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

Optimal. Leaf size=144 \[ \frac {3 \sqrt [3]{1-x^7}}{2 x}-\frac {\log \left (2^{2/3} \sqrt [3]{1-x^7}-x\right )}{2\ 2^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2\ 2^{2/3} \sqrt [3]{1-x^7}+x}\right )}{2\ 2^{2/3}}+\frac {\log \left (2^{2/3} \sqrt [3]{1-x^7} x+2 \sqrt [3]{2} \left (1-x^7\right )^{2/3}+x^2\right )}{4\ 2^{2/3}} \]

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Rubi [F]  time = 1.60, 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 {\sqrt [3]{1-x^7} \left (-2+x^3+2 x^7\right ) \left (3+4 x^7\right )}{x^2 \left (-1+x^7\right ) \left (-4+x^3+4 x^7\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(3*Hypergeometric2F1[-1/7, 2/3, 6/7, x^7])/(2*x) - (x^2*Hypergeometric2F1[2/7, 2/3, 9/7, x^7])/4 - (x^6*Hyperg
eometric2F1[2/3, 6/7, 13/7, x^7])/3 - (7*Defer[Int][x/((1 - x^7)^(2/3)*(-4 + x^3 + 4*x^7)), x])/2 + Defer[Int]
[x^4/((1 - x^7)^(2/3)*(-4 + x^3 + 4*x^7)), x]/2

Rubi steps

\begin {align*} \int \frac {\sqrt [3]{1-x^7} \left (-2+x^3+2 x^7\right ) \left (3+4 x^7\right )}{x^2 \left (-1+x^7\right ) \left (-4+x^3+4 x^7\right )} \, dx &=-\int \frac {\left (-2+x^3+2 x^7\right ) \left (3+4 x^7\right )}{x^2 \left (1-x^7\right )^{2/3} \left (-4+x^3+4 x^7\right )} \, dx\\ &=-\int \left (\frac {3}{2 x^2 \left (1-x^7\right )^{2/3}}+\frac {x}{2 \left (1-x^7\right )^{2/3}}+\frac {2 x^5}{\left (1-x^7\right )^{2/3}}-\frac {x \left (-7+x^3\right )}{2 \left (1-x^7\right )^{2/3} \left (-4+x^3+4 x^7\right )}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {x}{\left (1-x^7\right )^{2/3}} \, dx\right )+\frac {1}{2} \int \frac {x \left (-7+x^3\right )}{\left (1-x^7\right )^{2/3} \left (-4+x^3+4 x^7\right )} \, dx-\frac {3}{2} \int \frac {1}{x^2 \left (1-x^7\right )^{2/3}} \, dx-2 \int \frac {x^5}{\left (1-x^7\right )^{2/3}} \, dx\\ &=\frac {3 \, _2F_1\left (-\frac {1}{7},\frac {2}{3};\frac {6}{7};x^7\right )}{2 x}-\frac {1}{4} x^2 \, _2F_1\left (\frac {2}{7},\frac {2}{3};\frac {9}{7};x^7\right )-\frac {1}{3} x^6 \, _2F_1\left (\frac {2}{3},\frac {6}{7};\frac {13}{7};x^7\right )+\frac {1}{2} \int \left (-\frac {7 x}{\left (1-x^7\right )^{2/3} \left (-4+x^3+4 x^7\right )}+\frac {x^4}{\left (1-x^7\right )^{2/3} \left (-4+x^3+4 x^7\right )}\right ) \, dx\\ &=\frac {3 \, _2F_1\left (-\frac {1}{7},\frac {2}{3};\frac {6}{7};x^7\right )}{2 x}-\frac {1}{4} x^2 \, _2F_1\left (\frac {2}{7},\frac {2}{3};\frac {9}{7};x^7\right )-\frac {1}{3} x^6 \, _2F_1\left (\frac {2}{3},\frac {6}{7};\frac {13}{7};x^7\right )+\frac {1}{2} \int \frac {x^4}{\left (1-x^7\right )^{2/3} \left (-4+x^3+4 x^7\right )} \, dx-\frac {7}{2} \int \frac {x}{\left (1-x^7\right )^{2/3} \left (-4+x^3+4 x^7\right )} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.42, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{1-x^7} \left (-2+x^3+2 x^7\right ) \left (3+4 x^7\right )}{x^2 \left (-1+x^7\right ) \left (-4+x^3+4 x^7\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 18.09, size = 144, normalized size = 1.00 \begin {gather*} \frac {3 \sqrt [3]{1-x^7}}{2 x}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2\ 2^{2/3} \sqrt [3]{1-x^7}}\right )}{2\ 2^{2/3}}-\frac {\log \left (-x+2^{2/3} \sqrt [3]{1-x^7}\right )}{2\ 2^{2/3}}+\frac {\log \left (x^2+2^{2/3} x \sqrt [3]{1-x^7}+2 \sqrt [3]{2} \left (1-x^7\right )^{2/3}\right )}{4\ 2^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(1 - x^7)^(1/3))/(2*x) - (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*2^(2/3)*(1 - x^7)^(1/3))])/(2*2^(2/3)) - Log[-x
+ 2^(2/3)*(1 - x^7)^(1/3)]/(2*2^(2/3)) + Log[x^2 + 2^(2/3)*x*(1 - x^7)^(1/3) + 2*2^(1/3)*(1 - x^7)^(2/3)]/(4*
2^(2/3))

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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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 (2 \, x^{7} + x^{3} - 2\right )} {\left (-x^{7} + 1\right )}^{\frac {1}{3}}}{{\left (4 \, x^{7} + x^{3} - 4\right )} {\left (x^{7} - 1\right )} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 61.10, size = 1486, normalized size = 10.32 \[\text {Expression too large to display}\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2*(x^7-1)/x/(-x^7+1)^(2/3)+(1/4*RootOf(_Z^3+2)*ln((4*RootOf(_Z^3+2)*x^14+4*RootOf(RootOf(_Z^3+2)^2+2*_Z*Roo
tOf(_Z^3+2)+4*_Z^2)*x^14+RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^3*x^10+RootOf(Root
Of(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^2*x^10-3*(x^14-2*x^7+1)^(1/3)*RootOf(_Z^3+2)^2*x^8-R
ootOf(_Z^3+2)*x^10-RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x^10-8*RootOf(_Z^3+2)*x^7-8*RootOf(Root
Of(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x^7-RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2
)^3*x^3-RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^2*x^3-3*(x^14-2*x^7+1)^(2/3)*Root
Of(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^2*x^2+3*(x^14-2*x^7+1)^(1/3)*RootOf(_Z^3+2)^2*x
+RootOf(_Z^3+2)*x^3+RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x^3+4*RootOf(_Z^3+2)+4*RootOf(RootOf(_
Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2))/(-1+x)/(4*x^7+x^3-4)/(x^6+x^5+x^4+x^3+x^2+x+1))-1/4*ln((-4*RootOf(_Z^3+2
)*x^14-4*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x^14+RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+
4*_Z^2)*RootOf(_Z^3+2)^3*x^10+RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^2*x^10-6*(x
^14-2*x^7+1)^(1/3)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)*x^8+8*RootOf(_Z^3+2)*x^7
+8*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x^7-RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)
*RootOf(_Z^3+2)^3*x^3-RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^2*x^3+3*(x^14-2*x^7
+1)^(2/3)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^2*x^2+6*(x^14-2*x^7+1)^(1/3)*Root
Of(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)*x-3*(x^14-2*x^7+1)^(2/3)*x^2-4*RootOf(_Z^3+2)-4
*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2))/(-1+x)/(4*x^7+x^3-4)/(x^6+x^5+x^4+x^3+x^2+x+1))*RootOf(_
Z^3+2)-1/2*ln((-4*RootOf(_Z^3+2)*x^14-4*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x^14+RootOf(RootOf
(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^3*x^10+RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^
2)^2*RootOf(_Z^3+2)^2*x^10-6*(x^14-2*x^7+1)^(1/3)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_
Z^3+2)*x^8+8*RootOf(_Z^3+2)*x^7+8*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x^7-RootOf(RootOf(_Z^3+2
)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^3*x^3-RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)^2*Roo
tOf(_Z^3+2)^2*x^3+3*(x^14-2*x^7+1)^(2/3)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^2*
x^2+6*(x^14-2*x^7+1)^(1/3)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)*x-3*(x^14-2*x^7+
1)^(2/3)*x^2-4*RootOf(_Z^3+2)-4*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2))/(-1+x)/(4*x^7+x^3-4)/(x^6
+x^5+x^4+x^3+x^2+x+1))*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2))/(-x^7+1)^(2/3)*((x^7-1)^2)^(1/3)

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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 (2 \, x^{7} + x^{3} - 2\right )} {\left (-x^{7} + 1\right )}^{\frac {1}{3}}}{{\left (4 \, x^{7} + x^{3} - 4\right )} {\left (x^{7} - 1\right )} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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