\(\int \frac {\sqrt [3]{1+\sqrt [4]{x}}}{\sqrt {x}} \, dx\) [214]

Optimal result

Integrand size = 17, antiderivative size = 29 \[ \int \frac {\sqrt [3]{1+\sqrt [4]{x}}}{\sqrt {x}} \, dx=-3 \left (1+\sqrt [4]{x}\right )^{4/3}+\frac {12}{7} \left (1+\sqrt [4]{x}\right )^{7/3} \]

[Out]

-3*(1+x^(1/4))^(4/3)+12/7*(1+x^(1/4))^(7/3)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {272, 45} \[ \int \frac {\sqrt [3]{1+\sqrt [4]{x}}}{\sqrt {x}} \, dx=\frac {12}{7} \left (\sqrt [4]{x}+1\right )^{7/3}-3 \left (\sqrt [4]{x}+1\right )^{4/3} \]

[In]

Int[(1 + x^(1/4))^(1/3)/Sqrt[x],x]

[Out]

-3*(1 + x^(1/4))^(4/3) + (12*(1 + x^(1/4))^(7/3))/7

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}& = 4 \text {Subst}\left (\int x \sqrt [3]{1+x} \, dx,x,\sqrt [4]{x}\right ) \\ & = 4 \text {Subst}\left (\int \left (-\sqrt [3]{1+x}+(1+x)^{4/3}\right ) \, dx,x,\sqrt [4]{x}\right ) \\ & = -3 \left (1+\sqrt [4]{x}\right )^{4/3}+\frac {12}{7} \left (1+\sqrt [4]{x}\right )^{7/3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt [3]{1+\sqrt [4]{x}}}{\sqrt {x}} \, dx=\frac {3}{7} \left (1+\sqrt [4]{x}\right )^{4/3} \left (-3+4 \sqrt [4]{x}\right ) \]

[In]

Integrate[(1 + x^(1/4))^(1/3)/Sqrt[x],x]

[Out]

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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 2.

Time = 0.20 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.59

method	result	size
meijerg	\(2 \sqrt {x}\, {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (-\frac {1}{3},2;3;-x^{\frac {1}{4}}\right )\)	\(17\)
derivativedivides	\(-3 \left (1+x^{\frac {1}{4}}\right )^{\frac {4}{3}}+\frac {12 \left (1+x^{\frac {1}{4}}\right )^{\frac {7}{3}}}{7}\)	\(20\)
default	\(-3 \left (1+x^{\frac {1}{4}}\right )^{\frac {4}{3}}+\frac {12 \left (1+x^{\frac {1}{4}}\right )^{\frac {7}{3}}}{7}\)	\(20\)

[In]

int((1+x^(1/4))^(1/3)/x^(1/2),x,method=_RETURNVERBOSE)

[Out]

2*x^(1/2)*hypergeom([-1/3,2],[3],-x^(1/4))

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.66 \[ \int \frac {\sqrt [3]{1+\sqrt [4]{x}}}{\sqrt {x}} \, dx=\frac {3}{7} \, {\left (4 \, \sqrt {x} + x^{\frac {1}{4}} - 3\right )} {\left (x^{\frac {1}{4}} + 1\right )}^{\frac {1}{3}} \]

[In]

integrate((1+x^(1/4))^(1/3)/x^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (24) = 48\).

Time = 0.68 (sec) , antiderivative size = 134, normalized size of antiderivative = 4.62 \[ \int \frac {\sqrt [3]{1+\sqrt [4]{x}}}{\sqrt {x}} \, dx=\frac {12 x^{\frac {7}{4}} \sqrt [3]{\sqrt [4]{x} + 1}}{7 x^{\frac {5}{4}} + 7 x} - \frac {6 x^{\frac {5}{4}} \sqrt [3]{\sqrt [4]{x} + 1}}{7 x^{\frac {5}{4}} + 7 x} + \frac {9 x^{\frac {5}{4}}}{7 x^{\frac {5}{4}} + 7 x} + \frac {15 x^{\frac {3}{2}} \sqrt [3]{\sqrt [4]{x} + 1}}{7 x^{\frac {5}{4}} + 7 x} - \frac {9 x \sqrt [3]{\sqrt [4]{x} + 1}}{7 x^{\frac {5}{4}} + 7 x} + \frac {9 x}{7 x^{\frac {5}{4}} + 7 x} \]

[In]

integrate((1+x**(1/4))**(1/3)/x**(1/2),x)

[Out]

12*x**(7/4)*(x**(1/4) + 1)**(1/3)/(7*x**(5/4) + 7*x) - 6*x**(5/4)*(x**(1/4) + 1)**(1/3)/(7*x**(5/4) + 7*x) + 9
*x**(5/4)/(7*x**(5/4) + 7*x) + 15*x**(3/2)*(x**(1/4) + 1)**(1/3)/(7*x**(5/4) + 7*x) - 9*x*(x**(1/4) + 1)**(1/3
)/(7*x**(5/4) + 7*x) + 9*x/(7*x**(5/4) + 7*x)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.66 \[ \int \frac {\sqrt [3]{1+\sqrt [4]{x}}}{\sqrt {x}} \, dx=\frac {12}{7} \, {\left (x^{\frac {1}{4}} + 1\right )}^{\frac {7}{3}} - 3 \, {\left (x^{\frac {1}{4}} + 1\right )}^{\frac {4}{3}} \]

[In]

integrate((1+x^(1/4))^(1/3)/x^(1/2),x, algorithm="maxima")

[Out]

12/7*(x^(1/4) + 1)^(7/3) - 3*(x^(1/4) + 1)^(4/3)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.66 \[ \int \frac {\sqrt [3]{1+\sqrt [4]{x}}}{\sqrt {x}} \, dx=\frac {12}{7} \, {\left (x^{\frac {1}{4}} + 1\right )}^{\frac {7}{3}} - 3 \, {\left (x^{\frac {1}{4}} + 1\right )}^{\frac {4}{3}} \]

[In]

integrate((1+x^(1/4))^(1/3)/x^(1/2),x, algorithm="giac")

[Out]

12/7*(x^(1/4) + 1)^(7/3) - 3*(x^(1/4) + 1)^(4/3)

Mupad [B] (verification not implemented)

Time = 0.58 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.55 \[ \int \frac {\sqrt [3]{1+\sqrt [4]{x}}}{\sqrt {x}} \, dx=\frac {3\,{\left (x^{1/4}+1\right )}^{4/3}\,\left (4\,x^{1/4}-3\right )}{7} \]

[In]

int((x^(1/4) + 1)^(1/3)/x^(1/2),x)

[Out]

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