3.54 \(\int x^9 \left (a+b x^2\right )^5 \, dx\)

Optimal. Leaf size=69 \[ \frac{a^5 x^{10}}{10}+\frac{5}{12} a^4 b x^{12}+\frac{5}{7} a^3 b^2 x^{14}+\frac{5}{8} a^2 b^3 x^{16}+\frac{5}{18} a b^4 x^{18}+\frac{b^5 x^{20}}{20} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.114488, antiderivative size = 69, 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 \[ \frac{a^5 x^{10}}{10}+\frac{5}{12} a^4 b x^{12}+\frac{5}{7} a^3 b^2 x^{14}+\frac{5}{8} a^2 b^3 x^{16}+\frac{5}{18} a b^4 x^{18}+\frac{b^5 x^{20}}{20} \]

Antiderivative was successfully verified.

[In]  Int[x^9*(a + b*x^2)^5,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 16.2229, size = 66, normalized size = 0.96 \[ \frac{a^{5} x^{10}}{10} + \frac{5 a^{4} b x^{12}}{12} + \frac{5 a^{3} b^{2} x^{14}}{7} + \frac{5 a^{2} b^{3} x^{16}}{8} + \frac{5 a b^{4} x^{18}}{18} + \frac{b^{5} x^{20}}{20} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9*(b*x**2+a)**5,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.00401707, size = 69, normalized size = 1. \[ \frac{a^5 x^{10}}{10}+\frac{5}{12} a^4 b x^{12}+\frac{5}{7} a^3 b^2 x^{14}+\frac{5}{8} a^2 b^3 x^{16}+\frac{5}{18} a b^4 x^{18}+\frac{b^5 x^{20}}{20} \]

Antiderivative was successfully verified.

[In]  Integrate[x^9*(a + b*x^2)^5,x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 58, normalized size = 0.8 \[{\frac{{a}^{5}{x}^{10}}{10}}+{\frac{5\,{a}^{4}b{x}^{12}}{12}}+{\frac{5\,{a}^{3}{b}^{2}{x}^{14}}{7}}+{\frac{5\,{a}^{2}{b}^{3}{x}^{16}}{8}}+{\frac{5\,a{b}^{4}{x}^{18}}{18}}+{\frac{{b}^{5}{x}^{20}}{20}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9*(b*x^2+a)^5,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.35211, size = 77, normalized size = 1.12 \[ \frac{1}{20} \, b^{5} x^{20} + \frac{5}{18} \, a b^{4} x^{18} + \frac{5}{8} \, a^{2} b^{3} x^{16} + \frac{5}{7} \, a^{3} b^{2} x^{14} + \frac{5}{12} \, a^{4} b x^{12} + \frac{1}{10} \, a^{5} x^{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^5*x^9,x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.188498, size = 1, normalized size = 0.01 \[ \frac{1}{20} x^{20} b^{5} + \frac{5}{18} x^{18} b^{4} a + \frac{5}{8} x^{16} b^{3} a^{2} + \frac{5}{7} x^{14} b^{2} a^{3} + \frac{5}{12} x^{12} b a^{4} + \frac{1}{10} x^{10} a^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^5*x^9,x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.124101, size = 66, normalized size = 0.96 \[ \frac{a^{5} x^{10}}{10} + \frac{5 a^{4} b x^{12}}{12} + \frac{5 a^{3} b^{2} x^{14}}{7} + \frac{5 a^{2} b^{3} x^{16}}{8} + \frac{5 a b^{4} x^{18}}{18} + \frac{b^{5} x^{20}}{20} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9*(b*x**2+a)**5,x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.205086, size = 77, normalized size = 1.12 \[ \frac{1}{20} \, b^{5} x^{20} + \frac{5}{18} \, a b^{4} x^{18} + \frac{5}{8} \, a^{2} b^{3} x^{16} + \frac{5}{7} \, a^{3} b^{2} x^{14} + \frac{5}{12} \, a^{4} b x^{12} + \frac{1}{10} \, a^{5} x^{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^5*x^9,x, algorithm="giac")

[Out]

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