3.7.94 \(\int \frac {5+\sqrt [4]{x}}{-6+x} \, dx\) [694]

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

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 54, normalized size of antiderivative = 1.00, 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 = {1845, 266, 327, 218, 212, 209} \begin {gather*} -2 \sqrt [4]{6} \text {ArcTan}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{6}}\right )+4 \sqrt [4]{x}+5 \log (6-x)-2 \sqrt [4]{6} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{6}}\right ) \end {gather*}

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 209

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

Rule 212

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

Rule 218

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] &&  !Gt
Q[a/b, 0]

Rule 266

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 327

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 1845

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

Rubi steps

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

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Mathematica [A]
time = 0.08, size = 52, normalized size = 0.96 \begin {gather*} 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 {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(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]

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Maple [A]
time = 5.08, size = 50, normalized size = 0.93

method	result	size
derivativedivides	\(4 x^{\frac {1}{4}}-6^{\frac {1}{4}} \left (\ln \left (\frac {x^{\frac {1}{4}}+6^{\frac {1}{4}}}{x^{\frac {1}{4}}-6^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {x^{\frac {1}{4}} 6^{\frac {3}{4}}}{6}\right )\right )+5 \ln \left (-6+x \right )\)	\(50\)
default	\(4 x^{\frac {1}{4}}-6^{\frac {1}{4}} \left (\ln \left (\frac {x^{\frac {1}{4}}+6^{\frac {1}{4}}}{x^{\frac {1}{4}}-6^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {x^{\frac {1}{4}} 6^{\frac {3}{4}}}{6}\right )\right )+5 \ln \left (-6+x \right )\)	\(50\)
meijerg	\(5 \ln \left (1-\frac {x}{6}\right )-6^{\frac {1}{4}} \left (-1\right )^{\frac {3}{4}} \left (\frac {2 \,8^{\frac {1}{4}} 3^{\frac {3}{4}} x^{\frac {1}{4}} \left (-1\right )^{\frac {1}{4}}}{3}+\left (-1\right )^{\frac {1}{4}} \left (\ln \left (1-\frac {x^{\frac {1}{4}} 6^{\frac {3}{4}}}{6}\right )-\ln \left (1+\frac {x^{\frac {1}{4}} 6^{\frac {3}{4}}}{6}\right )-2 \arctan \left (\frac {x^{\frac {1}{4}} 6^{\frac {3}{4}}}{6}\right )\right )\right )\)	\(73\)
trager	\(\text {Expression too large to display}\)	\(3981\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.51, size = 67, normalized size = 1.24 \begin {gather*} -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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (42) = 84\).
time = 0.40, size = 86, normalized size = 1.59 \begin {gather*} -{\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 {gather*}

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 = 0.64, size = 100, normalized size = 1.85 \begin {gather*} 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 \cdot \sqrt [4]{6} \operatorname {atan}{\left (\frac {6^{\frac {3}{4}} \sqrt [4]{x}}{6} \right )} \end {gather*}

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 = 2.32, size = 55, normalized size = 1.02 \begin {gather*} -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 {gather*}

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))

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Mupad [B]
time = 0.09, size = 162, normalized size = 3.00 \begin {gather*} \ln \left (11520\,x^{1/4}-\left (6^{1/4}+5\right )\,\left (2304\,x^{1/4}-2304\,6^{1/4}+11520\right )+57600\right )\,\left (6^{1/4}+5\right )-\ln \left (\left (6^{1/4}-5\right )\,\left (2304\,6^{1/4}+2304\,x^{1/4}+11520\right )+11520\,x^{1/4}+57600\right )\,\left (6^{1/4}-5\right )-\ln \left (11520\,x^{1/4}+\left (\sqrt {-\sqrt {6}}-5\right )\,\left (2304\,\sqrt {-\sqrt {6}}+2304\,x^{1/4}+11520\right )+57600\right )\,\left (\sqrt {-\sqrt {6}}-5\right )+\ln \left (11520\,x^{1/4}-\left (\sqrt {-\sqrt {6}}+5\right )\,\left (2304\,x^{1/4}-2304\,\sqrt {-\sqrt {6}}+11520\right )+57600\right )\,\left (\sqrt {-\sqrt {6}}+5\right )+4\,x^{1/4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^(1/4) + 5)/(x - 6),x)

[Out]

log(11520*x^(1/4) - (6^(1/4) + 5)*(2304*x^(1/4) - 2304*6^(1/4) + 11520) + 57600)*(6^(1/4) + 5) - log((6^(1/4)
- 5)*(2304*6^(1/4) + 2304*x^(1/4) + 11520) + 11520*x^(1/4) + 57600)*(6^(1/4) - 5) - log(11520*x^(1/4) + ((-6^(
1/2))^(1/2) - 5)*(2304*(-6^(1/2))^(1/2) + 2304*x^(1/4) + 11520) + 57600)*((-6^(1/2))^(1/2) - 5) + log(11520*x^
(1/4) - ((-6^(1/2))^(1/2) + 5)*(2304*x^(1/4) - 2304*(-6^(1/2))^(1/2) + 11520) + 57600)*((-6^(1/2))^(1/2) + 5)
+ 4*x^(1/4)

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