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

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 = 2.73, size = 112, normalized size = 1.00 \begin {gather*} \frac {3 \sqrt [3]{-1+2 x^3+x^8}}{x}+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+2 x^3+x^8}}\right )+\log \left (-x+\sqrt [3]{-1+2 x^3+x^8}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{-1+2 x^3+x^8}+\left (-1+2 x^3+x^8\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 = 17.48, 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 = 57.87, size = 611, normalized size = 5.46

method	result	size
trager	\(\frac {3 \left (x^{8}+2 x^{3}-1\right )^{\frac {1}{3}}}{x}-3 \ln \left (\frac {5377901481395524539 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{8}-10267890788505543624 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{8}+4455082089494573732 x^{8}-487912174285505454 \left (x^{8}+2 x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -487912174285505454 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{8}+2 x^{3}-1\right )^{\frac {1}{3}} x^{2}-12548436789922890591 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+11735249832780381501 \left (x^{8}+2 x^{3}-1\right )^{\frac {2}{3}} x +11735249832780381501 \left (x^{8}+2 x^{3}-1\right )^{\frac {1}{3}} x^{2}+15592787313231008062 x^{3}-5377901481395524539 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+10267890788505543624 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-4455082089494573732}{x^{8}+x^{3}-1}\right ) \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (\frac {5377901481395524539 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{8}+13853158442769226650 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{8}+8475256961373702111 x^{8}+487912174285505454 \left (x^{8}+2 x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x +487912174285505454 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{8}+2 x^{3}-1\right )^{\frac {1}{3}} x^{2}+12548436789922890591 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+11897887224208883319 \left (x^{8}+2 x^{3}-1\right )^{\frac {2}{3}} x +11897887224208883319 \left (x^{8}+2 x^{3}-1\right )^{\frac {1}{3}} x^{2}+19775599576538638259 x^{3}-5377901481395524539 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-13853158442769226650 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-8475256961373702111}{x^{8}+x^{3}-1}\right )-\ln \left (\frac {5377901481395524539 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{8}-10267890788505543624 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{8}+4455082089494573732 x^{8}-487912174285505454 \left (x^{8}+2 x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -487912174285505454 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{8}+2 x^{3}-1\right )^{\frac {1}{3}} x^{2}-12548436789922890591 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+11735249832780381501 \left (x^{8}+2 x^{3}-1\right )^{\frac {2}{3}} x +11735249832780381501 \left (x^{8}+2 x^{3}-1\right )^{\frac {1}{3}} x^{2}+15592787313231008062 x^{3}-5377901481395524539 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+10267890788505543624 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-4455082089494573732}{x^{8}+x^{3}-1}\right )\)	\(611\)
risch	\(\frac {3 \left (x^{8}+2 x^{3}-1\right )^{\frac {1}{3}}}{x}+\frac {\left (\ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{16}+x^{16}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{11}-5 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{11}+4 x^{11}-3 \left (x^{16}+4 x^{11}-2 x^{8}+4 x^{6}-4 x^{3}+1\right )^{\frac {1}{3}} x^{9}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{8}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}-2 x^{8}-6 x^{6} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+4 x^{6}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{16}+4 x^{11}-2 x^{8}+4 x^{6}-4 x^{3}+1\right )^{\frac {2}{3}} x^{2}-6 \left (x^{16}+4 x^{11}-2 x^{8}+4 x^{6}-4 x^{3}+1\right )^{\frac {1}{3}} x^{4}+5 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 \left (x^{16}+4 x^{11}-2 x^{8}+4 x^{6}-4 x^{3}+1\right )^{\frac {2}{3}} x^{2}-4 x^{3}+3 \left (x^{16}+4 x^{11}-2 x^{8}+4 x^{6}-4 x^{3}+1\right )^{\frac {1}{3}} x -\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1}{\left (x^{8}+2 x^{3}-1\right ) \left (x^{8}+x^{3}-1\right )}\right )+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{16}-x^{16}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{11}+4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{11}+3 \left (x^{16}+4 x^{11}-2 x^{8}+4 x^{6}-4 x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{9}-5 x^{11}+3 \left (x^{16}+4 x^{11}-2 x^{8}+4 x^{6}-4 x^{3}+1\right )^{\frac {1}{3}} x^{9}-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{8}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}+2 x^{8}+4 x^{6} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+6 \left (x^{16}+4 x^{11}-2 x^{8}+4 x^{6}-4 x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}-6 x^{6}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{16}+4 x^{11}-2 x^{8}+4 x^{6}-4 x^{3}+1\right )^{\frac {2}{3}} x^{2}+6 \left (x^{16}+4 x^{11}-2 x^{8}+4 x^{6}-4 x^{3}+1\right )^{\frac {1}{3}} x^{4}-4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-3 \left (x^{16}+4 x^{11}-2 x^{8}+4 x^{6}-4 x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +5 x^{3}-3 \left (x^{16}+4 x^{11}-2 x^{8}+4 x^{6}-4 x^{3}+1\right )^{\frac {1}{3}} x +\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1}{\left (x^{8}+2 x^{3}-1\right ) \left (x^{8}+x^{3}-1\right )}\right )\right ) \left (\left (x^{8}+2 x^{3}-1\right )^{2}\right )^{\frac {1}{3}}}{\left (x^{8}+2 x^{3}-1\right )^{\frac {2}{3}}}\)	\(787\)

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,method=_RETURNVERBOSE)

[Out]

3*(x^8+2*x^3-1)^(1/3)/x-3*ln((5377901481395524539*RootOf(9*_Z^2+3*_Z+1)^2*x^8-10267890788505543624*RootOf(9*_Z
^2+3*_Z+1)*x^8+4455082089494573732*x^8-487912174285505454*(x^8+2*x^3-1)^(2/3)*RootOf(9*_Z^2+3*_Z+1)*x-48791217
4285505454*RootOf(9*_Z^2+3*_Z+1)*(x^8+2*x^3-1)^(1/3)*x^2-12548436789922890591*RootOf(9*_Z^2+3*_Z+1)*x^3+117352
49832780381501*(x^8+2*x^3-1)^(2/3)*x+11735249832780381501*(x^8+2*x^3-1)^(1/3)*x^2+15592787313231008062*x^3-537
7901481395524539*RootOf(9*_Z^2+3*_Z+1)^2+10267890788505543624*RootOf(9*_Z^2+3*_Z+1)-4455082089494573732)/(x^8+
x^3-1))*RootOf(9*_Z^2+3*_Z+1)+3*RootOf(9*_Z^2+3*_Z+1)*ln((5377901481395524539*RootOf(9*_Z^2+3*_Z+1)^2*x^8+1385
3158442769226650*RootOf(9*_Z^2+3*_Z+1)*x^8+8475256961373702111*x^8+487912174285505454*(x^8+2*x^3-1)^(2/3)*Root
Of(9*_Z^2+3*_Z+1)*x+487912174285505454*RootOf(9*_Z^2+3*_Z+1)*(x^8+2*x^3-1)^(1/3)*x^2+12548436789922890591*Root
Of(9*_Z^2+3*_Z+1)*x^3+11897887224208883319*(x^8+2*x^3-1)^(2/3)*x+11897887224208883319*(x^8+2*x^3-1)^(1/3)*x^2+
19775599576538638259*x^3-5377901481395524539*RootOf(9*_Z^2+3*_Z+1)^2-13853158442769226650*RootOf(9*_Z^2+3*_Z+1
)-8475256961373702111)/(x^8+x^3-1))-ln((5377901481395524539*RootOf(9*_Z^2+3*_Z+1)^2*x^8-10267890788505543624*R
ootOf(9*_Z^2+3*_Z+1)*x^8+4455082089494573732*x^8-487912174285505454*(x^8+2*x^3-1)^(2/3)*RootOf(9*_Z^2+3*_Z+1)*
x-487912174285505454*RootOf(9*_Z^2+3*_Z+1)*(x^8+2*x^3-1)^(1/3)*x^2-12548436789922890591*RootOf(9*_Z^2+3*_Z+1)*
x^3+11735249832780381501*(x^8+2*x^3-1)^(2/3)*x+11735249832780381501*(x^8+2*x^3-1)^(1/3)*x^2+155927873132310080
62*x^3-5377901481395524539*RootOf(9*_Z^2+3*_Z+1)^2+10267890788505543624*RootOf(9*_Z^2+3*_Z+1)-4455082089494573
732)/(x^8+x^3-1))

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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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