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

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

[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)))

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Rubi [F]
time = 0.83, 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 ) \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*}

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Mathematica [F]
time = 10.12, 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]

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 14.09, size = 339, normalized size = 2.08

method	result	size
trager	\(\ln \left (-\frac {89530993272594509387852 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{7}-9935189275233670145246431 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{7}-10335580785254469493615216 x^{7}-89530993272594509387852 \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}+9935189275233670145246431 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}+20449832047033328657637351 x^{2} \left (x^{4}-x \right )^{\frac {2}{3}}+10335580785254469493615216 x^{4}+10825503288547863351371853 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}-x \right )^{\frac {1}{3}} x -10027471246530585051439424 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-9624328758485465306265498 x \left (x^{4}-x \right )^{\frac {1}{3}}-20452583025057649054442492 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-10427862756551384399808209}{x^{7}-x^{4}-1}\right )+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {92281971296914906192993 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{7}+10517393749823978909196061 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{7}+10117002239803179560827276 x^{7}-92281971296914906192993 \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}-10517393749823978909196061 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}-10825503288547863351371853 x^{2} \left (x^{4}-x \right )^{\frac {2}{3}}-10117002239803179560827276 x^{4}-20449832047033328657637351 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}-x \right )^{\frac {1}{3}} x -10335580785254469493615216 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-9624328758485465306265498 x \left (x^{4}-x \right )^{\frac {1}{3}}-9935189275233670145246431 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+89530993272594509387852}{x^{7}-x^{4}-1}\right )\)	\(339\)

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]

ln(-(89530993272594509387852*RootOf(_Z^2+_Z+1)^2*x^7-9935189275233670145246431*RootOf(_Z^2+_Z+1)*x^7-103355807
85254469493615216*x^7-89530993272594509387852*RootOf(_Z^2+_Z+1)^2*x^4+9624328758485465306265498*RootOf(_Z^2+_Z
+1)*(x^4-x)^(2/3)*x^2+9935189275233670145246431*RootOf(_Z^2+_Z+1)*x^4+20449832047033328657637351*x^2*(x^4-x)^(
2/3)+10335580785254469493615216*x^4+10825503288547863351371853*RootOf(_Z^2+_Z+1)*(x^4-x)^(1/3)*x-1002747124653
0585051439424*RootOf(_Z^2+_Z+1)^2-9624328758485465306265498*x*(x^4-x)^(1/3)-20452583025057649054442492*RootOf(
_Z^2+_Z+1)-10427862756551384399808209)/(x^7-x^4-1))+RootOf(_Z^2+_Z+1)*ln((92281971296914906192993*RootOf(_Z^2+
_Z+1)^2*x^7+10517393749823978909196061*RootOf(_Z^2+_Z+1)*x^7+10117002239803179560827276*x^7-922819712969149061
92993*RootOf(_Z^2+_Z+1)^2*x^4+9624328758485465306265498*RootOf(_Z^2+_Z+1)*(x^4-x)^(2/3)*x^2-105173937498239789
09196061*RootOf(_Z^2+_Z+1)*x^4-10825503288547863351371853*x^2*(x^4-x)^(2/3)-10117002239803179560827276*x^4-204
49832047033328657637351*RootOf(_Z^2+_Z+1)*(x^4-x)^(1/3)*x-10335580785254469493615216*RootOf(_Z^2+_Z+1)^2-96243
28758485465306265498*x*(x^4-x)^(1/3)-9935189275233670145246431*RootOf(_Z^2+_Z+1)+89530993272594509387852)/(x^7
-x^4-1))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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)

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Fricas [A]
time = 1.90, 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))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \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)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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)

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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)

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