3.20.96 \(\int \frac {(3+2 x) (1+x+3 x^3)^{2/3}}{x^3 (1+x+x^3)} \, dx\) [1996]

Optimal. Leaf size=141 \[ -\frac {3 \left (1+x+3 x^3\right )^{2/3}}{2 x^2}+2^{2/3} \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{1+x+3 x^3}}\right )-2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{1+x+3 x^3}\right )+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{1+x+3 x^3}+\sqrt [3]{2} \left (1+x+3 x^3\right )^{2/3}\right )}{\sqrt [3]{2}} \]

[Out]

-3/2*(3*x^3+x+1)^(2/3)/x^2+2^(2/3)*3^(1/2)*arctan(3^(1/2)*x/(x+2^(2/3)*(3*x^3+x+1)^(1/3)))-2^(2/3)*ln(-2*x+2^(
2/3)*(3*x^3+x+1)^(1/3))+1/2*ln(2*x^2+2^(2/3)*x*(3*x^3+x+1)^(1/3)+2^(1/3)*(3*x^3+x+1)^(2/3))*2^(2/3)

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Rubi [F]
time = 1.11, 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+3 x^3\right )^{2/3}}{x^3 \left (1+x+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(3*(1 + x + 3*x^3)^(2/3)*Defer[Int][(((2/(-9 + Sqrt[85]))^(1/3) - ((-9 + Sqrt[85])/2)^(1/3) + 3*x)^(2/3)*(1 +
(2/(-9 + Sqrt[85]))^(2/3) + ((-9 + Sqrt[85])/2)^(2/3) - 3*((2/(-9 + Sqrt[85]))^(1/3) - ((-9 + Sqrt[85])/2)^(1/
3))*x + 9*x^2)^(2/3))/x^3, x])/(((2/(-9 + Sqrt[85]))^(1/3) - ((-9 + Sqrt[85])/2)^(1/3) + 3*x)^(2/3)*(1 + (2/(-
9 + Sqrt[85]))^(2/3) + ((-9 + Sqrt[85])/2)^(2/3) - 3*((2/(-9 + Sqrt[85]))^(1/3) - ((-9 + Sqrt[85])/2)^(1/3))*x
+ 9*x^2)^(2/3)) - ((1 + x + 3*x^3)^(2/3)*Defer[Int][(((2/(-9 + Sqrt[85]))^(1/3) - ((-9 + Sqrt[85])/2)^(1/3) +
3*x)^(2/3)*(1 + (2/(-9 + Sqrt[85]))^(2/3) + ((-9 + Sqrt[85])/2)^(2/3) - 3*((2/(-9 + Sqrt[85]))^(1/3) - ((-9 +
Sqrt[85])/2)^(1/3))*x + 9*x^2)^(2/3))/x^2, x])/(((2/(-9 + Sqrt[85]))^(1/3) - ((-9 + Sqrt[85])/2)^(1/3) + 3*x)
^(2/3)*(1 + (2/(-9 + Sqrt[85]))^(2/3) + ((-9 + Sqrt[85])/2)^(2/3) - 3*((2/(-9 + Sqrt[85]))^(1/3) - ((-9 + Sqrt
[85])/2)^(1/3))*x + 9*x^2)^(2/3)) + ((1 + x + 3*x^3)^(2/3)*Defer[Int][(((2/(-9 + Sqrt[85]))^(1/3) - ((-9 + Sqr
t[85])/2)^(1/3) + 3*x)^(2/3)*(1 + (2/(-9 + Sqrt[85]))^(2/3) + ((-9 + Sqrt[85])/2)^(2/3) - 3*((2/(-9 + Sqrt[85]
))^(1/3) - ((-9 + Sqrt[85])/2)^(1/3))*x + 9*x^2)^(2/3))/x, x])/(((2/(-9 + Sqrt[85]))^(1/3) - ((-9 + Sqrt[85])/
2)^(1/3) + 3*x)^(2/3)*(1 + (2/(-9 + Sqrt[85]))^(2/3) + ((-9 + Sqrt[85])/2)^(2/3) - 3*((2/(-9 + Sqrt[85]))^(1/3
) - ((-9 + Sqrt[85])/2)^(1/3))*x + 9*x^2)^(2/3)) - 4*Defer[Int][(1 + x + 3*x^3)^(2/3)/(1 + x + x^3), x] + Defe
r[Int][(x*(1 + x + 3*x^3)^(2/3))/(1 + x + x^3), x] - Defer[Int][(x^2*(1 + x + 3*x^3)^(2/3))/(1 + x + x^3), x]

Rubi steps

\begin {align*} \int \frac {(3+2 x) \left (1+x+3 x^3\right )^{2/3}}{x^3 \left (1+x+x^3\right )} \, dx &=\int \left (\frac {3 \left (1+x+3 x^3\right )^{2/3}}{x^3}-\frac {\left (1+x+3 x^3\right )^{2/3}}{x^2}+\frac {\left (1+x+3 x^3\right )^{2/3}}{x}+\frac {\left (-4+x-x^2\right ) \left (1+x+3 x^3\right )^{2/3}}{1+x+x^3}\right ) \, dx\\ &=3 \int \frac {\left (1+x+3 x^3\right )^{2/3}}{x^3} \, dx-\int \frac {\left (1+x+3 x^3\right )^{2/3}}{x^2} \, dx+\int \frac {\left (1+x+3 x^3\right )^{2/3}}{x} \, dx+\int \frac {\left (-4+x-x^2\right ) \left (1+x+3 x^3\right )^{2/3}}{1+x+x^3} \, dx\\ &=-\frac {\left (1+x+3 x^3\right )^{2/3} \int \frac {\left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+3 x\right )^{2/3} \left (1+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+9 x^2\right )^{2/3}}{x^2} \, dx}{\left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+3 x\right )^{2/3} \left (1+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+9 x^2\right )^{2/3}}+\frac {\left (1+x+3 x^3\right )^{2/3} \int \frac {\left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+3 x\right )^{2/3} \left (1+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+9 x^2\right )^{2/3}}{x} \, dx}{\left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+3 x\right )^{2/3} \left (1+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+9 x^2\right )^{2/3}}+\frac {\left (3 \left (1+x+3 x^3\right )^{2/3}\right ) \int \frac {\left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+3 x\right )^{2/3} \left (1+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+9 x^2\right )^{2/3}}{x^3} \, dx}{\left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+3 x\right )^{2/3} \left (1+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+9 x^2\right )^{2/3}}+\int \left (-\frac {4 \left (1+x+3 x^3\right )^{2/3}}{1+x+x^3}+\frac {x \left (1+x+3 x^3\right )^{2/3}}{1+x+x^3}-\frac {x^2 \left (1+x+3 x^3\right )^{2/3}}{1+x+x^3}\right ) \, dx\\ &=-\left (4 \int \frac {\left (1+x+3 x^3\right )^{2/3}}{1+x+x^3} \, dx\right )-\frac {\left (1+x+3 x^3\right )^{2/3} \int \frac {\left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+3 x\right )^{2/3} \left (1+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+9 x^2\right )^{2/3}}{x^2} \, dx}{\left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+3 x\right )^{2/3} \left (1+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+9 x^2\right )^{2/3}}+\frac {\left (1+x+3 x^3\right )^{2/3} \int \frac {\left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+3 x\right )^{2/3} \left (1+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+9 x^2\right )^{2/3}}{x} \, dx}{\left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+3 x\right )^{2/3} \left (1+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+9 x^2\right )^{2/3}}+\frac {\left (3 \left (1+x+3 x^3\right )^{2/3}\right ) \int \frac {\left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+3 x\right )^{2/3} \left (1+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+9 x^2\right )^{2/3}}{x^3} \, dx}{\left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+3 x\right )^{2/3} \left (1+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+9 x^2\right )^{2/3}}+\int \frac {x \left (1+x+3 x^3\right )^{2/3}}{1+x+x^3} \, dx-\int \frac {x^2 \left (1+x+3 x^3\right )^{2/3}}{1+x+x^3} \, dx\\ \end {align*}

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Mathematica [A]
time = 0.40, size = 141, normalized size = 1.00 \begin {gather*} -\frac {3 \left (1+x+3 x^3\right )^{2/3}}{2 x^2}+2^{2/3} \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{1+x+3 x^3}}\right )-2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{1+x+3 x^3}\right )+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{1+x+3 x^3}+\sqrt [3]{2} \left (1+x+3 x^3\right )^{2/3}\right )}{\sqrt [3]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 11.03, size = 981, normalized size = 6.96

method	result	size
trager	\(\text {Expression too large to display}\)	\(981\)
risch	\(\text {Expression too large to display}\)	\(1026\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2*(3*x^3+x+1)^(2/3)/x^2+6*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*ln(-(984*RootOf(RootOf(_Z^3+
4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootOf(_Z^3+4)^3*x^3+11916*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z
^2)^2*RootOf(_Z^3+4)^2*x^3-3033*(3*x^3+x+1)^(2/3)*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootOf(
_Z^3+4)^2*x-492*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootOf(_Z^3+4)^3*x-5958*RootOf(RootOf(_Z^
3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)^2*RootOf(_Z^3+4)^2*x-1011*RootOf(_Z^3+4)^2*(3*x^3+x+1)^(1/3)*x^2+14814*Roo
tOf(_Z^3+4)*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*(3*x^3+x+1)^(1/3)*x^2-492*RootOf(RootOf(_Z^3+
4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootOf(_Z^3+4)^3-5958*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)^2
*RootOf(_Z^3+4)^2+4100*RootOf(_Z^3+4)*x^3+49650*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*x^3+6960*
(3*x^3+x+1)^(2/3)*x+1148*RootOf(_Z^3+4)*x+13902*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*x+1148*Ro
otOf(_Z^3+4)+13902*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2))/(x^3+x+1))+RootOf(_Z^3+4)*ln((1986*Ro
otOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootOf(_Z^3+4)^3*x^3+5904*RootOf(RootOf(_Z^3+4)^2+6*_Z*Root
Of(_Z^3+4)+36*_Z^2)^2*RootOf(_Z^3+4)^2*x^3-3033*(3*x^3+x+1)^(2/3)*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+
36*_Z^2)*RootOf(_Z^3+4)^2*x-993*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootOf(_Z^3+4)^3*x-2952*R
ootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)^2*RootOf(_Z^3+4)^2*x-1011*RootOf(_Z^3+4)^2*(3*x^3+x+1)^(1/
3)*x^2-20880*RootOf(_Z^3+4)*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*(3*x^3+x+1)^(1/3)*x^2-993*Roo
tOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2)*RootOf(_Z^3+4)^3-2952*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z
^3+4)+36*_Z^2)^2*RootOf(_Z^3+4)^2-9599*RootOf(_Z^3+4)*x^3-28536*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36
*_Z^2)*x^3-4938*(3*x^3+x+1)^(2/3)*x-1655*RootOf(_Z^3+4)*x-4920*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*
_Z^2)*x-1655*RootOf(_Z^3+4)-4920*RootOf(RootOf(_Z^3+4)^2+6*_Z*RootOf(_Z^3+4)+36*_Z^2))/(x^3+x+1))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 380 vs. \(2 (112) = 224\).
time = 6.15, size = 380, normalized size = 2.70 \begin {gather*} \frac {2 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} x^{2} \arctan \left (\frac {3 \, \sqrt {3} \left (-4\right )^{\frac {2}{3}} {\left (7 \, x^{7} + 8 \, x^{5} + 8 \, x^{4} + x^{3} + 2 \, x^{2} + x\right )} {\left (3 \, x^{3} + x + 1\right )}^{\frac {2}{3}} - 6 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} {\left (55 \, x^{8} + 20 \, x^{6} + 20 \, x^{5} + x^{4} + 2 \, x^{3} + x^{2}\right )} {\left (3 \, x^{3} + x + 1\right )}^{\frac {1}{3}} + \sqrt {3} {\left (433 \, x^{9} + 255 \, x^{7} + 255 \, x^{6} + 39 \, x^{5} + 78 \, x^{4} + 40 \, x^{3} + 3 \, x^{2} + 3 \, x + 1\right )}}{3 \, {\left (323 \, x^{9} + 105 \, x^{7} + 105 \, x^{6} - 3 \, x^{5} - 6 \, x^{4} - 4 \, x^{3} - 3 \, x^{2} - 3 \, x - 1\right )}}\right ) + 2 \, \left (-4\right )^{\frac {1}{3}} x^{2} \log \left (\frac {3 \, \left (-4\right )^{\frac {2}{3}} {\left (3 \, x^{3} + x + 1\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (3 \, x^{3} + x + 1\right )}^{\frac {2}{3}} x - \left (-4\right )^{\frac {1}{3}} {\left (x^{3} + x + 1\right )}}{x^{3} + x + 1}\right ) - \left (-4\right )^{\frac {1}{3}} x^{2} \log \left (-\frac {6 \, \left (-4\right )^{\frac {1}{3}} {\left (7 \, x^{4} + x^{2} + x\right )} {\left (3 \, x^{3} + x + 1\right )}^{\frac {2}{3}} - \left (-4\right )^{\frac {2}{3}} {\left (55 \, x^{6} + 20 \, x^{4} + 20 \, x^{3} + x^{2} + 2 \, x + 1\right )} - 24 \, {\left (4 \, x^{5} + x^{3} + x^{2}\right )} {\left (3 \, x^{3} + x + 1\right )}^{\frac {1}{3}}}{x^{6} + 2 \, x^{4} + 2 \, x^{3} + x^{2} + 2 \, x + 1}\right ) - 9 \, {\left (3 \, x^{3} + x + 1\right )}^{\frac {2}{3}}}{6 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*sqrt(3)*(-4)^(1/3)*x^2*arctan(1/3*(3*sqrt(3)*(-4)^(2/3)*(7*x^7 + 8*x^5 + 8*x^4 + x^3 + 2*x^2 + x)*(3*x^
3 + x + 1)^(2/3) - 6*sqrt(3)*(-4)^(1/3)*(55*x^8 + 20*x^6 + 20*x^5 + x^4 + 2*x^3 + x^2)*(3*x^3 + x + 1)^(1/3) +
sqrt(3)*(433*x^9 + 255*x^7 + 255*x^6 + 39*x^5 + 78*x^4 + 40*x^3 + 3*x^2 + 3*x + 1))/(323*x^9 + 105*x^7 + 105*
x^6 - 3*x^5 - 6*x^4 - 4*x^3 - 3*x^2 - 3*x - 1)) + 2*(-4)^(1/3)*x^2*log((3*(-4)^(2/3)*(3*x^3 + x + 1)^(1/3)*x^2
- 6*(3*x^3 + x + 1)^(2/3)*x - (-4)^(1/3)*(x^3 + x + 1))/(x^3 + x + 1)) - (-4)^(1/3)*x^2*log(-(6*(-4)^(1/3)*(7
*x^4 + x^2 + x)*(3*x^3 + x + 1)^(2/3) - (-4)^(2/3)*(55*x^6 + 20*x^4 + 20*x^3 + x^2 + 2*x + 1) - 24*(4*x^5 + x^
3 + x^2)*(3*x^3 + x + 1)^(1/3))/(x^6 + 2*x^4 + 2*x^3 + x^2 + 2*x + 1)) - 9*(3*x^3 + x + 1)^(2/3))/x^2

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Sympy [F(-1)] Timed out
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((3+2*x)*(3*x**3+x+1)**(2/3)/x**3/(x**3+x+1),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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