + b*x^3)^2]*(10*a^3 + 20*a^2*b*x^3 + 15*a*b^2*x^6 + 4*b^3*x^9))/(60
*(a + b*x^3))

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Maple [A]  time = 0.01, size = 58, normalized size = 0.7 \[{\frac{{x}^{6} \left ( 4\,{b}^{3}{x}^{9}+15\,a{b}^{2}{x}^{6}+20\,{a}^{2}b{x}^{3}+10\,{a}^{3} \right ) }{60\, \left ( b{x}^{3}+a \right ) ^{3}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/60*x^6*(4*b^3*x^9+15*a*b^2*x^6+20*a^2*b*x^3+10*a^3)*((b*x^3+a)^2)^(3/2)/(b*x^3
+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^6 + 2*a*b*x^3 + a^2)^(3/2)*x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.249463, size = 47, normalized size = 0.6 \[ \frac{1}{15} \, b^{3} x^{15} + \frac{1}{4} \, a b^{2} x^{12} + \frac{1}{3} \, a^{2} b x^{9} + \frac{1}{6} \, a^{3} x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.298897, size = 61, normalized size = 0.78 \[ \frac{1}{60} \,{\left (4 \, b^{3} x^{15} + 15 \, a b^{2} x^{12} + 20 \, a^{2} b x^{9} + 10 \, a^{3} x^{6}\right )}{\rm sign}\left (b x^{3} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/60*(4*b^3*x^15 + 15*a*b^2*x^12 + 20*a^2*b*x^9 + 10*a^3*x^6)*sign(b*x^3 + a)