\(\int \frac {x^2 (4+7 x^3)}{\sqrt [3]{x+x^4} (-1+x^4+x^7)} \, dx\) [2054]

Optimal result

Integrand size = 30, antiderivative size = 147 \[ \int \frac {x^2 \left (4+7 x^3\right )}{\sqrt [3]{x+x^4} \left (-1+x^4+x^7\right )} \, dx=-\sqrt {3} \arctan \left (\frac {3 \sqrt {3} x \sqrt [3]{x+x^4}-3 x^2 \sqrt [3]{x+x^4}}{-6+2 \sqrt {3} x-3 x \sqrt [3]{x+x^4}+\sqrt {3} x^2 \sqrt [3]{x+x^4}}\right )+2 \text {arctanh}\left (1-2 x \sqrt [3]{x+x^4}\right )-\text {arctanh}\left (\frac {1+x \sqrt [3]{x+x^4}}{1+x \sqrt [3]{x+x^4}+2 x^2 \left (x+x^4\right )^{2/3}}\right ) \]

[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 = 6.57 (sec) , antiderivative size = 485, normalized size of antiderivative = 3.30

method	result	size
trager	\(\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {-25169216 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{7}-17080755 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{7}[phone] x^{7}-25169216 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{4}[phone] \left (x^{4}+x \right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-17080755 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}[phone] x^{2} \left (x^{4}+x \right )^{\frac {2}{3}}[phone] x^{4}[phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}+x \right )^{\frac {1}{3}} x [phone] x \left (x^{4}+x \right )^{\frac {1}{3}}[phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}[phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )[phone]}{x^{7}+x^{4}-1}\right )-\ln \left (-\frac {25169216 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{7}[phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{7}-126749913 x^{7}[phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{4}[phone] \left (x^{4}+x \right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}[phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}-143830668 x^{2} \left (x^{4}+x \right )^{\frac {2}{3}}-126749913 x^{4}[phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}+x \right )^{\frac {1}{3}} x -143830668 x \left (x^{4}+x \right )^{\frac {1}{3}}-50338432 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}[phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-42249971}{x^{7}+x^{4}-1}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-\ln \left (-\frac {25169216 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{7}[phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{7}-126749913 x^{7}[phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{4}[phone] \left (x^{4}+x \right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}[phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}-143830668 x^{2} \left (x^{4}+x \right )^{\frac {2}{3}}-126749913 x^{4}[phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}+x \right )^{\frac {1}{3}} x -143830668 x \left (x^{4}+x \right )^{\frac {1}{3}}-50338432 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}[phone] \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-42249971}{x^{7}+x^{4}-1}\right )\)	\(485\)

[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((-25169216*RootOf(_Z^2+_Z+1)^2*x^7-17080755*RootOf(_Z^2+_Z+1)*x^7+134838374*x^7-25169216*
RootOf(_Z^2+_Z+1)^2*x^4+168096051*(x^4+x)^(2/3)*RootOf(_Z^2+_Z+1)*x^2-17080755*RootOf(_Z^2+_Z+1)*x^4+311926719
*x^2*(x^4+x)^(2/3)[phone]*x^4+168096051*RootOf(_Z^2+_Z+1)*(x^4+x)^(1/3)*x+311926719*x*(x^4+x)^(1/3)+5033843
2*RootOf(_Z^2+_Z+1)^2+210346022*RootOf(_Z^2+_Z+1)[phone])/(x^7+x^4-1))-ln(-(25169216*RootOf(_Z^2+_Z+1)^2*x^
7+33257677*RootOf(_Z^2+_Z+1)*x^7-126749913*x^7+25169216*RootOf(_Z^2+_Z+1)^2*x^4+168096051*(x^4+x)^(2/3)*RootOf
(_Z^2+_Z+1)*x^2+33257677*RootOf(_Z^2+_Z+1)*x^4-143830668*x^2*(x^4+x)^(2/3)-126749913*x^4+168096051*RootOf(_Z^2
+_Z+1)*(x^4+x)^(1/3)*x-143830668*x*(x^4+x)^(1/3)-50338432*RootOf(_Z^2+_Z+1)^2+109669158*RootOf(_Z^2+_Z+1)-4224
9971)/(x^7+x^4-1))*RootOf(_Z^2+_Z+1)-ln(-(25169216*RootOf(_Z^2+_Z+1)^2*x^7+33257677*RootOf(_Z^2+_Z+1)*x^7-1267
49913*x^7+25169216*RootOf(_Z^2+_Z+1)^2*x^4+168096051*(x^4+x)^(2/3)*RootOf(_Z^2+_Z+1)*x^2+33257677*RootOf(_Z^2+
_Z+1)*x^4-143830668*x^2*(x^4+x)^(2/3)-126749913*x^4+168096051*RootOf(_Z^2+_Z+1)*(x^4+x)^(1/3)*x-143830668*x*(x
^4+x)^(1/3)-50338432*RootOf(_Z^2+_Z+1)^2+109669158*RootOf(_Z^2+_Z+1)-42249971)/(x^7+x^4-1))

Fricas [A] (verification not implemented)

none

Time = 2.03 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.70 \[ \int \frac {x^2 \left (4+7 x^3\right )}{\sqrt [3]{x+x^4} \left (-1+x^4+x^7\right )} \, dx=-\sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} {\left (x^{4} + x\right )}^{\frac {2}{3}} x^{2} - 4 \, \sqrt {3} {\left (x^{4} + x\right )}^{\frac {1}{3}} x - \sqrt {3} {\left (x^{7} + x^{4}\right )}}{x^{7} + x^{4} + 8}\right ) + \frac {1}{2} \, \log \left (\frac {x^{7} + x^{4} - 3 \, {\left (x^{4} + x\right )}^{\frac {2}{3}} x^{2} + 3 \, {\left (x^{4} + x\right )}^{\frac {1}{3}} x - 1}{x^{7} + x^{4} - 1}\right ) \]

[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 + x^4)^(1/3)*(x^4 + x^7 - 1)),x)

[Out]

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