3.7.22 \(\int x^4 (a+b x^4)^2 \, dx\) [622]

Optimal. Leaf size=30 \[ \frac {a^2 x^5}{5}+\frac {2}{9} a b x^9+\frac {b^2 x^{13}}{13} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 30, 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 = {276} \begin {gather*} \frac {a^2 x^5}{5}+\frac {2}{9} a b x^9+\frac {b^2 x^{13}}{13} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4*(a + b*x^4)^2,x]

[Out]

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

Rule 276

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

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 30, normalized size = 1.00 \begin {gather*} \frac {a^2 x^5}{5}+\frac {2}{9} a b x^9+\frac {b^2 x^{13}}{13} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4*(a + b*x^4)^2,x]

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.14, size = 25, normalized size = 0.83

method	result	size
gosper	\(\frac {1}{5} a^{2} x^{5}+\frac {2}{9} a b \,x^{9}+\frac {1}{13} b^{2} x^{13}\)	\(25\)
default	\(\frac {1}{5} a^{2} x^{5}+\frac {2}{9} a b \,x^{9}+\frac {1}{13} b^{2} x^{13}\)	\(25\)
norman	\(\frac {1}{5} a^{2} x^{5}+\frac {2}{9} a b \,x^{9}+\frac {1}{13} b^{2} x^{13}\)	\(25\)
risch	\(\frac {1}{5} a^{2} x^{5}+\frac {2}{9} a b \,x^{9}+\frac {1}{13} b^{2} x^{13}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(b*x^4+a)^2,x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.30, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{13} \, b^{2} x^{13} + \frac {2}{9} \, a b x^{9} + \frac {1}{5} \, a^{2} x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.35, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{13} \, b^{2} x^{13} + \frac {2}{9} \, a b x^{9} + \frac {1}{5} \, a^{2} x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]
time = 0.01, size = 26, normalized size = 0.87 \begin {gather*} \frac {a^{2} x^{5}}{5} + \frac {2 a b x^{9}}{9} + \frac {b^{2} x^{13}}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(b*x**4+a)**2,x)

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 0.81, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{13} \, b^{2} x^{13} + \frac {2}{9} \, a b x^{9} + \frac {1}{5} \, a^{2} x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 0.03, size = 24, normalized size = 0.80 \begin {gather*} \frac {a^2\,x^5}{5}+\frac {2\,a\,b\,x^9}{9}+\frac {b^2\,x^{13}}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a + b*x^4)^2,x)

[Out]

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

________________________________________________________________________________________