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

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

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Rubi [F]  time = 0.41, 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 {(3+2 x) \left (1+x+x^3\right )^{2/3}}{1+2 x+x^2+x^3+x^4+x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.15, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(3+2 x) \left (1+x+x^3\right )^{2/3}}{1+2 x+x^2+x^3+x^4+x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.30, size = 59, normalized size = 1.00 \begin {gather*} -\text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}^2+\log \left (\sqrt [3]{1+x+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-1+2 \text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-RootSum[1 - #1^3 + #1^6 & , (-(Log[x]*#1^2) + Log[(1 + x + x^3)^(1/3) - x*#1]*#1^2)/(-1 + 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((3+2*x)*(x^3+x+1)^(2/3)/(x^6+x^4+x^3+x^2+2*x+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 + 1\right )}^{\frac {2}{3}} {\left (2 \, x + 3\right )}}{x^{6} + x^{4} + x^{3} + x^{2} + 2 \, x + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 6.10, size = 1598, normalized size = 27.08

method	result	size
trager	\(-9 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{5} \ln \left (-\frac {18 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{5} x^{3}+9 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{4} \left (x^{3}+x +1\right )^{\frac {1}{3}} x^{2}+3 \left (x^{3}+x +1\right )^{\frac {2}{3}} \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{3} x +9 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{5} x +9 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{5}-4 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{2} x^{3}-2 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right ) \left (x^{3}+x +1\right )^{\frac {1}{3}} x^{2}-\left (x^{3}+x +1\right )^{\frac {2}{3}} x -2 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{2} x -2 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{2}}{9 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{3} x^{3}-x^{3}+x +1}\right )+18 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{5} \ln \left (-\frac {-81 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{7} x^{3}+27 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{4} x^{3}+3 \left (x^{3}+x +1\right )^{\frac {2}{3}} \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{2} x +9 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{4} x +9 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{4}-2 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right ) x^{3}+\left (x^{3}+x +1\right )^{\frac {1}{3}} x^{2}-\RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right ) x -\RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )}{9 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{3} x^{3}-x^{3}+x +1}\right )-9 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{4} \ln \left (\frac {54 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{7} x^{3}+9 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{5} \left (x^{3}+x +1\right )^{\frac {1}{3}} x^{2}-15 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{4} x^{3}+3 \left (x^{3}+x +1\right )^{\frac {2}{3}} \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{3} x -3 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{2} \left (x^{3}+x +1\right )^{\frac {1}{3}} x^{2}+6 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{4} x +6 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{4}+\RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right ) x^{3}-\left (x^{3}+x +1\right )^{\frac {2}{3}} x -\RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right ) x -\RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )}{9 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{3} x^{3}-2 x^{3}-x -1}\right )+3 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{2} \ln \left (-\frac {18 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{5} x^{3}+9 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{4} \left (x^{3}+x +1\right )^{\frac {1}{3}} x^{2}+3 \left (x^{3}+x +1\right )^{\frac {2}{3}} \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{3} x +9 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{5} x +9 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{5}-4 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{2} x^{3}-2 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right ) \left (x^{3}+x +1\right )^{\frac {1}{3}} x^{2}-\left (x^{3}+x +1\right )^{\frac {2}{3}} x -2 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{2} x -2 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{2}}{9 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{3} x^{3}-x^{3}+x +1}\right )-3 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{2} \ln \left (-\frac {-81 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{7} x^{3}+27 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{4} x^{3}+3 \left (x^{3}+x +1\right )^{\frac {2}{3}} \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{2} x +9 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{4} x +9 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{4}-2 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right ) x^{3}+\left (x^{3}+x +1\right )^{\frac {1}{3}} x^{2}-\RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right ) x -\RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )}{9 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{3} x^{3}-x^{3}+x +1}\right )+\RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right ) \ln \left (\frac {3 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{2} \left (x^{3}+x +1\right )^{\frac {1}{3}} x^{2}+2 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right ) x^{3}+\left (x^{3}+x +1\right )^{\frac {2}{3}} x +\RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right ) x +\RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )}{9 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{3} x^{3}-2 x^{3}-x -1}\right )+\RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right ) \ln \left (\frac {54 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{7} x^{3}+9 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{5} \left (x^{3}+x +1\right )^{\frac {1}{3}} x^{2}-15 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{4} x^{3}+3 \left (x^{3}+x +1\right )^{\frac {2}{3}} \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{3} x -3 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{2} \left (x^{3}+x +1\right )^{\frac {1}{3}} x^{2}+6 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{4} x +6 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{4}+\RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right ) x^{3}-\left (x^{3}+x +1\right )^{\frac {2}{3}} x -\RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right ) x -\RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )}{9 \RootOf \left (27 \textit {\_Z}^{6}-9 \textit {\_Z}^{3}+1\right )^{3} x^{3}-2 x^{3}-x -1}\right )\)	\(1598\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9*RootOf(27*_Z^6-9*_Z^3+1)^5*ln(-(18*RootOf(27*_Z^6-9*_Z^3+1)^5*x^3+9*RootOf(27*_Z^6-9*_Z^3+1)^4*(x^3+x+1)^(1
/3)*x^2+3*(x^3+x+1)^(2/3)*RootOf(27*_Z^6-9*_Z^3+1)^3*x+9*RootOf(27*_Z^6-9*_Z^3+1)^5*x+9*RootOf(27*_Z^6-9*_Z^3+
1)^5-4*RootOf(27*_Z^6-9*_Z^3+1)^2*x^3-2*RootOf(27*_Z^6-9*_Z^3+1)*(x^3+x+1)^(1/3)*x^2-(x^3+x+1)^(2/3)*x-2*RootO
f(27*_Z^6-9*_Z^3+1)^2*x-2*RootOf(27*_Z^6-9*_Z^3+1)^2)/(9*RootOf(27*_Z^6-9*_Z^3+1)^3*x^3-x^3+x+1))+18*RootOf(27
*_Z^6-9*_Z^3+1)^5*ln(-(-81*RootOf(27*_Z^6-9*_Z^3+1)^7*x^3+27*RootOf(27*_Z^6-9*_Z^3+1)^4*x^3+3*(x^3+x+1)^(2/3)*
RootOf(27*_Z^6-9*_Z^3+1)^2*x+9*RootOf(27*_Z^6-9*_Z^3+1)^4*x+9*RootOf(27*_Z^6-9*_Z^3+1)^4-2*RootOf(27*_Z^6-9*_Z
^3+1)*x^3+(x^3+x+1)^(1/3)*x^2-RootOf(27*_Z^6-9*_Z^3+1)*x-RootOf(27*_Z^6-9*_Z^3+1))/(9*RootOf(27*_Z^6-9*_Z^3+1)
^3*x^3-x^3+x+1))-9*RootOf(27*_Z^6-9*_Z^3+1)^4*ln((54*RootOf(27*_Z^6-9*_Z^3+1)^7*x^3+9*RootOf(27*_Z^6-9*_Z^3+1)
^5*(x^3+x+1)^(1/3)*x^2-15*RootOf(27*_Z^6-9*_Z^3+1)^4*x^3+3*(x^3+x+1)^(2/3)*RootOf(27*_Z^6-9*_Z^3+1)^3*x-3*Root
Of(27*_Z^6-9*_Z^3+1)^2*(x^3+x+1)^(1/3)*x^2+6*RootOf(27*_Z^6-9*_Z^3+1)^4*x+6*RootOf(27*_Z^6-9*_Z^3+1)^4+RootOf(
27*_Z^6-9*_Z^3+1)*x^3-(x^3+x+1)^(2/3)*x-RootOf(27*_Z^6-9*_Z^3+1)*x-RootOf(27*_Z^6-9*_Z^3+1))/(9*RootOf(27*_Z^6
-9*_Z^3+1)^3*x^3-2*x^3-x-1))+3*RootOf(27*_Z^6-9*_Z^3+1)^2*ln(-(18*RootOf(27*_Z^6-9*_Z^3+1)^5*x^3+9*RootOf(27*_
Z^6-9*_Z^3+1)^4*(x^3+x+1)^(1/3)*x^2+3*(x^3+x+1)^(2/3)*RootOf(27*_Z^6-9*_Z^3+1)^3*x+9*RootOf(27*_Z^6-9*_Z^3+1)^
5*x+9*RootOf(27*_Z^6-9*_Z^3+1)^5-4*RootOf(27*_Z^6-9*_Z^3+1)^2*x^3-2*RootOf(27*_Z^6-9*_Z^3+1)*(x^3+x+1)^(1/3)*x
^2-(x^3+x+1)^(2/3)*x-2*RootOf(27*_Z^6-9*_Z^3+1)^2*x-2*RootOf(27*_Z^6-9*_Z^3+1)^2)/(9*RootOf(27*_Z^6-9*_Z^3+1)^
3*x^3-x^3+x+1))-3*RootOf(27*_Z^6-9*_Z^3+1)^2*ln(-(-81*RootOf(27*_Z^6-9*_Z^3+1)^7*x^3+27*RootOf(27*_Z^6-9*_Z^3+
1)^4*x^3+3*(x^3+x+1)^(2/3)*RootOf(27*_Z^6-9*_Z^3+1)^2*x+9*RootOf(27*_Z^6-9*_Z^3+1)^4*x+9*RootOf(27*_Z^6-9*_Z^3
+1)^4-2*RootOf(27*_Z^6-9*_Z^3+1)*x^3+(x^3+x+1)^(1/3)*x^2-RootOf(27*_Z^6-9*_Z^3+1)*x-RootOf(27*_Z^6-9*_Z^3+1))/
(9*RootOf(27*_Z^6-9*_Z^3+1)^3*x^3-x^3+x+1))+RootOf(27*_Z^6-9*_Z^3+1)*ln((3*RootOf(27*_Z^6-9*_Z^3+1)^2*(x^3+x+1
)^(1/3)*x^2+2*RootOf(27*_Z^6-9*_Z^3+1)*x^3+(x^3+x+1)^(2/3)*x+RootOf(27*_Z^6-9*_Z^3+1)*x+RootOf(27*_Z^6-9*_Z^3+
1))/(9*RootOf(27*_Z^6-9*_Z^3+1)^3*x^3-2*x^3-x-1))+RootOf(27*_Z^6-9*_Z^3+1)*ln((54*RootOf(27*_Z^6-9*_Z^3+1)^7*x
^3+9*RootOf(27*_Z^6-9*_Z^3+1)^5*(x^3+x+1)^(1/3)*x^2-15*RootOf(27*_Z^6-9*_Z^3+1)^4*x^3+3*(x^3+x+1)^(2/3)*RootOf
(27*_Z^6-9*_Z^3+1)^3*x-3*RootOf(27*_Z^6-9*_Z^3+1)^2*(x^3+x+1)^(1/3)*x^2+6*RootOf(27*_Z^6-9*_Z^3+1)^4*x+6*RootO
f(27*_Z^6-9*_Z^3+1)^4+RootOf(27*_Z^6-9*_Z^3+1)*x^3-(x^3+x+1)^(2/3)*x-RootOf(27*_Z^6-9*_Z^3+1)*x-RootOf(27*_Z^6
-9*_Z^3+1))/(9*RootOf(27*_Z^6-9*_Z^3+1)^3*x^3-2*x^3-x-1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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