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

Optimal. Leaf size=147 \[ 2 \tanh ^{-1}\left (1-2 x \sqrt [3]{x^4+x}\right )-\sqrt {3} \tan ^{-1}\left (\frac {3 \sqrt {3} x \sqrt [3]{x^4+x}-3 x^2 \sqrt [3]{x^4+x}}{-3 \sqrt [3]{x^4+x} x+\sqrt {3} \sqrt [3]{x^4+x} x^2+2 \sqrt {3} x-6}\right )-\tanh ^{-1}\left (\frac {\sqrt [3]{x^4+x} x+1}{\sqrt [3]{x^4+x} x+2 \left (x^4+x\right )^{2/3} x^2+1}\right ) \]

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Rubi [F]  time = 1.24, 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*}

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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]

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IntegrateAlgebraic [A]  time = 9.91, size = 147, 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))]

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fricas [A]  time = 3.60, size = 103, normalized size = 0.70 \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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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)

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maple [C]  time = 11.02, size = 319, normalized size = 2.17

method	result	size
trager	\(\ln \left (\frac {67419187 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{7}-92588403 x^{7} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-84499942 x^{7}[phone] \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{4}[phone] \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}+x \right )^{\frac {2}{3}} x^{2}-92588403 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}[phone] x^{2} \left (x^{4}+x \right )^{\frac {2}{3}}-84499942 x^{4}[phone] \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}+x \right )^{\frac {1}{3}} x -134838374 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-168096051 x \left (x^{4}+x \right )^{\frac {1}{3}}-286757503 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-126749913}{x^{7}+x^{4}-1}\right )+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {42249971 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{7}[phone] x^{7} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )[phone] x^{7}[phone] \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{4}[phone] \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}+x \right )^{\frac {2}{3}} x^{2}[phone] \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}-143830668 x^{2} \left (x^{4}+x \right )^{\frac {2}{3}}[phone] x^{4}-311926719 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{4}+x \right )^{\frac {1}{3}} x -84499942 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-168096051 x \left (x^{4}+x \right )^{\frac {1}{3}}-92588403 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )[phone]}{x^{7}+x^{4}-1}\right )\)	\(319\)

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((67419187*RootOf(_Z^2+_Z+1)^2*x^7-92588403*x^7*RootOf(_Z^2+_Z+1)-84499942*x^7+67419187*RootOf(_Z^2+_Z+1)^2*
x^4+168096051*RootOf(_Z^2+_Z+1)*(x^4+x)^(2/3)*x^2-92588403*RootOf(_Z^2+_Z+1)*x^4+311926719*x^2*(x^4+x)^(2/3)-8
4499942*x^4+143830668*RootOf(_Z^2+_Z+1)*(x^4+x)^(1/3)*x-134838374*RootOf(_Z^2+_Z+1)^2-168096051*x*(x^4+x)^(1/3
)-286757503*RootOf(_Z^2+_Z+1)-126749913)/(x^7+x^4-1))+RootOf(_Z^2+_Z+1)*ln(-(42249971*RootOf(_Z^2+_Z+1)^2*x^7+
194169100*x^7*RootOf(_Z^2+_Z+1)[phone]*x^7+42249971*RootOf(_Z^2+_Z+1)^2*x^4+168096051*RootOf(_Z^2+_Z+1)*(x^
4+x)^(2/3)*x^2+194169100*RootOf(_Z^2+_Z+1)*x^4-143830668*x^2*(x^4+x)^(2/3)[phone]*x^4-311926719*RootOf(_Z^2
+_Z+1)*(x^4+x)^(1/3)*x-84499942*RootOf(_Z^2+_Z+1)^2-168096051*x*(x^4+x)^(1/3)-92588403*RootOf(_Z^2+_Z+1)+67419
187)/(x^7+x^4-1))

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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)

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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 + 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)

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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)

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