3.23.4 \(\int \frac {x^2 (-4+7 x^3)}{\sqrt [3]{-x+x^4} (-1-x^4+x^7)} \, dx\)

Optimal. Leaf size=163 \[ 2 \tanh ^{-1}\left (1-2 x \sqrt [3]{x^4-x}\right )-\sqrt {3} \tan ^{-1}\left (\frac {3 \sqrt {3} x \sqrt [3]{x^4-x}-3 x^2 \sqrt [3]{x^4-x}}{-3 \sqrt [3]{x^4-x} x+\sqrt {3} \sqrt [3]{x^4-x} x^2+2 \sqrt {3} x-6}\right )-\tanh ^{-1}\left (\frac {\sqrt [3]{x^4-x} x+1}{\sqrt [3]{x^4-x} x+2 \left (x^4-x\right )^{2/3} x^2+1}\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 1.33, 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+7 x^3\right )}{\sqrt [3]{-x+x^4} \left (-1-x^4+x^7\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^2*(-4 + 7*x^3))/((-x + x^4)^(1/3)*(-1 - x^4 + x^7)),x]

[Out]

(-12*x^(1/3)*(-1 + x^3)^(1/3)*Defer[Subst][Defer[Int][x^7/((-1 + x^9)^(1/3)*(-1 - x^12 + x^21)), x], x, x^(1/3
)])/(-x + x^4)^(1/3) + (21*x^(1/3)*(-1 + x^3)^(1/3)*Defer[Subst][Defer[Int][x^16/((-1 + x^9)^(1/3)*(-1 - x^12
+ x^21)), x], x, x^(1/3)])/(-x + x^4)^(1/3)

Rubi steps

\begin {align*} \int \frac {x^2 \left (-4+7 x^3\right )}{\sqrt [3]{-x+x^4} \left (-1-x^4+x^7\right )} \, dx &=\frac {\left (\sqrt [3]{x} \sqrt [3]{-1+x^3}\right ) \int \frac {x^{5/3} \left (-4+7 x^3\right )}{\sqrt [3]{-1+x^3} \left (-1-x^4+x^7\right )} \, dx}{\sqrt [3]{-x+x^4}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {x^7 \left (-4+7 x^9\right )}{\sqrt [3]{-1+x^9} \left (-1-x^{12}+x^{21}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^4}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{-1+x^3}\right ) \operatorname {Subst}\left (\int \left (-\frac {4 x^7}{\sqrt [3]{-1+x^9} \left (-1-x^{12}+x^{21}\right )}+\frac {7 x^{16}}{\sqrt [3]{-1+x^9} \left (-1-x^{12}+x^{21}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^4}}\\ &=-\frac {\left (12 \sqrt [3]{x} \sqrt [3]{-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {x^7}{\sqrt [3]{-1+x^9} \left (-1-x^{12}+x^{21}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^4}}+\frac {\left (21 \sqrt [3]{x} \sqrt [3]{-1+x^3}\right ) \operatorname {Subst}\left (\int \frac {x^{16}}{\sqrt [3]{-1+x^9} \left (-1-x^{12}+x^{21}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^4}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 0.21, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2 \left (-4+7 x^3\right )}{\sqrt [3]{-x+x^4} \left (-1-x^4+x^7\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(x^2*(-4 + 7*x^3))/((-x + x^4)^(1/3)*(-1 - x^4 + x^7)),x]

[Out]

Integrate[(x^2*(-4 + 7*x^3))/((-x + x^4)^(1/3)*(-1 - x^4 + x^7)), x]

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 10.15, size = 163, normalized size = 1.00 \begin {gather*} -\sqrt {3} \tan ^{-1}\left (\frac {3 \sqrt {3} x \sqrt [3]{-x+x^4}-3 x^2 \sqrt [3]{-x+x^4}}{-6+2 \sqrt {3} x-3 x \sqrt [3]{-x+x^4}+\sqrt {3} x^2 \sqrt [3]{-x+x^4}}\right )+2 \tanh ^{-1}\left (1-2 x \sqrt [3]{-x+x^4}\right )-\tanh ^{-1}\left (\frac {1+x \sqrt [3]{-x+x^4}}{1+x \sqrt [3]{-x+x^4}+2 x^2 \left (-x+x^4\right )^{2/3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^2*(-4 + 7*x^3))/((-x + x^4)^(1/3)*(-1 - x^4 + x^7)),x]

[Out]

-(Sqrt[3]*ArcTan[(3*Sqrt[3]*x*(-x + x^4)^(1/3) - 3*x^2*(-x + x^4)^(1/3))/(-6 + 2*Sqrt[3]*x - 3*x*(-x + x^4)^(1
/3) + Sqrt[3]*x^2*(-x + x^4)^(1/3))]) + 2*ArcTanh[1 - 2*x*(-x + x^4)^(1/3)] - ArcTanh[(1 + x*(-x + x^4)^(1/3))
/(1 + x*(-x + x^4)^(1/3) + 2*x^2*(-x + x^4)^(2/3))]

________________________________________________________________________________________

fricas [A]  time = 3.46, size = 119, normalized size = 0.73 \begin {gather*} -\sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} {\left (x^{4} - x\right )}^{\frac {2}{3}} x^{2} - 4 \, \sqrt {3} {\left (x^{4} - x\right )}^{\frac {1}{3}} x - \sqrt {3} {\left (x^{7} - x^{4}\right )}}{x^{7} - x^{4} + 8}\right ) + \frac {1}{2} \, \log \left (\frac {x^{7} - x^{4} - 3 \, {\left (x^{4} - x\right )}^{\frac {2}{3}} x^{2} + 3 \, {\left (x^{4} - x\right )}^{\frac {1}{3}} x - 1}{x^{7} - x^{4} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(7*x^3-4)/(x^4-x)^(1/3)/(x^7-x^4-1),x, algorithm="fricas")

[Out]

-sqrt(3)*arctan((2*sqrt(3)*(x^4 - x)^(2/3)*x^2 - 4*sqrt(3)*(x^4 - x)^(1/3)*x - sqrt(3)*(x^7 - x^4))/(x^7 - x^4
+ 8)) + 1/2*log((x^7 - x^4 - 3*(x^4 - x)^(2/3)*x^2 + 3*(x^4 - x)^(1/3)*x - 1)/(x^7 - x^4 - 1))

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (7 \, x^{3} - 4\right )} x^{2}}{{\left (x^{7} - x^{4} - 1\right )} {\left (x^{4} - x\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(7*x^3-4)/(x^4-x)^(1/3)/(x^7-x^4-1),x, algorithm="giac")

[Out]

integrate((7*x^3 - 4)*x^2/((x^7 - x^4 - 1)*(x^4 - x)^(1/3)), x)

________________________________________________________________________________________

maple [C]  time = 17.80, size = 516, normalized size = 3.17

method	result	size
trager	\(\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {2750978024320396805141 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{7}-397640531996478951563644 x^{7} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+10027471246530585051439424 x^{7}-2750978024320396805141 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{4}+9624328758485465306265498 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}-x \right )^{\frac {2}{3}} x^{2}+397640531996478951563644 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}+20449832047033328657637351 x^{2} \left (x^{4}-x \right )^{\frac {2}{3}}-10027471246530585051439424 x^{4}+9624328758485465306265498 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}-x \right )^{\frac {1}{3}} x -308109538723884442175792 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}+20449832047033328657637351 x \left (x^{4}-x \right )^{\frac {1}{3}}+9716610729782380212458491 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+10117002239803179560827276}{x^{7}-x^{4}-1}\right )-\ln \left (-\frac {2750978024320396805141 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{7}+403142488045119745173926 x^{7} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+10427862756551384399808209 x^{7}-2750978024320396805141 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{4}-9624328758485465306265498 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}-x \right )^{\frac {2}{3}} x^{2}-403142488045119745173926 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}+10825503288547863351371853 x^{2} \left (x^{4}-x \right )^{\frac {2}{3}}-10427862756551384399808209 x^{4}-9624328758485465306265498 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}-x \right )^{\frac {1}{3}} x -308109538723884442175792 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}+10825503288547863351371853 x \left (x^{4}-x \right )^{\frac {1}{3}}-10332829807230149096810075 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+92281971296914906192993}{x^{7}-x^{4}-1}\right ) \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-\ln \left (-\frac {2750978024320396805141 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{7}+403142488045119745173926 x^{7} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+10427862756551384399808209 x^{7}-2750978024320396805141 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{4}-9624328758485465306265498 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}-x \right )^{\frac {2}{3}} x^{2}-403142488045119745173926 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}+10825503288547863351371853 x^{2} \left (x^{4}-x \right )^{\frac {2}{3}}-10427862756551384399808209 x^{4}-9624328758485465306265498 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}-x \right )^{\frac {1}{3}} x -308109538723884442175792 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}+10825503288547863351371853 x \left (x^{4}-x \right )^{\frac {1}{3}}-10332829807230149096810075 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+92281971296914906192993}{x^{7}-x^{4}-1}\right )\)	\(516\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(7*x^3-4)/(x^4-x)^(1/3)/(x^7-x^4-1),x,method=_RETURNVERBOSE)

[Out]

RootOf(_Z^2+_Z+1)*ln(-(2750978024320396805141*RootOf(_Z^2+_Z+1)^2*x^7-397640531996478951563644*x^7*RootOf(_Z^2
+_Z+1)+10027471246530585051439424*x^7-2750978024320396805141*RootOf(_Z^2+_Z+1)^2*x^4+9624328758485465306265498
*RootOf(_Z^2+_Z+1)*(x^4-x)^(2/3)*x^2+397640531996478951563644*RootOf(_Z^2+_Z+1)*x^4+20449832047033328657637351
*x^2*(x^4-x)^(2/3)-10027471246530585051439424*x^4+9624328758485465306265498*RootOf(_Z^2+_Z+1)*(x^4-x)^(1/3)*x-
308109538723884442175792*RootOf(_Z^2+_Z+1)^2+20449832047033328657637351*x*(x^4-x)^(1/3)+9716610729782380212458
491*RootOf(_Z^2+_Z+1)+10117002239803179560827276)/(x^7-x^4-1))-ln(-(2750978024320396805141*RootOf(_Z^2+_Z+1)^2
*x^7+403142488045119745173926*x^7*RootOf(_Z^2+_Z+1)+10427862756551384399808209*x^7-2750978024320396805141*Root
Of(_Z^2+_Z+1)^2*x^4-9624328758485465306265498*RootOf(_Z^2+_Z+1)*(x^4-x)^(2/3)*x^2-403142488045119745173926*Roo
tOf(_Z^2+_Z+1)*x^4+10825503288547863351371853*x^2*(x^4-x)^(2/3)-10427862756551384399808209*x^4-962432875848546
5306265498*RootOf(_Z^2+_Z+1)*(x^4-x)^(1/3)*x-308109538723884442175792*RootOf(_Z^2+_Z+1)^2+10825503288547863351
371853*x*(x^4-x)^(1/3)-10332829807230149096810075*RootOf(_Z^2+_Z+1)+92281971296914906192993)/(x^7-x^4-1))*Root
Of(_Z^2+_Z+1)-ln(-(2750978024320396805141*RootOf(_Z^2+_Z+1)^2*x^7+403142488045119745173926*x^7*RootOf(_Z^2+_Z+
1)+10427862756551384399808209*x^7-2750978024320396805141*RootOf(_Z^2+_Z+1)^2*x^4-9624328758485465306265498*Roo
tOf(_Z^2+_Z+1)*(x^4-x)^(2/3)*x^2-403142488045119745173926*RootOf(_Z^2+_Z+1)*x^4+10825503288547863351371853*x^2
*(x^4-x)^(2/3)-10427862756551384399808209*x^4-9624328758485465306265498*RootOf(_Z^2+_Z+1)*(x^4-x)^(1/3)*x-3081
09538723884442175792*RootOf(_Z^2+_Z+1)^2+10825503288547863351371853*x*(x^4-x)^(1/3)-10332829807230149096810075
*RootOf(_Z^2+_Z+1)+92281971296914906192993)/(x^7-x^4-1))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (7 \, x^{3} - 4\right )} x^{2}}{{\left (x^{7} - x^{4} - 1\right )} {\left (x^{4} - x\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(7*x^3-4)/(x^4-x)^(1/3)/(x^7-x^4-1),x, algorithm="maxima")

[Out]

integrate((7*x^3 - 4)*x^2/((x^7 - x^4 - 1)*(x^4 - x)^(1/3)), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {x^2\,\left (7\,x^3-4\right )}{{\left (x^4-x\right )}^{1/3}\,\left (-x^7+x^4+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^2*(7*x^3 - 4))/((x^4 - x)^(1/3)*(x^4 - x^7 + 1)),x)

[Out]

-int((x^2*(7*x^3 - 4))/((x^4 - x)^(1/3)*(x^4 - x^7 + 1)), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} \left (7 x^{3} - 4\right )}{\sqrt [3]{x \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x^{7} - x^{4} - 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(7*x**3-4)/(x**4-x)**(1/3)/(x**7-x**4-1),x)

[Out]

Integral(x**2*(7*x**3 - 4)/((x*(x - 1)*(x**2 + x + 1))**(1/3)*(x**7 - x**4 - 1)), x)

________________________________________________________________________________________