3.21.67 \(\int \frac {\sqrt [3]{-1+2 x^3+x^8} (3+5 x^8)}{x^2 (-1+x^8)} \, dx\)
Optimal. Leaf size=149 \[ \frac {3 \sqrt [3]{x^8+2 x^3-1}}{x}+\sqrt [3]{2} \log \left (2^{2/3} \sqrt [3]{x^8+2 x^3-1}-2 x\right )+\sqrt [3]{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^8+2 x^3-1}+x}\right )-\frac {\log \left (2 x^2+2^{2/3} \sqrt [3]{x^8+2 x^3-1} x+\sqrt [3]{2} \left (x^8+2 x^3-1\right )^{2/3}\right )}{2^{2/3}} \]
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Rubi [F] time = 1.59, 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 [F] time = 0.72, size = 0, normalized size = 0.00 \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]
Integrate[((-1 + 2*x^3 + x^8)^(1/3)*(3 + 5*x^8))/(x^2*(-1 + x^8)),x]
[Out]
Integrate[((-1 + 2*x^3 + x^8)^(1/3)*(3 + 5*x^8))/(x^2*(-1 + x^8)), x]
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IntegrateAlgebraic [A] time = 2.84, size = 149, normalized size = 1.00 \begin {gather*} \frac {3 \sqrt [3]{-1+2 x^3+x^8}}{x}+\sqrt [3]{2} \sqrt {3} \tan ^{-1}\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]
IntegrateAlgebraic[((-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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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^8+2*x^3-1)^(1/3)*(5*x^8+3)/x^2/(x^8-1),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 (5 \, x^{8} + 3\right )} {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {1}{3}}}{{\left (x^{8} - 1\right )} x^{2}}\,{d 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="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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maple [C] time = 9.58, size = 2498, normalized size = 16.77 \[\text {Expression too large to display}\]
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+4*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+8*RootOf(RootOf(_Z^3-2)^2+2*_Z*Root
Of(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x^11+6*(x^16+4*x^11-2*x^8+4*x^6-4*x^3+1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+2
*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)*x^9+6*RootOf(_Z^3-2)*x^11+24*x^11*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootO
f(_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+16*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^8-8*x^8*RootOf(RootOf(_Z^3-2)^
2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)-2*RootOf(_Z^3-2)^3*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^3-8*Roo
tOf(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+12*(x^16+4*x^11-2*x^8+4*x^6-4*x^3
+1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)*x^4+8*RootOf(_Z^3-2)*x^6+32*x^6*R
ootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)-6*(x^16+4*x^11-2*x^8+4*x^6-4*x^3+1)^(1/3)*RootOf(RootOf(_Z^
3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)*x-6*RootOf(_Z^3-2)*x^3-24*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootO
f(_Z^3-2)+4*_Z^2)*x^3+RootOf(_Z^3-2)+4*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(_Z^3-2)+2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*ln((-4*RootOf(_
Z^3-2)*x^11+2*RootOf(_Z^3-2)*x^8-4*RootOf(_Z^3-2)*x^6+4*RootOf(_Z^3-2)*x^3+16*RootOf(RootOf(_Z^3-2)^2+2*_Z*Roo
tOf(_Z^3-2)+4*_Z^2)*x^3-RootOf(_Z^3-2)*x^16-4*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)-8*RootOf(Roo
tOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x^3-RootOf(_Z^3-2)-6*(x^16+4*x^11-2*x^8+4*x^6-4*x
^3+1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)*x^9-2*RootOf(_Z^3-2)^3*RootOf(R
ootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^3-16*x^6*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)+16*
RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x^6+4*RootOf(RootOf(_Z^3-2)^2+2*_Z*Root
Of(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x^6-6*(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^2-12*(x^16+4*x^11-2*x^8+4*x^6-4*x^3+1)^(1/3)*RootOf(RootOf(_Z^3-2)^
2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)*x^4+6*(x^16+4*x^11-2*x^8+4*x^6-4*x^3+1)^(1/3)*RootOf(RootOf(_Z^3-
2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)*x+8*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*Root
Of(_Z^3-2)^2*x^11+2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x^11-6*(x^16+4*x^11-2
*x^8+4*x^6-4*x^3+1)^(2/3)*x^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^9-6*(x^16+4*x^11-2*
x^8+4*x^6-4*x^3+1)^(1/3)*RootOf(_Z^3-2)^2*x^4+3*(x^16+4*x^11-2*x^8+4*x^6-4*x^3+1)^(1/3)*RootOf(_Z^3-2)^2*x-16*
x^11*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)+8*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)*x^16)/(x^8+2*x^3-1)/(-1+x)/(1+x)/(x^2+1)/(x^4+1))-2
*ln((RootOf(_Z^3-2)*x^16+4*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+8*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*Root
Of(_Z^3-2)^2*x^11+6*(x^16+4*x^11-2*x^8+4*x^6-4*x^3+1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2
)*RootOf(_Z^3-2)*x^9+6*RootOf(_Z^3-2)*x^11+24*x^11*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)+4*RootO
f(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x^6+16*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^8-8*x^8*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_
Z^2)-2*RootOf(_Z^3-2)^3*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^3-8*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+12*(x^16+4*x^11-2*x^8+4*x^6-4*x^3+1)^(1/3)*RootOf(RootOf(_
Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)*x^4+8*RootOf(_Z^3-2)*x^6+32*x^6*RootOf(RootOf(_Z^3-2)^2+2*
_Z*RootOf(_Z^3-2)+4*_Z^2)-6*(x^16+4*x^11-2*x^8+4*x^6-4*x^3+1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2
)+4*_Z^2)*RootOf(_Z^3-2)*x-6*RootOf(_Z^3-2)*x^3-24*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^3+Roo
tOf(_Z^3-2)+4*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))/(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*} \int \frac {{\left (5 \, x^{8} + 3\right )} {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {1}{3}}}{{\left (x^{8} - 1\right )} x^{2}}\,{d 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="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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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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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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