3.634 \(\int x^4 (a+b x^4)^3 \, dx\)

Optimal. Leaf size=43 \[ \frac {a^3 x^5}{5}+\frac {1}{3} a^2 b x^9+\frac {3}{13} a b^2 x^{13}+\frac {b^3 x^{17}}{17} \]

[Out]

1/5*a^3*x^5+1/3*a^2*b*x^9+3/13*a*b^2*x^13+1/17*b^3*x^17

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {270} \[ \frac {1}{3} a^2 b x^9+\frac {a^3 x^5}{5}+\frac {3}{13} a b^2 x^{13}+\frac {b^3 x^{17}}{17} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(a + b*x^4)^3,x]

[Out]

(a^3*x^5)/5 + (a^2*b*x^9)/3 + (3*a*b^2*x^13)/13 + (b^3*x^17)/17

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int x^4 \left (a+b x^4\right )^3 \, dx &=\int \left (a^3 x^4+3 a^2 b x^8+3 a b^2 x^{12}+b^3 x^{16}\right ) \, dx\\ &=\frac {a^3 x^5}{5}+\frac {1}{3} a^2 b x^9+\frac {3}{13} a b^2 x^{13}+\frac {b^3 x^{17}}{17}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 43, normalized size = 1.00 \[ \frac {a^3 x^5}{5}+\frac {1}{3} a^2 b x^9+\frac {3}{13} a b^2 x^{13}+\frac {b^3 x^{17}}{17} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(a + b*x^4)^3,x]

[Out]

(a^3*x^5)/5 + (a^2*b*x^9)/3 + (3*a*b^2*x^13)/13 + (b^3*x^17)/17

________________________________________________________________________________________

fricas [A]  time = 0.56, size = 35, normalized size = 0.81 \[ \frac {1}{17} x^{17} b^{3} + \frac {3}{13} x^{13} b^{2} a + \frac {1}{3} x^{9} b a^{2} + \frac {1}{5} x^{5} a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^4+a)^3,x, algorithm="fricas")

[Out]

1/17*x^17*b^3 + 3/13*x^13*b^2*a + 1/3*x^9*b*a^2 + 1/5*x^5*a^3

________________________________________________________________________________________

giac [A]  time = 0.15, size = 35, normalized size = 0.81 \[ \frac {1}{17} \, b^{3} x^{17} + \frac {3}{13} \, a b^{2} x^{13} + \frac {1}{3} \, a^{2} b x^{9} + \frac {1}{5} \, a^{3} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^4+a)^3,x, algorithm="giac")

[Out]

1/17*b^3*x^17 + 3/13*a*b^2*x^13 + 1/3*a^2*b*x^9 + 1/5*a^3*x^5

________________________________________________________________________________________

maple [A]  time = 0.00, size = 36, normalized size = 0.84 \[ \frac {1}{17} b^{3} x^{17}+\frac {3}{13} a \,b^{2} x^{13}+\frac {1}{3} a^{2} b \,x^{9}+\frac {1}{5} a^{3} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(b*x^4+a)^3,x)

[Out]

1/5*a^3*x^5+1/3*a^2*b*x^9+3/13*a*b^2*x^13+1/17*b^3*x^17

________________________________________________________________________________________

maxima [A]  time = 1.34, size = 35, normalized size = 0.81 \[ \frac {1}{17} \, b^{3} x^{17} + \frac {3}{13} \, a b^{2} x^{13} + \frac {1}{3} \, a^{2} b x^{9} + \frac {1}{5} \, a^{3} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^4+a)^3,x, algorithm="maxima")

[Out]

1/17*b^3*x^17 + 3/13*a*b^2*x^13 + 1/3*a^2*b*x^9 + 1/5*a^3*x^5

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 35, normalized size = 0.81 \[ \frac {a^3\,x^5}{5}+\frac {a^2\,b\,x^9}{3}+\frac {3\,a\,b^2\,x^{13}}{13}+\frac {b^3\,x^{17}}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a + b*x^4)^3,x)

[Out]

(a^3*x^5)/5 + (b^3*x^17)/17 + (a^2*b*x^9)/3 + (3*a*b^2*x^13)/13

________________________________________________________________________________________

sympy [A]  time = 0.13, size = 37, normalized size = 0.86 \[ \frac {a^{3} x^{5}}{5} + \frac {a^{2} b x^{9}}{3} + \frac {3 a b^{2} x^{13}}{13} + \frac {b^{3} x^{17}}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(b*x**4+a)**3,x)

[Out]

a**3*x**5/5 + a**2*b*x**9/3 + 3*a*b**2*x**13/13 + b**3*x**17/17

________________________________________________________________________________________