3.16.20 \(\int \frac {(1+x^3+x^8)^{2/3} (-3+5 x^8)}{x^3 (1+x^8)} \, dx\) [1520]

Optimal. Leaf size=105 \[ \frac {3 \left (1+x^3+x^8\right )^{2/3}}{2 x^2}-\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^3+x^8}}\right )+\log \left (-x+\sqrt [3]{1+x^3+x^8}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{1+x^3+x^8}+\left (1+x^3+x^8\right )^{2/3}\right ) \]

[Out]

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

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Rubi [F]
time = 1.92, 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^3+x^8\right )^{2/3} \left (-3+5 x^8\right )}{x^3 \left (1+x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-1)^(3/4)*Defer[Int][(1 + x^3 + x^8)^(2/3)/((-1)^(1/8) - x), x] + (-1)^(1/4)*Defer[Int][(1 + x^3 + x^8)^(2/3)
/((-1)^(3/8) - x), x] - (-1)^(3/4)*Defer[Int][(1 + x^3 + x^8)^(2/3)/(-(-1)^(5/8) - x), x] - (-1)^(1/4)*Defer[I
nt][(1 + x^3 + x^8)^(2/3)/(-(-1)^(7/8) - x), x] - 3*Defer[Int][(1 + x^3 + x^8)^(2/3)/x^3, x] - (-1)^(3/4)*Defe
r[Int][(1 + x^3 + x^8)^(2/3)/((-1)^(1/8) + x), x] - (-1)^(1/4)*Defer[Int][(1 + x^3 + x^8)^(2/3)/((-1)^(3/8) +
x), x] + (-1)^(3/4)*Defer[Int][(1 + x^3 + x^8)^(2/3)/(-(-1)^(5/8) + x), x] + (-1)^(1/4)*Defer[Int][(1 + x^3 +
x^8)^(2/3)/(-(-1)^(7/8) + x), x]

Rubi steps

\begin {align*} \int \frac {\left (1+x^3+x^8\right )^{2/3} \left (-3+5 x^8\right )}{x^3 \left (1+x^8\right )} \, dx &=\int \left (-\frac {3 \left (1+x^3+x^8\right )^{2/3}}{x^3}+\frac {8 x^5 \left (1+x^3+x^8\right )^{2/3}}{1+x^8}\right ) \, dx\\ &=-\left (3 \int \frac {\left (1+x^3+x^8\right )^{2/3}}{x^3} \, dx\right )+8 \int \frac {x^5 \left (1+x^3+x^8\right )^{2/3}}{1+x^8} \, dx\\ &=-\left (3 \int \frac {\left (1+x^3+x^8\right )^{2/3}}{x^3} \, dx\right )+8 \int \left (\frac {x \left (1+x^3+x^8\right )^{2/3}}{2 \left (-i+x^4\right )}+\frac {x \left (1+x^3+x^8\right )^{2/3}}{2 \left (i+x^4\right )}\right ) \, dx\\ &=-\left (3 \int \frac {\left (1+x^3+x^8\right )^{2/3}}{x^3} \, dx\right )+4 \int \frac {x \left (1+x^3+x^8\right )^{2/3}}{-i+x^4} \, dx+4 \int \frac {x \left (1+x^3+x^8\right )^{2/3}}{i+x^4} \, dx\\ &=-\left (3 \int \frac {\left (1+x^3+x^8\right )^{2/3}}{x^3} \, dx\right )+4 \int \left (-\frac {(-1)^{3/4} x \left (1+x^3+x^8\right )^{2/3}}{2 \left (-\sqrt [4]{-1}+x^2\right )}+\frac {(-1)^{3/4} x \left (1+x^3+x^8\right )^{2/3}}{2 \left (\sqrt [4]{-1}+x^2\right )}\right ) \, dx+4 \int \left (-\frac {\sqrt [4]{-1} x \left (1+x^3+x^8\right )^{2/3}}{2 \left (-(-1)^{3/4}+x^2\right )}+\frac {\sqrt [4]{-1} x \left (1+x^3+x^8\right )^{2/3}}{2 \left ((-1)^{3/4}+x^2\right )}\right ) \, dx\\ &=-\left (3 \int \frac {\left (1+x^3+x^8\right )^{2/3}}{x^3} \, dx\right )-\left (2 \sqrt [4]{-1}\right ) \int \frac {x \left (1+x^3+x^8\right )^{2/3}}{-(-1)^{3/4}+x^2} \, dx+\left (2 \sqrt [4]{-1}\right ) \int \frac {x \left (1+x^3+x^8\right )^{2/3}}{(-1)^{3/4}+x^2} \, dx-\left (2 (-1)^{3/4}\right ) \int \frac {x \left (1+x^3+x^8\right )^{2/3}}{-\sqrt [4]{-1}+x^2} \, dx+\left (2 (-1)^{3/4}\right ) \int \frac {x \left (1+x^3+x^8\right )^{2/3}}{\sqrt [4]{-1}+x^2} \, dx\\ &=-\left (3 \int \frac {\left (1+x^3+x^8\right )^{2/3}}{x^3} \, dx\right )-\left (2 \sqrt [4]{-1}\right ) \int \left (-\frac {\left (1+x^3+x^8\right )^{2/3}}{2 \left ((-1)^{3/8}-x\right )}+\frac {\left (1+x^3+x^8\right )^{2/3}}{2 \left ((-1)^{3/8}+x\right )}\right ) \, dx+\left (2 \sqrt [4]{-1}\right ) \int \left (-\frac {\left (1+x^3+x^8\right )^{2/3}}{2 \left (-(-1)^{7/8}-x\right )}+\frac {\left (1+x^3+x^8\right )^{2/3}}{2 \left (-(-1)^{7/8}+x\right )}\right ) \, dx-\left (2 (-1)^{3/4}\right ) \int \left (-\frac {\left (1+x^3+x^8\right )^{2/3}}{2 \left (\sqrt [8]{-1}-x\right )}+\frac {\left (1+x^3+x^8\right )^{2/3}}{2 \left (\sqrt [8]{-1}+x\right )}\right ) \, dx+\left (2 (-1)^{3/4}\right ) \int \left (-\frac {\left (1+x^3+x^8\right )^{2/3}}{2 \left (-(-1)^{5/8}-x\right )}+\frac {\left (1+x^3+x^8\right )^{2/3}}{2 \left (-(-1)^{5/8}+x\right )}\right ) \, dx\\ &=-\left (3 \int \frac {\left (1+x^3+x^8\right )^{2/3}}{x^3} \, dx\right )+\sqrt [4]{-1} \int \frac {\left (1+x^3+x^8\right )^{2/3}}{(-1)^{3/8}-x} \, dx-\sqrt [4]{-1} \int \frac {\left (1+x^3+x^8\right )^{2/3}}{-(-1)^{7/8}-x} \, dx-\sqrt [4]{-1} \int \frac {\left (1+x^3+x^8\right )^{2/3}}{(-1)^{3/8}+x} \, dx+\sqrt [4]{-1} \int \frac {\left (1+x^3+x^8\right )^{2/3}}{-(-1)^{7/8}+x} \, dx+(-1)^{3/4} \int \frac {\left (1+x^3+x^8\right )^{2/3}}{\sqrt [8]{-1}-x} \, dx-(-1)^{3/4} \int \frac {\left (1+x^3+x^8\right )^{2/3}}{-(-1)^{5/8}-x} \, dx-(-1)^{3/4} \int \frac {\left (1+x^3+x^8\right )^{2/3}}{\sqrt [8]{-1}+x} \, dx+(-1)^{3/4} \int \frac {\left (1+x^3+x^8\right )^{2/3}}{-(-1)^{5/8}+x} \, dx\\ \end {align*}

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Mathematica [A]
time = 1.38, size = 105, normalized size = 1.00 \begin {gather*} \frac {3 \left (1+x^3+x^8\right )^{2/3}}{2 x^2}-\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^3+x^8}}\right )+\log \left (-x+\sqrt [3]{1+x^3+x^8}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{1+x^3+x^8}+\left (1+x^3+x^8\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 28.50, size = 280, normalized size = 2.67

method	result	size
risch	\(\frac {3 \left (x^{8}+x^{3}+1\right )^{\frac {2}{3}}}{2 x^{2}}+\ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{8}-x^{8}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+\left (x^{8}+x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{8}+x^{3}+1\right )^{\frac {1}{3}} x^{2}-3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+2 \left (x^{8}+x^{3}+1\right )^{\frac {2}{3}} x -\left (x^{8}+x^{3}+1\right )^{\frac {1}{3}} x^{2}-2 x^{3}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1}{x^{8}+1}\right )+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{8}+x^{8}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+\left (x^{8}+x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{8}+x^{3}+1\right )^{\frac {1}{3}} x^{2}-\left (x^{8}+x^{3}+1\right )^{\frac {2}{3}} x -\left (x^{8}+x^{3}+1\right )^{\frac {1}{3}} x^{2}+x^{3}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1}{x^{8}+1}\right )\)	\(280\)
trager	\(\text {Expression too large to display}\)	\(632\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Fricas [A]
time = 6.54, size = 142, normalized size = 1.35 \begin {gather*} -\frac {2 \, \sqrt {3} x^{2} \arctan \left (-\frac {137873421075913623962723091849713877803864238548587911957688 \, \sqrt {3} {\left (x^{8} + x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 404258375252242985308203241426570926701619857965304026905546 \, \sqrt {3} {\left (x^{8} + x^{3} + 1\right )}^{\frac {2}{3}} x - \sqrt {3} {\left (82882407811392064917283059085655866224123024545593970500905 \, x^{8} + 133192477088164680672740074788428524448877809708358057473929 \, x^{3} + 82882407811392064917283059085655866224123024545593970500905\right )}}{3 \, {\left (260722961671046910462256771296925520157489755605248242108289 \, x^{8} + 271065898164078304635463166638142402252742048256945969431617 \, x^{3} + 260722961671046910462256771296925520157489755605248242108289\right )}}\right ) - x^{2} \log \left (\frac {x^{8} + 3 \, {\left (x^{8} + x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{8} + x^{3} + 1\right )}^{\frac {2}{3}} x + 1}{x^{8} + 1}\right ) - 3 \, {\left (x^{8} + x^{3} + 1\right )}^{\frac {2}{3}}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*sqrt(3)*x^2*arctan(-1/3*(137873421075913623962723091849713877803864238548587911957688*sqrt(3)*(x^8 + x
^3 + 1)^(1/3)*x^2 - 404258375252242985308203241426570926701619857965304026905546*sqrt(3)*(x^8 + x^3 + 1)^(2/3)
*x - sqrt(3)*(82882407811392064917283059085655866224123024545593970500905*x^8 + 133192477088164680672740074788
428524448877809708358057473929*x^3 + 82882407811392064917283059085655866224123024545593970500905))/(2607229616
71046910462256771296925520157489755605248242108289*x^8 + 27106589816407830463546316663814240225274204825694596
9431617*x^3 + 260722961671046910462256771296925520157489755605248242108289)) - x^2*log((x^8 + 3*(x^8 + x^3 + 1
)^(1/3)*x^2 - 3*(x^8 + x^3 + 1)^(2/3)*x + 1)/(x^8 + 1)) - 3*(x^8 + x^3 + 1)^(2/3))/x^2

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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 ) \left (x^{8} + x^{3} + 1\right )^{\frac {2}{3}}}{x^{3} \left (x^{8} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate((5*x^8 - 3)*(x^8 + x^3 + 1)^(2/3)/((x^8 + 1)*x^3), 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+x^3+1\right )}^{2/3}}{x^3\,\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)*(x^3 + x^8 + 1)^(2/3))/(x^3*(x^8 + 1)),x)

[Out]

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

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