3.4.51 \(\int \frac {x^2 (-4+x^3)}{(-1+x^3)^{3/4} (1-x^3+x^4)} \, dx\)

Optimal. Leaf size=29 \[ 2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^3-1}}\right )-2 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^3-1}}\right ) \]

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Rubi [F]  time = 1.04, 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 {x^2 \left (-4+x^3\right )}{\left (-1+x^3\right )^{3/4} \left (1-x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(x*(1 - x^3)^(3/4)*Hypergeometric2F1[1/3, 3/4, 4/3, x^3])/(-1 + x^3)^(3/4) + (x^2*(1 - x^3)^(3/4)*Hypergeometr
ic2F1[2/3, 3/4, 5/3, x^3])/(2*(-1 + x^3)^(3/4)) - Defer[Int][1/((-1 + x^3)^(3/4)*(1 - x^3 + x^4)), x] - Defer[
Int][x/((-1 + x^3)^(3/4)*(1 - x^3 + x^4)), x] - 4*Defer[Int][x^2/((-1 + x^3)^(3/4)*(1 - x^3 + x^4)), x] + Defe
r[Int][x^3/((-1 + x^3)^(3/4)*(1 - x^3 + x^4)), x]

Rubi steps

\begin {align*} \int \frac {x^2 \left (-4+x^3\right )}{\left (-1+x^3\right )^{3/4} \left (1-x^3+x^4\right )} \, dx &=\int \left (\frac {1}{\left (-1+x^3\right )^{3/4}}+\frac {x}{\left (-1+x^3\right )^{3/4}}-\frac {1+x+4 x^2-x^3}{\left (-1+x^3\right )^{3/4} \left (1-x^3+x^4\right )}\right ) \, dx\\ &=\int \frac {1}{\left (-1+x^3\right )^{3/4}} \, dx+\int \frac {x}{\left (-1+x^3\right )^{3/4}} \, dx-\int \frac {1+x+4 x^2-x^3}{\left (-1+x^3\right )^{3/4} \left (1-x^3+x^4\right )} \, dx\\ &=\frac {\left (1-x^3\right )^{3/4} \int \frac {1}{\left (1-x^3\right )^{3/4}} \, dx}{\left (-1+x^3\right )^{3/4}}+\frac {\left (1-x^3\right )^{3/4} \int \frac {x}{\left (1-x^3\right )^{3/4}} \, dx}{\left (-1+x^3\right )^{3/4}}-\int \left (\frac {1}{\left (-1+x^3\right )^{3/4} \left (1-x^3+x^4\right )}+\frac {x}{\left (-1+x^3\right )^{3/4} \left (1-x^3+x^4\right )}+\frac {4 x^2}{\left (-1+x^3\right )^{3/4} \left (1-x^3+x^4\right )}-\frac {x^3}{\left (-1+x^3\right )^{3/4} \left (1-x^3+x^4\right )}\right ) \, dx\\ &=\frac {x \left (1-x^3\right )^{3/4} \, _2F_1\left (\frac {1}{3},\frac {3}{4};\frac {4}{3};x^3\right )}{\left (-1+x^3\right )^{3/4}}+\frac {x^2 \left (1-x^3\right )^{3/4} \, _2F_1\left (\frac {2}{3},\frac {3}{4};\frac {5}{3};x^3\right )}{2 \left (-1+x^3\right )^{3/4}}-4 \int \frac {x^2}{\left (-1+x^3\right )^{3/4} \left (1-x^3+x^4\right )} \, dx-\int \frac {1}{\left (-1+x^3\right )^{3/4} \left (1-x^3+x^4\right )} \, dx-\int \frac {x}{\left (-1+x^3\right )^{3/4} \left (1-x^3+x^4\right )} \, dx+\int \frac {x^3}{\left (-1+x^3\right )^{3/4} \left (1-x^3+x^4\right )} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.20, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2 \left (-4+x^3\right )}{\left (-1+x^3\right )^{3/4} \left (1-x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 2.47, size = 29, normalized size = 1.00 \begin {gather*} 2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^3}}\right )-2 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.46, size = 48, normalized size = 1.66 \begin {gather*} -2 \, \arctan \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \log \left (\frac {x + {\left (x^{3} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \log \left (-\frac {x - {\left (x^{3} - 1\right )}^{\frac {1}{4}}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.87, size = 153, normalized size = 5.28

method	result	size
trager	\(\ln \left (\frac {2 \left (x^{3}-1\right )^{\frac {3}{4}} x -2 x^{2} \sqrt {x^{3}-1}+2 \left (x^{3}-1\right )^{\frac {1}{4}} x^{3}-x^{4}-x^{3}+1}{x^{4}-x^{3}+1}\right )+\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{3}-1}\, x^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}-2 \left (x^{3}-1\right )^{\frac {3}{4}} x +2 \left (x^{3}-1\right )^{\frac {1}{4}} x^{3}+\RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{4}-x^{3}+1}\right )\)	\(153\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln((2*(x^3-1)^(3/4)*x-2*x^2*(x^3-1)^(1/2)+2*(x^3-1)^(1/4)*x^3-x^4-x^3+1)/(x^4-x^3+1))+RootOf(_Z^2+1)*ln(-(2*Ro
otOf(_Z^2+1)*(x^3-1)^(1/2)*x^2-RootOf(_Z^2+1)*x^4-RootOf(_Z^2+1)*x^3-2*(x^3-1)^(3/4)*x+2*(x^3-1)^(1/4)*x^3+Roo
tOf(_Z^2+1))/(x^4-x^3+1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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