3.72 \(\int x^4 \left (a+b x^2\right )^5 \, dx\)

Optimal. Leaf size=69 \[ \frac{a^5 x^5}{5}+\frac{5}{7} a^4 b x^7+\frac{10}{9} a^3 b^2 x^9+\frac{10}{11} a^2 b^3 x^{11}+\frac{5}{13} a b^4 x^{13}+\frac{b^5 x^{15}}{15} \]

[Out]

(a^5*x^5)/5 + (5*a^4*b*x^7)/7 + (10*a^3*b^2*x^9)/9 + (10*a^2*b^3*x^11)/11 + (5*a
*b^4*x^13)/13 + (b^5*x^15)/15

_______________________________________________________________________________________

Rubi [A]  time = 0.0708871, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{a^5 x^5}{5}+\frac{5}{7} a^4 b x^7+\frac{10}{9} a^3 b^2 x^9+\frac{10}{11} a^2 b^3 x^{11}+\frac{5}{13} a b^4 x^{13}+\frac{b^5 x^{15}}{15} \]

Antiderivative was successfully verified.

[In]  Int[x^4*(a + b*x^2)^5,x]

[Out]

(a^5*x^5)/5 + (5*a^4*b*x^7)/7 + (10*a^3*b^2*x^9)/9 + (10*a^2*b^3*x^11)/11 + (5*a
*b^4*x^13)/13 + (b^5*x^15)/15

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 11.9829, size = 66, normalized size = 0.96 \[ \frac{a^{5} x^{5}}{5} + \frac{5 a^{4} b x^{7}}{7} + \frac{10 a^{3} b^{2} x^{9}}{9} + \frac{10 a^{2} b^{3} x^{11}}{11} + \frac{5 a b^{4} x^{13}}{13} + \frac{b^{5} x^{15}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(b*x**2+a)**5,x)

[Out]

a**5*x**5/5 + 5*a**4*b*x**7/7 + 10*a**3*b**2*x**9/9 + 10*a**2*b**3*x**11/11 + 5*
a*b**4*x**13/13 + b**5*x**15/15

_______________________________________________________________________________________

Mathematica [A]  time = 0.00382316, size = 69, normalized size = 1. \[ \frac{a^5 x^5}{5}+\frac{5}{7} a^4 b x^7+\frac{10}{9} a^3 b^2 x^9+\frac{10}{11} a^2 b^3 x^{11}+\frac{5}{13} a b^4 x^{13}+\frac{b^5 x^{15}}{15} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4*(a + b*x^2)^5,x]

[Out]

(a^5*x^5)/5 + (5*a^4*b*x^7)/7 + (10*a^3*b^2*x^9)/9 + (10*a^2*b^3*x^11)/11 + (5*a
*b^4*x^13)/13 + (b^5*x^15)/15

_______________________________________________________________________________________

Maple [A]  time = 0.002, size = 58, normalized size = 0.8 \[{\frac{{a}^{5}{x}^{5}}{5}}+{\frac{5\,{a}^{4}b{x}^{7}}{7}}+{\frac{10\,{a}^{3}{b}^{2}{x}^{9}}{9}}+{\frac{10\,{a}^{2}{b}^{3}{x}^{11}}{11}}+{\frac{5\,a{b}^{4}{x}^{13}}{13}}+{\frac{{b}^{5}{x}^{15}}{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(b*x^2+a)^5,x)

[Out]

1/5*a^5*x^5+5/7*a^4*b*x^7+10/9*a^3*b^2*x^9+10/11*a^2*b^3*x^11+5/13*a*b^4*x^13+1/
15*b^5*x^15

_______________________________________________________________________________________

Maxima [A]  time = 1.34451, size = 77, normalized size = 1.12 \[ \frac{1}{15} \, b^{5} x^{15} + \frac{5}{13} \, a b^{4} x^{13} + \frac{10}{11} \, a^{2} b^{3} x^{11} + \frac{10}{9} \, a^{3} b^{2} x^{9} + \frac{5}{7} \, a^{4} b x^{7} + \frac{1}{5} \, a^{5} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^5*x^4,x, algorithm="maxima")

[Out]

1/15*b^5*x^15 + 5/13*a*b^4*x^13 + 10/11*a^2*b^3*x^11 + 10/9*a^3*b^2*x^9 + 5/7*a^
4*b*x^7 + 1/5*a^5*x^5

_______________________________________________________________________________________

Fricas [A]  time = 0.175377, size = 1, normalized size = 0.01 \[ \frac{1}{15} x^{15} b^{5} + \frac{5}{13} x^{13} b^{4} a + \frac{10}{11} x^{11} b^{3} a^{2} + \frac{10}{9} x^{9} b^{2} a^{3} + \frac{5}{7} x^{7} b a^{4} + \frac{1}{5} x^{5} a^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^5*x^4,x, algorithm="fricas")

[Out]

1/15*x^15*b^5 + 5/13*x^13*b^4*a + 10/11*x^11*b^3*a^2 + 10/9*x^9*b^2*a^3 + 5/7*x^
7*b*a^4 + 1/5*x^5*a^5

_______________________________________________________________________________________

Sympy [A]  time = 0.122299, size = 66, normalized size = 0.96 \[ \frac{a^{5} x^{5}}{5} + \frac{5 a^{4} b x^{7}}{7} + \frac{10 a^{3} b^{2} x^{9}}{9} + \frac{10 a^{2} b^{3} x^{11}}{11} + \frac{5 a b^{4} x^{13}}{13} + \frac{b^{5} x^{15}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(b*x**2+a)**5,x)

[Out]

a**5*x**5/5 + 5*a**4*b*x**7/7 + 10*a**3*b**2*x**9/9 + 10*a**2*b**3*x**11/11 + 5*
a*b**4*x**13/13 + b**5*x**15/15

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.20979, size = 77, normalized size = 1.12 \[ \frac{1}{15} \, b^{5} x^{15} + \frac{5}{13} \, a b^{4} x^{13} + \frac{10}{11} \, a^{2} b^{3} x^{11} + \frac{10}{9} \, a^{3} b^{2} x^{9} + \frac{5}{7} \, a^{4} b x^{7} + \frac{1}{5} \, a^{5} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^5*x^4,x, algorithm="giac")

[Out]

1/15*b^5*x^15 + 5/13*a*b^4*x^13 + 10/11*a^2*b^3*x^11 + 10/9*a^3*b^2*x^9 + 5/7*a^
4*b*x^7 + 1/5*a^5*x^5