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*x^(5/4))/5

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Rubi [A]  time = 0.0015401, antiderivative size = 11, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0003907, size = 11, normalized size = 1. \[ x-\frac{4 x^{5/4}}{5} \]

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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Maple [C]  time = 0.009, size = 44, normalized size = 4. \begin{align*} -{\frac{4}{5}{x}^{{\frac{5}{4}}}}+x+2\,\ln \left ( 1+\sqrt [4]{x} \right ) +2\,\ln \left ( \sqrt [4]{x}-1 \right ) -\ln \left ( x-1 \right ) -\ln \left ( -1+\sqrt{x} \right ) +\ln \left ( 1+\sqrt{x} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.999827, size = 9, normalized size = 0.82 \begin{align*} -\frac{4}{5} \, x^{\frac{5}{4}} + x \end{align*}

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 = 1.42162, size = 24, normalized size = 2.18 \begin{align*} -\frac{4}{5} \, x^{\frac{5}{4}} + x \end{align*}

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 = 5.64295, size = 8, normalized size = 0.73 \begin{align*} - \frac{4 x^{\frac{5}{4}}}{5} + x \end{align*}

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 = 1.20375, size = 9, normalized size = 0.82 \begin{align*} -\frac{4}{5} \, x^{\frac{5}{4}} + x \end{align*}

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