3.10.46 \(\int \frac {\sqrt [3]{1+2 x+x^2}}{4+x+x^2+x^3} \, dx\)

Optimal. Leaf size=72 \[ \frac {\sqrt [3]{(x+1)^2} \text {RootSum}\left [\text {$\#$1}^9-2 \text {$\#$1}^6+2 \text {$\#$1}^3+3\& ,\frac {\text {$\#$1}^2 \log \left (\sqrt [3]{x+1}-\text {$\#$1}\right )}{3 \text {$\#$1}^6-4 \text {$\#$1}^3+2}\& \right ]}{(x+1)^{2/3}} \]

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Rubi [F]  time = 0.10, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\sqrt [3]{1+2 x+x^2}}{4+x+x^2+x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1 + 2*x + x^2)^(1/3)/(4 + x + x^2 + x^3),x]

[Out]

Defer[Int][(1 + 2*x + x^2)^(1/3)/(4 + x + x^2 + x^3), x]

Rubi steps

\begin {align*} \int \frac {\sqrt [3]{1+2 x+x^2}}{4+x+x^2+x^3} \, dx &=\int \frac {\sqrt [3]{1+2 x+x^2}}{4+x+x^2+x^3} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 72, normalized size = 1.00 \begin {gather*} \frac {\sqrt [3]{(x+1)^2} \text {RootSum}\left [\text {$\#$1}^9-2 \text {$\#$1}^6+2 \text {$\#$1}^3+3\&,\frac {\text {$\#$1}^2 \log \left (\sqrt [3]{x+1}-\text {$\#$1}\right )}{3 \text {$\#$1}^6-4 \text {$\#$1}^3+2}\&\right ]}{(x+1)^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + 2*x + x^2)^(1/3)/(4 + x + x^2 + x^3),x]

[Out]

(((1 + x)^2)^(1/3)*RootSum[3 + 2*#1^3 - 2*#1^6 + #1^9 & , (Log[(1 + x)^(1/3) - #1]*#1^2)/(2 - 4*#1^3 + 3*#1^6)
& ])/(1 + x)^(2/3)

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IntegrateAlgebraic [A]  time = 7.12, size = 72, normalized size = 1.00 \begin {gather*} \frac {\sqrt [3]{(1+x)^2} \text {RootSum}\left [3+2 \text {$\#$1}^3-2 \text {$\#$1}^6+\text {$\#$1}^9\&,\frac {\log \left (\sqrt [3]{1+x}-\text {$\#$1}\right ) \text {$\#$1}^2}{2-4 \text {$\#$1}^3+3 \text {$\#$1}^6}\&\right ]}{(1+x)^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 + 2*x + x^2)^(1/3)/(4 + x + x^2 + x^3),x]

[Out]

(((1 + x)^2)^(1/3)*RootSum[3 + 2*#1^3 - 2*#1^6 + #1^9 & , (Log[(1 + x)^(1/3) - #1]*#1^2)/(2 - 4*#1^3 + 3*#1^6)
& ])/(1 + x)^(2/3)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+2*x+1)^(1/3)/(x^3+x^2+x+4),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{2} + 2 \, x + 1\right )}^{\frac {1}{3}}}{x^{3} + x^{2} + x + 4}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+2*x+1)^(1/3)/(x^3+x^2+x+4),x, algorithm="giac")

[Out]

integrate((x^2 + 2*x + 1)^(1/3)/(x^3 + x^2 + x + 4), x)

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{2}+2 x +1\right )^{\frac {1}{3}}}{x^{3}+x^{2}+x +4}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+2*x+1)^(1/3)/(x^3+x^2+x+4),x)

[Out]

int((x^2+2*x+1)^(1/3)/(x^3+x^2+x+4),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{2} + 2 \, x + 1\right )}^{\frac {1}{3}}}{x^{3} + x^{2} + x + 4}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+2*x+1)^(1/3)/(x^3+x^2+x+4),x, algorithm="maxima")

[Out]

integrate((x^2 + 2*x + 1)^(1/3)/(x^3 + x^2 + x + 4), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^2+2\,x+1\right )}^{1/3}}{x^3+x^2+x+4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + x^2 + 1)^(1/3)/(x + x^2 + x^3 + 4),x)

[Out]

int((2*x + x^2 + 1)^(1/3)/(x + x^2 + x^3 + 4), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{\left (x + 1\right )^{2}}}{x^{3} + x^{2} + x + 4}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2+2*x+1)**(1/3)/(x**3+x**2+x+4),x)

[Out]

Integral(((x + 1)**2)**(1/3)/(x**3 + x**2 + x + 4), x)

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