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

Optimal. Leaf size=121 \[ \frac {\log \left (2^{2/3} \sqrt [3]{x^4+x^2}-2 x\right )}{\sqrt [3]{2}}-\frac {\log \left (2 x^2+2^{2/3} \sqrt [3]{x^4+x^2} x+\sqrt [3]{2} \left (x^4+x^2\right )^{2/3}\right )}{2 \sqrt [3]{2}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^4+x^2}+x}\right )}{\sqrt [3]{2}} \]

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Rubi [C]  time = 0.65, antiderivative size = 127, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2056, 6733, 6725, 245, 1438, 429, 465, 510} \begin {gather*} -\frac {3 \sqrt [3]{x^2+1} x^2 F_1\left (\frac {2}{3};1,\frac {1}{3};\frac {5}{3};x^2,-x^2\right )}{2 \sqrt [3]{x^4+x^2}}-\frac {6 \sqrt [3]{x^2+1} x F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};x^2,-x^2\right )}{\sqrt [3]{x^4+x^2}}+\frac {3 \sqrt [3]{x^2+1} x \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^2\right )}{\sqrt [3]{x^4+x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-6*x*(1 + x^2)^(1/3)*AppellF1[1/6, 1, 1/3, 7/6, x^2, -x^2])/(x^2 + x^4)^(1/3) - (3*x^2*(1 + x^2)^(1/3)*Appell
F1[2/3, 1, 1/3, 5/3, x^2, -x^2])/(2*(x^2 + x^4)^(1/3)) + (3*x*(1 + x^2)^(1/3)*Hypergeometric2F1[1/6, 1/3, 7/6,
-x^2])/(x^2 + x^4)^(1/3)

Rule 245

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-p, 1/n, 1/n + 1, -((b*x^n)/a)],
x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p
] || GtQ[a, 0])

Rule 429

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, -((b*x^n)/a), -((d*x^n)/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 465

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = GCD[m + 1,
n]}, Dist[1/k, Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q, x], x, x^k], x] /; k != 1] /;
FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegerQ[m]

Rule 510

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a^p*c^q
*(e*x)^(m + 1)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 1438

Int[((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + c*x^(2
*n))^p, (d/(d^2 - e^2*x^(2*n)) - (e*x^n)/(d^2 - e^2*x^(2*n)))^(-q), x], x] /; FreeQ[{a, c, d, e, n, p}, x] &&
EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[q, 0]

Rule 2056

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rule 6733

Int[(u_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k, Subst[Int[x^(k*(m + 1) - 1)*(u /. x -> x^k
), x], x, x^(1/k)], x]] /; FractionQ[m]

Rubi steps

\begin {align*} \int \frac {1+x}{(-1+x) \sqrt [3]{x^2+x^4}} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {1+x}{(-1+x) x^{2/3} \sqrt [3]{1+x^2}} \, dx}{\sqrt [3]{x^2+x^4}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1+x^3}{\left (-1+x^3\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt [3]{1+x^6}}+\frac {2}{\left (-1+x^3\right ) \sqrt [3]{1+x^6}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}}+\frac {\left (6 x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+x^3\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}}\\ &=\frac {3 x \sqrt [3]{1+x^2} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^2\right )}{\sqrt [3]{x^2+x^4}}+\frac {\left (6 x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\left (-1+x^6\right ) \sqrt [3]{1+x^6}}+\frac {x^3}{\left (-1+x^6\right ) \sqrt [3]{1+x^6}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}}\\ &=\frac {3 x \sqrt [3]{1+x^2} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^2\right )}{\sqrt [3]{x^2+x^4}}+\frac {\left (6 x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+x^6\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}}+\frac {\left (6 x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (-1+x^6\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^4}}\\ &=-\frac {6 x \sqrt [3]{1+x^2} F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};x^2,-x^2\right )}{\sqrt [3]{x^2+x^4}}+\frac {3 x \sqrt [3]{1+x^2} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^2\right )}{\sqrt [3]{x^2+x^4}}+\frac {\left (3 x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (-1+x^3\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x^2+x^4}}\\ &=-\frac {6 x \sqrt [3]{1+x^2} F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};x^2,-x^2\right )}{\sqrt [3]{x^2+x^4}}-\frac {3 x^2 \sqrt [3]{1+x^2} F_1\left (\frac {2}{3};1,\frac {1}{3};\frac {5}{3};x^2,-x^2\right )}{2 \sqrt [3]{x^2+x^4}}+\frac {3 x \sqrt [3]{1+x^2} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^2\right )}{\sqrt [3]{x^2+x^4}}\\ \end {align*}

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Mathematica [F]  time = 0.20, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x}{(-1+x) \sqrt [3]{x^2+x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.43, size = 121, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{x^2+x^4}}\right )}{\sqrt [3]{2}}+\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{x^2+x^4}\right )}{\sqrt [3]{2}}-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{x^2+x^4}+\sqrt [3]{2} \left (x^2+x^4\right )^{2/3}\right )}{2 \sqrt [3]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 + x)/((-1 + x)*(x^2 + x^4)^(1/3)),x]

[Out]

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

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fricas [B]  time = 3.81, size = 307, normalized size = 2.54 \begin {gather*} -\frac {1}{6} \cdot 4^{\frac {1}{3}} \sqrt {3} \arctan \left (\frac {3 \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (x^{4} + 2 \, x^{3} - 6 \, x^{2} + 2 \, x + 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}} + 6 \cdot 4^{\frac {1}{3}} \sqrt {3} {\left (x^{5} + 14 \, x^{4} + 6 \, x^{3} + 14 \, x^{2} + x\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} + \sqrt {3} {\left (x^{7} + 30 \, x^{6} + 51 \, x^{5} + 52 \, x^{4} + 51 \, x^{3} + 30 \, x^{2} + x\right )}}{3 \, {\left (x^{7} - 6 \, x^{6} - 93 \, x^{5} - 20 \, x^{4} - 93 \, x^{3} - 6 \, x^{2} + x\right )}}\right ) - \frac {1}{12} \cdot 4^{\frac {1}{3}} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}} {\left (x^{2} + 4 \, x + 1\right )} + 4^{\frac {2}{3}} {\left (x^{5} + 14 \, x^{4} + 6 \, x^{3} + 14 \, x^{2} + x\right )} + 24 \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{3} + x^{2} + x\right )}}{x^{5} - 4 \, x^{4} + 6 \, x^{3} - 4 \, x^{2} + x}\right ) + \frac {1}{6} \cdot 4^{\frac {1}{3}} \log \left (-\frac {3 \cdot 4^{\frac {2}{3}} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} x + 4^{\frac {1}{3}} {\left (x^{3} - 2 \, x^{2} + x\right )} - 6 \, {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}}}{x^{3} - 2 \, x^{2} + x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*4^(1/3)*sqrt(3)*arctan(1/3*(3*4^(2/3)*sqrt(3)*(x^4 + 2*x^3 - 6*x^2 + 2*x + 1)*(x^4 + x^2)^(2/3) + 6*4^(1/
3)*sqrt(3)*(x^5 + 14*x^4 + 6*x^3 + 14*x^2 + x)*(x^4 + x^2)^(1/3) + sqrt(3)*(x^7 + 30*x^6 + 51*x^5 + 52*x^4 + 5
1*x^3 + 30*x^2 + x))/(x^7 - 6*x^6 - 93*x^5 - 20*x^4 - 93*x^3 - 6*x^2 + x)) - 1/12*4^(1/3)*log((6*4^(1/3)*(x^4
+ x^2)^(2/3)*(x^2 + 4*x + 1) + 4^(2/3)*(x^5 + 14*x^4 + 6*x^3 + 14*x^2 + x) + 24*(x^4 + x^2)^(1/3)*(x^3 + x^2 +
x))/(x^5 - 4*x^4 + 6*x^3 - 4*x^2 + x)) + 1/6*4^(1/3)*log(-(3*4^(2/3)*(x^4 + x^2)^(1/3)*x + 4^(1/3)*(x^3 - 2*x
^2 + x) - 6*(x^4 + x^2)^(2/3))/(x^3 - 2*x^2 + x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(-1+x)/(x^4+x^2)^(1/3),x, algorithm="giac")

[Out]

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

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maple [C]  time = 10.00, size = 1448, normalized size = 11.97

method	result	size
trager	\(\text {Expression too large to display}\)	\(1448\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*RootOf(_Z^3-4)*ln(-(8332*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^3*x^3-1334*Roo
tOf(4*RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)^2*RootOf(_Z^3-4)^2*x^3-20830*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*R
ootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^3*x^2+3335*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)^2*RootOf(_Z^
3-4)^2*x^2+8332*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^3*x-52824*(x^4+x^2)^(2/3)*R
ootOf(4*RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^2-1334*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*RootOf
(_Z^3-4)+_Z^2)^2*RootOf(_Z^3-4)^2*x+91368*(x^4+x^2)^(1/3)*RootOf(_Z^3-4)^2*x+151332*(x^4+x^2)^(1/3)*RootOf(_Z^
3-4)*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)*x+149976*RootOf(_Z^3-4)*x^3-24012*RootOf(4*RootOf(_Z^
3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)*x^3+133312*RootOf(_Z^3-4)*x^2-21344*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^
3-4)+_Z^2)*x^2+149976*RootOf(_Z^3-4)*x-605328*(x^4+x^2)^(2/3)-24012*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3
-4)+_Z^2)*x)/(-1+x)^2/x)-1/2*ln(-(4166*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^3*x^
3+2750*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)^2*RootOf(_Z^3-4)^2*x^3-10415*RootOf(4*RootOf(_Z^3-4
)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^3*x^2-6875*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)^2*
RootOf(_Z^3-4)^2*x^2+4166*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^3*x+26412*(x^4+x^
2)^(2/3)*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^2+2750*RootOf(4*RootOf(_Z^3-4)^2+2
*_Z*RootOf(_Z^3-4)+_Z^2)^2*RootOf(_Z^3-4)^2*x+151332*(x^4+x^2)^(1/3)*RootOf(_Z^3-4)^2*x+22842*(x^4+x^2)^(1/3)*
RootOf(_Z^3-4)*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)*x-41660*RootOf(_Z^3-4)*x^3-27500*RootOf(4*R
ootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)*x^3-149976*RootOf(_Z^3-4)*x^2-99000*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*R
ootOf(_Z^3-4)+_Z^2)*x^2-41660*RootOf(_Z^3-4)*x-91368*(x^4+x^2)^(2/3)-27500*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*Root
Of(_Z^3-4)+_Z^2)*x)/(-1+x)^2/x)*RootOf(_Z^3-4)-1/4*ln(-(4166*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^
2)*RootOf(_Z^3-4)^3*x^3+2750*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)^2*RootOf(_Z^3-4)^2*x^3-10415*
RootOf(4*RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^3*x^2-6875*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*R
ootOf(_Z^3-4)+_Z^2)^2*RootOf(_Z^3-4)^2*x^2+4166*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^
3-4)^3*x+26412*(x^4+x^2)^(2/3)*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)*RootOf(_Z^3-4)^2+2750*RootO
f(4*RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)^2*RootOf(_Z^3-4)^2*x+151332*(x^4+x^2)^(1/3)*RootOf(_Z^3-4)^2*x+
22842*(x^4+x^2)^(1/3)*RootOf(_Z^3-4)*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)*x-41660*RootOf(_Z^3-4
)*x^3-27500*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)*x^3-149976*RootOf(_Z^3-4)*x^2-99000*RootOf(4*R
ootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)*x^2-41660*RootOf(_Z^3-4)*x-91368*(x^4+x^2)^(2/3)-27500*RootOf(4*Root
Of(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)*x)/(-1+x)^2/x)*RootOf(4*RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+_Z^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(-1+x)/(x^4+x^2)^(1/3),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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