3.10.44 \(\int \frac {1}{\sqrt [4]{1+x}+\sqrt {1+x}} \, dx\) [944]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, 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 = {2037, 1607, 272, 45} \begin {gather*} 2 \sqrt {x+1}-4 \sqrt [4]{x+1}+4 \log \left (\sqrt [4]{x+1}+1\right ) \end {gather*}

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 45

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])

Rule 272

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 1607

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 2037

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]

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 31, normalized size = 1.00 \begin {gather*} 2 \sqrt [4]{1+x} \left (-2+\sqrt [4]{1+x}\right )+4 \log \left (1+\sqrt [4]{1+x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.02, size = 26, normalized size = 0.84

method	result	size
derivativedivides	\(-4 \left (1+x \right )^{\frac {1}{4}}+4 \ln \left (1+\left (1+x \right )^{\frac {1}{4}}\right )+2 \sqrt {1+x}\)	\(26\)
default	\(-4 \left (1+x \right )^{\frac {1}{4}}+4 \ln \left (1+\left (1+x \right )^{\frac {1}{4}}\right )+2 \sqrt {1+x}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.37, size = 25, normalized size = 0.81 \begin {gather*} 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 {gather*}

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 = 0.32, size = 25, normalized size = 0.81 \begin {gather*} 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 {gather*}

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 = 0.10, size = 27, normalized size = 0.87 \begin {gather*} - 4 \sqrt [4]{x + 1} + 2 \sqrt {x + 1} + 4 \log {\left (\sqrt [4]{x + 1} + 1 \right )} \end {gather*}

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 = 4.10, size = 25, normalized size = 0.81 \begin {gather*} 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 {gather*}

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)

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Mupad [B]
time = 0.07, size = 25, normalized size = 0.81 \begin {gather*} 4\,\ln \left ({\left (x+1\right )}^{1/4}+1\right )+2\,\sqrt {x+1}-4\,{\left (x+1\right )}^{1/4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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