3.21.67 \(\int \frac {\sqrt [3]{-1+2 x^3+x^8} (3+5 x^8)}{x^2 (-1+x^8)} \, dx\) [2067]

Optimal. Leaf size=149 \[ \frac {3 \sqrt [3]{-1+2 x^3+x^8}}{x}+\sqrt [3]{2} \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-1+2 x^3+x^8}}\right )+\sqrt [3]{2} \log \left (-2 x+2^{2/3} \sqrt [3]{-1+2 x^3+x^8}\right )-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-1+2 x^3+x^8}+\sqrt [3]{2} \left (-1+2 x^3+x^8\right )^{2/3}\right )}{2^{2/3}} \]

[Out]

3*(x^8+2*x^3-1)^(1/3)/x+2^(1/3)*3^(1/2)*arctan(3^(1/2)*x/(x+2^(2/3)*(x^8+2*x^3-1)^(1/3)))+2^(1/3)*ln(-2*x+2^(2
/3)*(x^8+2*x^3-1)^(1/3))-1/2*ln(2*x^2+2^(2/3)*x*(x^8+2*x^3-1)^(1/3)+2^(1/3)*(x^8+2*x^3-1)^(2/3))*2^(1/3)

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Rubi [F]
time = 1.05, 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+2 x^3+x^8} \left (3+5 x^8\right )}{x^2 \left (-1+x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-1 + 2*x^3 + x^8)^(1/3)*(3 + 5*x^8))/(x^2*(-1 + x^8)),x]

[Out]

Defer[Int][(-1 + 2*x^3 + x^8)^(1/3)/(-1 - x), x] + I*Defer[Int][(-1 + 2*x^3 + x^8)^(1/3)/(I - x), x] + (-1)^(3
/4)*Defer[Int][(-1 + 2*x^3 + x^8)^(1/3)/((-1)^(1/4) - x), x] - (-1)^(1/4)*Defer[Int][(-1 + 2*x^3 + x^8)^(1/3)/
(-(-1)^(3/4) - x), x] + Defer[Int][(-1 + 2*x^3 + x^8)^(1/3)/(-1 + x), x] - 3*Defer[Int][(-1 + 2*x^3 + x^8)^(1/
3)/x^2, x] + I*Defer[Int][(-1 + 2*x^3 + x^8)^(1/3)/(I + x), x] + (-1)^(3/4)*Defer[Int][(-1 + 2*x^3 + x^8)^(1/3
)/((-1)^(1/4) + x), x] - (-1)^(1/4)*Defer[Int][(-1 + 2*x^3 + x^8)^(1/3)/(-(-1)^(3/4) + x), x]

Rubi steps

\begin {align*} \int \frac {\sqrt [3]{-1+2 x^3+x^8} \left (3+5 x^8\right )}{x^2 \left (-1+x^8\right )} \, dx &=\int \left (\frac {\sqrt [3]{-1+2 x^3+x^8}}{-1-x}+\frac {\sqrt [3]{-1+2 x^3+x^8}}{-1+x}-\frac {3 \sqrt [3]{-1+2 x^3+x^8}}{x^2}+\frac {2 \sqrt [3]{-1+2 x^3+x^8}}{1+x^2}+\frac {4 x^2 \sqrt [3]{-1+2 x^3+x^8}}{1+x^4}\right ) \, dx\\ &=2 \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{1+x^2} \, dx-3 \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{x^2} \, dx+4 \int \frac {x^2 \sqrt [3]{-1+2 x^3+x^8}}{1+x^4} \, dx+\int \frac {\sqrt [3]{-1+2 x^3+x^8}}{-1-x} \, dx+\int \frac {\sqrt [3]{-1+2 x^3+x^8}}{-1+x} \, dx\\ &=2 \int \left (\frac {i \sqrt [3]{-1+2 x^3+x^8}}{2 (i-x)}+\frac {i \sqrt [3]{-1+2 x^3+x^8}}{2 (i+x)}\right ) \, dx-3 \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{x^2} \, dx+4 \int \left (-\frac {\sqrt [3]{-1+2 x^3+x^8}}{2 \left (i-x^2\right )}+\frac {\sqrt [3]{-1+2 x^3+x^8}}{2 \left (i+x^2\right )}\right ) \, dx+\int \frac {\sqrt [3]{-1+2 x^3+x^8}}{-1-x} \, dx+\int \frac {\sqrt [3]{-1+2 x^3+x^8}}{-1+x} \, dx\\ &=i \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{i-x} \, dx+i \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{i+x} \, dx-2 \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{i-x^2} \, dx+2 \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{i+x^2} \, dx-3 \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{x^2} \, dx+\int \frac {\sqrt [3]{-1+2 x^3+x^8}}{-1-x} \, dx+\int \frac {\sqrt [3]{-1+2 x^3+x^8}}{-1+x} \, dx\\ &=i \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{i-x} \, dx+i \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{i+x} \, dx-2 \int \left (-\frac {(-1)^{3/4} \sqrt [3]{-1+2 x^3+x^8}}{2 \left (\sqrt [4]{-1}-x\right )}-\frac {(-1)^{3/4} \sqrt [3]{-1+2 x^3+x^8}}{2 \left (\sqrt [4]{-1}+x\right )}\right ) \, dx+2 \int \left (-\frac {\sqrt [4]{-1} \sqrt [3]{-1+2 x^3+x^8}}{2 \left (-(-1)^{3/4}-x\right )}-\frac {\sqrt [4]{-1} \sqrt [3]{-1+2 x^3+x^8}}{2 \left (-(-1)^{3/4}+x\right )}\right ) \, dx-3 \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{x^2} \, dx+\int \frac {\sqrt [3]{-1+2 x^3+x^8}}{-1-x} \, dx+\int \frac {\sqrt [3]{-1+2 x^3+x^8}}{-1+x} \, dx\\ &=i \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{i-x} \, dx+i \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{i+x} \, dx-3 \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{x^2} \, dx-\sqrt [4]{-1} \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{-(-1)^{3/4}-x} \, dx-\sqrt [4]{-1} \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{-(-1)^{3/4}+x} \, dx+(-1)^{3/4} \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{\sqrt [4]{-1}-x} \, dx+(-1)^{3/4} \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{\sqrt [4]{-1}+x} \, dx+\int \frac {\sqrt [3]{-1+2 x^3+x^8}}{-1-x} \, dx+\int \frac {\sqrt [3]{-1+2 x^3+x^8}}{-1+x} \, dx\\ \end {align*}

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Mathematica [A]
time = 1.50, size = 149, normalized size = 1.00 \begin {gather*} \frac {3 \sqrt [3]{-1+2 x^3+x^8}}{x}+\sqrt [3]{2} \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-1+2 x^3+x^8}}\right )+\sqrt [3]{2} \log \left (-2 x+2^{2/3} \sqrt [3]{-1+2 x^3+x^8}\right )-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-1+2 x^3+x^8}+\sqrt [3]{2} \left (-1+2 x^3+x^8\right )^{2/3}\right )}{2^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-1 + 2*x^3 + x^8)^(1/3)*(3 + 5*x^8))/(x^2*(-1 + x^8)),x]

[Out]

(3*(-1 + 2*x^3 + x^8)^(1/3))/x + 2^(1/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(-1 + 2*x^3 + x^8)^(1/3))] +
2^(1/3)*Log[-2*x + 2^(2/3)*(-1 + 2*x^3 + x^8)^(1/3)] - Log[2*x^2 + 2^(2/3)*x*(-1 + 2*x^3 + x^8)^(1/3) + 2^(1/3
)*(-1 + 2*x^3 + x^8)^(2/3)]/2^(2/3)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 5.28, size = 2336, normalized size = 15.68 \[\text {Expression too large to display}\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^8+2*x^3-1)^(1/3)*(5*x^8+3)/x^2/(x^8-1),x)

[Out]

3*(x^8+2*x^3-1)^(1/3)/x+(-ln(-(-RootOf(_Z^3-2)*x^16-RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^16+2
*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x^11+2*RootOf(RootOf(_Z^3-2)^2+2*_Z*Root
Of(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x^11-3*(x^16+4*x^11-2*x^8+4*x^6-4*x^3+1)^(1/3)*RootOf(_Z^3-2)*RootOf(Roo
tOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^9-4*RootOf(_Z^3-2)*x^11-4*x^11*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf
(_Z^3-2)+4*_Z^2)+4*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x^6+4*RootOf(RootOf(_Z
^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x^6+2*RootOf(_Z^3-2)*x^8+2*x^8*RootOf(RootOf(_Z^3-2)^2+
2*_Z*RootOf(_Z^3-2)+4*_Z^2)-2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x^3-2*RootO
f(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x^3+3*(x^16+4*x^11-2*x^8+4*x^6-4*x^3+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^16+4*x^11-2*x^8+4*x^6-4*x^3+1)
^(1/3)*RootOf(_Z^3-2)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^4-4*RootOf(_Z^3-2)*x^6-4*x^6*RootO
f(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)+3*(x^16+4*x^11-2*x^8+4*x^6-4*x^3+1)^(1/3)*RootOf(_Z^3-2)*RootOf
(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x+4*RootOf(_Z^3-2)*x^3+4*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^
3-2)+4*_Z^2)*x^3+3*(x^16+4*x^11-2*x^8+4*x^6-4*x^3+1)^(2/3)*x^2-RootOf(_Z^3-2)-RootOf(RootOf(_Z^3-2)^2+2*_Z*Roo
tOf(_Z^3-2)+4*_Z^2))/(x^8+2*x^3-1)/(-1+x)/(1+x)/(x^2+1)/(x^4+1))*RootOf(_Z^3-2)-2*ln(-(-RootOf(_Z^3-2)*x^16-Ro
otOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^16+2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*R
ootOf(_Z^3-2)^3*x^11+2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x^11-3*(x^16+4*x
^11-2*x^8+4*x^6-4*x^3+1)^(1/3)*RootOf(_Z^3-2)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^9-4*RootOf
(_Z^3-2)*x^11-4*x^11*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)+4*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf
(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x^6+4*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2
*x^6+2*RootOf(_Z^3-2)*x^8+2*x^8*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)-2*RootOf(RootOf(_Z^3-2)^2+
2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x^3-2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootO
f(_Z^3-2)^2*x^3+3*(x^16+4*x^11-2*x^8+4*x^6-4*x^3+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^16+4*x^11-2*x^8+4*x^6-4*x^3+1)^(1/3)*RootOf(_Z^3-2)*RootOf(RootOf(_Z^3-2)^2+2*_Z*Roo
tOf(_Z^3-2)+4*_Z^2)*x^4-4*RootOf(_Z^3-2)*x^6-4*x^6*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)+3*(x^16
+4*x^11-2*x^8+4*x^6-4*x^3+1)^(1/3)*RootOf(_Z^3-2)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x+4*Root
Of(_Z^3-2)*x^3+4*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^3+3*(x^16+4*x^11-2*x^8+4*x^6-4*x^3+1)^(
2/3)*x^2-RootOf(_Z^3-2)-RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2))/(x^8+2*x^3-1)/(-1+x)/(1+x)/(x^2+1
)/(x^4+1))*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)+RootOf(_Z^3-2)*ln(-(2*RootOf(_Z^3-2)*x^16+2*Roo
tOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^16+4*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*Ro
otOf(_Z^3-2)^3*x^11+4*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x^11-3*(x^16+4*x^
11-2*x^8+4*x^6-4*x^3+1)^(1/3)*RootOf(_Z^3-2)^2*x^9+12*RootOf(_Z^3-2)*x^11+12*x^11*RootOf(RootOf(_Z^3-2)^2+2*_Z
*RootOf(_Z^3-2)+4*_Z^2)+8*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x^6+8*RootOf(Ro
otOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x^6-4*RootOf(_Z^3-2)*x^8-4*x^8*RootOf(RootOf(_Z^
3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)-4*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x^3-
4*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x^3-6*(x^16+4*x^11-2*x^8+4*x^6-4*x^3+
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^16+4*x^11-2*x^8+4*x^6-4
*x^3+1)^(1/3)*RootOf(_Z^3-2)^2*x^4+16*RootOf(_Z^3-2)*x^6+16*x^6*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*
_Z^2)+3*(x^16+4*x^11-2*x^8+4*x^6-4*x^3+1)^(1/3)*RootOf(_Z^3-2)^2*x-12*RootOf(_Z^3-2)*x^3-12*RootOf(RootOf(_Z^3
-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^3+2*RootOf(_Z^3-2)+2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2))/
(x^8+2*x^3-1)/(-1+x)/(1+x)/(x^2+1)/(x^4+1)))/(x^8+2*x^3-1)^(2/3)*((x^8+2*x^3-1)^2)^(1/3)

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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^8+2*x^3-1)^(1/3)*(5*x^8+3)/x^2/(x^8-1),x, algorithm="maxima")

[Out]

integrate((5*x^8 + 3)*(x^8 + 2*x^3 - 1)^(1/3)/((x^8 - 1)*x^2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 369 vs. \(2 (121) = 242\).
time = 231.98, size = 369, normalized size = 2.48 \begin {gather*} \frac {2 \, \sqrt {3} 2^{\frac {1}{3}} x \arctan \left (\frac {6 \, \sqrt {3} 2^{\frac {2}{3}} {\left (x^{18} + 18 \, x^{13} - 2 \, x^{10} + 36 \, x^{8} - 18 \, x^{5} + x^{2}\right )} {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {1}{3}} + 6 \, \sqrt {3} 2^{\frac {1}{3}} {\left (x^{17} + 6 \, x^{12} - 2 \, x^{9} - 6 \, x^{4} + x\right )} {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {2}{3}} + \sqrt {3} {\left (x^{24} + 36 \, x^{19} - 3 \, x^{16} + 180 \, x^{14} - 72 \, x^{11} + 216 \, x^{9} + 3 \, x^{8} - 180 \, x^{6} + 36 \, x^{3} - 1\right )}}{3 \, {\left (x^{24} - 3 \, x^{16} - 108 \, x^{14} - 216 \, x^{9} + 3 \, x^{8} + 108 \, x^{6} - 1\right )}}\right ) + 2 \cdot 2^{\frac {1}{3}} x \log \left (-\frac {6 \cdot 2^{\frac {1}{3}} {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 2^{\frac {2}{3}} {\left (x^{8} - 1\right )} - 6 \, {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {2}{3}} x}{x^{8} - 1}\right ) - 2^{\frac {1}{3}} x \log \left (\frac {3 \cdot 2^{\frac {2}{3}} {\left (x^{9} + 6 \, x^{4} - x\right )} {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (x^{16} + 18 \, x^{11} - 2 \, x^{8} + 36 \, x^{6} - 18 \, x^{3} + 1\right )} + 12 \, {\left (x^{10} + 3 \, x^{5} - x^{2}\right )} {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {1}{3}}}{x^{16} - 2 \, x^{8} + 1}\right ) + 18 \, {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {1}{3}}}{6 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^8+2*x^3-1)^(1/3)*(5*x^8+3)/x^2/(x^8-1),x, algorithm="fricas")

[Out]

1/6*(2*sqrt(3)*2^(1/3)*x*arctan(1/3*(6*sqrt(3)*2^(2/3)*(x^18 + 18*x^13 - 2*x^10 + 36*x^8 - 18*x^5 + x^2)*(x^8
+ 2*x^3 - 1)^(1/3) + 6*sqrt(3)*2^(1/3)*(x^17 + 6*x^12 - 2*x^9 - 6*x^4 + x)*(x^8 + 2*x^3 - 1)^(2/3) + sqrt(3)*(
x^24 + 36*x^19 - 3*x^16 + 180*x^14 - 72*x^11 + 216*x^9 + 3*x^8 - 180*x^6 + 36*x^3 - 1))/(x^24 - 3*x^16 - 108*x
^14 - 216*x^9 + 3*x^8 + 108*x^6 - 1)) + 2*2^(1/3)*x*log(-(6*2^(1/3)*(x^8 + 2*x^3 - 1)^(1/3)*x^2 + 2^(2/3)*(x^8
- 1) - 6*(x^8 + 2*x^3 - 1)^(2/3)*x)/(x^8 - 1)) - 2^(1/3)*x*log((3*2^(2/3)*(x^9 + 6*x^4 - x)*(x^8 + 2*x^3 - 1)
^(2/3) + 2^(1/3)*(x^16 + 18*x^11 - 2*x^8 + 36*x^6 - 18*x^3 + 1) + 12*(x^10 + 3*x^5 - x^2)*(x^8 + 2*x^3 - 1)^(1
/3))/(x^16 - 2*x^8 + 1)) + 18*(x^8 + 2*x^3 - 1)^(1/3))/x

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (5 x^{8} + 3\right ) \sqrt [3]{x^{8} + 2 x^{3} - 1}}{x^{2} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**8+2*x**3-1)**(1/3)*(5*x**8+3)/x**2/(x**8-1),x)

[Out]

Integral((5*x**8 + 3)*(x**8 + 2*x**3 - 1)**(1/3)/(x**2*(x - 1)*(x + 1)*(x**2 + 1)*(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^8+2*x^3-1)^(1/3)*(5*x^8+3)/x^2/(x^8-1),x, algorithm="giac")

[Out]

integrate((5*x^8 + 3)*(x^8 + 2*x^3 - 1)^(1/3)/((x^8 - 1)*x^2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (5\,x^8+3\right )\,{\left (x^8+2\,x^3-1\right )}^{1/3}}{x^2\,\left (x^8-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((5*x^8 + 3)*(2*x^3 + x^8 - 1)^(1/3))/(x^2*(x^8 - 1)),x)

[Out]

int(((5*x^8 + 3)*(2*x^3 + x^8 - 1)^(1/3))/(x^2*(x^8 - 1)), x)

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