3.10.54 \(\int \frac {1-\sqrt {x}}{1+\sqrt [4]{x}} \, dx\) [954]

Optimal. Leaf size=11 \[ x-\frac {4 x^{5/4}}{5} \]

[Out]

x-4/5*x^(5/4)

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Rubi [A]
time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {26} \begin {gather*} x-\frac {4 x^{5/4}}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - Sqrt[x])/(1 + x^(1/4)),x]

[Out]

x - (4*x^(5/4))/5

Rule 26

Int[(u_.)*((a_) + (b_.)*(x_)^(n_.))^(m_.)*((c_) + (d_.)*(x_)^(j_))^(p_.), x_Symbol] :> Dist[(-b^2/d)^m, Int[u/
(a - b*x^n)^m, x], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[j, 2*n] && EqQ[p, -m] && EqQ[b^2*c + a^2*d, 0]
&& GtQ[a, 0] && LtQ[d, 0]

Rubi steps

\begin {align*} \int \frac {1-\sqrt {x}}{1+\sqrt [4]{x}} \, dx &=\int \left (1-\sqrt [4]{x}\right ) \, dx\\ &=x-\frac {4 x^{5/4}}{5}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 11, normalized size = 1.00 \begin {gather*} x-\frac {4 x^{5/4}}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x - (4*x^(5/4))/5

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Maple [C] Result contains higher order function than in optimal. Order 3 vs. order 2.
time = 0.22, size = 46, normalized size = 4.18

method	result	size
derivativedivides	\(x -\frac {4 x^{\frac {5}{4}}}{5}\)	\(8\)
meijerg	\(\frac {x^{\frac {1}{4}} \left (4 \sqrt {x}-6 x^{\frac {1}{4}}+12\right )}{3}-\frac {x^{\frac {1}{4}} \left (12 x -15 x^{\frac {3}{4}}+20 \sqrt {x}-30 x^{\frac {1}{4}}+60\right )}{15}\)	\(44\)
default	\(-\frac {4 x^{\frac {5}{4}}}{5}+x +2 \ln \left (1+x^{\frac {1}{4}}\right )-\ln \left (1-x \right )-\ln \left (-1+\sqrt {x}\right )+\ln \left (1+\sqrt {x}\right )+2 \ln \left (-1+x^{\frac {1}{4}}\right )\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 7, normalized size = 0.64 \begin {gather*} -\frac {4}{5} \, x^{\frac {5}{4}} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/5*x^(5/4) + x

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Fricas [A]
time = 0.32, size = 7, normalized size = 0.64 \begin {gather*} -\frac {4}{5} \, x^{\frac {5}{4}} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/5*x^(5/4) + x

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Sympy [A]
time = 2.14, size = 8, normalized size = 0.73 \begin {gather*} - \frac {4 x^{\frac {5}{4}}}{5} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*x**(5/4)/5 + x

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Giac [A]
time = 5.44, size = 7, normalized size = 0.64 \begin {gather*} -\frac {4}{5} \, x^{\frac {5}{4}} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/5*x^(5/4) + x

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Mupad [B]
time = 0.03, size = 7, normalized size = 0.64 \begin {gather*} x-\frac {4\,x^{5/4}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^(1/2) - 1)/(x^(1/4) + 1),x)

[Out]

x - (4*x^(5/4))/5

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