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

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} \[ x-\frac {4 x^{5/4}}{5} \]

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 \[ x-\frac {4 x^{5/4}}{5} \]

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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fricas [A]  time = 0.76, size = 7, normalized size = 0.64 \[ -\frac {4}{5} \, x^{\frac {5}{4}} + x \]

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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giac [A]  time = 0.35, size = 7, normalized size = 0.64 \[ -\frac {4}{5} \, x^{\frac {5}{4}} + x \]

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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maple [C]  time = 0.01, size = 44, normalized size = 4.00 \[ -\frac {4 x^{\frac {5}{4}}}{5}+x -\ln \left (x -1\right )-\ln \left (\sqrt {x}-1\right )+\ln \left (\sqrt {x}+1\right )+2 \ln \left (x^{\frac {1}{4}}+1\right )+2 \ln \left (x^{\frac {1}{4}}-1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.61, size = 7, normalized size = 0.64 \[ -\frac {4}{5} \, x^{\frac {5}{4}} + x \]

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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mupad [B]  time = 0.03, size = 7, normalized size = 0.64 \[ x-\frac {4\,x^{5/4}}{5} \]

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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sympy [A]  time = 4.36, size = 8, normalized size = 0.73 \[ - \frac {4 x^{\frac {5}{4}}}{5} + x \]

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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