\(\int \frac {x^2 (-4+7 x^3)}{\sqrt [3]{-x+x^4} (-1-x^4+x^7)} \, dx\) [2204]

Optimal result

Integrand size = 34, antiderivative size = 163 \[ \int \frac {x^2 \left (-4+7 x^3\right )}{\sqrt [3]{-x+x^4} \left (-1-x^4+x^7\right )} \, dx=-\sqrt {3} \arctan \left (\frac {3 \sqrt {3} x \sqrt [3]{-x+x^4}-3 x^2 \sqrt [3]{-x+x^4}}{-6+2 \sqrt {3} x-3 x \sqrt [3]{-x+x^4}+\sqrt {3} x^2 \sqrt [3]{-x+x^4}}\right )+2 \text {arctanh}\left (1-2 x \sqrt [3]{-x+x^4}\right )-\text {arctanh}\left (\frac {1+x \sqrt [3]{-x+x^4}}{1+x \sqrt [3]{-x+x^4}+2 x^2 \left (-x+x^4\right )^{2/3}}\right ) \]

[Out]

-3^(1/2)*arctan((3*3^(1/2)*x*(x^4-x)^(1/3)-3*x^2*(x^4-x)^(1/3))/(-6+2*x*3^(1/2)-3*x*(x^4-x)^(1/3)+3^(1/2)*x^2*
(x^4-x)^(1/3)))-2*arctanh(-1+2*x*(x^4-x)^(1/3))-arctanh((1+x*(x^4-x)^(1/3))/(1+x*(x^4-x)^(1/3)+2*x^2*(x^4-x)^(
2/3)))

Rubi [F]

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

[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*} \text {integral}& = \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 ) \text {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 ) \text {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 ) \text {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 ) \text {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]

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

[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]

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 13.81 (sec) , antiderivative size = 516, normalized size of antiderivative = 3.17

method	result	size
trager	\(\text {Expression too large to display}\)	\(516\)

[In]

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

[Out]

-ln(-(2750978024320396805141*RootOf(_Z^2+_Z+1)^2*x^7+403142488045119745173926*RootOf(_Z^2+_Z+1)*x^7+1042786275
6551384399808209*x^7-2750978024320396805141*RootOf(_Z^2+_Z+1)^2*x^4-9624328758485465306265498*RootOf(_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-30810953872388444
2175792*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))*RootOf(_Z^2+_Z+1)+RootOf(_Z^2+_Z+1)*ln(-(2750978024320396805141*Ro
otOf(_Z^2+_Z+1)^2*x^7-397640531996478951563644*RootOf(_Z^2+_Z+1)*x^7+10027471246530585051439424*x^7-2750978024
320396805141*RootOf(_Z^2+_Z+1)^2*x^4+9624328758485465306265498*RootOf(_Z^2+_Z+1)*(x^4-x)^(2/3)*x^2+39764053199
6478951563644*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+204
49832047033328657637351*x*(x^4-x)^(1/3)+9716610729782380212458491*RootOf(_Z^2+_Z+1)+10117002239803179560827276
)/(x^7-x^4-1))-ln(-(2750978024320396805141*RootOf(_Z^2+_Z+1)^2*x^7+403142488045119745173926*RootOf(_Z^2+_Z+1)*
x^7+10427862756551384399808209*x^7-2750978024320396805141*RootOf(_Z^2+_Z+1)^2*x^4-9624328758485465306265498*Ro
otOf(_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-308
109538723884442175792*RootOf(_Z^2+_Z+1)^2+10825503288547863351371853*x*(x^4-x)^(1/3)-1033282980723014909681007
5*RootOf(_Z^2+_Z+1)+92281971296914906192993)/(x^7-x^4-1))

Fricas [A] (verification not implemented)

none

Time = 1.97 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.73 \[ \int \frac {x^2 \left (-4+7 x^3\right )}{\sqrt [3]{-x+x^4} \left (-1-x^4+x^7\right )} \, dx=-\sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} {\left (x^{4} - x\right )}^{\frac {2}{3}} x^{2} - 4 \, \sqrt {3} {\left (x^{4} - x\right )}^{\frac {1}{3}} x - \sqrt {3} {\left (x^{7} - x^{4}\right )}}{x^{7} - x^{4} + 8}\right ) + \frac {1}{2} \, \log \left (\frac {x^{7} - x^{4} - 3 \, {\left (x^{4} - x\right )}^{\frac {2}{3}} x^{2} + 3 \, {\left (x^{4} - x\right )}^{\frac {1}{3}} x - 1}{x^{7} - x^{4} - 1}\right ) \]

[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))

Sympy [F]

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

[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)

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \left (-4+7 x^3\right )}{\sqrt [3]{-x+x^4} \left (-1-x^4+x^7\right )} \, dx=-\int \frac {x^2\,\left (7\,x^3-4\right )}{{\left (x^4-x\right )}^{1/3}\,\left (-x^7+x^4+1\right )} \,d x \]

[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)