3.23.59 \(\int \frac {-5+x}{\sqrt [3]{-2-x+x^2} (-3+4 x+x^2)} \, dx\)

Optimal. Leaf size=171 \[ \frac {\log \left (2 \sqrt [3]{x^2-x-2}+2^{2/3} x+2^{2/3}\right )}{2^{2/3}}-\frac {\log \left (-\sqrt [3]{2} x^2-2 \left (x^2-x-2\right )^{2/3}+\left (2^{2/3} x+2^{2/3}\right ) \sqrt [3]{x^2-x-2}-2 \sqrt [3]{2} x-\sqrt [3]{2}\right )}{2\ 2^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x^2-x-2}}{\sqrt [3]{x^2-x-2}-2^{2/3} x-2^{2/3}}\right )}{2^{2/3}} \]

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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 {-5+x}{\sqrt [3]{-2-x+x^2} \left (-3+4 x+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.17, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-5+x}{\sqrt [3]{-2-x+x^2} \left (-3+4 x+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.22, size = 171, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{-2-x+x^2}}{-2^{2/3}-2^{2/3} x+\sqrt [3]{-2-x+x^2}}\right )}{2^{2/3}}+\frac {\log \left (2^{2/3}+2^{2/3} x+2 \sqrt [3]{-2-x+x^2}\right )}{2^{2/3}}-\frac {\log \left (-\sqrt [3]{2}-2 \sqrt [3]{2} x-\sqrt [3]{2} x^2+\left (2^{2/3}+2^{2/3} x\right ) \sqrt [3]{-2-x+x^2}-2 \left (-2-x+x^2\right )^{2/3}\right )}{2\ 2^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 7.40, size = 300, normalized size = 1.75 \begin {gather*} -\frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {3} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (12 \cdot 4^{\frac {2}{3}} {\left (x^{4} + 5 \, x^{3} + 4 \, x^{2} + 9 \, x - 9\right )} {\left (x^{2} - x - 2\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (x^{6} + 30 \, x^{5} + 3 \, x^{4} + 100 \, x^{3} - 45 \, x^{2} - 306 \, x - 351\right )} + 12 \, {\left (x^{5} - 9 \, x^{4} + 40 \, x^{2} + 75 \, x + 45\right )} {\left (x^{2} - x - 2\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (x^{6} - 42 \, x^{5} - 69 \, x^{4} + 100 \, x^{3} + 315 \, x^{2} + 486 \, x + 81\right )}}\right ) - \frac {1}{24} \cdot 4^{\frac {2}{3}} \log \left (\frac {6 \cdot 4^{\frac {2}{3}} {\left (x^{2} + x + 3\right )} {\left (x^{2} - x - 2\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (x^{4} - 10 \, x^{3} + 10 \, x^{2} + 30 \, x + 45\right )} - 6 \, {\left (x^{3} - x^{2} + 7 \, x + 9\right )} {\left (x^{2} - x - 2\right )}^{\frac {1}{3}}}{x^{4} + 8 \, x^{3} + 10 \, x^{2} - 24 \, x + 9}\right ) + \frac {1}{12} \cdot 4^{\frac {2}{3}} \log \left (\frac {4^{\frac {2}{3}} {\left (x^{2} + 4 \, x - 3\right )} + 6 \cdot 4^{\frac {1}{3}} {\left (x^{2} - x - 2\right )}^{\frac {1}{3}} {\left (x + 1\right )} + 12 \, {\left (x^{2} - x - 2\right )}^{\frac {2}{3}}}{x^{2} + 4 \, x - 3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*4^(1/6)*sqrt(3)*arctan(1/6*4^(1/6)*sqrt(3)*(12*4^(2/3)*(x^4 + 5*x^3 + 4*x^2 + 9*x - 9)*(x^2 - x - 2)^(2/3
) + 4^(1/3)*(x^6 + 30*x^5 + 3*x^4 + 100*x^3 - 45*x^2 - 306*x - 351) + 12*(x^5 - 9*x^4 + 40*x^2 + 75*x + 45)*(x
^2 - x - 2)^(1/3))/(x^6 - 42*x^5 - 69*x^4 + 100*x^3 + 315*x^2 + 486*x + 81)) - 1/24*4^(2/3)*log((6*4^(2/3)*(x^
2 + x + 3)*(x^2 - x - 2)^(2/3) + 4^(1/3)*(x^4 - 10*x^3 + 10*x^2 + 30*x + 45) - 6*(x^3 - x^2 + 7*x + 9)*(x^2 -
x - 2)^(1/3))/(x^4 + 8*x^3 + 10*x^2 - 24*x + 9)) + 1/12*4^(2/3)*log((4^(2/3)*(x^2 + 4*x - 3) + 6*4^(1/3)*(x^2
- x - 2)^(1/3)*(x + 1) + 12*(x^2 - x - 2)^(2/3))/(x^2 + 4*x - 3))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 9.61, size = 1404, normalized size = 8.21

method	result	size
trager	\(\text {Expression too large to display}\)	\(1404\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*RootOf(_Z^3-2)*ln(-(452*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x^2+192*RootO
f(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x^2+1500*(x^2-x-2)^(2/3)*RootOf(RootOf(_Z^3-
2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^2+1130*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*Roo
tOf(_Z^3-2)^3*x+480*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x+123*(x^2-x-2)^(1/
3)*RootOf(_Z^3-2)^2*x+1746*(x^2-x-2)^(1/3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)*
x+123*(x^2-x-2)^(1/3)*RootOf(_Z^3-2)^2+1746*(x^2-x-2)^(1/3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2
)*RootOf(_Z^3-2)-113*RootOf(_Z^3-2)*x^2-48*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^2+1746*(x^2-x
-2)^(2/3)+904*RootOf(_Z^3-2)*x+384*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x-2373*RootOf(_Z^3-2)-1
008*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2))/(x^2+4*x-3))-1/2*ln((-452*RootOf(RootOf(_Z^3-2)^2+2*_
Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x^2-712*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf
(_Z^3-2)^2*x^2+1500*(x^2-x-2)^(2/3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^2-1130*
RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x-1780*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootO
f(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x-873*(x^2-x-2)^(1/3)*RootOf(_Z^3-2)^2*x-246*(x^2-x-2)^(1/3)*RootOf(RootO
f(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)*x-873*(x^2-x-2)^(1/3)*RootOf(_Z^3-2)^2-246*(x^2-x-2)^(1
/3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)-565*RootOf(_Z^3-2)*x^2-890*RootOf(RootO
f(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^2-246*(x^2-x-2)^(2/3)-226*RootOf(_Z^3-2)*x-356*RootOf(RootOf(_Z^3-2)
^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x-2373*RootOf(_Z^3-2)-3738*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2))
/(x^2+4*x-3))*RootOf(_Z^3-2)-ln((-452*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x^2
-712*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x^2+1500*(x^2-x-2)^(2/3)*RootOf(Ro
otOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^2-1130*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4
*_Z^2)*RootOf(_Z^3-2)^3*x-1780*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x-873*(x
^2-x-2)^(1/3)*RootOf(_Z^3-2)^2*x-246*(x^2-x-2)^(1/3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootO
f(_Z^3-2)*x-873*(x^2-x-2)^(1/3)*RootOf(_Z^3-2)^2-246*(x^2-x-2)^(1/3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-
2)+4*_Z^2)*RootOf(_Z^3-2)-565*RootOf(_Z^3-2)*x^2-890*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^2-2
46*(x^2-x-2)^(2/3)-226*RootOf(_Z^3-2)*x-356*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x-2373*RootOf(
_Z^3-2)-3738*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2))/(x^2+4*x-3))*RootOf(RootOf(_Z^3-2)^2+2*_Z*Ro
otOf(_Z^3-2)+4*_Z^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x-5}{{\left (x^2-x-2\right )}^{1/3}\,\left (x^2+4\,x-3\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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