3.8.71 \(\int \frac {(3+x^2) (1+x^2+x^3)^{2/3}}{-1-2 x^2+x^3-x^4+x^5+x^6} \, dx\)

Optimal. Leaf size=59 \[ \text {RootSum}\left [\text {$\#$1}^6-3 \text {$\#$1}^3+1\& ,\frac {\text {$\#$1}^2 \log \left (\sqrt [3]{x^3+x^2+1}-\text {$\#$1} x\right )-\text {$\#$1}^2 \log (x)}{2 \text {$\#$1}^3-3}\& \right ] \]

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Rubi [F]  time = 0.61, 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 {\left (3+x^2\right ) \left (1+x^2+x^3\right )^{2/3}}{-1-2 x^2+x^3-x^4+x^5+x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((3 + x^2)*(1 + x^2 + x^3)^(2/3))/(-1 - 2*x^2 + x^3 - x^4 + x^5 + x^6),x]

[Out]

3*Defer[Int][(1 + x^2 + x^3)^(2/3)/(-1 - 2*x^2 + x^3 - x^4 + x^5 + x^6), x] + Defer[Int][(x^2*(1 + x^2 + x^3)^
(2/3))/(-1 - 2*x^2 + x^3 - x^4 + x^5 + x^6), x]

Rubi steps

\begin {align*} \int \frac {\left (3+x^2\right ) \left (1+x^2+x^3\right )^{2/3}}{-1-2 x^2+x^3-x^4+x^5+x^6} \, dx &=\int \left (\frac {3 \left (1+x^2+x^3\right )^{2/3}}{-1-2 x^2+x^3-x^4+x^5+x^6}+\frac {x^2 \left (1+x^2+x^3\right )^{2/3}}{-1-2 x^2+x^3-x^4+x^5+x^6}\right ) \, dx\\ &=3 \int \frac {\left (1+x^2+x^3\right )^{2/3}}{-1-2 x^2+x^3-x^4+x^5+x^6} \, dx+\int \frac {x^2 \left (1+x^2+x^3\right )^{2/3}}{-1-2 x^2+x^3-x^4+x^5+x^6} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.18, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (3+x^2\right ) \left (1+x^2+x^3\right )^{2/3}}{-1-2 x^2+x^3-x^4+x^5+x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((3 + x^2)*(1 + x^2 + x^3)^(2/3))/(-1 - 2*x^2 + x^3 - x^4 + x^5 + x^6),x]

[Out]

Integrate[((3 + x^2)*(1 + x^2 + x^3)^(2/3))/(-1 - 2*x^2 + x^3 - x^4 + x^5 + x^6), x]

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IntegrateAlgebraic [A]  time = 0.31, size = 59, normalized size = 1.00 \begin {gather*} \text {RootSum}\left [1-3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}^2+\log \left (\sqrt [3]{1+x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-3+2 \text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((3 + x^2)*(1 + x^2 + x^3)^(2/3))/(-1 - 2*x^2 + x^3 - x^4 + x^5 + x^6),x]

[Out]

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

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+3)*(x^3+x^2+1)^(2/3)/(x^6+x^5-x^4+x^3-2*x^2-1),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (tr
ace 0)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} + x^{2} + 1\right )}^{\frac {2}{3}} {\left (x^{2} + 3\right )}}{x^{6} + x^{5} - x^{4} + x^{3} - 2 \, x^{2} - 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+3)*(x^3+x^2+1)^(2/3)/(x^6+x^5-x^4+x^3-2*x^2-1),x, algorithm="giac")

[Out]

integrate((x^3 + x^2 + 1)^(2/3)*(x^2 + 3)/(x^6 + x^5 - x^4 + x^3 - 2*x^2 - 1), x)

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maple [B]  time = 31.33, size = 9017, normalized size = 152.83

method	result	size
trager	\(\text {Expression too large to display}\)	\(9017\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+3)*(x^3+x^2+1)^(2/3)/(x^6+x^5-x^4+x^3-2*x^2-1),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} + x^{2} + 1\right )}^{\frac {2}{3}} {\left (x^{2} + 3\right )}}{x^{6} + x^{5} - x^{4} + x^{3} - 2 \, x^{2} - 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+3)*(x^3+x^2+1)^(2/3)/(x^6+x^5-x^4+x^3-2*x^2-1),x, algorithm="maxima")

[Out]

integrate((x^3 + x^2 + 1)^(2/3)*(x^2 + 3)/(x^6 + x^5 - x^4 + x^3 - 2*x^2 - 1), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int -\frac {\left (x^2+3\right )\,{\left (x^3+x^2+1\right )}^{2/3}}{-x^6-x^5+x^4-x^3+2\,x^2+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((x^2 + 3)*(x^2 + x^3 + 1)^(2/3))/(2*x^2 - x^3 + x^4 - x^5 - x^6 + 1),x)

[Out]

int(-((x^2 + 3)*(x^2 + x^3 + 1)^(2/3))/(2*x^2 - x^3 + x^4 - x^5 - x^6 + 1), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{2} + 3\right ) \left (x^{3} + x^{2} + 1\right )^{\frac {2}{3}}}{x^{6} + x^{5} - x^{4} + x^{3} - 2 x^{2} - 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2+3)*(x**3+x**2+1)**(2/3)/(x**6+x**5-x**4+x**3-2*x**2-1),x)

[Out]

Integral((x**2 + 3)*(x**3 + x**2 + 1)**(2/3)/(x**6 + x**5 - x**4 + x**3 - 2*x**2 - 1), x)

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