3.19.27 \(\int \frac {(2+x^3+4 x^6) \sqrt [3]{x+2 x^3-x^4-x^7}}{(-1-2 x^2+x^3+x^6) (-1-x^2+x^3+x^6)} \, dx\) [1827]

Optimal. Leaf size=124 \[ -\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+2 x^3-x^4-x^7}}\right )-\log \left (-x+\sqrt [3]{x+2 x^3-x^4-x^7}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{x+2 x^3-x^4-x^7}+\left (x+2 x^3-x^4-x^7\right )^{2/3}\right ) \]

[Out]

-3^(1/2)*arctan(3^(1/2)*x/(x+2*(-x^7-x^4+2*x^3+x)^(1/3)))-ln(-x+(-x^7-x^4+2*x^3+x)^(1/3))+1/2*ln(x^2+x*(-x^7-x
^4+2*x^3+x)^(1/3)+(-x^7-x^4+2*x^3+x)^(2/3))

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Rubi [F]
time = 2.95, 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 (2+x^3+4 x^6\right ) \sqrt [3]{x+2 x^3-x^4-x^7}}{\left (-1-2 x^2+x^3+x^6\right ) \left (-1-x^2+x^3+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((2 + x^3 + 4*x^6)*(x + 2*x^3 - x^4 - x^7)^(1/3))/((-1 - 2*x^2 + x^3 + x^6)*(-1 - x^2 + x^3 + x^6)),x]

[Out]

-(((x + 2*x^3 - x^4 - x^7)^(1/3)*Defer[Subst][Defer[Int][1/((-1 + x)*(1 + 2*x^6 - x^9 - x^18)^(2/3)), x], x, x
^(1/3)])/(x^(1/3)*(1 + 2*x^2 - x^3 - x^6)^(1/3))) - (12*(x + 2*x^3 - x^4 - x^7)^(1/3)*Defer[Subst][Defer[Int][
x^3/(1 + 2*x^6 - x^9 - x^18)^(2/3), x], x, x^(1/3)])/(x^(1/3)*(1 + 2*x^2 - x^3 - x^6)^(1/3)) + ((1 - I*Sqrt[3]
)*(x + 2*x^3 - x^4 - x^7)^(1/3)*Defer[Subst][Defer[Int][1/((1 - I*Sqrt[3] + 2*x)*(1 + 2*x^6 - x^9 - x^18)^(2/3
)), x], x, x^(1/3)])/(x^(1/3)*(1 + 2*x^2 - x^3 - x^6)^(1/3)) + ((1 + I*Sqrt[3])*(x + 2*x^3 - x^4 - x^7)^(1/3)*
Defer[Subst][Defer[Int][1/((1 + I*Sqrt[3] + 2*x)*(1 + 2*x^6 - x^9 - x^18)^(2/3)), x], x, x^(1/3)])/(x^(1/3)*(1
+ 2*x^2 - x^3 - x^6)^(1/3)) - (3*(x + 2*x^3 - x^4 - x^7)^(1/3)*Defer[Subst][Defer[Int][1/((1 + x^3 + 2*x^6 +
x^9 + x^12 + x^15)*(1 + 2*x^6 - x^9 - x^18)^(2/3)), x], x, x^(1/3)])/(x^(1/3)*(1 + 2*x^2 - x^3 - x^6)^(1/3)) +
(12*(x + 2*x^3 - x^4 - x^7)^(1/3)*Defer[Subst][Defer[Int][x^3/((1 + x^3 + 2*x^6 + x^9 + x^12 + x^15)*(1 + 2*x
^6 - x^9 - x^18)^(2/3)), x], x, x^(1/3)])/(x^(1/3)*(1 + 2*x^2 - x^3 - x^6)^(1/3)) + (6*(x + 2*x^3 - x^4 - x^7)
^(1/3)*Defer[Subst][Defer[Int][x^6/((1 + x^3 + 2*x^6 + x^9 + x^12 + x^15)*(1 + 2*x^6 - x^9 - x^18)^(2/3)), x],
x, x^(1/3)])/(x^(1/3)*(1 + 2*x^2 - x^3 - x^6)^(1/3)) + (15*(x + 2*x^3 - x^4 - x^7)^(1/3)*Defer[Subst][Defer[I
nt][x^9/((1 + x^3 + 2*x^6 + x^9 + x^12 + x^15)*(1 + 2*x^6 - x^9 - x^18)^(2/3)), x], x, x^(1/3)])/(x^(1/3)*(1 +
2*x^2 - x^3 - x^6)^(1/3)) + (3*(x + 2*x^3 - x^4 - x^7)^(1/3)*Defer[Subst][Defer[Int][x^12/((1 + x^3 + 2*x^6 +
x^9 + x^12 + x^15)*(1 + 2*x^6 - x^9 - x^18)^(2/3)), x], x, x^(1/3)])/(x^(1/3)*(1 + 2*x^2 - x^3 - x^6)^(1/3))

Rubi steps

\begin {align*} \int \frac {\left (2+x^3+4 x^6\right ) \sqrt [3]{x+2 x^3-x^4-x^7}}{\left (-1-2 x^2+x^3+x^6\right ) \left (-1-x^2+x^3+x^6\right )} \, dx &=\frac {\sqrt [3]{x+2 x^3-x^4-x^7} \int \frac {\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6} \left (2+x^3+4 x^6\right )}{\left (-1-2 x^2+x^3+x^6\right ) \left (-1-x^2+x^3+x^6\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}\\ &=-\frac {\sqrt [3]{x+2 x^3-x^4-x^7} \int \frac {\sqrt [3]{x} \left (2+x^3+4 x^6\right )}{\left (1+2 x^2-x^3-x^6\right )^{2/3} \left (-1-x^2+x^3+x^6\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}\\ &=-\frac {\left (3 \sqrt [3]{x+2 x^3-x^4-x^7}\right ) \text {Subst}\left (\int \frac {x^3 \left (2+x^9+4 x^{18}\right )}{\left (1+2 x^6-x^9-x^{18}\right )^{2/3} \left (-1-x^6+x^9+x^{18}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}\\ &=-\frac {\left (3 \sqrt [3]{x+2 x^3-x^4-x^7}\right ) \text {Subst}\left (\int \left (\frac {4 x^3}{\left (1+2 x^6-x^9-x^{18}\right )^{2/3}}+\frac {x^3 \left (6+4 x^6-3 x^9\right )}{\left (1+2 x^6-x^9-x^{18}\right )^{2/3} \left (-1-x^6+x^9+x^{18}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}\\ &=-\frac {\left (3 \sqrt [3]{x+2 x^3-x^4-x^7}\right ) \text {Subst}\left (\int \frac {x^3 \left (6+4 x^6-3 x^9\right )}{\left (1+2 x^6-x^9-x^{18}\right )^{2/3} \left (-1-x^6+x^9+x^{18}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}-\frac {\left (12 \sqrt [3]{x+2 x^3-x^4-x^7}\right ) \text {Subst}\left (\int \frac {x^3}{\left (1+2 x^6-x^9-x^{18}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}\\ &=-\frac {\left (3 \sqrt [3]{x+2 x^3-x^4-x^7}\right ) \text {Subst}\left (\int \left (\frac {1}{3 (-1+x) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}}+\frac {-2-x}{3 \left (1+x+x^2\right ) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}}+\frac {1-4 x^3-2 x^6-5 x^9-x^{12}}{\left (1+x^3+2 x^6+x^9+x^{12}+x^{15}\right ) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}-\frac {\left (12 \sqrt [3]{x+2 x^3-x^4-x^7}\right ) \text {Subst}\left (\int \frac {x^3}{\left (1+2 x^6-x^9-x^{18}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}\\ &=-\frac {\sqrt [3]{x+2 x^3-x^4-x^7} \text {Subst}\left (\int \frac {1}{(-1+x) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}-\frac {\sqrt [3]{x+2 x^3-x^4-x^7} \text {Subst}\left (\int \frac {-2-x}{\left (1+x+x^2\right ) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}-\frac {\left (3 \sqrt [3]{x+2 x^3-x^4-x^7}\right ) \text {Subst}\left (\int \frac {1-4 x^3-2 x^6-5 x^9-x^{12}}{\left (1+x^3+2 x^6+x^9+x^{12}+x^{15}\right ) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}-\frac {\left (12 \sqrt [3]{x+2 x^3-x^4-x^7}\right ) \text {Subst}\left (\int \frac {x^3}{\left (1+2 x^6-x^9-x^{18}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}\\ &=-\frac {\sqrt [3]{x+2 x^3-x^4-x^7} \text {Subst}\left (\int \frac {1}{(-1+x) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}-\frac {\sqrt [3]{x+2 x^3-x^4-x^7} \text {Subst}\left (\int \left (\frac {-1+i \sqrt {3}}{\left (1-i \sqrt {3}+2 x\right ) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}}+\frac {-1-i \sqrt {3}}{\left (1+i \sqrt {3}+2 x\right ) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}-\frac {\left (3 \sqrt [3]{x+2 x^3-x^4-x^7}\right ) \text {Subst}\left (\int \left (\frac {1}{\left (1+x^3+2 x^6+x^9+x^{12}+x^{15}\right ) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}}-\frac {4 x^3}{\left (1+x^3+2 x^6+x^9+x^{12}+x^{15}\right ) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}}-\frac {2 x^6}{\left (1+x^3+2 x^6+x^9+x^{12}+x^{15}\right ) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}}-\frac {5 x^9}{\left (1+x^3+2 x^6+x^9+x^{12}+x^{15}\right ) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}}-\frac {x^{12}}{\left (1+x^3+2 x^6+x^9+x^{12}+x^{15}\right ) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}-\frac {\left (12 \sqrt [3]{x+2 x^3-x^4-x^7}\right ) \text {Subst}\left (\int \frac {x^3}{\left (1+2 x^6-x^9-x^{18}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}\\ &=-\frac {\sqrt [3]{x+2 x^3-x^4-x^7} \text {Subst}\left (\int \frac {1}{(-1+x) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}-\frac {\left (3 \sqrt [3]{x+2 x^3-x^4-x^7}\right ) \text {Subst}\left (\int \frac {1}{\left (1+x^3+2 x^6+x^9+x^{12}+x^{15}\right ) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}+\frac {\left (3 \sqrt [3]{x+2 x^3-x^4-x^7}\right ) \text {Subst}\left (\int \frac {x^{12}}{\left (1+x^3+2 x^6+x^9+x^{12}+x^{15}\right ) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}+\frac {\left (6 \sqrt [3]{x+2 x^3-x^4-x^7}\right ) \text {Subst}\left (\int \frac {x^6}{\left (1+x^3+2 x^6+x^9+x^{12}+x^{15}\right ) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}-\frac {\left (12 \sqrt [3]{x+2 x^3-x^4-x^7}\right ) \text {Subst}\left (\int \frac {x^3}{\left (1+2 x^6-x^9-x^{18}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}+\frac {\left (12 \sqrt [3]{x+2 x^3-x^4-x^7}\right ) \text {Subst}\left (\int \frac {x^3}{\left (1+x^3+2 x^6+x^9+x^{12}+x^{15}\right ) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}+\frac {\left (15 \sqrt [3]{x+2 x^3-x^4-x^7}\right ) \text {Subst}\left (\int \frac {x^9}{\left (1+x^3+2 x^6+x^9+x^{12}+x^{15}\right ) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}-\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x+2 x^3-x^4-x^7}\right ) \text {Subst}\left (\int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}-\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x+2 x^3-x^4-x^7}\right ) \text {Subst}\left (\int \frac {1}{\left (1-i \sqrt {3}+2 x\right ) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}\\ \end {align*}

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Mathematica [A]
time = 1.10, size = 169, normalized size = 1.36 \begin {gather*} -\frac {x^{2/3} \left (-1-2 x^2+x^3+x^6\right )^{2/3} \left (2 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}-2 \sqrt [3]{-1-2 x^2+x^3+x^6}}\right )+2 \log \left (x^{2/3}+\sqrt [3]{-1-2 x^2+x^3+x^6}\right )-\log \left (x^{4/3}-x^{2/3} \sqrt [3]{-1-2 x^2+x^3+x^6}+\left (-1-2 x^2+x^3+x^6\right )^{2/3}\right )\right )}{2 \left (-x \left (-1-2 x^2+x^3+x^6\right )\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((2 + x^3 + 4*x^6)*(x + 2*x^3 - x^4 - x^7)^(1/3))/((-1 - 2*x^2 + x^3 + x^6)*(-1 - x^2 + x^3 + x^6)),
x]

[Out]

-1/2*(x^(2/3)*(-1 - 2*x^2 + x^3 + x^6)^(2/3)*(2*Sqrt[3]*ArcTan[(Sqrt[3]*x^(2/3))/(x^(2/3) - 2*(-1 - 2*x^2 + x^
3 + x^6)^(1/3))] + 2*Log[x^(2/3) + (-1 - 2*x^2 + x^3 + x^6)^(1/3)] - Log[x^(4/3) - x^(2/3)*(-1 - 2*x^2 + x^3 +
x^6)^(1/3) + (-1 - 2*x^2 + x^3 + x^6)^(2/3)]))/(-(x*(-1 - 2*x^2 + x^3 + x^6)))^(2/3)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 15.72, size = 809, normalized size = 6.52

method	result	size
trager	\(\text {Expression too large to display}\)	\(809\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^6+x^3+2)*(-x^7-x^4+2*x^3+x)^(1/3)/(x^6+x^3-2*x^2-1)/(x^6+x^3-x^2-1),x,method=_RETURNVERBOSE)

[Out]

RootOf(_Z^2-_Z+1)*ln(-(730028245413184482054075083395532*RootOf(_Z^2-_Z+1)^2*x^6+84223449014871390439295018407
03761*RootOf(_Z^2-_Z+1)*x^6-34597182845770647978929558180859956*x^6+730028245413184482054075083395532*RootOf(_
Z^2-_Z+1)^2*x^3-12958001356084024556459832730270693*RootOf(_Z^2-_Z+1)^2*x^2+8422344901487139043929501840703761
*RootOf(_Z^2-_Z+1)*x^3+17022464797383185409016479416056902*RootOf(_Z^2-_Z+1)*(-x^7-x^4+2*x^3+x)^(2/3)[phone]
797383185409016479416056902*RootOf(_Z^2-_Z+1)*(-x^7-x^4+2*x^3+x)^(1/3)*x+20828093006566886439492735222228302*x
^2*RootOf(_Z^2-_Z+1)-34597182845770647978929558180859956*x^3-730028245413184482054075083395532*RootOf(_Z^2-_Z+
1)^2+44479584238084155986967210188354781*(-x^7-x^4+2*x^3+x)^(2/3)+44479584238084155986967210188354781*x*(-x^7-
x^4+2*x^3+x)^(1/3)+66848793973183963891491010722339576*x^2-8422344901487139043929501840703761*RootOf(_Z^2-_Z+1
)+34597182845770647978929558180859956)/(-1+x)/(x^5+x^4+x^3+2*x^2+x+1))-ln(-(3400786525179064264005064770847424
*RootOf(_Z^2-_Z+1)^2*x^6-56723962910512233760212677940167972*RootOf(_Z^2-_Z+1)*x^6+497948084383360527335932978
39306443*x^6+3400786525179064264005064770847424*RootOf(_Z^2-_Z+1)^2*x^3-60363960821928390686089899682541776*Ro
otOf(_Z^2-_Z+1)^2*x^2-56723962910512233760212677940167972*RootOf(_Z^2-_Z+1)*x^3+997171982984901579657958462377
79567*RootOf(_Z^2-_Z+1)*(-x^7-x^4+2*x^3+x)^(2/3)+99717198298490157965795846237779567*RootOf(_Z^2-_Z+1)*(-x^7-x
^4+2*x^3+x)^(1/3)*x+213404335505751718148093359089641891*x^2*RootOf(_Z^2-_Z+1)+4979480843833605273359329783930
6443*x^3-3400786525179064264005064770847424*RootOf(_Z^2-_Z+1)^2-39464867440980807442968852967597490*(-x^7-x^4+
2*x^3+x)^(2/3)-39464867440980807442968852967597490*x*(-x^7-x^4+2*x^3+x)^(1/3)-14622285017606618659864698571859
8285*x^2+56723962910512233760212677940167972*RootOf(_Z^2-_Z+1)-49794808438336052733593297839306443)/(-1+x)/(x^
5+x^4+x^3+2*x^2+x+1))*RootOf(_Z^2-_Z+1)+ln(-(3400786525179064264005064770847424*RootOf(_Z^2-_Z+1)^2*x^6-567239
62910512233760212677940167972*RootOf(_Z^2-_Z+1)*x^6+49794808438336052733593297839306443*x^6+340078652517906426
4005064770847424*RootOf(_Z^2-_Z+1)^2*x^3-60363960821928390686089899682541776*RootOf(_Z^2-_Z+1)^2*x^2-567239629
10512233760212677940167972*RootOf(_Z^2-_Z+1)*x^3+99717198298490157965795846237779567*RootOf(_Z^2-_Z+1)*(-x^7-x
^4+2*x^3+x)^(2/3)+99717198298490157965795846237779567*RootOf(_Z^2-_Z+1)*(-x^7-x^4+2*x^3+x)^(1/3)*x+21340433550
5751718148093359089641891*x^2*RootOf(_Z^2-_Z+1)+49794808438336052733593297839306443*x^3-3400786525179064264005
064770847424*RootOf(_Z^2-_Z+1)^2-39464867440980807442968852967597490*(-x^7-x^4+2*x^3+x)^(2/3)-3946486744098080
7442968852967597490*x*(-x^7-x^4+2*x^3+x)^(1/3)-146222850176066186598646985718598285*x^2+5672396291051223376021
2677940167972*RootOf(_Z^2-_Z+1)-49794808438336052733593297839306443)/(-1+x)/(x^5+x^4+x^3+2*x^2+x+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((4*x^6+x^3+2)*(-x^7-x^4+2*x^3+x)^(1/3)/(x^6+x^3-2*x^2-1)/(x^6+x^3-x^2-1),x, algorithm="maxima")

[Out]

integrate((-x^7 - x^4 + 2*x^3 + x)^(1/3)*(4*x^6 + x^3 + 2)/((x^6 + x^3 - x^2 - 1)*(x^6 + x^3 - 2*x^2 - 1)), x)

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Fricas [A]
time = 1.63, size = 174, normalized size = 1.40 \begin {gather*} \sqrt {3} \arctan \left (-\frac {70 \, \sqrt {3} {\left (-x^{7} - x^{4} + 2 \, x^{3} + x\right )}^{\frac {1}{3}} x - \sqrt {3} {\left (32 \, x^{6} + 32 \, x^{3} - 39 \, x^{2} - 32\right )} - 56 \, \sqrt {3} {\left (-x^{7} - x^{4} + 2 \, x^{3} + x\right )}^{\frac {2}{3}}}{64 \, x^{6} + 64 \, x^{3} - 253 \, x^{2} - 64}\right ) - \frac {1}{2} \, \log \left (\frac {x^{6} + x^{3} - x^{2} - 3 \, {\left (-x^{7} - x^{4} + 2 \, x^{3} + x\right )}^{\frac {1}{3}} x + 3 \, {\left (-x^{7} - x^{4} + 2 \, x^{3} + x\right )}^{\frac {2}{3}} - 1}{x^{6} + x^{3} - x^{2} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^6+x^3+2)*(-x^7-x^4+2*x^3+x)^(1/3)/(x^6+x^3-2*x^2-1)/(x^6+x^3-x^2-1),x, algorithm="fricas")

[Out]

sqrt(3)*arctan(-(70*sqrt(3)*(-x^7 - x^4 + 2*x^3 + x)^(1/3)*x - sqrt(3)*(32*x^6 + 32*x^3 - 39*x^2 - 32) - 56*sq
rt(3)*(-x^7 - x^4 + 2*x^3 + x)^(2/3))/(64*x^6 + 64*x^3 - 253*x^2 - 64)) - 1/2*log((x^6 + x^3 - x^2 - 3*(-x^7 -
x^4 + 2*x^3 + x)^(1/3)*x + 3*(-x^7 - x^4 + 2*x^3 + x)^(2/3) - 1)/(x^6 + x^3 - x^2 - 1))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{- x \left (x^{6} + x^{3} - 2 x^{2} - 1\right )} \left (4 x^{6} + x^{3} + 2\right )}{\left (x - 1\right ) \left (x^{6} + x^{3} - 2 x^{2} - 1\right ) \left (x^{5} + x^{4} + x^{3} + 2 x^{2} + x + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**6+x**3+2)*(-x**7-x**4+2*x**3+x)**(1/3)/(x**6+x**3-2*x**2-1)/(x**6+x**3-x**2-1),x)

[Out]

Integral((-x*(x**6 + x**3 - 2*x**2 - 1))**(1/3)*(4*x**6 + x**3 + 2)/((x - 1)*(x**6 + x**3 - 2*x**2 - 1)*(x**5
+ x**4 + x**3 + 2*x**2 + x + 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((4*x^6+x^3+2)*(-x^7-x^4+2*x^3+x)^(1/3)/(x^6+x^3-2*x^2-1)/(x^6+x^3-x^2-1),x, algorithm="giac")

[Out]

integrate((-x^7 - x^4 + 2*x^3 + x)^(1/3)*(4*x^6 + x^3 + 2)/((x^6 + x^3 - x^2 - 1)*(x^6 + x^3 - 2*x^2 - 1)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (4\,x^6+x^3+2\right )\,{\left (-x^7-x^4+2\,x^3+x\right )}^{1/3}}{\left (-x^6-x^3+2\,x^2+1\right )\,\left (-x^6-x^3+x^2+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3 + 4*x^6 + 2)*(x + 2*x^3 - x^4 - x^7)^(1/3))/((2*x^2 - x^3 - x^6 + 1)*(x^2 - x^3 - x^6 + 1)),x)

[Out]

int(((x^3 + 4*x^6 + 2)*(x + 2*x^3 - x^4 - x^7)^(1/3))/((2*x^2 - x^3 - x^6 + 1)*(x^2 - x^3 - x^6 + 1)), x)

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