3.55 \(\int x^8 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2} \, dx\)

Optimal. Leaf size=119 \[ \frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^7}{24 b^3}-\frac{2 a \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^6}{21 b^3}+\frac{a^2 \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^5}{18 b^3} \]

[Out]

(a^2*(a + b*x^3)^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(18*b^3) - (2*a*(a + b*x^3)^
6*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(21*b^3) + ((a + b*x^3)^7*Sqrt[a^2 + 2*a*b*x^
3 + b^2*x^6])/(24*b^3)

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Rubi [A]  time = 0.207968, antiderivative size = 119, 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 \[ \frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^7}{24 b^3}-\frac{2 a \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^6}{21 b^3}+\frac{a^2 \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^5}{18 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^8*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]

[Out]

(a^2*(a + b*x^3)^5*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(18*b^3) - (2*a*(a + b*x^3)^
6*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])/(21*b^3) + ((a + b*x^3)^7*Sqrt[a^2 + 2*a*b*x^
3 + b^2*x^6])/(24*b^3)

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Rubi in Sympy [A]  time = 21.9823, size = 107, normalized size = 0.9 \[ \frac{a^{2} \left (2 a + 2 b x^{3}\right ) \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{5}{2}}}{144 b^{3}} - \frac{a \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{7}{2}}}{84 b^{3}} + \frac{x^{6} \left (2 a + 2 b x^{3}\right ) \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{5}{2}}}{48 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8*(b**2*x**6+2*a*b*x**3+a**2)**(5/2),x)

[Out]

a**2*(2*a + 2*b*x**3)*(a**2 + 2*a*b*x**3 + b**2*x**6)**(5/2)/(144*b**3) - a*(a**
2 + 2*a*b*x**3 + b**2*x**6)**(7/2)/(84*b**3) + x**6*(2*a + 2*b*x**3)*(a**2 + 2*a
*b*x**3 + b**2*x**6)**(5/2)/(48*b)

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Mathematica [A]  time = 0.0367037, size = 83, normalized size = 0.7 \[ \frac{x^9 \sqrt{\left (a+b x^3\right )^2} \left (56 a^5+210 a^4 b x^3+336 a^3 b^2 x^6+280 a^2 b^3 x^9+120 a b^4 x^{12}+21 b^5 x^{15}\right )}{504 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]

[Out]

(x^9*Sqrt[(a + b*x^3)^2]*(56*a^5 + 210*a^4*b*x^3 + 336*a^3*b^2*x^6 + 280*a^2*b^3
*x^9 + 120*a*b^4*x^12 + 21*b^5*x^15))/(504*(a + b*x^3))

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Maple [A]  time = 0.009, size = 80, normalized size = 0.7 \[{\frac{{x}^{9} \left ( 21\,{b}^{5}{x}^{15}+120\,a{b}^{4}{x}^{12}+280\,{a}^{2}{b}^{3}{x}^{9}+336\,{a}^{3}{b}^{2}{x}^{6}+210\,{a}^{4}b{x}^{3}+56\,{a}^{5} \right ) }{504\, \left ( b{x}^{3}+a \right ) ^{5}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8*(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x)

[Out]

1/504*x^9*(21*b^5*x^15+120*a*b^4*x^12+280*a^2*b^3*x^9+336*a^3*b^2*x^6+210*a^4*b*
x^3+56*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^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.250057, size = 77, normalized size = 0.65 \[ \frac{1}{24} \, b^{5} x^{24} + \frac{5}{21} \, a b^{4} x^{21} + \frac{5}{9} \, a^{2} b^{3} x^{18} + \frac{2}{3} \, a^{3} b^{2} x^{15} + \frac{5}{12} \, a^{4} b x^{12} + \frac{1}{9} \, a^{5} x^{9} \]

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^8,x, algorithm="fricas")

[Out]

1/24*b^5*x^24 + 5/21*a*b^4*x^21 + 5/9*a^2*b^3*x^18 + 2/3*a^3*b^2*x^15 + 5/12*a^4
*b*x^12 + 1/9*a^5*x^9

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{8} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8*(b**2*x**6+2*a*b*x**3+a**2)**(5/2),x)

[Out]

Integral(x**8*((a + b*x**3)**2)**(5/2), x)

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GIAC/XCAS [A]  time = 0.288693, size = 142, normalized size = 1.19 \[ \frac{1}{24} \, b^{5} x^{24}{\rm sign}\left (b x^{3} + a\right ) + \frac{5}{21} \, a b^{4} x^{21}{\rm sign}\left (b x^{3} + a\right ) + \frac{5}{9} \, a^{2} b^{3} x^{18}{\rm sign}\left (b x^{3} + a\right ) + \frac{2}{3} \, a^{3} b^{2} x^{15}{\rm sign}\left (b x^{3} + a\right ) + \frac{5}{12} \, a^{4} b x^{12}{\rm sign}\left (b x^{3} + a\right ) + \frac{1}{9} \, a^{5} x^{9}{\rm sign}\left (b x^{3} + a\right ) \]

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^8,x, algorithm="giac")

[Out]

1/24*b^5*x^24*sign(b*x^3 + a) + 5/21*a*b^4*x^21*sign(b*x^3 + a) + 5/9*a^2*b^3*x^
18*sign(b*x^3 + a) + 2/3*a^3*b^2*x^15*sign(b*x^3 + a) + 5/12*a^4*b*x^12*sign(b*x
^3 + a) + 1/9*a^5*x^9*sign(b*x^3 + a)