3.6.1 \(\int \frac {-3+2 x}{\sqrt [4]{-x+x^2} (1-x+x^3)} \, dx\)

Optimal. Leaf size=39 \[ 2 \tan ^{-1}\left (\frac {\sqrt [4]{x^2-x}}{x}\right )-2 \tanh ^{-1}\left (\frac {\left (x^2-x\right )^{3/4}}{x-1}\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 1.15, 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}{\sqrt [4]{-x+x^2} \left (1-x+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-12*(-1 + x)^(1/4)*x^(1/4)*Defer[Subst][Defer[Int][x^2/((-1 + x^4)^(1/4)*(1 - x^4 + x^12)), x], x, x^(1/4)])/
(-x + x^2)^(1/4) + (8*(-1 + x)^(1/4)*x^(1/4)*Defer[Subst][Defer[Int][x^6/((-1 + x^4)^(1/4)*(1 - x^4 + x^12)),
x], x, x^(1/4)])/(-x + x^2)^(1/4)

Rubi steps

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

________________________________________________________________________________________

Mathematica [F]  time = 0.54, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-3+2 x}{\sqrt [4]{-x+x^2} \left (1-x+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.10, size = 39, normalized size = 1.00 \begin {gather*} 2 \tan ^{-1}\left (\frac {\sqrt [4]{-x+x^2}}{x}\right )-2 \tanh ^{-1}\left (\frac {\left (-x+x^2\right )^{3/4}}{-1+x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*ArcTan[(-x + x^2)^(1/4)/x] - 2*ArcTanh[(-x + x^2)^(3/4)/(-1 + x)]

________________________________________________________________________________________

fricas [B]  time = 7.78, size = 93, normalized size = 2.38 \begin {gather*} \arctan \left (\frac {2 \, {\left ({\left (x^{2} - x\right )}^{\frac {1}{4}} x^{2} + {\left (x^{2} - x\right )}^{\frac {3}{4}}\right )}}{x^{3} - x + 1}\right ) + \log \left (-\frac {x^{3} - 2 \, {\left (x^{2} - x\right )}^{\frac {1}{4}} x^{2} + 2 \, \sqrt {x^{2} - x} x + x - 2 \, {\left (x^{2} - x\right )}^{\frac {3}{4}} - 1}{x^{3} - x + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(2*((x^2 - x)^(1/4)*x^2 + (x^2 - x)^(3/4))/(x^3 - x + 1)) + log(-(x^3 - 2*(x^2 - x)^(1/4)*x^2 + 2*sqrt(x
^2 - x)*x + x - 2*(x^2 - x)^(3/4) - 1)/(x^3 - x + 1))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [C]  time = 1.80, size = 150, normalized size = 3.85

method	result	size
trager	\(-\ln \left (-\frac {2 \left (x^{2}-x \right )^{\frac {3}{4}}+2 x \sqrt {x^{2}-x}+2 \left (x^{2}-x \right )^{\frac {1}{4}} x^{2}+x^{3}+x -1}{x^{3}-x +1}\right )+\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{2}-x}\, x -\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{2}-x \right )^{\frac {3}{4}}-2 \left (x^{2}-x \right )^{\frac {1}{4}} x^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x +\RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{3}-x +1}\right )\)	\(150\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-ln(-(2*(x^2-x)^(3/4)+2*x*(x^2-x)^(1/2)+2*(x^2-x)^(1/4)*x^2+x^3+x-1)/(x^3-x+1))+RootOf(_Z^2+1)*ln(-(2*RootOf(_
Z^2+1)*(x^2-x)^(1/2)*x-RootOf(_Z^2+1)*x^3+2*(x^2-x)^(3/4)-2*(x^2-x)^(1/4)*x^2-RootOf(_Z^2+1)*x+RootOf(_Z^2+1))
/(x^3-x+1))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {2\,x-3}{{\left (x^2-x\right )}^{1/4}\,\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^2 - x)^(1/4)*(x^3 - x + 1)),x)

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  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)/(x**2-x)**(1/4)/(x**3-x+1),x)

[Out]

Timed out

________________________________________________________________________________________