3.89 \(\int x^5 \left (a+b x^2\right )^8 \, dx\)

Optimal. Leaf size=53 \[ \frac{a^2 \left (a+b x^2\right )^9}{18 b^3}+\frac{\left (a+b x^2\right )^{11}}{22 b^3}-\frac{a \left (a+b x^2\right )^{10}}{10 b^3} \]

[Out]

(a^2*(a + b*x^2)^9)/(18*b^3) - (a*(a + b*x^2)^10)/(10*b^3) + (a + b*x^2)^11/(22*
b^3)

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Rubi [A]  time = 0.205004, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{a^2 \left (a+b x^2\right )^9}{18 b^3}+\frac{\left (a+b x^2\right )^{11}}{22 b^3}-\frac{a \left (a+b x^2\right )^{10}}{10 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^5*(a + b*x^2)^8,x]

[Out]

(a^2*(a + b*x^2)^9)/(18*b^3) - (a*(a + b*x^2)^10)/(10*b^3) + (a + b*x^2)^11/(22*
b^3)

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Rubi in Sympy [A]  time = 17.4834, size = 44, normalized size = 0.83 \[ \frac{a^{2} \left (a + b x^{2}\right )^{9}}{18 b^{3}} - \frac{a \left (a + b x^{2}\right )^{10}}{10 b^{3}} + \frac{\left (a + b x^{2}\right )^{11}}{22 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*(a + b*x**2)**9/(18*b**3) - a*(a + b*x**2)**10/(10*b**3) + (a + b*x**2)**11
/(22*b**3)

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Mathematica [A]  time = 0.00456712, size = 103, normalized size = 1.94 \[ \frac{a^8 x^6}{6}+a^7 b x^8+\frac{14}{5} a^6 b^2 x^{10}+\frac{14}{3} a^5 b^3 x^{12}+5 a^4 b^4 x^{14}+\frac{7}{2} a^3 b^5 x^{16}+\frac{14}{9} a^2 b^6 x^{18}+\frac{2}{5} a b^7 x^{20}+\frac{b^8 x^{22}}{22} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5*(a + b*x^2)^8,x]

[Out]

(a^8*x^6)/6 + a^7*b*x^8 + (14*a^6*b^2*x^10)/5 + (14*a^5*b^3*x^12)/3 + 5*a^4*b^4*
x^14 + (7*a^3*b^5*x^16)/2 + (14*a^2*b^6*x^18)/9 + (2*a*b^7*x^20)/5 + (b^8*x^22)/
22

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Maple [A]  time = 0.001, size = 90, normalized size = 1.7 \[{\frac{{b}^{8}{x}^{22}}{22}}+{\frac{2\,a{b}^{7}{x}^{20}}{5}}+{\frac{14\,{a}^{2}{b}^{6}{x}^{18}}{9}}+{\frac{7\,{a}^{3}{b}^{5}{x}^{16}}{2}}+5\,{a}^{4}{b}^{4}{x}^{14}+{\frac{14\,{a}^{5}{b}^{3}{x}^{12}}{3}}+{\frac{14\,{a}^{6}{b}^{2}{x}^{10}}{5}}+{a}^{7}b{x}^{8}+{\frac{{a}^{8}{x}^{6}}{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(b*x^2+a)^8,x)

[Out]

1/22*b^8*x^22+2/5*a*b^7*x^20+14/9*a^2*b^6*x^18+7/2*a^3*b^5*x^16+5*a^4*b^4*x^14+1
4/3*a^5*b^3*x^12+14/5*a^6*b^2*x^10+a^7*b*x^8+1/6*a^8*x^6

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Maxima [A]  time = 1.34853, size = 120, normalized size = 2.26 \[ \frac{1}{22} \, b^{8} x^{22} + \frac{2}{5} \, a b^{7} x^{20} + \frac{14}{9} \, a^{2} b^{6} x^{18} + \frac{7}{2} \, a^{3} b^{5} x^{16} + 5 \, a^{4} b^{4} x^{14} + \frac{14}{3} \, a^{5} b^{3} x^{12} + \frac{14}{5} \, a^{6} b^{2} x^{10} + a^{7} b x^{8} + \frac{1}{6} \, a^{8} x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^8*x^5,x, algorithm="maxima")

[Out]

1/22*b^8*x^22 + 2/5*a*b^7*x^20 + 14/9*a^2*b^6*x^18 + 7/2*a^3*b^5*x^16 + 5*a^4*b^
4*x^14 + 14/3*a^5*b^3*x^12 + 14/5*a^6*b^2*x^10 + a^7*b*x^8 + 1/6*a^8*x^6

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Fricas [A]  time = 0.174142, size = 1, normalized size = 0.02 \[ \frac{1}{22} x^{22} b^{8} + \frac{2}{5} x^{20} b^{7} a + \frac{14}{9} x^{18} b^{6} a^{2} + \frac{7}{2} x^{16} b^{5} a^{3} + 5 x^{14} b^{4} a^{4} + \frac{14}{3} x^{12} b^{3} a^{5} + \frac{14}{5} x^{10} b^{2} a^{6} + x^{8} b a^{7} + \frac{1}{6} x^{6} a^{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^8*x^5,x, algorithm="fricas")

[Out]

1/22*x^22*b^8 + 2/5*x^20*b^7*a + 14/9*x^18*b^6*a^2 + 7/2*x^16*b^5*a^3 + 5*x^14*b
^4*a^4 + 14/3*x^12*b^3*a^5 + 14/5*x^10*b^2*a^6 + x^8*b*a^7 + 1/6*x^6*a^8

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Sympy [A]  time = 0.157364, size = 102, normalized size = 1.92 \[ \frac{a^{8} x^{6}}{6} + a^{7} b x^{8} + \frac{14 a^{6} b^{2} x^{10}}{5} + \frac{14 a^{5} b^{3} x^{12}}{3} + 5 a^{4} b^{4} x^{14} + \frac{7 a^{3} b^{5} x^{16}}{2} + \frac{14 a^{2} b^{6} x^{18}}{9} + \frac{2 a b^{7} x^{20}}{5} + \frac{b^{8} x^{22}}{22} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(b*x**2+a)**8,x)

[Out]

a**8*x**6/6 + a**7*b*x**8 + 14*a**6*b**2*x**10/5 + 14*a**5*b**3*x**12/3 + 5*a**4
*b**4*x**14 + 7*a**3*b**5*x**16/2 + 14*a**2*b**6*x**18/9 + 2*a*b**7*x**20/5 + b*
*8*x**22/22

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GIAC/XCAS [A]  time = 0.205505, size = 120, normalized size = 2.26 \[ \frac{1}{22} \, b^{8} x^{22} + \frac{2}{5} \, a b^{7} x^{20} + \frac{14}{9} \, a^{2} b^{6} x^{18} + \frac{7}{2} \, a^{3} b^{5} x^{16} + 5 \, a^{4} b^{4} x^{14} + \frac{14}{3} \, a^{5} b^{3} x^{12} + \frac{14}{5} \, a^{6} b^{2} x^{10} + a^{7} b x^{8} + \frac{1}{6} \, a^{8} x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^8*x^5,x, algorithm="giac")

[Out]

1/22*b^8*x^22 + 2/5*a*b^7*x^20 + 14/9*a^2*b^6*x^18 + 7/2*a^3*b^5*x^16 + 5*a^4*b^
4*x^14 + 14/3*a^5*b^3*x^12 + 14/5*a^6*b^2*x^10 + a^7*b*x^8 + 1/6*a^8*x^6