3.916 \(\int \frac{1+\sqrt{x}}{x^{5/6}+x^{7/6}} \, dx\)

Optimal. Leaf size=26 \[ 3 \sqrt [3]{x}-3 \log \left (\sqrt [3]{x}+1\right )+6 \tan ^{-1}\left (\sqrt [6]{x}\right ) \]

[Out]

3*x^(1/3) + 6*ArcTan[x^(1/6)] - 3*Log[1 + x^(1/3)]

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Rubi [A]  time = 0.0400841, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {1593, 1819, 1810, 635, 203, 260} \[ 3 \sqrt [3]{x}-3 \log \left (\sqrt [3]{x}+1\right )+6 \tan ^{-1}\left (\sqrt [6]{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 + Sqrt[x])/(x^(5/6) + x^(7/6)),x]

[Out]

3*x^(1/3) + 6*ArcTan[x^(1/6)] - 3*Log[1 + x^(1/3)]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 1819

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1)
, Pq, x]*(a + b*x^Simplify[n/(m + 1)])^p, x], x, x^(m + 1)], x] /; FreeQ[{a, b, m, n, p}, x] && NeQ[m, -1] &&
IGtQ[Simplify[n/(m + 1)], 0] && PolyQ[Pq, x^(m + 1)]

Rule 1810

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a,
b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 260

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

Rubi steps

\begin{align*} \int \frac{1+\sqrt{x}}{x^{5/6}+x^{7/6}} \, dx &=\int \frac{1+\sqrt{x}}{\left (1+\sqrt [3]{x}\right ) x^{5/6}} \, dx\\ &=6 \operatorname{Subst}\left (\int \frac{1+x^3}{1+x^2} \, dx,x,\sqrt [6]{x}\right )\\ &=6 \operatorname{Subst}\left (\int \left (x+\frac{1-x}{1+x^2}\right ) \, dx,x,\sqrt [6]{x}\right )\\ &=3 \sqrt [3]{x}+6 \operatorname{Subst}\left (\int \frac{1-x}{1+x^2} \, dx,x,\sqrt [6]{x}\right )\\ &=3 \sqrt [3]{x}+6 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt [6]{x}\right )-6 \operatorname{Subst}\left (\int \frac{x}{1+x^2} \, dx,x,\sqrt [6]{x}\right )\\ &=3 \sqrt [3]{x}+6 \tan ^{-1}\left (\sqrt [6]{x}\right )-3 \log \left (1+\sqrt [3]{x}\right )\\ \end{align*}

Mathematica [C]  time = 0.024554, size = 38, normalized size = 1.46 \[ 3 \sqrt [3]{x}+(-3-3 i) \log \left (-\sqrt [6]{x}+i\right )-(3-3 i) \log \left (\sqrt [6]{x}+i\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Sqrt[x])/(x^(5/6) + x^(7/6)),x]

[Out]

3*x^(1/3) - (3 + 3*I)*Log[I - x^(1/6)] - (3 - 3*I)*Log[I + x^(1/6)]

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Maple [A]  time = 0.003, size = 21, normalized size = 0.8 \begin{align*} 3\,\sqrt [3]{x}+6\,\arctan \left ( \sqrt [6]{x} \right ) -3\,\ln \left ( 1+\sqrt [3]{x} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x^(1/2))/(x^(5/6)+x^(7/6)),x)

[Out]

3*x^(1/3)+6*arctan(x^(1/6))-3*ln(1+x^(1/3))

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Maxima [A]  time = 1.78767, size = 27, normalized size = 1.04 \begin{align*} 3 \, x^{\frac{1}{3}} + 6 \, \arctan \left (x^{\frac{1}{6}}\right ) - 3 \, \log \left (x^{\frac{1}{3}} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x^(1/2))/(x^(5/6)+x^(7/6)),x, algorithm="maxima")

[Out]

3*x^(1/3) + 6*arctan(x^(1/6)) - 3*log(x^(1/3) + 1)

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Fricas [A]  time = 1.75033, size = 70, normalized size = 2.69 \begin{align*} 3 \, x^{\frac{1}{3}} + 6 \, \arctan \left (x^{\frac{1}{6}}\right ) - 3 \, \log \left (x^{\frac{1}{3}} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x^(1/2))/(x^(5/6)+x^(7/6)),x, algorithm="fricas")

[Out]

3*x^(1/3) + 6*arctan(x^(1/6)) - 3*log(x^(1/3) + 1)

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Sympy [A]  time = 4.88963, size = 24, normalized size = 0.92 \begin{align*} 3 \sqrt [3]{x} - 3 \log{\left (\sqrt [3]{x} + 1 \right )} + 6 \operatorname{atan}{\left (\sqrt [6]{x} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x**(1/2))/(x**(5/6)+x**(7/6)),x)

[Out]

3*x**(1/3) - 3*log(x**(1/3) + 1) + 6*atan(x**(1/6))

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Giac [A]  time = 1.09701, size = 27, normalized size = 1.04 \begin{align*} 3 \, x^{\frac{1}{3}} + 6 \, \arctan \left (x^{\frac{1}{6}}\right ) - 3 \, \log \left (x^{\frac{1}{3}} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x^(1/2))/(x^(5/6)+x^(7/6)),x, algorithm="giac")

[Out]

3*x^(1/3) + 6*arctan(x^(1/6)) - 3*log(x^(1/3) + 1)