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

Optimal. Leaf size=161 \[ \frac {3 \left (2 x^3+2 x^2+1\right )^{2/3}}{2 x^2}+3^{2/3} \log \left (3^{2/3} \sqrt [3]{2 x^3+2 x^2+1}-3 x\right )-\frac {1}{2} 3^{2/3} \log \left (3 x^2+3^{2/3} \sqrt [3]{2 x^3+2 x^2+1} x+\sqrt [3]{3} \left (2 x^3+2 x^2+1\right )^{2/3}\right )-3 \sqrt [6]{3} \tan ^{-1}\left (\frac {3^{5/6} x}{2 \sqrt [3]{2 x^3+2 x^2+1}+\sqrt [3]{3} x}\right ) \]

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Rubi [F]  time = 2.59, 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+2 x^2\right ) \left (1+2 x^2+2 x^3\right )^{2/3}}{x^3 \left (-1-2 x^2+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

3*Defer[Int][(1 + 2*x^2 + 2*x^3)^(2/3)/(-1 - 2*x^2 + x^3), x] + 8*Defer[Int][(x*(1 + 2*x^2 + 2*x^3)^(2/3))/(-1
- 2*x^2 + x^3), x] - 4*Defer[Int][(x^2*(1 + 2*x^2 + 2*x^3)^(2/3))/(-1 - 2*x^2 + x^3), x] - (27*2^(2/3)*(1 + 2
*x^2 + 2*x^3)^(2/3)*Defer[Subst][Defer[Int][((((31 + 3*Sqrt[105])^(1/3)*(2*2^(1/3) + (31 - 3*Sqrt[105])^(2/3))
)/6 + 2*x)^(2/3)*((-4 + (62 - 6*Sqrt[105])^(2/3) + (8*2^(1/3))/(31 - 3*Sqrt[105])^(2/3))/9 - (2*((62 - 6*Sqrt[
105])^(1/3) + (2*2^(2/3))/(31 - 3*Sqrt[105])^(1/3))*x)/3 + 4*x^2)^(2/3))/(-1/3 + x)^3, x], x, 1/3 + x])/((4 +
(31 + 3*Sqrt[105])^(1/3)*(2*2^(1/3) + (31 - 3*Sqrt[105])^(2/3)) + 12*x)^(2/3)*(-4 + (62 - 6*Sqrt[105])^(2/3) +
(8*2^(1/3))/(31 - 3*Sqrt[105])^(2/3) - 2*((62 - 6*Sqrt[105])^(1/3) + (2*2^(2/3))/(31 - 3*Sqrt[105])^(1/3))*(1
+ 3*x) + 4*(1 + 3*x)^2)^(2/3)) + (36*2^(2/3)*(1 + 2*x^2 + 2*x^3)^(2/3)*Defer[Subst][Defer[Int][((((31 + 3*Sqr
t[105])^(1/3)*(2*2^(1/3) + (31 - 3*Sqrt[105])^(2/3)))/6 + 2*x)^(2/3)*((-4 + (62 - 6*Sqrt[105])^(2/3) + (8*2^(1
/3))/(31 - 3*Sqrt[105])^(2/3))/9 - (2*((62 - 6*Sqrt[105])^(1/3) + (2*2^(2/3))/(31 - 3*Sqrt[105])^(1/3))*x)/3 +
4*x^2)^(2/3))/(-1/3 + x), x], x, 1/3 + x])/((4 + (31 + 3*Sqrt[105])^(1/3)*(2*2^(1/3) + (31 - 3*Sqrt[105])^(2/
3)) + 12*x)^(2/3)*(-4 + (62 - 6*Sqrt[105])^(2/3) + (8*2^(1/3))/(31 - 3*Sqrt[105])^(2/3) - 2*((62 - 6*Sqrt[105]
)^(1/3) + (2*2^(2/3))/(31 - 3*Sqrt[105])^(1/3))*(1 + 3*x) + 4*(1 + 3*x)^2)^(2/3))

Rubi steps

\begin {align*} \int \frac {\left (3+2 x^2\right ) \left (1+2 x^2+2 x^3\right )^{2/3}}{x^3 \left (-1-2 x^2+x^3\right )} \, dx &=\int \left (-\frac {3 \left (1+2 x^2+2 x^3\right )^{2/3}}{x^3}+\frac {4 \left (1+2 x^2+2 x^3\right )^{2/3}}{x}+\frac {\left (3+8 x-4 x^2\right ) \left (1+2 x^2+2 x^3\right )^{2/3}}{-1-2 x^2+x^3}\right ) \, dx\\ &=-\left (3 \int \frac {\left (1+2 x^2+2 x^3\right )^{2/3}}{x^3} \, dx\right )+4 \int \frac {\left (1+2 x^2+2 x^3\right )^{2/3}}{x} \, dx+\int \frac {\left (3+8 x-4 x^2\right ) \left (1+2 x^2+2 x^3\right )^{2/3}}{-1-2 x^2+x^3} \, dx\\ &=-\left (3 \operatorname {Subst}\left (\int \frac {\left (\frac {31}{27}-\frac {2 x}{3}+2 x^3\right )^{2/3}}{\left (-\frac {1}{3}+x\right )^3} \, dx,x,\frac {1}{3}+x\right )\right )+4 \operatorname {Subst}\left (\int \frac {\left (\frac {31}{27}-\frac {2 x}{3}+2 x^3\right )^{2/3}}{-\frac {1}{3}+x} \, dx,x,\frac {1}{3}+x\right )+\int \left (\frac {3 \left (1+2 x^2+2 x^3\right )^{2/3}}{-1-2 x^2+x^3}+\frac {8 x \left (1+2 x^2+2 x^3\right )^{2/3}}{-1-2 x^2+x^3}-\frac {4 x^2 \left (1+2 x^2+2 x^3\right )^{2/3}}{-1-2 x^2+x^3}\right ) \, dx\\ &=3 \int \frac {\left (1+2 x^2+2 x^3\right )^{2/3}}{-1-2 x^2+x^3} \, dx-4 \int \frac {x^2 \left (1+2 x^2+2 x^3\right )^{2/3}}{-1-2 x^2+x^3} \, dx+8 \int \frac {x \left (1+2 x^2+2 x^3\right )^{2/3}}{-1-2 x^2+x^3} \, dx-\frac {\left (9\ 2^{2/3} \sqrt [3]{3} \left (1+2 x^2+2 x^3\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {1}{6} \sqrt [3]{31+3 \sqrt {105}} \left (2 \sqrt [3]{2}+\left (31-3 \sqrt {105}\right )^{2/3}\right )+2 x\right )^{2/3} \left (\frac {1}{9} \left (-4+\left (62-6 \sqrt {105}\right )^{2/3}+\frac {8 \sqrt [3]{2}}{\left (31-3 \sqrt {105}\right )^{2/3}}\right )-\frac {2}{3} \left (\sqrt [3]{62-6 \sqrt {105}}+\frac {2\ 2^{2/3}}{\sqrt [3]{31-3 \sqrt {105}}}\right ) x+4 x^2\right )^{2/3}}{\left (-\frac {1}{3}+x\right )^3} \, dx,x,\frac {1}{3}+x\right )}{\left (4+\sqrt [3]{31+3 \sqrt {105}} \left (2 \sqrt [3]{2}+\left (31-3 \sqrt {105}\right )^{2/3}\right )+12 x\right )^{2/3} \left (\sqrt [3]{3 \left (9+\sqrt {105}\right )} \left (2-\sqrt [3]{62-6 \sqrt {105}}\right )+2 \left (4-\sqrt [3]{62-6 \sqrt {105}}-\frac {2\ 2^{2/3}}{\sqrt [3]{31-3 \sqrt {105}}}\right ) x+12 x^2\right )^{2/3}}+\frac {\left (12\ 2^{2/3} \sqrt [3]{3} \left (1+2 x^2+2 x^3\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {1}{6} \sqrt [3]{31+3 \sqrt {105}} \left (2 \sqrt [3]{2}+\left (31-3 \sqrt {105}\right )^{2/3}\right )+2 x\right )^{2/3} \left (\frac {1}{9} \left (-4+\left (62-6 \sqrt {105}\right )^{2/3}+\frac {8 \sqrt [3]{2}}{\left (31-3 \sqrt {105}\right )^{2/3}}\right )-\frac {2}{3} \left (\sqrt [3]{62-6 \sqrt {105}}+\frac {2\ 2^{2/3}}{\sqrt [3]{31-3 \sqrt {105}}}\right ) x+4 x^2\right )^{2/3}}{-\frac {1}{3}+x} \, dx,x,\frac {1}{3}+x\right )}{\left (4+\sqrt [3]{31+3 \sqrt {105}} \left (2 \sqrt [3]{2}+\left (31-3 \sqrt {105}\right )^{2/3}\right )+12 x\right )^{2/3} \left (\sqrt [3]{3 \left (9+\sqrt {105}\right )} \left (2-\sqrt [3]{62-6 \sqrt {105}}\right )+2 \left (4-\sqrt [3]{62-6 \sqrt {105}}-\frac {2\ 2^{2/3}}{\sqrt [3]{31-3 \sqrt {105}}}\right ) x+12 x^2\right )^{2/3}}\\ \end {align*}

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Mathematica [F]  time = 0.38, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (3+2 x^2\right ) \left (1+2 x^2+2 x^3\right )^{2/3}}{x^3 \left (-1-2 x^2+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.45, size = 161, normalized size = 1.00 \begin {gather*} \frac {3 \left (1+2 x^2+2 x^3\right )^{2/3}}{2 x^2}-3 \sqrt [6]{3} \tan ^{-1}\left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{1+2 x^2+2 x^3}}\right )+3^{2/3} \log \left (-3 x+3^{2/3} \sqrt [3]{1+2 x^2+2 x^3}\right )-\frac {1}{2} 3^{2/3} \log \left (3 x^2+3^{2/3} x \sqrt [3]{1+2 x^2+2 x^3}+\sqrt [3]{3} \left (1+2 x^2+2 x^3\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(1 + 2*x^2 + 2*x^3)^(2/3))/(2*x^2) - 3*3^(1/6)*ArcTan[(3^(5/6)*x)/(3^(1/3)*x + 2*(1 + 2*x^2 + 2*x^3)^(1/3))
] + 3^(2/3)*Log[-3*x + 3^(2/3)*(1 + 2*x^2 + 2*x^3)^(1/3)] - (3^(2/3)*Log[3*x^2 + 3^(2/3)*x*(1 + 2*x^2 + 2*x^3)
^(1/3) + 3^(1/3)*(1 + 2*x^2 + 2*x^3)^(2/3)])/2

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fricas [B]  time = 19.12, size = 438, normalized size = 2.72 \begin {gather*} -\frac {2 \cdot 9^{\frac {1}{3}} \sqrt {3} x^{2} \arctan \left (\frac {2 \cdot 9^{\frac {2}{3}} \sqrt {3} {\left (8 \, x^{7} - 14 \, x^{6} - 4 \, x^{5} - 7 \, x^{4} - 4 \, x^{3} - x\right )} {\left (2 \, x^{3} + 2 \, x^{2} + 1\right )}^{\frac {2}{3}} - 6 \cdot 9^{\frac {1}{3}} \sqrt {3} {\left (55 \, x^{8} + 50 \, x^{7} + 4 \, x^{6} + 25 \, x^{5} + 4 \, x^{4} + x^{2}\right )} {\left (2 \, x^{3} + 2 \, x^{2} + 1\right )}^{\frac {1}{3}} - \sqrt {3} {\left (377 \, x^{9} + 600 \, x^{8} + 204 \, x^{7} + 308 \, x^{6} + 204 \, x^{5} + 12 \, x^{4} + 51 \, x^{3} + 6 \, x^{2} + 1\right )}}{3 \, {\left (487 \, x^{9} + 480 \, x^{8} + 12 \, x^{7} + 232 \, x^{6} + 12 \, x^{5} - 12 \, x^{4} + 3 \, x^{3} - 6 \, x^{2} - 1\right )}}\right ) - 2 \cdot 9^{\frac {1}{3}} x^{2} \log \left (\frac {3 \cdot 9^{\frac {2}{3}} {\left (2 \, x^{3} + 2 \, x^{2} + 1\right )}^{\frac {1}{3}} x^{2} - 9 \, {\left (2 \, x^{3} + 2 \, x^{2} + 1\right )}^{\frac {2}{3}} x - 9^{\frac {1}{3}} {\left (x^{3} - 2 \, x^{2} - 1\right )}}{x^{3} - 2 \, x^{2} - 1}\right ) + 9^{\frac {1}{3}} x^{2} \log \left (\frac {9 \cdot 9^{\frac {1}{3}} {\left (8 \, x^{4} + 2 \, x^{3} + x\right )} {\left (2 \, x^{3} + 2 \, x^{2} + 1\right )}^{\frac {2}{3}} + 9^{\frac {2}{3}} {\left (55 \, x^{6} + 50 \, x^{5} + 4 \, x^{4} + 25 \, x^{3} + 4 \, x^{2} + 1\right )} + 27 \, {\left (7 \, x^{5} + 4 \, x^{4} + 2 \, x^{2}\right )} {\left (2 \, x^{3} + 2 \, x^{2} + 1\right )}^{\frac {1}{3}}}{x^{6} - 4 \, x^{5} + 4 \, x^{4} - 2 \, x^{3} + 4 \, x^{2} + 1}\right ) - 9 \, {\left (2 \, x^{3} + 2 \, x^{2} + 1\right )}^{\frac {2}{3}}}{6 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(2*9^(1/3)*sqrt(3)*x^2*arctan(1/3*(2*9^(2/3)*sqrt(3)*(8*x^7 - 14*x^6 - 4*x^5 - 7*x^4 - 4*x^3 - x)*(2*x^3
+ 2*x^2 + 1)^(2/3) - 6*9^(1/3)*sqrt(3)*(55*x^8 + 50*x^7 + 4*x^6 + 25*x^5 + 4*x^4 + x^2)*(2*x^3 + 2*x^2 + 1)^(1
/3) - sqrt(3)*(377*x^9 + 600*x^8 + 204*x^7 + 308*x^6 + 204*x^5 + 12*x^4 + 51*x^3 + 6*x^2 + 1))/(487*x^9 + 480*
x^8 + 12*x^7 + 232*x^6 + 12*x^5 - 12*x^4 + 3*x^3 - 6*x^2 - 1)) - 2*9^(1/3)*x^2*log((3*9^(2/3)*(2*x^3 + 2*x^2 +
1)^(1/3)*x^2 - 9*(2*x^3 + 2*x^2 + 1)^(2/3)*x - 9^(1/3)*(x^3 - 2*x^2 - 1))/(x^3 - 2*x^2 - 1)) + 9^(1/3)*x^2*lo
g((9*9^(1/3)*(8*x^4 + 2*x^3 + x)*(2*x^3 + 2*x^2 + 1)^(2/3) + 9^(2/3)*(55*x^6 + 50*x^5 + 4*x^4 + 25*x^3 + 4*x^2
+ 1) + 27*(7*x^5 + 4*x^4 + 2*x^2)*(2*x^3 + 2*x^2 + 1)^(1/3))/(x^6 - 4*x^5 + 4*x^4 - 2*x^3 + 4*x^2 + 1)) - 9*(
2*x^3 + 2*x^2 + 1)^(2/3))/x^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 19.48, size = 749, normalized size = 4.65

method	result	size
risch	\(\frac {3 \left (2 x^{3}+2 x^{2}+1\right )^{\frac {2}{3}}}{2 x^{2}}+3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+3 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+9 \textit {\_Z}^{2}\right ) \ln \left (\frac {-5 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+3 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+9 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-9\right )^{3} x^{3}-9 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+3 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+9 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-9\right )^{2} x^{3}+7 \left (2 x^{3}+2 x^{2}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3}-9\right )^{2} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+3 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+9 \textit {\_Z}^{2}\right ) x -7 \left (2 x^{3}+2 x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-9\right )^{2} x^{2}-24 \left (2 x^{3}+2 x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+3 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+9 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-9\right ) x^{2}+10 \RootOf \left (\textit {\_Z}^{3}-9\right ) x^{3}+18 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+3 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+9 \textit {\_Z}^{2}\right ) x^{3}-3 \left (2 x^{3}+2 x^{2}+1\right )^{\frac {2}{3}} x +10 \RootOf \left (\textit {\_Z}^{3}-9\right ) x^{2}+18 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+3 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+9 \textit {\_Z}^{2}\right ) x^{2}+5 \RootOf \left (\textit {\_Z}^{3}-9\right )+9 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+3 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+9 \textit {\_Z}^{2}\right )}{x^{3}-2 x^{2}-1}\right )+\RootOf \left (\textit {\_Z}^{3}-9\right ) \ln \left (-\frac {-3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+3 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+9 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-9\right )^{3} x^{3}-15 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+3 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+9 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-9\right )^{2} x^{3}+7 \left (2 x^{3}+2 x^{2}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3}-9\right )^{2} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+3 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+9 \textit {\_Z}^{2}\right ) x -7 \left (2 x^{3}+2 x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-9\right )^{2} x^{2}+3 \left (2 x^{3}+2 x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+3 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+9 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-9\right ) x^{2}-15 \RootOf \left (\textit {\_Z}^{3}-9\right ) x^{3}-75 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+3 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+9 \textit {\_Z}^{2}\right ) x^{3}+24 \left (2 x^{3}+2 x^{2}+1\right )^{\frac {2}{3}} x -6 \RootOf \left (\textit {\_Z}^{3}-9\right ) x^{2}-30 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+3 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+9 \textit {\_Z}^{2}\right ) x^{2}-3 \RootOf \left (\textit {\_Z}^{3}-9\right )-15 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+3 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+9 \textit {\_Z}^{2}\right )}{x^{3}-2 x^{2}-1}\right )\)	\(749\)
trager	\(\text {Expression too large to display}\)	\(1561\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*(2*x^3+2*x^2+1)^(2/3)/x^2+3*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*ln((-5*RootOf(RootOf(_Z^3-
9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^3*x^3-9*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)^2*
RootOf(_Z^3-9)^2*x^3+7*(2*x^3+2*x^2+1)^(2/3)*RootOf(_Z^3-9)^2*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z
^2)*x-7*(2*x^3+2*x^2+1)^(1/3)*RootOf(_Z^3-9)^2*x^2-24*(2*x^3+2*x^2+1)^(1/3)*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootO
f(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)*x^2+10*RootOf(_Z^3-9)*x^3+18*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z
^2)*x^3-3*(2*x^3+2*x^2+1)^(2/3)*x+10*RootOf(_Z^3-9)*x^2+18*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)
*x^2+5*RootOf(_Z^3-9)+9*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2))/(x^3-2*x^2-1))+RootOf(_Z^3-9)*ln(
-(-3*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)^3*x^3-15*RootOf(RootOf(_Z^3-9)^2+3*_Z*
RootOf(_Z^3-9)+9*_Z^2)^2*RootOf(_Z^3-9)^2*x^3+7*(2*x^3+2*x^2+1)^(2/3)*RootOf(_Z^3-9)^2*RootOf(RootOf(_Z^3-9)^2
+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*x-7*(2*x^3+2*x^2+1)^(1/3)*RootOf(_Z^3-9)^2*x^2+3*(2*x^3+2*x^2+1)^(1/3)*RootOf(Roo
tOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2)*RootOf(_Z^3-9)*x^2-15*RootOf(_Z^3-9)*x^3-75*RootOf(RootOf(_Z^3-9)^2+
3*_Z*RootOf(_Z^3-9)+9*_Z^2)*x^3+24*(2*x^3+2*x^2+1)^(2/3)*x-6*RootOf(_Z^3-9)*x^2-30*RootOf(RootOf(_Z^3-9)^2+3*_
Z*RootOf(_Z^3-9)+9*_Z^2)*x^2-3*RootOf(_Z^3-9)-15*RootOf(RootOf(_Z^3-9)^2+3*_Z*RootOf(_Z^3-9)+9*_Z^2))/(x^3-2*x
^2-1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((2*x^3 + 2*x^2 + 1)^(2/3)*(2*x^2 + 3)/((x^3 - 2*x^2 - 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^2+3\right )\,{\left (2\,x^3+2\,x^2+1\right )}^{2/3}}{x^3\,\left (-x^3+2\,x^2+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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