3.13.49 \(\int \frac {(-1+x^7)^{2/3} (3+4 x^7)}{x^3 (-1+x^3+x^7)} \, dx\)

Optimal. Leaf size=90 \[ \log \left (\sqrt [3]{x^7-1}+x\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^7-1}-x}\right )+\frac {3 \left (x^7-1\right )^{2/3}}{2 x^2}-\frac {1}{2} \log \left (-\sqrt [3]{x^7-1} x+\left (x^7-1\right )^{2/3}+x^2\right ) \]

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Rubi [F]  time = 0.72, 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 {\left (-1+x^7\right )^{2/3} \left (3+4 x^7\right )}{x^3 \left (-1+x^3+x^7\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(3*(-1 + x^7)^(2/3)*Hypergeometric2F1[-2/3, -2/7, 5/7, x^7])/(2*x^2*(1 - x^7)^(2/3)) + 3*Defer[Int][(-1 + x^7)
^(2/3)/(-1 + x^3 + x^7), x] + 7*Defer[Int][(x^4*(-1 + x^7)^(2/3))/(-1 + x^3 + x^7), x]

Rubi steps

\begin {align*} \int \frac {\left (-1+x^7\right )^{2/3} \left (3+4 x^7\right )}{x^3 \left (-1+x^3+x^7\right )} \, dx &=\int \left (-\frac {3 \left (-1+x^7\right )^{2/3}}{x^3}+\frac {\left (3+7 x^4\right ) \left (-1+x^7\right )^{2/3}}{-1+x^3+x^7}\right ) \, dx\\ &=-\left (3 \int \frac {\left (-1+x^7\right )^{2/3}}{x^3} \, dx\right )+\int \frac {\left (3+7 x^4\right ) \left (-1+x^7\right )^{2/3}}{-1+x^3+x^7} \, dx\\ &=-\frac {\left (3 \left (-1+x^7\right )^{2/3}\right ) \int \frac {\left (1-x^7\right )^{2/3}}{x^3} \, dx}{\left (1-x^7\right )^{2/3}}+\int \left (\frac {3 \left (-1+x^7\right )^{2/3}}{-1+x^3+x^7}+\frac {7 x^4 \left (-1+x^7\right )^{2/3}}{-1+x^3+x^7}\right ) \, dx\\ &=\frac {3 \left (-1+x^7\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {2}{7};\frac {5}{7};x^7\right )}{2 x^2 \left (1-x^7\right )^{2/3}}+3 \int \frac {\left (-1+x^7\right )^{2/3}}{-1+x^3+x^7} \, dx+7 \int \frac {x^4 \left (-1+x^7\right )^{2/3}}{-1+x^3+x^7} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.27, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^7\right )^{2/3} \left (3+4 x^7\right )}{x^3 \left (-1+x^3+x^7\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 17.79, size = 90, normalized size = 1.00 \begin {gather*} \frac {3 \left (-1+x^7\right )^{2/3}}{2 x^2}+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-1+x^7}}\right )+\log \left (x+\sqrt [3]{-1+x^7}\right )-\frac {1}{2} \log \left (x^2-x \sqrt [3]{-1+x^7}+\left (-1+x^7\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(-1 + x^7)^(2/3))/(2*x^2) + Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2*(-1 + x^7)^(1/3))] + Log[x + (-1 + x^7)^(1/3
)] - Log[x^2 - x*(-1 + x^7)^(1/3) + (-1 + x^7)^(2/3)]/2

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fricas [A]  time = 16.20, size = 131, normalized size = 1.46 \begin {gather*} \frac {2 \, \sqrt {3} x^{2} \arctan \left (-\frac {26962 \, \sqrt {3} {\left (x^{7} - 1\right )}^{\frac {1}{3}} x^{2} - 60268 \, \sqrt {3} {\left (x^{7} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (34656 \, x^{7} - 8959 \, x^{3} - 34656\right )}}{54872 \, x^{7} + 4913 \, x^{3} - 54872}\right ) + x^{2} \log \left (\frac {x^{7} + x^{3} + 3 \, {\left (x^{7} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{7} - 1\right )}^{\frac {2}{3}} x - 1}{x^{7} + x^{3} - 1}\right ) + 3 \, {\left (x^{7} - 1\right )}^{\frac {2}{3}}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*sqrt(3)*x^2*arctan(-(26962*sqrt(3)*(x^7 - 1)^(1/3)*x^2 - 60268*sqrt(3)*(x^7 - 1)^(2/3)*x + sqrt(3)*(346
56*x^7 - 8959*x^3 - 34656))/(54872*x^7 + 4913*x^3 - 54872)) + x^2*log((x^7 + x^3 + 3*(x^7 - 1)^(1/3)*x^2 + 3*(
x^7 - 1)^(2/3)*x - 1)/(x^7 + x^3 - 1)) + 3*(x^7 - 1)^(2/3))/x^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 59.83, size = 264, normalized size = 2.93

method	result	size
risch	\(\frac {3 \left (x^{7}-1\right )^{\frac {2}{3}}}{2 x^{2}}+\ln \left (\frac {-x^{7} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-x^{7}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-\left (x^{7}-1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{7}-1\right )^{\frac {1}{3}} x^{2}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-2 \left (x^{7}-1\right )^{\frac {2}{3}} x -\left (x^{7}-1\right )^{\frac {1}{3}} x^{2}+x^{3}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1}{x^{7}+x^{3}-1}\right )+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {x^{7} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+x^{7}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-\left (x^{7}-1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{7}-1\right )^{\frac {1}{3}} x^{2}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+\left (x^{7}-1\right )^{\frac {2}{3}} x -\left (x^{7}-1\right )^{\frac {1}{3}} x^{2}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1}{x^{7}+x^{3}-1}\right )\)	\(264\)
trager	\(\frac {3 \left (x^{7}-1\right )^{\frac {2}{3}}}{2 x^{2}}-12 \ln \left (-\frac {66066186469498767988246012779075919590528 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{7}-16844938467913844130160220890990438495796 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{7}-80634406416788660822330883626933520827419 x^{7}-1048800710203292941813405452867830223499632 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{3}-928417484526511344275296327342965046304892 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{7}-1\right )^{\frac {2}{3}} x +928417484526511344275296327342965046304892 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{7}-1\right )^{\frac {1}{3}} x^{2}-1038167997717157830888927503704530996724868 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{3}+82955737212302297388569874434225445140826 \left (x^{7}-1\right )^{\frac {2}{3}} x -82955737212302297388569874434225445140826 \left (x^{7}-1\right )^{\frac {1}{3}} x^{2}-5420800431380750307383588815256035013608 x^{3}-66066186469498767988246012779075919590528 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}+16844938467913844130160220890990438495796 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+80634406416788660822330883626933520827419}{x^{7}+x^{3}-1}\right ) \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+12 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \ln \left (-\frac {198198559408496303964738038337227758771584 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{7}+83567908638490916384603669062509275282652 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{7}-236315605748606297101364137058488871200172 x^{7}-3146402130609878825440216358603490670498896 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{3}+2785252453579534032825888982028895138914676 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{7}-1\right )^{\frac {2}{3}} x -2785252453579534032825888982028895138914676 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{7}-1\right )^{\frac {1}{3}} x^{2}+2590103638049827021760079784679677878424788 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{3}+480971582768534728234533705138417596998701 \left (x^{7}-1\right )^{\frac {2}{3}} x -480971582768534728234533705138417596998701 \left (x^{7}-1\right )^{\frac {1}{3}} x^{2}+221429583339245270512301829212284847817484 x^{3}-198198559408496303964738038337227758771584 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}-83567908638490916384603669062509275282652 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+236315605748606297101364137058488871200172}{x^{7}+x^{3}-1}\right )-\ln \left (-\frac {66066186469498767988246012779075919590528 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{7}-16844938467913844130160220890990438495796 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{7}-80634406416788660822330883626933520827419 x^{7}-1048800710203292941813405452867830223499632 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{3}-928417484526511344275296327342965046304892 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{7}-1\right )^{\frac {2}{3}} x +928417484526511344275296327342965046304892 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{7}-1\right )^{\frac {1}{3}} x^{2}-1038167997717157830888927503704530996724868 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{3}+82955737212302297388569874434225445140826 \left (x^{7}-1\right )^{\frac {2}{3}} x -82955737212302297388569874434225445140826 \left (x^{7}-1\right )^{\frac {1}{3}} x^{2}-5420800431380750307383588815256035013608 x^{3}-66066186469498767988246012779075919590528 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}+16844938467913844130160220890990438495796 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+80634406416788660822330883626933520827419}{x^{7}+x^{3}-1}\right )\)	\(603\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*(x^7-1)^(2/3)/x^2+ln((-x^7*RootOf(_Z^2+_Z+1)-x^7+RootOf(_Z^2+_Z+1)^2*x^3-(x^7-1)^(2/3)*RootOf(_Z^2+_Z+1)*x
+RootOf(_Z^2+_Z+1)*(x^7-1)^(1/3)*x^2+2*RootOf(_Z^2+_Z+1)*x^3-2*(x^7-1)^(2/3)*x-(x^7-1)^(1/3)*x^2+x^3+RootOf(_Z
^2+_Z+1)+1)/(x^7+x^3-1))+RootOf(_Z^2+_Z+1)*ln(-(x^7*RootOf(_Z^2+_Z+1)+x^7+RootOf(_Z^2+_Z+1)^2*x^3-(x^7-1)^(2/3
)*RootOf(_Z^2+_Z+1)*x-2*RootOf(_Z^2+_Z+1)*(x^7-1)^(1/3)*x^2+RootOf(_Z^2+_Z+1)*x^3+(x^7-1)^(2/3)*x-(x^7-1)^(1/3
)*x^2-RootOf(_Z^2+_Z+1)-1)/(x^7+x^3-1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\left (x - 1\right ) \left (x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (4 x^{7} + 3\right )}{x^{3} \left (x^{7} + x^{3} - 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(((x - 1)*(x**6 + x**5 + x**4 + x**3 + x**2 + x + 1))**(2/3)*(4*x**7 + 3)/(x**3*(x**7 + x**3 - 1)), x)

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