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

Optimal. Leaf size=53 \[ \frac{a^2 \left (a+b x^2\right )^6}{12 b^3}+\frac{\left (a+b x^2\right )^8}{16 b^3}-\frac{a \left (a+b x^2\right )^7}{7 b^3} \]

[Out]

(a^2*(a + b*x^2)^6)/(12*b^3) - (a*(a + b*x^2)^7)/(7*b^3) + (a + b*x^2)^8/(16*b^3
)

_______________________________________________________________________________________

Rubi [A]  time = 0.16092, 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 \[ \frac{a^2 \left (a+b x^2\right )^6}{12 b^3}+\frac{\left (a+b x^2\right )^8}{16 b^3}-\frac{a \left (a+b x^2\right )^7}{7 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^5*(a + b*x^2)^5,x]

[Out]

(a^2*(a + b*x^2)^6)/(12*b^3) - (a*(a + b*x^2)^7)/(7*b^3) + (a + b*x^2)^8/(16*b^3
)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 14.4351, size = 44, normalized size = 0.83 \[ \frac{a^{2} \left (a + b x^{2}\right )^{6}}{12 b^{3}} - \frac{a \left (a + b x^{2}\right )^{7}}{7 b^{3}} + \frac{\left (a + b x^{2}\right )^{8}}{16 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*(a + b*x**2)**6/(12*b**3) - a*(a + b*x**2)**7/(7*b**3) + (a + b*x**2)**8/(1
6*b**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.00368588, size = 66, normalized size = 1.25 \[ \frac{a^5 x^6}{6}+\frac{5}{8} a^4 b x^8+a^3 b^2 x^{10}+\frac{5}{6} a^2 b^3 x^{12}+\frac{5}{14} a b^4 x^{14}+\frac{b^5 x^{16}}{16} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5*(a + b*x^2)^5,x]

[Out]

(a^5*x^6)/6 + (5*a^4*b*x^8)/8 + a^3*b^2*x^10 + (5*a^2*b^3*x^12)/6 + (5*a*b^4*x^1
4)/14 + (b^5*x^16)/16

_______________________________________________________________________________________

Maple [A]  time = 0.002, size = 57, normalized size = 1.1 \[{\frac{{b}^{5}{x}^{16}}{16}}+{\frac{5\,a{b}^{4}{x}^{14}}{14}}+{\frac{5\,{a}^{2}{b}^{3}{x}^{12}}{6}}+{a}^{3}{b}^{2}{x}^{10}+{\frac{5\,{a}^{4}b{x}^{8}}{8}}+{\frac{{a}^{5}{x}^{6}}{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(b*x^2+a)^5,x)

[Out]

1/16*b^5*x^16+5/14*a*b^4*x^14+5/6*a^2*b^3*x^12+a^3*b^2*x^10+5/8*a^4*b*x^8+1/6*a^
5*x^6

_______________________________________________________________________________________

Maxima [A]  time = 1.36766, size = 76, normalized size = 1.43 \[ \frac{1}{16} \, b^{5} x^{16} + \frac{5}{14} \, a b^{4} x^{14} + \frac{5}{6} \, a^{2} b^{3} x^{12} + a^{3} b^{2} x^{10} + \frac{5}{8} \, a^{4} b x^{8} + \frac{1}{6} \, a^{5} x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*b^5*x^16 + 5/14*a*b^4*x^14 + 5/6*a^2*b^3*x^12 + a^3*b^2*x^10 + 5/8*a^4*b*x^
8 + 1/6*a^5*x^6

_______________________________________________________________________________________

Fricas [A]  time = 0.187868, size = 1, normalized size = 0.02 \[ \frac{1}{16} x^{16} b^{5} + \frac{5}{14} x^{14} b^{4} a + \frac{5}{6} x^{12} b^{3} a^{2} + x^{10} b^{2} a^{3} + \frac{5}{8} x^{8} b a^{4} + \frac{1}{6} x^{6} a^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*x^16*b^5 + 5/14*x^14*b^4*a + 5/6*x^12*b^3*a^2 + x^10*b^2*a^3 + 5/8*x^8*b*a^
4 + 1/6*x^6*a^5

_______________________________________________________________________________________

Sympy [A]  time = 0.122172, size = 63, normalized size = 1.19 \[ \frac{a^{5} x^{6}}{6} + \frac{5 a^{4} b x^{8}}{8} + a^{3} b^{2} x^{10} + \frac{5 a^{2} b^{3} x^{12}}{6} + \frac{5 a b^{4} x^{14}}{14} + \frac{b^{5} x^{16}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(b*x**2+a)**5,x)

[Out]

a**5*x**6/6 + 5*a**4*b*x**8/8 + a**3*b**2*x**10 + 5*a**2*b**3*x**12/6 + 5*a*b**4
*x**14/14 + b**5*x**16/16

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.207026, size = 76, normalized size = 1.43 \[ \frac{1}{16} \, b^{5} x^{16} + \frac{5}{14} \, a b^{4} x^{14} + \frac{5}{6} \, a^{2} b^{3} x^{12} + a^{3} b^{2} x^{10} + \frac{5}{8} \, a^{4} b x^{8} + \frac{1}{6} \, a^{5} x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*b^5*x^16 + 5/14*a*b^4*x^14 + 5/6*a^2*b^3*x^12 + a^3*b^2*x^10 + 5/8*a^4*b*x^
8 + 1/6*a^5*x^6