\(\int x^7 \sqrt [3]{1+x^4} \, dx\) [985]

Optimal result

Integrand size = 13, antiderivative size = 27 \[ \int x^7 \sqrt [3]{1+x^4} \, dx=-\frac {3}{16} \left (1+x^4\right )^{4/3}+\frac {3}{28} \left (1+x^4\right )^{7/3} \]

[Out]

-3/16*(x^4+1)^(4/3)+3/28*(x^4+1)^(7/3)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \[ \int x^7 \sqrt [3]{1+x^4} \, dx=\frac {3}{28} \left (x^4+1\right )^{7/3}-\frac {3}{16} \left (x^4+1\right )^{4/3} \]

[In]

Int[x^7*(1 + x^4)^(1/3),x]

[Out]

(-3*(1 + x^4)^(4/3))/16 + (3*(1 + x^4)^(7/3))/28

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} \text {Subst}\left (\int x \sqrt [3]{1+x} \, dx,x,x^4\right ) \\ & = \frac {1}{4} \text {Subst}\left (\int \left (-\sqrt [3]{1+x}+(1+x)^{4/3}\right ) \, dx,x,x^4\right ) \\ & = -\frac {3}{16} \left (1+x^4\right )^{4/3}+\frac {3}{28} \left (1+x^4\right )^{7/3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int x^7 \sqrt [3]{1+x^4} \, dx=\frac {3}{112} \left (1+x^4\right )^{4/3} \left (-3+4 x^4\right ) \]

[In]

Integrate[x^7*(1 + x^4)^(1/3),x]

[Out]

(3*(1 + x^4)^(4/3)*(-3 + 4*x^4))/112

Maple [A] (verified)

Time = 4.16 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.63

method	result	size
gosper	\(\frac {3 \left (x^{4}+1\right )^{\frac {4}{3}} \left (4 x^{4}-3\right )}{112}\)	\(17\)
meijerg	\(\frac {x^{8} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (-\frac {1}{3},2;3;-x^{4}\right )}{8}\)	\(17\)
pseudoelliptic	\(\frac {3 \left (x^{4}+1\right )^{\frac {4}{3}} \left (4 x^{4}-3\right )}{112}\)	\(17\)
risch	\(\frac {3 \left (x^{4}+1\right )^{\frac {1}{3}} \left (4 x^{8}+x^{4}-3\right )}{112}\)	\(20\)
trager	\(\left (\frac {3}{28} x^{8}+\frac {3}{112} x^{4}-\frac {9}{112}\right ) \left (x^{4}+1\right )^{\frac {1}{3}}\)	\(21\)

[In]

int(x^7*(x^4+1)^(1/3),x,method=_RETURNVERBOSE)

[Out]

3/112*(x^4+1)^(4/3)*(4*x^4-3)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int x^7 \sqrt [3]{1+x^4} \, dx=\frac {3}{112} \, {\left (4 \, x^{8} + x^{4} - 3\right )} {\left (x^{4} + 1\right )}^{\frac {1}{3}} \]

[In]

integrate(x^7*(x^4+1)^(1/3),x, algorithm="fricas")

[Out]

3/112*(4*x^8 + x^4 - 3)*(x^4 + 1)^(1/3)

Sympy [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.52 \[ \int x^7 \sqrt [3]{1+x^4} \, dx=\frac {3 x^{8} \sqrt [3]{x^{4} + 1}}{28} + \frac {3 x^{4} \sqrt [3]{x^{4} + 1}}{112} - \frac {9 \sqrt [3]{x^{4} + 1}}{112} \]

[In]

integrate(x**7*(x**4+1)**(1/3),x)

[Out]

3*x**8*(x**4 + 1)**(1/3)/28 + 3*x**4*(x**4 + 1)**(1/3)/112 - 9*(x**4 + 1)**(1/3)/112

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int x^7 \sqrt [3]{1+x^4} \, dx=\frac {3}{28} \, {\left (x^{4} + 1\right )}^{\frac {7}{3}} - \frac {3}{16} \, {\left (x^{4} + 1\right )}^{\frac {4}{3}} \]

[In]

integrate(x^7*(x^4+1)^(1/3),x, algorithm="maxima")

[Out]

3/28*(x^4 + 1)^(7/3) - 3/16*(x^4 + 1)^(4/3)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int x^7 \sqrt [3]{1+x^4} \, dx=\frac {3}{28} \, {\left (x^{4} + 1\right )}^{\frac {7}{3}} - \frac {3}{16} \, {\left (x^{4} + 1\right )}^{\frac {4}{3}} \]

[In]

integrate(x^7*(x^4+1)^(1/3),x, algorithm="giac")

[Out]

3/28*(x^4 + 1)^(7/3) - 3/16*(x^4 + 1)^(4/3)

Mupad [B] (verification not implemented)

Time = 5.49 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.59 \[ \int x^7 \sqrt [3]{1+x^4} \, dx=\frac {3\,{\left (x^4+1\right )}^{4/3}\,\left (4\,x^4-3\right )}{112} \]

[In]

int(x^7*(x^4 + 1)^(1/3),x)

[Out]

(3*(x^4 + 1)^(4/3)*(4*x^4 - 3))/112