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

Optimal. Leaf size=80 \[ \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^3-1}}{\sqrt {x^3-1}-x^2}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^3-1}}{\sqrt {x^3-1}+x^2}\right ) \]

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Rubi [F]  time = 1.14, 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)*Hypergeom
etric2F1[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] + De
fer[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]
+ Defer[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 = 2.73, size = 80, normalized size = 1.00 \begin {gather*} \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{-1+x^3}}{-x^2+\sqrt {-1+x^3}}\right )-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{-1+x^3}}{x^2+\sqrt {-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]

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

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fricas [B]  time = 0.46, size = 189, normalized size = 2.36 \begin {gather*} 2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt {\frac {x^{2} + \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {1}{4}} x + \sqrt {x^{3} - 1}}{x^{2}}} - x - \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {1}{4}}}{x}\right ) + 2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt {\frac {x^{2} - \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {1}{4}} x + \sqrt {x^{3} - 1}}{x^{2}}} + x - \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \sqrt {2} \log \left (\frac {4 \, {\left (x^{2} + \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {1}{4}} x + \sqrt {x^{3} - 1}\right )}}{x^{2}}\right ) + \frac {1}{2} \, \sqrt {2} \log \left (\frac {4 \, {\left (x^{2} - \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {1}{4}} x + \sqrt {x^{3} - 1}\right )}}{x^{2}}\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*sqrt(2)*arctan((sqrt(2)*x*sqrt((x^2 + sqrt(2)*(x^3 - 1)^(1/4)*x + sqrt(x^3 - 1))/x^2) - x - sqrt(2)*(x^3 - 1
)^(1/4))/x) + 2*sqrt(2)*arctan((sqrt(2)*x*sqrt((x^2 - sqrt(2)*(x^3 - 1)^(1/4)*x + sqrt(x^3 - 1))/x^2) + x - sq
rt(2)*(x^3 - 1)^(1/4))/x) - 1/2*sqrt(2)*log(4*(x^2 + sqrt(2)*(x^3 - 1)^(1/4)*x + sqrt(x^3 - 1))/x^2) + 1/2*sqr
t(2)*log(4*(x^2 - sqrt(2)*(x^3 - 1)^(1/4)*x + sqrt(x^3 - 1))/x^2)

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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 = 1.40, size = 206, normalized size = 2.58

method	result	size
trager	\(-\RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{4}+2 \left (x^{3}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{3}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{3}+2 \sqrt {x^{3}-1}\, \RootOf \left (\textit {\_Z}^{4}+1\right ) x^{2}+2 \left (x^{3}-1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{4}+1\right )^{3}}{x^{4}+x^{3}-1}\right )-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {2 \sqrt {x^{3}-1}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{2}-2 \left (x^{3}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{4}+2 \left (x^{3}-1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{3}+\RootOf \left (\textit {\_Z}^{4}+1\right )}{x^{4}+x^{3}-1}\right )\)	\(206\)

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]

-RootOf(_Z^4+1)*ln(-(RootOf(_Z^4+1)^3*x^4+2*(x^3-1)^(1/4)*RootOf(_Z^4+1)^2*x^3-RootOf(_Z^4+1)^3*x^3+2*(x^3-1)^
(1/2)*RootOf(_Z^4+1)*x^2+2*(x^3-1)^(3/4)*x+RootOf(_Z^4+1)^3)/(x^4+x^3-1))-RootOf(_Z^4+1)^3*ln(-(2*(x^3-1)^(1/2
)*RootOf(_Z^4+1)^3*x^2-2*(x^3-1)^(1/4)*RootOf(_Z^4+1)^2*x^3+RootOf(_Z^4+1)*x^4+2*(x^3-1)^(3/4)*x-RootOf(_Z^4+1
)*x^3+RootOf(_Z^4+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 = -0.01 \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^3 + x^4 - 1)),x)

[Out]

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