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

Optimal. Leaf size=78 \[ \frac{\left (a+b x^3\right )^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}{21 b^2}-\frac{a \left (a+b x^3\right )^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{18 b^2} \]

[Out]

-(a*(a + b*x^3)^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(18*b^2) + ((a + b*x^3)^6*Sqr
t[a^2 + 2*a*b*x^3 + b^2*x^6])/(21*b^2)

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Rubi [A]  time = 0.139693, 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 \[ \frac{\left (a+b x^3\right )^6 \sqrt{a^2+2 a b x^3+b^2 x^6}}{21 b^2}-\frac{a \left (a+b x^3\right )^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{18 b^2} \]

Antiderivative was successfully verified.

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

[Out]

-(a*(a + b*x^3)^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(18*b^2) + ((a + b*x^3)^6*Sqr
t[a^2 + 2*a*b*x^3 + b^2*x^6])/(21*b^2)

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Rubi in Sympy [A]  time = 14.7412, size = 65, normalized size = 0.83 \[ - \frac{a \left (2 a + 2 b x^{3}\right ) \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{5}{2}}}{36 b^{2}} + \frac{\left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{7}{2}}}{21 b^{2}} \]

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)**(5/2),x)

[Out]

-a*(2*a + 2*b*x**3)*(a**2 + 2*a*b*x**3 + b**2*x**6)**(5/2)/(36*b**2) + (a**2 + 2
*a*b*x**3 + b**2*x**6)**(7/2)/(21*b**2)

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Mathematica [A]  time = 0.0361341, size = 83, normalized size = 1.06 \[ \frac{x^6 \sqrt{\left (a+b x^3\right )^2} \left (21 a^5+70 a^4 b x^3+105 a^3 b^2 x^6+84 a^2 b^3 x^9+35 a b^4 x^{12}+6 b^5 x^{15}\right )}{126 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

(x^6*Sqrt[(a + b*x^3)^2]*(21*a^5 + 70*a^4*b*x^3 + 105*a^3*b^2*x^6 + 84*a^2*b^3*x
^9 + 35*a*b^4*x^12 + 6*b^5*x^15))/(126*(a + b*x^3))

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Maple [A]  time = 0.009, size = 80, normalized size = 1. \[{\frac{{x}^{6} \left ( 6\,{b}^{5}{x}^{15}+35\,a{b}^{4}{x}^{12}+84\,{a}^{2}{b}^{3}{x}^{9}+105\,{a}^{3}{b}^{2}{x}^{6}+70\,{a}^{4}b{x}^{3}+21\,{a}^{5} \right ) }{126\, \left ( b{x}^{3}+a \right ) ^{5}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/126*x^6*(6*b^5*x^15+35*a*b^4*x^12+84*a^2*b^3*x^9+105*a^3*b^2*x^6+70*a^4*b*x^3+
21*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^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.249862, size = 77, normalized size = 0.99 \[ \frac{1}{21} \, b^{5} x^{21} + \frac{5}{18} \, a b^{4} x^{18} + \frac{2}{3} \, a^{2} b^{3} x^{15} + \frac{5}{6} \, a^{3} b^{2} x^{12} + \frac{5}{9} \, a^{4} b x^{9} + \frac{1}{6} \, a^{5} x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/21*b^5*x^21 + 5/18*a*b^4*x^18 + 2/3*a^2*b^3*x^15 + 5/6*a^3*b^2*x^12 + 5/9*a^4*
b*x^9 + 1/6*a^5*x^6

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

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)**(5/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.267288, size = 90, normalized size = 1.15 \[ \frac{1}{126} \,{\left (6 \, b^{5} x^{21} + 35 \, a b^{4} x^{18} + 84 \, a^{2} b^{3} x^{15} + 105 \, a^{3} b^{2} x^{12} + 70 \, a^{4} b x^{9} + 21 \, a^{5} x^{6}\right )}{\rm sign}\left (b x^{3} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/126*(6*b^5*x^21 + 35*a*b^4*x^18 + 84*a^2*b^3*x^15 + 105*a^3*b^2*x^12 + 70*a^4*
b*x^9 + 21*a^5*x^6)*sign(b*x^3 + a)