∫ (−2+x2) 3√___x+x3 _________________________

3.16.53 $\int \frac {(-2+x^2) \sqrt [3]{x+x^3}}{x^2 (4-2 x^2+x^4)} \, dx$

Optimal. Leaf size=107 \[ \frac {3 \sqrt [3]{x^3+x}}{4 x}-\frac {1}{8} \text {RootSum}\left [4 \text {$\#$1}^6-10 \text {$\#$1}^3+7\& ,\frac {-4 \text {$\#$1}^3 \log \left (\sqrt [3]{x^3+x}-\text {$\#$1} x\right )+4 \text {$\#$1}^3 \log (x)+7 \log \left (\sqrt [3]{x^3+x}-\text {$\#$1} x\right )-7 \log (x)}{4 \text {$\#$1}^5-5 \text {$\#$1}^2}\& \right ] \]

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Rubi [C] time = 0.89, antiderivative size = 253, normalized size of antiderivative = 2.36, number of steps used = 9, number of rules used = 5, integrand size = 30, $\frac {\text {number of rules}}{\text {integrand size}}$ = 0.167, Rules used = {2056, 6728, 466, 465, 510} \begin {gather*} \frac {\left (\sqrt {3}+3 i\right ) \sqrt [3]{1-\frac {x^2}{1-i \sqrt {3}}} \sqrt [3]{x^3+x} \, _2F_1\left (-\frac {1}{3},-\frac {1}{3};\frac {2}{3};-\frac {\frac {x^2}{1-i \sqrt {3}}+x^2}{1-\frac {x^2}{1-i \sqrt {3}}}\right )}{4 \left (\sqrt {3}+i\right ) x \sqrt [3]{x^2+1}}+\frac {\left (-\sqrt {3}+3 i\right ) \sqrt [3]{1-\frac {x^2}{1+i \sqrt {3}}} \sqrt [3]{x^3+x} \, _2F_1\left (-\frac {1}{3},-\frac {1}{3};\frac {2}{3};-\frac {\frac {x^2}{1+i \sqrt {3}}+x^2}{1-\frac {x^2}{1+i \sqrt {3}}}\right )}{4 \left (-\sqrt {3}+i\right ) x \sqrt [3]{x^2+1}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((3*I + Sqrt[3])*(1 - x^2/(1 - I*Sqrt[3]))^(1/3)*(x + x^3)^(1/3)*Hypergeometric2F1[-1/3, -1/3, 2/3, -((x^2 + x

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

^2/(1 + I*Sqrt[3]))^(1/3)*(x + x^3)^(1/3)*Hypergeometric2F1[-1/3, -1/3, 2/3, -((x^2 + x^2/(1 + I*Sqrt[3]))/(1

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

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 466

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno

minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/e^n)^p*(c + (d*x^(k*n))/e^n)^q, x], x, (e*

x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege

rQ[p]

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 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 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +

b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

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

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Mathematica [C] time = 0.25, size = 198, normalized size = 1.85 \begin {gather*} \frac {\sqrt [3]{x^3+x} \left (\left (3+i \sqrt {3}\right ) \sqrt [3]{\frac {-i x^2+\sqrt {3}+i}{\sqrt {3}+i}} \, _2F_1\left (-\frac {1}{3},-\frac {1}{3};\frac {2}{3};-\frac {\left (2 i+\sqrt {3}\right ) x^2}{-i x^2+\sqrt {3}+i}\right )+\left (3-i \sqrt {3}\right ) \sqrt [3]{\frac {i x^2+\sqrt {3}-i}{\sqrt {3}-i}} \, _2F_1\left (-\frac {1}{3},-\frac {1}{3};\frac {2}{3};-\frac {\left (-2 i+\sqrt {3}\right ) x^2}{i x^2+\sqrt {3}-i}\right )\right )}{8 x \sqrt [3]{x^2+1}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((x + x^3)^(1/3)*((3 + I*Sqrt[3])*((I + Sqrt[3] - I*x^2)/(I + Sqrt[3]))^(1/3)*Hypergeometric2F1[-1/3, -1/3, 2/

3, -(((2*I + Sqrt[3])*x^2)/(I + Sqrt[3] - I*x^2))] + (3 - I*Sqrt[3])*((-I + Sqrt[3] + I*x^2)/(-I + Sqrt[3]))^(

1/3)*Hypergeometric2F1[-1/3, -1/3, 2/3, -(((-2*I + Sqrt[3])*x^2)/(-I + Sqrt[3] + I*x^2))]))/(8*x*(1 + x^2)^(1/

3))

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IntegrateAlgebraic [A] time = 0.26, size = 107, normalized size = 1.00 \begin {gather*} \frac {3 \sqrt [3]{x+x^3}}{4 x}-\frac {1}{8} \text {RootSum}\left [7-10 \text {$\#$1}^3+4 \text {$\#$1}^6\&,\frac {-7 \log (x)+7 \log \left (\sqrt [3]{x+x^3}-x \text {$\#$1}\right )+4 \log (x) \text {$\#$1}^3-4 \log \left (\sqrt [3]{x+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-5 \text {$\#$1}^2+4 \text {$\#$1}^5}\&\right ] \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(3*(x + x^3)^(1/3))/(4*x) - RootSum[7 - 10*#1^3 + 4*#1^6 & , (-7*Log[x] + 7*Log[(x + x^3)^(1/3) - x*#1] + 4*Lo

g[x]*#1^3 - 4*Log[(x + x^3)^(1/3) - x*#1]*#1^3)/(-5*#1^2 + 4*#1^5) & ]/8

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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> Error detected within library code: integrate: implementation incomplete (tr

ace 0)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B] time = 368.43, size = 2048, normalized size = 19.14


method	result	size

trager	$\text {Expression too large to display}$	$2048$
risch	$\text {Expression too large to display}$	$8598$

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*(x^3+x)^(1/3)/x+1/24*RootOf(_Z^3+46656*RootOf(5038848*_Z^6+3888*_Z^3+7)^3+36)*ln(-(40310784*RootOf(_Z^3+46

656*RootOf(5038848*_Z^6+3888*_Z^3+7)^3+36)^2*RootOf(5038848*_Z^6+3888*_Z^3+7)^6*x^2-40310784*RootOf(5038848*_Z

^6+3888*_Z^3+7)^6*RootOf(_Z^3+46656*RootOf(5038848*_Z^6+3888*_Z^3+7)^3+36)^2+360288*RootOf(_Z^3+46656*RootOf(5

038848*_Z^6+3888*_Z^3+7)^3+36)^2*RootOf(5038848*_Z^6+3888*_Z^3+7)^3*x^2-917568*(x^3+x)^(1/3)*RootOf(5038848*_Z

^6+3888*_Z^3+7)^3*RootOf(_Z^3+46656*RootOf(5038848*_Z^6+3888*_Z^3+7)^3+36)*x+145152*RootOf(5038848*_Z^6+3888*_

Z^3+7)^3*RootOf(_Z^3+46656*RootOf(5038848*_Z^6+3888*_Z^3+7)^3+36)^2+886464*(x^3+x)^(2/3)*RootOf(5038848*_Z^6+3

888*_Z^3+7)^3-47*RootOf(_Z^3+46656*RootOf(5038848*_Z^6+3888*_Z^3+7)^3+36)^2*x^2+846*(x^3+x)^(1/3)*RootOf(_Z^3+

46656*RootOf(5038848*_Z^6+3888*_Z^3+7)^3+36)*x-18*RootOf(_Z^3+46656*RootOf(5038848*_Z^6+3888*_Z^3+7)^3+36)^2-5

028*(x^3+x)^(2/3))/(2592*x^2*RootOf(5038848*_Z^6+3888*_Z^3+7)^3-2592*RootOf(5038848*_Z^6+3888*_Z^3+7)^3+x^2+4)

)-162/5*ln(-(3359232*RootOf(_Z^3+46656*RootOf(5038848*_Z^6+3888*_Z^3+7)^3+36)^2*RootOf(5038848*_Z^6+3888*_Z^3+

7)^6*x^2-3359232*RootOf(5038848*_Z^6+3888*_Z^3+7)^6*RootOf(_Z^3+46656*RootOf(5038848*_Z^6+3888*_Z^3+7)^3+36)^2

-21168*RootOf(_Z^3+46656*RootOf(5038848*_Z^6+3888*_Z^3+7)^3+36)^2*RootOf(5038848*_Z^6+3888*_Z^3+7)^3*x^2+25920

*(x^3+x)^(1/3)*RootOf(5038848*_Z^6+3888*_Z^3+7)^3*RootOf(_Z^3+46656*RootOf(5038848*_Z^6+3888*_Z^3+7)^3+36)*x+1

728*RootOf(5038848*_Z^6+3888*_Z^3+7)^3*RootOf(_Z^3+46656*RootOf(5038848*_Z^6+3888*_Z^3+7)^3+36)^2+233280*(x^3+

x)^(2/3)*RootOf(5038848*_Z^6+3888*_Z^3+7)^3+33*RootOf(_Z^3+46656*RootOf(5038848*_Z^6+3888*_Z^3+7)^3+36)^2*x^2-

190*(x^3+x)^(1/3)*RootOf(_Z^3+46656*RootOf(5038848*_Z^6+3888*_Z^3+7)^3+36)*x+22*RootOf(_Z^3+46656*RootOf(50388

48*_Z^6+3888*_Z^3+7)^3+36)^2+740*(x^3+x)^(2/3))/(2592*x^2*RootOf(5038848*_Z^6+3888*_Z^3+7)^3-2592*RootOf(50388

48*_Z^6+3888*_Z^3+7)^3+x^2+4))*RootOf(5038848*_Z^6+3888*_Z^3+7)^3*RootOf(_Z^3+46656*RootOf(5038848*_Z^6+3888*_

Z^3+7)^3+36)-1/30*ln(-(3359232*RootOf(_Z^3+46656*RootOf(5038848*_Z^6+3888*_Z^3+7)^3+36)^2*RootOf(5038848*_Z^6+

3888*_Z^3+7)^6*x^2-3359232*RootOf(5038848*_Z^6+3888*_Z^3+7)^6*RootOf(_Z^3+46656*RootOf(5038848*_Z^6+3888*_Z^3+

7)^3+36)^2-21168*RootOf(_Z^3+46656*RootOf(5038848*_Z^6+3888*_Z^3+7)^3+36)^2*RootOf(5038848*_Z^6+3888*_Z^3+7)^3

*x^2+25920*(x^3+x)^(1/3)*RootOf(5038848*_Z^6+3888*_Z^3+7)^3*RootOf(_Z^3+46656*RootOf(5038848*_Z^6+3888*_Z^3+7)

^3+36)*x+1728*RootOf(5038848*_Z^6+3888*_Z^3+7)^3*RootOf(_Z^3+46656*RootOf(5038848*_Z^6+3888*_Z^3+7)^3+36)^2+23

3280*(x^3+x)^(2/3)*RootOf(5038848*_Z^6+3888*_Z^3+7)^3+33*RootOf(_Z^3+46656*RootOf(5038848*_Z^6+3888*_Z^3+7)^3+

36)^2*x^2-190*(x^3+x)^(1/3)*RootOf(_Z^3+46656*RootOf(5038848*_Z^6+3888*_Z^3+7)^3+36)*x+22*RootOf(_Z^3+46656*Ro

otOf(5038848*_Z^6+3888*_Z^3+7)^3+36)^2+740*(x^3+x)^(2/3))/(2592*x^2*RootOf(5038848*_Z^6+3888*_Z^3+7)^3-2592*Ro

otOf(5038848*_Z^6+3888*_Z^3+7)^3+x^2+4))*RootOf(_Z^3+46656*RootOf(5038848*_Z^6+3888*_Z^3+7)^3+36)-5832/5*ln((5

44195584*x^2*RootOf(5038848*_Z^6+3888*_Z^3+7)^8-544195584*RootOf(5038848*_Z^6+3888*_Z^3+7)^8+2344464*RootOf(50

38848*_Z^6+3888*_Z^3+7)^5*x^2+5219640*(x^3+x)^(1/3)*RootOf(5038848*_Z^6+3888*_Z^3+7)^4*x-9167904*RootOf(503884

8*_Z^6+3888*_Z^3+7)^5+92340*(x^3+x)^(2/3)*RootOf(5038848*_Z^6+3888*_Z^3+7)^3-56889*RootOf(5038848*_Z^6+3888*_Z

^3+7)^2*x^2+945*(x^3+x)^(1/3)*RootOf(5038848*_Z^6+3888*_Z^3+7)*x-29106*RootOf(5038848*_Z^6+3888*_Z^3+7)^2+595*

(x^3+x)^(2/3))/(2592*x^2*RootOf(5038848*_Z^6+3888*_Z^3+7)^3-2592*RootOf(5038848*_Z^6+3888*_Z^3+7)^3+x^2-6))*Ro

otOf(5038848*_Z^6+3888*_Z^3+7)^4-6/5*ln((544195584*x^2*RootOf(5038848*_Z^6+3888*_Z^3+7)^8-544195584*RootOf(503

8848*_Z^6+3888*_Z^3+7)^8+2344464*RootOf(5038848*_Z^6+3888*_Z^3+7)^5*x^2+5219640*(x^3+x)^(1/3)*RootOf(5038848*_

Z^6+3888*_Z^3+7)^4*x-9167904*RootOf(5038848*_Z^6+3888*_Z^3+7)^5+92340*(x^3+x)^(2/3)*RootOf(5038848*_Z^6+3888*_

Z^3+7)^3-56889*RootOf(5038848*_Z^6+3888*_Z^3+7)^2*x^2+945*(x^3+x)^(1/3)*RootOf(5038848*_Z^6+3888*_Z^3+7)*x-291

06*RootOf(5038848*_Z^6+3888*_Z^3+7)^2+595*(x^3+x)^(2/3))/(2592*x^2*RootOf(5038848*_Z^6+3888*_Z^3+7)^3-2592*Roo

tOf(5038848*_Z^6+3888*_Z^3+7)^3+x^2-6))*RootOf(5038848*_Z^6+3888*_Z^3+7)+3/2*RootOf(5038848*_Z^6+3888*_Z^3+7)*

ln(-(544195584*x^2*RootOf(5038848*_Z^6+3888*_Z^3+7)^8-544195584*RootOf(5038848*_Z^6+3888*_Z^3+7)^8+1819584*Roo

tOf(5038848*_Z^6+3888*_Z^3+7)^5*x^2+379080*(x^3+x)^(1/3)*RootOf(5038848*_Z^6+3888*_Z^3+7)^4*x+1329696*RootOf(5

038848*_Z^6+3888*_Z^3+7)^5-14580*(x^3+x)^(2/3)*RootOf(5038848*_Z^6+3888*_Z^3+7)^3-2079*RootOf(5038848*_Z^6+388

8*_Z^3+7)^2*x^2+315*(x^3+x)^(1/3)*RootOf(5038848*_Z^6+3888*_Z^3+7)*x-756*RootOf(5038848*_Z^6+3888*_Z^3+7)^2+35

*(x^3+x)^(2/3))/(2592*x^2*RootOf(5038848*_Z^6+3888*_Z^3+7)^3-2592*RootOf(5038848*_Z^6+3888*_Z^3+7)^3+x^2-6))

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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {3 \, {\left ({\left (x^{3} + x\right )} x^{2} - 2 \, x^{3} - 2 \, x\right )} {\left (x^{2} + 1\right )}^{\frac {1}{3}}}{8 \, {\left (x^{\frac {17}{3}} - 2 \, x^{\frac {11}{3}} + 4 \, x^{\frac {5}{3}}\right )}} + \int \frac {6 \, {\left ({\left (x^{2} + 1\right )} x^{2} - x^{2} - 1\right )} {\left (x^{2} + 1\right )}^{\frac {1}{3}}}{x^{\frac {29}{3}} - 4 \, x^{\frac {23}{3}} + 12 \, x^{\frac {17}{3}} - 16 \, x^{\frac {11}{3}} + 16 \, x^{\frac {5}{3}}}\,{d x} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/8*((x^3 + x)*x^2 - 2*x^3 - 2*x)*(x^2 + 1)^(1/3)/(x^(17/3) - 2*x^(11/3) + 4*x^(5/3)) + integrate(6*((x^2 + 1

)*x^2 - x^2 - 1)*(x^2 + 1)^(1/3)/(x^(29/3) - 4*x^(23/3) + 12*x^(17/3) - 16*x^(11/3) + 16*x^(5/3)), x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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