3.26.4 \(\int \frac {3+x}{\sqrt [3]{-1+x^2} (5-x+2 x^2)} \, dx\)

Optimal. Leaf size=209 \[ \frac {\log \left (3 \sqrt [3]{x^2-1}+\sqrt [3]{2} 3^{2/3} x-\sqrt [3]{2} 3^{2/3}\right )}{\sqrt [3]{2} 3^{2/3}}-\frac {\log \left (2^{2/3} \sqrt [3]{3} x^2+3 \left (x^2-1\right )^{2/3}+\left (\sqrt [3]{2} 3^{2/3}-\sqrt [3]{2} 3^{2/3} x\right ) \sqrt [3]{x^2-1}-2\ 2^{2/3} \sqrt [3]{3} x+2^{2/3} \sqrt [3]{3}\right )}{2 \sqrt [3]{2} 3^{2/3}}+\frac {\tan ^{-1}\left (\frac {3^{5/6} \sqrt [3]{x^2-1}}{\sqrt [3]{3} \sqrt [3]{x^2-1}-2 \sqrt [3]{2} x+2 \sqrt [3]{2}}\right )}{\sqrt [3]{2} \sqrt [6]{3}} \]

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Rubi [F]  time = 0.01, 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+x}{\sqrt [3]{-1+x^2} \left (5-x+2 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.22, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3+x}{\sqrt [3]{-1+x^2} \left (5-x+2 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.28, size = 209, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {3^{5/6} \sqrt [3]{-1+x^2}}{2 \sqrt [3]{2}-2 \sqrt [3]{2} x+\sqrt [3]{3} \sqrt [3]{-1+x^2}}\right )}{\sqrt [3]{2} \sqrt [6]{3}}+\frac {\log \left (-\sqrt [3]{2} 3^{2/3}+\sqrt [3]{2} 3^{2/3} x+3 \sqrt [3]{-1+x^2}\right )}{\sqrt [3]{2} 3^{2/3}}-\frac {\log \left (2^{2/3} \sqrt [3]{3}-2\ 2^{2/3} \sqrt [3]{3} x+2^{2/3} \sqrt [3]{3} x^2+\left (\sqrt [3]{2} 3^{2/3}-\sqrt [3]{2} 3^{2/3} x\right ) \sqrt [3]{-1+x^2}+3 \left (-1+x^2\right )^{2/3}\right )}{2 \sqrt [3]{2} 3^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 8.70, size = 313, normalized size = 1.50 \begin {gather*} -\frac {1}{18} \cdot 18^{\frac {1}{6}} \sqrt {6} \arctan \left (\frac {18^{\frac {1}{6}} {\left (6 \cdot 18^{\frac {2}{3}} \sqrt {6} {\left (8 \, x^{4} - 26 \, x^{3} + 33 \, x^{2} - 56 \, x + 5\right )} {\left (x^{2} - 1\right )}^{\frac {2}{3}} + 18^{\frac {1}{3}} \sqrt {6} {\left (8 \, x^{6} + 96 \, x^{5} - 582 \, x^{4} + 155 \, x^{3} + 1029 \, x^{2} - 399 \, x - 91\right )} + 36 \, \sqrt {6} {\left (4 \, x^{5} - 62 \, x^{4} + 133 \, x^{3} - 31 \, x^{2} - 73 \, x + 29\right )} {\left (x^{2} - 1\right )}^{\frac {1}{3}}\right )}}{18 \, {\left (8 \, x^{6} - 336 \, x^{5} + 1038 \, x^{4} - 709 \, x^{3} - 483 \, x^{2} + 897 \, x - 199\right )}}\right ) - \frac {1}{108} \cdot 18^{\frac {2}{3}} \log \left (\frac {3 \cdot 18^{\frac {2}{3}} {\left (4 \, x^{2} - 11 \, x + 1\right )} {\left (x^{2} - 1\right )}^{\frac {2}{3}} + 18^{\frac {1}{3}} {\left (4 \, x^{4} - 58 \, x^{3} + 75 \, x^{2} + 44 \, x - 29\right )} - 36 \, {\left (x^{3} - 6 \, x^{2} + 3 \, x + 2\right )} {\left (x^{2} - 1\right )}^{\frac {1}{3}}}{4 \, x^{4} - 4 \, x^{3} + 21 \, x^{2} - 10 \, x + 25}\right ) + \frac {1}{54} \cdot 18^{\frac {2}{3}} \log \left (\frac {18^{\frac {2}{3}} {\left (2 \, x^{2} - x + 5\right )} + 18 \cdot 18^{\frac {1}{3}} {\left (x^{2} - 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} + 54 \, {\left (x^{2} - 1\right )}^{\frac {2}{3}}}{2 \, x^{2} - x + 5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/18*18^(1/6)*sqrt(6)*arctan(1/18*18^(1/6)*(6*18^(2/3)*sqrt(6)*(8*x^4 - 26*x^3 + 33*x^2 - 56*x + 5)*(x^2 - 1)
^(2/3) + 18^(1/3)*sqrt(6)*(8*x^6 + 96*x^5 - 582*x^4 + 155*x^3 + 1029*x^2 - 399*x - 91) + 36*sqrt(6)*(4*x^5 - 6
2*x^4 + 133*x^3 - 31*x^2 - 73*x + 29)*(x^2 - 1)^(1/3))/(8*x^6 - 336*x^5 + 1038*x^4 - 709*x^3 - 483*x^2 + 897*x
- 199)) - 1/108*18^(2/3)*log((3*18^(2/3)*(4*x^2 - 11*x + 1)*(x^2 - 1)^(2/3) + 18^(1/3)*(4*x^4 - 58*x^3 + 75*x
^2 + 44*x - 29) - 36*(x^3 - 6*x^2 + 3*x + 2)*(x^2 - 1)^(1/3))/(4*x^4 - 4*x^3 + 21*x^2 - 10*x + 25)) + 1/54*18^
(2/3)*log((18^(2/3)*(2*x^2 - x + 5) + 18*18^(1/3)*(x^2 - 1)^(1/3)*(x - 1) + 54*(x^2 - 1)^(2/3))/(2*x^2 - x + 5
))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 12.24, size = 910, normalized size = 4.35

method	result	size
trager	\(\frac {\RootOf \left (\textit {\_Z}^{3}-12\right ) \ln \left (-\frac {3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-12\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-12\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-12\right )^{3} x^{2}+1818 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-12\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-12\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-12\right )^{2} x^{2}-9 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-12\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-12\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-12\right )^{3} x -5454 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-12\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-12\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-12\right )^{2} x -1215 \left (x^{2}-1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-12\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-12\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-12\right )^{2}-405 \left (x^{2}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-12\right )^{2} x -3006 \left (x^{2}-1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-12\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-12\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-12\right ) x +405 \left (x^{2}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-12\right )^{2}+3006 \left (x^{2}-1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-12\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-12\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-12\right )+2 \RootOf \left (\textit {\_Z}^{3}-12\right ) x^{2}+1212 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-12\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-12\right )+36 \textit {\_Z}^{2}\right ) x^{2}-13 \RootOf \left (\textit {\_Z}^{3}-12\right ) x -7878 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-12\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-12\right )+36 \textit {\_Z}^{2}\right ) x +576 \left (x^{2}-1\right )^{\frac {2}{3}}-7 \RootOf \left (\textit {\_Z}^{3}-12\right )-4242 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-12\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-12\right )+36 \textit {\_Z}^{2}\right )}{2 x^{2}-x +5}\right )}{6}+\RootOf \left (\RootOf \left (\textit {\_Z}^{3}-12\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-12\right )+36 \textit {\_Z}^{2}\right ) \ln \left (\frac {303 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-12\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-12\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-12\right )^{3} x^{2}+18 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-12\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-12\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-12\right )^{2} x^{2}-909 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-12\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-12\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-12\right )^{3} x -54 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-12\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-12\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-12\right )^{2} x -1215 \left (x^{2}-1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-12\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-12\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-12\right )^{2}-405 \left (x^{2}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-12\right )^{2} x +576 \left (x^{2}-1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-12\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-12\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-12\right ) x +405 \left (x^{2}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-12\right )^{2}-576 \left (x^{2}-1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-12\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-12\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-12\right )+404 \RootOf \left (\textit {\_Z}^{3}-12\right ) x^{2}+24 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-12\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-12\right )+36 \textit {\_Z}^{2}\right ) x^{2}-505 \RootOf \left (\textit {\_Z}^{3}-12\right ) x -30 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-12\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-12\right )+36 \textit {\_Z}^{2}\right ) x -3006 \left (x^{2}-1\right )^{\frac {2}{3}}+707 \RootOf \left (\textit {\_Z}^{3}-12\right )+42 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-12\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-12\right )+36 \textit {\_Z}^{2}\right )}{2 x^{2}-x +5}\right )\)	\(910\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*RootOf(_Z^3-12)*ln(-(3*RootOf(RootOf(_Z^3-12)^2+6*_Z*RootOf(_Z^3-12)+36*_Z^2)*RootOf(_Z^3-12)^3*x^2+1818*R
ootOf(RootOf(_Z^3-12)^2+6*_Z*RootOf(_Z^3-12)+36*_Z^2)^2*RootOf(_Z^3-12)^2*x^2-9*RootOf(RootOf(_Z^3-12)^2+6*_Z*
RootOf(_Z^3-12)+36*_Z^2)*RootOf(_Z^3-12)^3*x-5454*RootOf(RootOf(_Z^3-12)^2+6*_Z*RootOf(_Z^3-12)+36*_Z^2)^2*Roo
tOf(_Z^3-12)^2*x-1215*(x^2-1)^(2/3)*RootOf(RootOf(_Z^3-12)^2+6*_Z*RootOf(_Z^3-12)+36*_Z^2)*RootOf(_Z^3-12)^2-4
05*(x^2-1)^(1/3)*RootOf(_Z^3-12)^2*x-3006*(x^2-1)^(1/3)*RootOf(RootOf(_Z^3-12)^2+6*_Z*RootOf(_Z^3-12)+36*_Z^2)
*RootOf(_Z^3-12)*x+405*(x^2-1)^(1/3)*RootOf(_Z^3-12)^2+3006*(x^2-1)^(1/3)*RootOf(RootOf(_Z^3-12)^2+6*_Z*RootOf
(_Z^3-12)+36*_Z^2)*RootOf(_Z^3-12)+2*RootOf(_Z^3-12)*x^2+1212*RootOf(RootOf(_Z^3-12)^2+6*_Z*RootOf(_Z^3-12)+36
*_Z^2)*x^2-13*RootOf(_Z^3-12)*x-7878*RootOf(RootOf(_Z^3-12)^2+6*_Z*RootOf(_Z^3-12)+36*_Z^2)*x+576*(x^2-1)^(2/3
)-7*RootOf(_Z^3-12)-4242*RootOf(RootOf(_Z^3-12)^2+6*_Z*RootOf(_Z^3-12)+36*_Z^2))/(2*x^2-x+5))+RootOf(RootOf(_Z
^3-12)^2+6*_Z*RootOf(_Z^3-12)+36*_Z^2)*ln((303*RootOf(RootOf(_Z^3-12)^2+6*_Z*RootOf(_Z^3-12)+36*_Z^2)*RootOf(_
Z^3-12)^3*x^2+18*RootOf(RootOf(_Z^3-12)^2+6*_Z*RootOf(_Z^3-12)+36*_Z^2)^2*RootOf(_Z^3-12)^2*x^2-909*RootOf(Roo
tOf(_Z^3-12)^2+6*_Z*RootOf(_Z^3-12)+36*_Z^2)*RootOf(_Z^3-12)^3*x-54*RootOf(RootOf(_Z^3-12)^2+6*_Z*RootOf(_Z^3-
12)+36*_Z^2)^2*RootOf(_Z^3-12)^2*x-1215*(x^2-1)^(2/3)*RootOf(RootOf(_Z^3-12)^2+6*_Z*RootOf(_Z^3-12)+36*_Z^2)*R
ootOf(_Z^3-12)^2-405*(x^2-1)^(1/3)*RootOf(_Z^3-12)^2*x+576*(x^2-1)^(1/3)*RootOf(RootOf(_Z^3-12)^2+6*_Z*RootOf(
_Z^3-12)+36*_Z^2)*RootOf(_Z^3-12)*x+405*(x^2-1)^(1/3)*RootOf(_Z^3-12)^2-576*(x^2-1)^(1/3)*RootOf(RootOf(_Z^3-1
2)^2+6*_Z*RootOf(_Z^3-12)+36*_Z^2)*RootOf(_Z^3-12)+404*RootOf(_Z^3-12)*x^2+24*RootOf(RootOf(_Z^3-12)^2+6*_Z*Ro
otOf(_Z^3-12)+36*_Z^2)*x^2-505*RootOf(_Z^3-12)*x-30*RootOf(RootOf(_Z^3-12)^2+6*_Z*RootOf(_Z^3-12)+36*_Z^2)*x-3
006*(x^2-1)^(2/3)+707*RootOf(_Z^3-12)+42*RootOf(RootOf(_Z^3-12)^2+6*_Z*RootOf(_Z^3-12)+36*_Z^2))/(2*x^2-x+5))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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