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

Optimal. Leaf size=112 \[ \frac {3 \sqrt [3]{x^8+2 x^3-1}}{x}+\log \left (\sqrt [3]{x^8+2 x^3-1}-x\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^8+2 x^3-1}+x}\right )-\frac {1}{2} \log \left (x^2+\sqrt [3]{x^8+2 x^3-1} x+\left (x^8+2 x^3-1\right )^{2/3}\right ) \]

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Rubi [F]  time = 0.99, 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^3+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^3 + x^8)),x]

[Out]

-3*Defer[Int][(-1 + 2*x^3 + x^8)^(1/3)/x^2, x] + 3*Defer[Int][(x*(-1 + 2*x^3 + x^8)^(1/3))/(-1 + x^3 + x^8), x
] + 8*Defer[Int][(x^6*(-1 + 2*x^3 + x^8)^(1/3))/(-1 + x^3 + x^8), 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^3+x^8\right )} \, dx &=\int \left (-\frac {3 \sqrt [3]{-1+2 x^3+x^8}}{x^2}+\frac {x \left (3+8 x^5\right ) \sqrt [3]{-1+2 x^3+x^8}}{-1+x^3+x^8}\right ) \, dx\\ &=-\left (3 \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{x^2} \, dx\right )+\int \frac {x \left (3+8 x^5\right ) \sqrt [3]{-1+2 x^3+x^8}}{-1+x^3+x^8} \, dx\\ &=-\left (3 \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{x^2} \, dx\right )+\int \left (\frac {3 x \sqrt [3]{-1+2 x^3+x^8}}{-1+x^3+x^8}+\frac {8 x^6 \sqrt [3]{-1+2 x^3+x^8}}{-1+x^3+x^8}\right ) \, dx\\ &=-\left (3 \int \frac {\sqrt [3]{-1+2 x^3+x^8}}{x^2} \, dx\right )+3 \int \frac {x \sqrt [3]{-1+2 x^3+x^8}}{-1+x^3+x^8} \, dx+8 \int \frac {x^6 \sqrt [3]{-1+2 x^3+x^8}}{-1+x^3+x^8} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.37, 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^3+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^3 + x^8)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 2.74, size = 112, normalized size = 1.00 \begin {gather*} \frac {3 \sqrt [3]{x^8+2 x^3-1}}{x}+\log \left (\sqrt [3]{x^8+2 x^3-1}-x\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^8+2 x^3-1}+x}\right )-\frac {1}{2} \log \left (x^2+\sqrt [3]{x^8+2 x^3-1} x+\left (x^8+2 x^3-1\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 17.21, size = 152, normalized size = 1.36 \begin {gather*} \frac {2 \, \sqrt {3} x \arctan \left (\frac {23155756059884469826063290091369873601204942180224 \, \sqrt {3} {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 61059012875773331838678659685174425801373874951458 \, \sqrt {3} {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (35248398304721470575821713544519821387080907584081 \, x^{8} + 77355782772550371408192688432791971088370316149922 \, x^{3} - 35248398304721470575821713544519821387080907584081\right )}}{3 \, {\left (20044909029062956675424368815298850195325332161233 \, x^{8} + 38996537437007387681732053612201126295409798546850 \, x^{3} - 20044909029062956675424368815298850195325332161233\right )}}\right ) + x \log \left (\frac {x^{8} + x^{3} + 3 \, {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {2}{3}} x - 1}{x^{8} + x^{3} - 1}\right ) + 6 \, {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {1}{3}}}{2 \, 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+x^3-1),x, algorithm="fricas")

[Out]

1/2*(2*sqrt(3)*x*arctan(1/3*(23155756059884469826063290091369873601204942180224*sqrt(3)*(x^8 + 2*x^3 - 1)^(1/3
)*x^2 + 61059012875773331838678659685174425801373874951458*sqrt(3)*(x^8 + 2*x^3 - 1)^(2/3)*x + sqrt(3)*(352483
98304721470575821713544519821387080907584081*x^8 + 77355782772550371408192688432791971088370316149922*x^3 - 35
248398304721470575821713544519821387080907584081))/(20044909029062956675424368815298850195325332161233*x^8 + 3
8996537437007387681732053612201126295409798546850*x^3 - 20044909029062956675424368815298850195325332161233)) +
x*log((x^8 + x^3 + 3*(x^8 + 2*x^3 - 1)^(1/3)*x^2 - 3*(x^8 + 2*x^3 - 1)^(2/3)*x - 1)/(x^8 + x^3 - 1)) + 6*(x^8
+ 2*x^3 - 1)^(1/3))/x

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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} + x^{3} - 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+x^3-1),x, algorithm="giac")

[Out]

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

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maple [C]  time = 6.58, size = 1136, normalized size = 10.14

result 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+x^3-1),x)

[Out]

3*(x^8+2*x^3-1)^(1/3)/x+(RootOf(_Z^2+_Z+1)*ln((-2*x^16*RootOf(_Z^2+_Z+1)-x^16+2*RootOf(_Z^2+_Z+1)^2*x^11-7*x^1
1*RootOf(_Z^2+_Z+1)-3*(x^16+4*x^11-2*x^8+4*x^6-4*x^3+1)^(1/3)*RootOf(_Z^2+_Z+1)*x^9-4*x^11-3*(x^16+4*x^11-2*x^
8+4*x^6-4*x^3+1)^(1/3)*x^9+4*RootOf(_Z^2+_Z+1)*x^8+4*RootOf(_Z^2+_Z+1)^2*x^6+2*x^8-6*RootOf(_Z^2+_Z+1)*x^6-6*(
x^16+4*x^11-2*x^8+4*x^6-4*x^3+1)^(1/3)*RootOf(_Z^2+_Z+1)*x^4-4*x^6-2*RootOf(_Z^2+_Z+1)^2*x^3-3*(x^16+4*x^11-2*
x^8+4*x^6-4*x^3+1)^(2/3)*RootOf(_Z^2+_Z+1)*x^2-6*(x^16+4*x^11-2*x^8+4*x^6-4*x^3+1)^(1/3)*x^4+7*RootOf(_Z^2+_Z+
1)*x^3-3*(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^2+_Z+
1)*x+4*x^3+3*(x^16+4*x^11-2*x^8+4*x^6-4*x^3+1)^(1/3)*x-2*RootOf(_Z^2+_Z+1)-1)/(x^8+2*x^3-1)/(x^8+x^3-1))-ln((2
*x^16*RootOf(_Z^2+_Z+1)+x^16+2*RootOf(_Z^2+_Z+1)^2*x^11+11*x^11*RootOf(_Z^2+_Z+1)+3*(x^16+4*x^11-2*x^8+4*x^6-4
*x^3+1)^(1/3)*RootOf(_Z^2+_Z+1)*x^9+5*x^11-4*RootOf(_Z^2+_Z+1)*x^8+4*RootOf(_Z^2+_Z+1)^2*x^6-2*x^8+14*RootOf(_
Z^2+_Z+1)*x^6+6*(x^16+4*x^11-2*x^8+4*x^6-4*x^3+1)^(1/3)*RootOf(_Z^2+_Z+1)*x^4+6*x^6-2*RootOf(_Z^2+_Z+1)^2*x^3+
3*(x^16+4*x^11-2*x^8+4*x^6-4*x^3+1)^(2/3)*RootOf(_Z^2+_Z+1)*x^2-11*RootOf(_Z^2+_Z+1)*x^3-3*(x^16+4*x^11-2*x^8+
4*x^6-4*x^3+1)^(1/3)*RootOf(_Z^2+_Z+1)*x-5*x^3+2*RootOf(_Z^2+_Z+1)+1)/(x^8+2*x^3-1)/(x^8+x^3-1))*RootOf(_Z^2+_
Z+1)-ln((2*x^16*RootOf(_Z^2+_Z+1)+x^16+2*RootOf(_Z^2+_Z+1)^2*x^11+11*x^11*RootOf(_Z^2+_Z+1)+3*(x^16+4*x^11-2*x
^8+4*x^6-4*x^3+1)^(1/3)*RootOf(_Z^2+_Z+1)*x^9+5*x^11-4*RootOf(_Z^2+_Z+1)*x^8+4*RootOf(_Z^2+_Z+1)^2*x^6-2*x^8+1
4*RootOf(_Z^2+_Z+1)*x^6+6*(x^16+4*x^11-2*x^8+4*x^6-4*x^3+1)^(1/3)*RootOf(_Z^2+_Z+1)*x^4+6*x^6-2*RootOf(_Z^2+_Z
+1)^2*x^3+3*(x^16+4*x^11-2*x^8+4*x^6-4*x^3+1)^(2/3)*RootOf(_Z^2+_Z+1)*x^2-11*RootOf(_Z^2+_Z+1)*x^3-3*(x^16+4*x
^11-2*x^8+4*x^6-4*x^3+1)^(1/3)*RootOf(_Z^2+_Z+1)*x-5*x^3+2*RootOf(_Z^2+_Z+1)+1)/(x^8+2*x^3-1)/(x^8+x^3-1)))/(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*} \int \frac {{\left (5 \, x^{8} + 3\right )} {\left (x^{8} + 2 \, x^{3} - 1\right )}^{\frac {1}{3}}}{{\left (x^{8} + x^{3} - 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+x^3-1),x, algorithm="maxima")

[Out]

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

[Out]

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

[Out]

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

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