3.213 \(\int \frac {1}{-\sqrt {1+x}+(1+x)^{2/3}} \, dx\)

Optimal. Leaf size=33 \[ 3 \sqrt [3]{x+1}+6 \sqrt [6]{x+1}+6 \log \left (1-\sqrt [6]{x+1}\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2012, 1593, 266, 43} \[ 3 \sqrt [3]{x+1}+6 \sqrt [6]{x+1}+6 \log \left (1-\sqrt [6]{x+1}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(-Sqrt[1 + x] + (1 + x)^(2/3))^(-1),x]

[Out]

6*(1 + x)^(1/6) + 3*(1 + x)^(1/3) + 6*Log[1 - (1 + x)^(1/6)]

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])

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 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 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]

Rubi steps

\begin {align*} \int \frac {1}{-\sqrt {1+x}+(1+x)^{2/3}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{-\sqrt {x}+x^{2/3}} \, dx,x,1+x\right )\\ &=\operatorname {Subst}\left (\int \frac {1}{\left (-1+\sqrt [6]{x}\right ) \sqrt {x}} \, dx,x,1+x\right )\\ &=6 \operatorname {Subst}\left (\int \frac {x^2}{-1+x} \, dx,x,\sqrt [6]{1+x}\right )\\ &=6 \operatorname {Subst}\left (\int \left (1+\frac {1}{-1+x}+x\right ) \, dx,x,\sqrt [6]{1+x}\right )\\ &=6 \sqrt [6]{1+x}+3 \sqrt [3]{1+x}+6 \log \left (1-\sqrt [6]{1+x}\right )\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 33, normalized size = 1.00 \[ 3 \left (\sqrt [3]{x+1}+2 \sqrt [6]{x+1}+2 \log \left (1-\sqrt [6]{x+1}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(-Sqrt[1 + x] + (1 + x)^(2/3))^(-1),x]

[Out]

3*(2*(1 + x)^(1/6) + (1 + x)^(1/3) + 2*Log[1 - (1 + x)^(1/6)])

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IntegrateAlgebraic [A]  time = 0.01, size = 31, normalized size = 0.94 \[ 3 \sqrt [3]{x+1}+6 \sqrt [6]{x+1}+6 \log \left (\sqrt [6]{x+1}-1\right ) \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-Sqrt[1 + x] + (1 + x)^(2/3))^(-1),x]

[Out]

6*(1 + x)^(1/6) + 3*(1 + x)^(1/3) + 6*Log[-1 + (1 + x)^(1/6)]

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fricas [A]  time = 0.72, size = 25, normalized size = 0.76 \[ 3 \, {\left (x + 1\right )}^{\frac {1}{3}} + 6 \, {\left (x + 1\right )}^{\frac {1}{6}} + 6 \, \log \left ({\left (x + 1\right )}^{\frac {1}{6}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((1+x)^(2/3)-(1+x)^(1/2)),x, algorithm="fricas")

[Out]

3*(x + 1)^(1/3) + 6*(x + 1)^(1/6) + 6*log((x + 1)^(1/6) - 1)

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giac [A]  time = 1.18, size = 26, normalized size = 0.79 \[ 3 \, {\left (x + 1\right )}^{\frac {1}{3}} + 6 \, {\left (x + 1\right )}^{\frac {1}{6}} + 6 \, \log \left ({\left | {\left (x + 1\right )}^{\frac {1}{6}} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((1+x)^(2/3)-(1+x)^(1/2)),x, algorithm="giac")

[Out]

3*(x + 1)^(1/3) + 6*(x + 1)^(1/6) + 6*log(abs((x + 1)^(1/6) - 1))

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maple [A]  time = 0.04, size = 26, normalized size = 0.79

method	result	size
derivativedivides	\(3 \left (1+x \right )^{\frac {1}{3}}+6 \left (1+x \right )^{\frac {1}{6}}+6 \ln \left (\left (1+x \right )^{\frac {1}{6}}-1\right )\)	\(26\)
default	\(6 \left (1+x \right )^{\frac {1}{6}}+3 \left (1+x \right )^{\frac {1}{3}}+\ln \relax (x )+2 \ln \left (\left (1+x \right )^{\frac {1}{6}}-1\right )-\ln \left (\left (1+x \right )^{\frac {1}{3}}+\left (1+x \right )^{\frac {1}{6}}+1\right )-2 \ln \left (1+\left (1+x \right )^{\frac {1}{6}}\right )+\ln \left (\left (1+x \right )^{\frac {1}{3}}-\left (1+x \right )^{\frac {1}{6}}+1\right )-\ln \left (1+\sqrt {1+x}\right )+\ln \left (-1+\sqrt {1+x}\right )+2 \ln \left (\left (1+x \right )^{\frac {1}{3}}-1\right )-\ln \left (\left (1+x \right )^{\frac {2}{3}}+\left (1+x \right )^{\frac {1}{3}}+1\right )\)	\(111\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((1+x)^(2/3)-(1+x)^(1/2)),x,method=_RETURNVERBOSE)

[Out]

3*(1+x)^(1/3)+6*(1+x)^(1/6)+6*ln((1+x)^(1/6)-1)

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maxima [A]  time = 0.52, size = 25, normalized size = 0.76 \[ 3 \, {\left (x + 1\right )}^{\frac {1}{3}} + 6 \, {\left (x + 1\right )}^{\frac {1}{6}} + 6 \, \log \left ({\left (x + 1\right )}^{\frac {1}{6}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((1+x)^(2/3)-(1+x)^(1/2)),x, algorithm="maxima")

[Out]

3*(x + 1)^(1/3) + 6*(x + 1)^(1/6) + 6*log((x + 1)^(1/6) - 1)

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mupad [B]  time = 0.24, size = 25, normalized size = 0.76 \[ 6\,\ln \left ({\left (x+1\right )}^{1/6}-1\right )+3\,{\left (x+1\right )}^{1/3}+6\,{\left (x+1\right )}^{1/6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/((x + 1)^(1/2) - (x + 1)^(2/3)),x)

[Out]

6*log((x + 1)^(1/6) - 1) + 3*(x + 1)^(1/3) + 6*(x + 1)^(1/6)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (x + 1\right )^{\frac {2}{3}} - \sqrt {x + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((1+x)**(2/3)-(1+x)**(1/2)),x)

[Out]

Integral(1/((x + 1)**(2/3) - sqrt(x + 1)), x)

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