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

Optimal. Leaf size=29 \[ \frac{12}{7} \left (\sqrt [4]{x}+1\right )^{7/3}-3 \left (\sqrt [4]{x}+1\right )^{4/3} \]

[Out]

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

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Rubi [A]  time = 0.0089935, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {266, 43} \[ \frac{12}{7} \left (\sqrt [4]{x}+1\right )^{7/3}-3 \left (\sqrt [4]{x}+1\right )^{4/3} \]

Antiderivative was successfully verified.

[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 266

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]]

Rule 43

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])

Rubi steps

\begin{align*} \int \frac{\sqrt [3]{1+\sqrt [4]{x}}}{\sqrt{x}} \, dx &=4 \operatorname{Subst}\left (\int x \sqrt [3]{1+x} \, dx,x,\sqrt [4]{x}\right )\\ &=4 \operatorname{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]  time = 0.0082711, size = 24, normalized size = 0.83 \[ \frac{3}{7} \left (\sqrt [4]{x}+1\right )^{4/3} \left (4 \sqrt [4]{x}-3\right ) \]

Antiderivative was successfully verified.

[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

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Maple [A]  time = 0.002, size = 20, normalized size = 0.7 \begin{align*} -3\, \left ( 1+\sqrt [4]{x} \right ) ^{4/3}+{\frac{12}{7} \left ( 1+\sqrt [4]{x} \right ) ^{{\frac{7}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.935212, size = 26, normalized size = 0.9 \begin{align*} \frac{12}{7} \,{\left (x^{\frac{1}{4}} + 1\right )}^{\frac{7}{3}} - 3 \,{\left (x^{\frac{1}{4}} + 1\right )}^{\frac{4}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.78556, size = 69, normalized size = 2.38 \begin{align*} \frac{3}{7} \,{\left (4 \, \sqrt{x} + x^{\frac{1}{4}} - 3\right )}{\left (x^{\frac{1}{4}} + 1\right )}^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [B]  time = 1.32971, size = 134, normalized size = 4.62 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 1.049, size = 26, normalized size = 0.9 \begin{align*} \frac{12}{7} \,{\left (x^{\frac{1}{4}} + 1\right )}^{\frac{7}{3}} - 3 \,{\left (x^{\frac{1}{4}} + 1\right )}^{\frac{4}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)