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

Optimal. Leaf size=31 \[ 2 \sqrt{x+1}-4 \sqrt [4]{x+1}+4 \log \left (\sqrt [4]{x+1}+1\right ) \]

[Out]

-4*(1 + x)^(1/4) + 2*Sqrt[1 + x] + 4*Log[1 + (1 + x)^(1/4)]

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Rubi [A]  time = 0.0152038, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {2012, 1593, 266, 43} \[ 2 \sqrt{x+1}-4 \sqrt [4]{x+1}+4 \log \left (\sqrt [4]{x+1}+1\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-4*(1 + x)^(1/4) + 2*Sqrt[1 + x] + 4*Log[1 + (1 + x)^(1/4)]

Rule 2012

Int[((a_.)*(u_)^(j_.) + (b_.)*(u_)^(n_.))^(p_), x_Symbol] :> Dist[1/Coefficient[u, x, 1], Subst[Int[(a*x^j + b
*x^n)^p, x], x, u], x] /; FreeQ[{a, b, j, n, p}, x] && LinearQ[u, x] && NeQ[u, x]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.011783, size = 31, normalized size = 1. \[ 2 \sqrt{x+1}-4 \sqrt [4]{x+1}+4 \log \left (\sqrt [4]{x+1}+1\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-4*(1 + x)^(1/4) + 2*Sqrt[1 + x] + 4*Log[1 + (1 + x)^(1/4)]

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Maple [A]  time = 0.008, size = 26, normalized size = 0.8 \begin{align*} -4\,\sqrt [4]{1+x}+4\,\ln \left ( 1+\sqrt [4]{1+x} \right ) +2\,\sqrt{1+x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.1067, size = 34, normalized size = 1.1 \begin{align*} 2 \, \sqrt{x + 1} - 4 \,{\left (x + 1\right )}^{\frac{1}{4}} + 4 \, \log \left ({\left (x + 1\right )}^{\frac{1}{4}} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x + 1) - 4*(x + 1)^(1/4) + 4*log((x + 1)^(1/4) + 1)

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Fricas [A]  time = 1.47412, size = 81, normalized size = 2.61 \begin{align*} 2 \, \sqrt{x + 1} - 4 \,{\left (x + 1\right )}^{\frac{1}{4}} + 4 \, \log \left ({\left (x + 1\right )}^{\frac{1}{4}} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x + 1) - 4*(x + 1)^(1/4) + 4*log((x + 1)^(1/4) + 1)

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Sympy [A]  time = 0.220459, size = 27, normalized size = 0.87 \begin{align*} - 4 \sqrt [4]{x + 1} + 2 \sqrt{x + 1} + 4 \log{\left (\sqrt [4]{x + 1} + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*(x + 1)**(1/4) + 2*sqrt(x + 1) + 4*log((x + 1)**(1/4) + 1)

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Giac [A]  time = 1.11456, size = 34, normalized size = 1.1 \begin{align*} 2 \, \sqrt{x + 1} - 4 \,{\left (x + 1\right )}^{\frac{1}{4}} + 4 \, \log \left ({\left (x + 1\right )}^{\frac{1}{4}} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x + 1) - 4*(x + 1)^(1/4) + 4*log((x + 1)^(1/4) + 1)