3.26.77 \(\int \frac {2}{(3+x) (2-8 x+8 x^2)^{2/3}} \, dx\)

Optimal. Leaf size=222 \[ -\frac {3 \sqrt [3]{2} \sqrt [3]{4 x^2-4 x+1}}{7 (2 x-1)}+\frac {1}{7} \sqrt [3]{\frac {2}{7}} \log \left (\left (4 x^2-4 x+1\right )^{2/3}+2 \sqrt [3]{7} x-\sqrt [3]{7}\right )-\frac {\log \left (\left (4 x^2-4 x+1\right )^{2/3}+7^{2/3} \sqrt [3]{4 x^2-4 x+1}-2 \sqrt [3]{7} x+\sqrt [3]{7}\right )}{7\ 2^{2/3} \sqrt [3]{7}}+\frac {1}{7} \sqrt [3]{\frac {2}{7}} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \left (4 x^2-4 x+1\right )^{2/3}}{\left (4 x^2-4 x+1\right )^{2/3}-4 \sqrt [3]{7} x+2 \sqrt [3]{7}}\right )-\frac {2}{21} \sqrt [3]{\frac {2}{7}} \log (2 x-1) \]

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Rubi [A]  time = 0.11, antiderivative size = 196, normalized size of antiderivative = 0.88, number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {12, 646, 51, 56, 617, 204, 31} \begin {gather*} \frac {3 \sqrt [3]{2} (1-2 x)}{7 \left (4 x^2-4 x+1\right )^{2/3}}-\frac {(2 x-1)^{4/3} \log (x+3)}{7\ 2^{2/3} \sqrt [3]{7} \left (4 x^2-4 x+1\right )^{2/3}}+\frac {3 (2 x-1)^{4/3} \log \left (\sqrt [3]{8 x-4}+2^{2/3} \sqrt [3]{7}\right )}{7\ 2^{2/3} \sqrt [3]{7} \left (4 x^2-4 x+1\right )^{2/3}}+\frac {\sqrt [3]{\frac {2}{7}} \sqrt {3} (2 x-1)^{4/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{2 x-1}}{\sqrt {3} \sqrt [3]{7}}\right )}{7 \left (4 x^2-4 x+1\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[2/((3 + x)*(2 - 8*x + 8*x^2)^(2/3)),x]

[Out]

(3*2^(1/3)*(1 - 2*x))/(7*(1 - 4*x + 4*x^2)^(2/3)) + ((2/7)^(1/3)*Sqrt[3]*(-1 + 2*x)^(4/3)*ArcTan[1/Sqrt[3] - (
2*(-1 + 2*x)^(1/3))/(Sqrt[3]*7^(1/3))])/(7*(1 - 4*x + 4*x^2)^(2/3)) - ((-1 + 2*x)^(4/3)*Log[3 + x])/(7*2^(2/3)
*7^(1/3)*(1 - 4*x + 4*x^2)^(2/3)) + (3*(-1 + 2*x)^(4/3)*Log[2^(2/3)*7^(1/3) + (-4 + 8*x)^(1/3)])/(7*2^(2/3)*7^
(1/3)*(1 - 4*x + 4*x^2)^(2/3))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 56

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[{q = Rt[-((b*c - a*d)/b), 3]}, Simp
[Log[RemoveContent[a + b*x, x]]/(2*b*q), x] + (Dist[3/(2*b), Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(
1/3)], x] - Dist[3/(2*b*q), Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && Ne
gQ[(b*c - a*d)/b]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 646

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

Rubi steps

\begin {align*} \int \frac {2}{(3+x) \left (2-8 x+8 x^2\right )^{2/3}} \, dx &=2 \int \frac {1}{(3+x) \left (2-8 x+8 x^2\right )^{2/3}} \, dx\\ &=\frac {\left (2 (-4+8 x)^{4/3}\right ) \int \frac {1}{(3+x) (-4+8 x)^{4/3}} \, dx}{\left (2-8 x+8 x^2\right )^{2/3}}\\ &=\frac {3 \sqrt [3]{2} (1-2 x)}{7 \left (1-4 x+4 x^2\right )^{2/3}}-\frac {(-4+8 x)^{4/3} \int \frac {1}{(3+x) \sqrt [3]{-4+8 x}} \, dx}{14 \left (2-8 x+8 x^2\right )^{2/3}}\\ &=\frac {3 \sqrt [3]{2} (1-2 x)}{7 \left (1-4 x+4 x^2\right )^{2/3}}-\frac {(-1+2 x)^{4/3} \log (3+x)}{7\ 2^{2/3} \sqrt [3]{7} \left (1-4 x+4 x^2\right )^{2/3}}-\frac {\left (3 (-4+8 x)^{4/3}\right ) \operatorname {Subst}\left (\int \frac {1}{2 \sqrt [3]{2} 7^{2/3}-2^{2/3} \sqrt [3]{7} x+x^2} \, dx,x,\sqrt [3]{-4+8 x}\right )}{28 \left (2-8 x+8 x^2\right )^{2/3}}+\frac {\left (3 (-4+8 x)^{4/3}\right ) \operatorname {Subst}\left (\int \frac {1}{2^{2/3} \sqrt [3]{7}+x} \, dx,x,\sqrt [3]{-4+8 x}\right )}{28\ 2^{2/3} \sqrt [3]{7} \left (2-8 x+8 x^2\right )^{2/3}}\\ &=\frac {3 \sqrt [3]{2} (1-2 x)}{7 \left (1-4 x+4 x^2\right )^{2/3}}-\frac {(-1+2 x)^{4/3} \log (3+x)}{7\ 2^{2/3} \sqrt [3]{7} \left (1-4 x+4 x^2\right )^{2/3}}+\frac {3 (-1+2 x)^{4/3} \log \left (2^{2/3} \sqrt [3]{7}+\sqrt [3]{-4+8 x}\right )}{7\ 2^{2/3} \sqrt [3]{7} \left (1-4 x+4 x^2\right )^{2/3}}-\frac {\left (3 (-4+8 x)^{4/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\sqrt [3]{\frac {2}{7}} \sqrt [3]{-4+8 x}\right )}{14\ 2^{2/3} \sqrt [3]{7} \left (2-8 x+8 x^2\right )^{2/3}}\\ &=\frac {3 \sqrt [3]{2} (1-2 x)}{7 \left (1-4 x+4 x^2\right )^{2/3}}+\frac {\sqrt [3]{\frac {2}{7}} \sqrt {3} (-1+2 x)^{4/3} \tan ^{-1}\left (\frac {7-2\ 7^{2/3} \sqrt [3]{-1+2 x}}{7 \sqrt {3}}\right )}{7 \left (1-4 x+4 x^2\right )^{2/3}}-\frac {(-1+2 x)^{4/3} \log (3+x)}{7\ 2^{2/3} \sqrt [3]{7} \left (1-4 x+4 x^2\right )^{2/3}}+\frac {3 (-1+2 x)^{4/3} \log \left (2^{2/3} \sqrt [3]{7}+\sqrt [3]{-4+8 x}\right )}{7\ 2^{2/3} \sqrt [3]{7} \left (1-4 x+4 x^2\right )^{2/3}}\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 40, normalized size = 0.18 \begin {gather*} -\frac {6 (2 x-1) \, _2F_1\left (-\frac {1}{3},1;\frac {2}{3};\frac {1}{7} (1-2 x)\right )}{7 \left (8 x^2-8 x+2\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*(-1 + 2*x)*Hypergeometric2F1[-1/3, 1, 2/3, (1 - 2*x)/7])/(7*(2 - 8*x + 8*x^2)^(2/3))

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IntegrateAlgebraic [A]  time = 0.63, size = 222, normalized size = 1.00 \begin {gather*} -\frac {3 \sqrt [3]{2} \sqrt [3]{1-4 x+4 x^2}}{7 (-1+2 x)}+\frac {1}{7} \sqrt [3]{\frac {2}{7}} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \left (1-4 x+4 x^2\right )^{2/3}}{2 \sqrt [3]{7}-4 \sqrt [3]{7} x+\left (1-4 x+4 x^2\right )^{2/3}}\right )-\frac {2}{21} \sqrt [3]{\frac {2}{7}} \log (-1+2 x)+\frac {1}{7} \sqrt [3]{\frac {2}{7}} \log \left (-\sqrt [3]{7}+2 \sqrt [3]{7} x+\left (1-4 x+4 x^2\right )^{2/3}\right )-\frac {\log \left (\sqrt [3]{7}-2 \sqrt [3]{7} x+7^{2/3} \sqrt [3]{1-4 x+4 x^2}+\left (1-4 x+4 x^2\right )^{2/3}\right )}{7\ 2^{2/3} \sqrt [3]{7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*2^(1/3)*(1 - 4*x + 4*x^2)^(1/3))/(7*(-1 + 2*x)) + ((2/7)^(1/3)*Sqrt[3]*ArcTan[(Sqrt[3]*(1 - 4*x + 4*x^2)^(
2/3))/(2*7^(1/3) - 4*7^(1/3)*x + (1 - 4*x + 4*x^2)^(2/3))])/7 - (2*(2/7)^(1/3)*Log[-1 + 2*x])/21 + ((2/7)^(1/3
)*Log[-7^(1/3) + 2*7^(1/3)*x + (1 - 4*x + 4*x^2)^(2/3)])/7 - Log[7^(1/3) - 2*7^(1/3)*x + 7^(2/3)*(1 - 4*x + 4*
x^2)^(1/3) + (1 - 4*x + 4*x^2)^(2/3)]/(7*2^(2/3)*7^(1/3))

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fricas [A]  time = 0.47, size = 220, normalized size = 0.99 \begin {gather*} \frac {2 \cdot 7^{\frac {2}{3}} \sqrt {3} 2^{\frac {1}{3}} {\left (2 \, x - 1\right )} \arctan \left (-\frac {7^{\frac {1}{6}} \sqrt {3} {\left (7^{\frac {5}{6}} {\left (2 \, x - 1\right )} - 7 \cdot 7^{\frac {1}{6}} 2^{\frac {2}{3}} {\left (8 \, x^{2} - 8 \, x + 2\right )}^{\frac {1}{3}}\right )}}{21 \, {\left (2 \, x - 1\right )}}\right ) - 7^{\frac {2}{3}} 2^{\frac {1}{3}} {\left (2 \, x - 1\right )} \log \left (-\frac {7^{\frac {2}{3}} 2^{\frac {1}{3}} {\left (8 \, x^{2} - 8 \, x + 2\right )}^{\frac {1}{3}} {\left (2 \, x - 1\right )} - 7^{\frac {1}{3}} 2^{\frac {2}{3}} {\left (4 \, x^{2} - 4 \, x + 1\right )} - 7 \, {\left (8 \, x^{2} - 8 \, x + 2\right )}^{\frac {2}{3}}}{4 \, x^{2} - 4 \, x + 1}\right ) + 2 \cdot 7^{\frac {2}{3}} 2^{\frac {1}{3}} {\left (2 \, x - 1\right )} \log \left (\frac {7^{\frac {2}{3}} 2^{\frac {1}{3}} {\left (2 \, x - 1\right )} + 7 \, {\left (8 \, x^{2} - 8 \, x + 2\right )}^{\frac {1}{3}}}{2 \, x - 1}\right ) - 42 \, {\left (8 \, x^{2} - 8 \, x + 2\right )}^{\frac {1}{3}}}{98 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/(3+x)/(8*x^2-8*x+2)^(2/3),x, algorithm="fricas")

[Out]

1/98*(2*7^(2/3)*sqrt(3)*2^(1/3)*(2*x - 1)*arctan(-1/21*7^(1/6)*sqrt(3)*(7^(5/6)*(2*x - 1) - 7*7^(1/6)*2^(2/3)*
(8*x^2 - 8*x + 2)^(1/3))/(2*x - 1)) - 7^(2/3)*2^(1/3)*(2*x - 1)*log(-(7^(2/3)*2^(1/3)*(8*x^2 - 8*x + 2)^(1/3)*
(2*x - 1) - 7^(1/3)*2^(2/3)*(4*x^2 - 4*x + 1) - 7*(8*x^2 - 8*x + 2)^(2/3))/(4*x^2 - 4*x + 1)) + 2*7^(2/3)*2^(1
/3)*(2*x - 1)*log((7^(2/3)*2^(1/3)*(2*x - 1) + 7*(8*x^2 - 8*x + 2)^(1/3))/(2*x - 1)) - 42*(8*x^2 - 8*x + 2)^(1
/3))/(2*x - 1)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2}{{\left (8 \, x^{2} - 8 \, x + 2\right )}^{\frac {2}{3}} {\left (x + 3\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/(3+x)/(8*x^2-8*x+2)^(2/3),x, algorithm="giac")

[Out]

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

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maple [A]  time = 3.09, size = 116, normalized size = 0.52

method	result	size
risch	\(-\frac {3 \left (-1+2 x \right ) 2^{\frac {1}{3}}}{7 \left (\left (-1+2 x \right )^{2}\right )^{\frac {2}{3}}}+\frac {\left (\frac {7^{\frac {2}{3}} \ln \left (\left (-1+2 x \right )^{\frac {1}{3}}+7^{\frac {1}{3}}\right )}{49}-\frac {7^{\frac {2}{3}} \ln \left (\left (-1+2 x \right )^{\frac {2}{3}}-7^{\frac {1}{3}} \left (-1+2 x \right )^{\frac {1}{3}}+7^{\frac {2}{3}}\right )}{98}-\frac {\sqrt {3}\, 7^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \,7^{\frac {2}{3}} \left (-1+2 x \right )^{\frac {1}{3}}}{7}-1\right )}{3}\right )}{49}\right ) 2^{\frac {1}{3}} \left (-1+2 x \right )^{\frac {4}{3}}}{\left (\left (-1+2 x \right )^{2}\right )^{\frac {2}{3}}}\)	\(116\)
trager	\(-\frac {3 \left (8 x^{2}-8 x +2\right )^{\frac {1}{3}}}{7 \left (-1+2 x \right )}+\frac {3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right ) \ln \left (\frac {-12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-98\right )^{4} x^{2}-420 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-98\right )^{3} x^{2}+6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-98\right )^{4} x +210 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-98\right )^{3} x +105 \left (8 x^{2}-8 x +2\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-98\right )^{2}-64 \RootOf \left (\textit {\_Z}^{3}-98\right )^{2} x^{2}-2240 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-98\right ) x^{2}-588 \left (8 x^{2}-8 x +2\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-98\right ) x -9408 \left (8 x^{2}-8 x +2\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right ) x +260 \RootOf \left (\textit {\_Z}^{3}-98\right )^{2} x +9100 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-98\right ) x -1568 \left (8 x^{2}-8 x +2\right )^{\frac {2}{3}}+294 \left (8 x^{2}-8 x +2\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-98\right )+4704 \left (8 x^{2}-8 x +2\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right )-114 \RootOf \left (\textit {\_Z}^{3}-98\right )^{2}-3990 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-98\right )}{\left (3+x \right ) \left (-1+2 x \right )}\right )}{7}+\frac {\RootOf \left (\textit {\_Z}^{3}-98\right ) \ln \left (-\frac {12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-98\right )^{4} x^{2}-168 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-98\right )^{3} x^{2}-6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-98\right )^{4} x +84 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-98\right )^{3} x +105 \left (8 x^{2}-8 x +2\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-98\right )^{2}-8 \RootOf \left (\textit {\_Z}^{3}-98\right )^{2} x^{2}+112 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-98\right ) x^{2}+448 \left (8 x^{2}-8 x +2\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-98\right ) x +12348 \left (8 x^{2}-8 x +2\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right ) x +232 \RootOf \left (\textit {\_Z}^{3}-98\right )^{2} x -3248 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-98\right ) x +2058 \left (8 x^{2}-8 x +2\right )^{\frac {2}{3}}-224 \left (8 x^{2}-8 x +2\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-98\right )-6174 \left (8 x^{2}-8 x +2\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right )-114 \RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+1596 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-98\right )^{2}+21 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-98\right )+441 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-98\right )}{\left (3+x \right ) \left (-1+2 x \right )}\right )}{49}\)	\(1008\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2/(3+x)/(8*x^2-8*x+2)^(2/3),x,method=_RETURNVERBOSE)

[Out]

-3/7*(-1+2*x)*2^(1/3)/((-1+2*x)^2)^(2/3)+(1/49*7^(2/3)*ln((-1+2*x)^(1/3)+7^(1/3))-1/98*7^(2/3)*ln((-1+2*x)^(2/
3)-7^(1/3)*(-1+2*x)^(1/3)+7^(2/3))-1/49*3^(1/2)*7^(2/3)*arctan(1/3*3^(1/2)*(2/7*7^(2/3)*(-1+2*x)^(1/3)-1)))*2^
(1/3)/((-1+2*x)^2)^(2/3)*(-1+2*x)^(4/3)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 2 \, \int \frac {1}{{\left (8 \, x^{2} - 8 \, x + 2\right )}^{\frac {2}{3}} {\left (x + 3\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/(3+x)/(8*x^2-8*x+2)^(2/3),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {2}{\left (x+3\right )\,{\left (8\,x^2-8\,x+2\right )}^{2/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2/((x + 3)*(8*x^2 - 8*x + 2)^(2/3)),x)

[Out]

int(2/((x + 3)*(8*x^2 - 8*x + 2)^(2/3)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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