3.694 \(\int \frac{5+\sqrt [4]{x}}{-6+x} \, dx\)

Optimal. Leaf size=54 \[ 4 \sqrt [4]{x}+5 \log (6-x)-2 \sqrt [4]{6} \tan ^{-1}\left (\frac{\sqrt [4]{x}}{\sqrt [4]{6}}\right )-2 \sqrt [4]{6} \tanh ^{-1}\left (\frac{\sqrt [4]{x}}{\sqrt [4]{6}}\right ) \]

[Out]

4*x^(1/4) - 2*6^(1/4)*ArcTan[x^(1/4)/6^(1/4)] - 2*6^(1/4)*ArcTanh[x^(1/4)/6^(1/4)] + 5*Log[6 - x]

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Rubi [A]  time = 0.0831094, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462, Rules used = {1831, 260, 321, 212, 206, 203} \[ 4 \sqrt [4]{x}+5 \log (6-x)-2 \sqrt [4]{6} \tan ^{-1}\left (\frac{\sqrt [4]{x}}{\sqrt [4]{6}}\right )-2 \sqrt [4]{6} \tanh ^{-1}\left (\frac{\sqrt [4]{x}}{\sqrt [4]{6}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(5 + x^(1/4))/(-6 + x),x]

[Out]

4*x^(1/4) - 2*6^(1/4)*ArcTan[x^(1/4)/6^(1/4)] - 2*6^(1/4)*ArcTanh[x^(1/4)/6^(1/4)] + 5*Log[6 - x]

Rule 1831

Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = Sum[((c*x)^(m + ii)*(Coeff[Pq,
x, ii] + Coeff[Pq, x, n/2 + ii]*x^(n/2)))/(c^ii*(a + b*x^n)), {ii, 0, n/2 - 1}]}, Int[v, x] /; SumQ[v]] /; Fr
eeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] && Expon[Pq, x] < n

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]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

Rule 206

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

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])

Rubi steps

\begin{align*} \int \frac{5+\sqrt [4]{x}}{-6+x} \, dx &=4 \operatorname{Subst}\left (\int \frac{x^3 (5+x)}{-6+x^4} \, dx,x,\sqrt [4]{x}\right )\\ &=4 \operatorname{Subst}\left (\int \left (\frac{5 x^3}{-6+x^4}+\frac{x^4}{-6+x^4}\right ) \, dx,x,\sqrt [4]{x}\right )\\ &=4 \operatorname{Subst}\left (\int \frac{x^4}{-6+x^4} \, dx,x,\sqrt [4]{x}\right )+20 \operatorname{Subst}\left (\int \frac{x^3}{-6+x^4} \, dx,x,\sqrt [4]{x}\right )\\ &=4 \sqrt [4]{x}+5 \log (6-x)+24 \operatorname{Subst}\left (\int \frac{1}{-6+x^4} \, dx,x,\sqrt [4]{x}\right )\\ &=4 \sqrt [4]{x}+5 \log (6-x)-\left (2 \sqrt{6}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{6}-x^2} \, dx,x,\sqrt [4]{x}\right )-\left (2 \sqrt{6}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{6}+x^2} \, dx,x,\sqrt [4]{x}\right )\\ &=4 \sqrt [4]{x}-2 \sqrt [4]{6} \tan ^{-1}\left (\frac{\sqrt [4]{x}}{\sqrt [4]{6}}\right )-2 \sqrt [4]{6} \tanh ^{-1}\left (\frac{\sqrt [4]{x}}{\sqrt [4]{6}}\right )+5 \log (6-x)\\ \end{align*}

Mathematica [C]  time = 0.0773881, size = 107, normalized size = 1.98 \[ 4 \sqrt [4]{x}+\left (5+\sqrt [4]{6}\right ) \log \left (\sqrt [4]{6}-\sqrt [4]{x}\right )+\left (5-i \sqrt [4]{6}\right ) \log \left (\sqrt [4]{6}-i \sqrt [4]{x}\right )+\left (5+i \sqrt [4]{6}\right ) \log \left (\sqrt [4]{6}+i \sqrt [4]{x}\right )-\left (\sqrt [4]{6}-5\right ) \log \left (\sqrt [4]{x}+\sqrt [4]{6}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(5 + x^(1/4))/(-6 + x),x]

[Out]

4*x^(1/4) + (5 + 6^(1/4))*Log[6^(1/4) - x^(1/4)] + (5 - I*6^(1/4))*Log[6^(1/4) - I*x^(1/4)] + (5 + I*6^(1/4))*
Log[6^(1/4) + I*x^(1/4)] - (-5 + 6^(1/4))*Log[6^(1/4) + x^(1/4)]

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Maple [A]  time = 0.004, size = 52, normalized size = 1. \begin{align*} 4\,\sqrt [4]{x}-2\,\sqrt [4]{6}\arctan \left ( 1/6\,\sqrt [4]{x}{6}^{3/4} \right ) -\sqrt [4]{6}\ln \left ({ \left ( \sqrt [4]{x}+\sqrt [4]{6} \right ) \left ( \sqrt [4]{x}-\sqrt [4]{6} \right ) ^{-1}} \right ) +5\,\ln \left ( -6+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5+x^(1/4))/(-6+x),x)

[Out]

4*x^(1/4)-2*6^(1/4)*arctan(1/6*x^(1/4)*6^(3/4))-6^(1/4)*ln((x^(1/4)+6^(1/4))/(x^(1/4)-6^(1/4)))+5*ln(-6+x)

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Maxima [A]  time = 1.48588, size = 90, normalized size = 1.67 \begin{align*} -2 \cdot 6^{\frac{1}{4}} \arctan \left (\frac{1}{6} \cdot 6^{\frac{3}{4}} x^{\frac{1}{4}}\right ) + 6^{\frac{1}{4}} \log \left (-\frac{6^{\frac{1}{4}} - x^{\frac{1}{4}}}{6^{\frac{1}{4}} + x^{\frac{1}{4}}}\right ) + 4 \, x^{\frac{1}{4}} + 5 \, \log \left (\sqrt{6} + \sqrt{x}\right ) + 5 \, \log \left (-\sqrt{6} + \sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5+x^(1/4))/(-6+x),x, algorithm="maxima")

[Out]

-2*6^(1/4)*arctan(1/6*6^(3/4)*x^(1/4)) + 6^(1/4)*log(-(6^(1/4) - x^(1/4))/(6^(1/4) + x^(1/4))) + 4*x^(1/4) + 5
*log(sqrt(6) + sqrt(x)) + 5*log(-sqrt(6) + sqrt(x))

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Fricas [B]  time = 1.5883, size = 281, normalized size = 5.2 \begin{align*} -{\left (6^{\frac{1}{4}} - 5\right )} \log \left (2 \cdot 6^{\frac{1}{4}} + 2 \, x^{\frac{1}{4}}\right ) +{\left (6^{\frac{1}{4}} + 5\right )} \log \left (-2 \cdot 6^{\frac{1}{4}} + 2 \, x^{\frac{1}{4}}\right ) + 4 \cdot 6^{\frac{1}{4}} \arctan \left (\frac{1}{6} \cdot 6^{\frac{3}{4}} \sqrt{\sqrt{6} + \sqrt{x}} - \frac{1}{6} \cdot 6^{\frac{3}{4}} x^{\frac{1}{4}}\right ) + 4 \, x^{\frac{1}{4}} + 5 \, \log \left (4 \, \sqrt{6} + 4 \, \sqrt{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5+x^(1/4))/(-6+x),x, algorithm="fricas")

[Out]

-(6^(1/4) - 5)*log(2*6^(1/4) + 2*x^(1/4)) + (6^(1/4) + 5)*log(-2*6^(1/4) + 2*x^(1/4)) + 4*6^(1/4)*arctan(1/6*6
^(3/4)*sqrt(sqrt(6) + sqrt(x)) - 1/6*6^(3/4)*x^(1/4)) + 4*x^(1/4) + 5*log(4*sqrt(6) + 4*sqrt(x))

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Sympy [A]  time = 2.489, size = 100, normalized size = 1.85 \begin{align*} 4 \sqrt [4]{x} + \sqrt [4]{6} \log{\left (\sqrt [4]{x} - \sqrt [4]{6} \right )} + 5 \log{\left (\sqrt [4]{x} - \sqrt [4]{6} \right )} - \sqrt [4]{6} \log{\left (\sqrt [4]{x} + \sqrt [4]{6} \right )} + 5 \log{\left (\sqrt [4]{x} + \sqrt [4]{6} \right )} + 5 \log{\left (\sqrt{x} + \sqrt{6} \right )} - 2 \sqrt [4]{6} \operatorname{atan}{\left (\frac{6^{\frac{3}{4}} \sqrt [4]{x}}{6} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5+x**(1/4))/(-6+x),x)

[Out]

4*x**(1/4) + 6**(1/4)*log(x**(1/4) - 6**(1/4)) + 5*log(x**(1/4) - 6**(1/4)) - 6**(1/4)*log(x**(1/4) + 6**(1/4)
) + 5*log(x**(1/4) + 6**(1/4)) + 5*log(sqrt(x) + sqrt(6)) - 2*6**(1/4)*atan(6**(3/4)*x**(1/4)/6)

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Giac [A]  time = 1.17942, size = 74, normalized size = 1.37 \begin{align*} -2 \cdot 6^{\frac{1}{4}} \arctan \left (\frac{1}{6} \cdot 6^{\frac{3}{4}} x^{\frac{1}{4}}\right ) - 6^{\frac{1}{4}} \log \left (6^{\frac{1}{4}} + x^{\frac{1}{4}}\right ) + 6^{\frac{1}{4}} \log \left ({\left | -6^{\frac{1}{4}} + x^{\frac{1}{4}} \right |}\right ) + 4 \, x^{\frac{1}{4}} + 5 \, \log \left ({\left | x - 6 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5+x^(1/4))/(-6+x),x, algorithm="giac")

[Out]

-2*6^(1/4)*arctan(1/6*6^(3/4)*x^(1/4)) - 6^(1/4)*log(6^(1/4) + x^(1/4)) + 6^(1/4)*log(abs(-6^(1/4) + x^(1/4)))
+ 4*x^(1/4) + 5*log(abs(x - 6))