3.7.35 \(\int x^3 (a+b x^4)^3 \, dx\) [635]

Optimal. Leaf size=16 \[ \frac {\left (a+b x^4\right )^4}{16 b} \]

[Out]

1/16*(b*x^4+a)^4/b

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {267} \begin {gather*} \frac {\left (a+b x^4\right )^4}{16 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x^4)^3,x]

[Out]

(a + b*x^4)^4/(16*b)

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin {align*} \int x^3 \left (a+b x^4\right )^3 \, dx &=\frac {\left (a+b x^4\right )^4}{16 b}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(43\) vs. \(2(16)=32\).
time = 0.00, size = 43, normalized size = 2.69 \begin {gather*} \frac {a^3 x^4}{4}+\frac {3}{8} a^2 b x^8+\frac {1}{4} a b^2 x^{12}+\frac {b^3 x^{16}}{16} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x^4)^3,x]

[Out]

(a^3*x^4)/4 + (3*a^2*b*x^8)/8 + (a*b^2*x^12)/4 + (b^3*x^16)/16

________________________________________________________________________________________

Maple [A]
time = 0.14, size = 15, normalized size = 0.94

method	result	size
default	\(\frac {\left (b \,x^{4}+a \right )^{4}}{16 b}\)	\(15\)
gosper	\(\frac {1}{4} a^{3} x^{4}+\frac {3}{8} a^{2} b \,x^{8}+\frac {1}{4} a \,b^{2} x^{12}+\frac {1}{16} b^{3} x^{16}\)	\(36\)
norman	\(\frac {1}{4} a^{3} x^{4}+\frac {3}{8} a^{2} b \,x^{8}+\frac {1}{4} a \,b^{2} x^{12}+\frac {1}{16} b^{3} x^{16}\)	\(36\)
risch	\(\frac {b^{3} x^{16}}{16}+\frac {a \,b^{2} x^{12}}{4}+\frac {3 a^{2} b \,x^{8}}{8}+\frac {a^{3} x^{4}}{4}+\frac {a^{4}}{16 b}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(b*x^4+a)^4/b

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 14, normalized size = 0.88 \begin {gather*} \frac {{\left (b x^{4} + a\right )}^{4}}{16 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(b*x^4 + a)^4/b

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (14) = 28\).
time = 0.35, size = 35, normalized size = 2.19 \begin {gather*} \frac {1}{16} \, b^{3} x^{16} + \frac {1}{4} \, a b^{2} x^{12} + \frac {3}{8} \, a^{2} b x^{8} + \frac {1}{4} \, a^{3} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*b^3*x^16 + 1/4*a*b^2*x^12 + 3/8*a^2*b*x^8 + 1/4*a^3*x^4

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (10) = 20\).
time = 0.01, size = 37, normalized size = 2.31 \begin {gather*} \frac {a^{3} x^{4}}{4} + \frac {3 a^{2} b x^{8}}{8} + \frac {a b^{2} x^{12}}{4} + \frac {b^{3} x^{16}}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*x**4/4 + 3*a**2*b*x**8/8 + a*b**2*x**12/4 + b**3*x**16/16

________________________________________________________________________________________

Giac [A]
time = 0.59, size = 14, normalized size = 0.88 \begin {gather*} \frac {{\left (b x^{4} + a\right )}^{4}}{16 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(b*x^4 + a)^4/b

________________________________________________________________________________________

Mupad [B]
time = 0.04, size = 35, normalized size = 2.19 \begin {gather*} \frac {a^3\,x^4}{4}+\frac {3\,a^2\,b\,x^8}{8}+\frac {a\,b^2\,x^{12}}{4}+\frac {b^3\,x^{16}}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*x^4)^3,x)

[Out]

(a^3*x^4)/4 + (b^3*x^16)/16 + (3*a^2*b*x^8)/8 + (a*b^2*x^12)/4

________________________________________________________________________________________