3.20.58 \(\int \frac {(-1+x) (-3+8 x-8 x^2+12 x^4)}{x (\frac {1-2 x^2}{1+2 x^2})^{2/3} (1+2 x^2) (3-7 x+7 x^2-6 x^3+2 x^4)} \, dx\)

Optimal. Leaf size=138 \[ \log \left (\sqrt [3]{\frac {1-2 x^2}{2 x^2+1}}+x-1\right )-\frac {1}{2} \log \left (x^2+\left (\frac {1-2 x^2}{2 x^2+1}\right )^{2/3}+(1-x) \sqrt [3]{\frac {1-2 x^2}{2 x^2+1}}-2 x+1\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x-\sqrt {3}}{-2 \sqrt [3]{\frac {1-2 x^2}{2 x^2+1}}+x-1}\right ) \]

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Rubi [F]  time = 10.10, 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 {(-1+x) \left (-3+8 x-8 x^2+12 x^4\right )}{x \left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3} \left (1+2 x^2\right ) \left (3-7 x+7 x^2-6 x^3+2 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-1 + x)*(-3 + 8*x - 8*x^2 + 12*x^4))/(x*((1 - 2*x^2)/(1 + 2*x^2))^(2/3)*(1 + 2*x^2)*(3 - 7*x + 7*x^2 - 6
*x^3 + 2*x^4)),x]

[Out]

(6*x*(1 - 2*x^2)^(2/3)*AppellF1[1/2, 2/3, 1/3, 3/2, 2*x^2, -2*x^2])/(((1 - 2*x^2)/(1 + 2*x^2))^(2/3)*(1 + 2*x^
2)^(2/3)) + (Sqrt[3]*(1 - 2*x^2)^(2/3)*ArcTan[1/Sqrt[3] + (2*(1 + 2*x^2)^(1/3))/(Sqrt[3]*(1 - 2*x^2)^(1/3))])/
(2*((1 - 2*x^2)/(1 + 2*x^2))^(2/3)*(1 + 2*x^2)^(2/3)) - ((1 - 2*x^2)^(2/3)*Log[x])/(2*((1 - 2*x^2)/(1 + 2*x^2)
)^(2/3)*(1 + 2*x^2)^(2/3)) + (3*(1 - 2*x^2)^(2/3)*Log[(1 - 2*x^2)^(1/3) - (1 + 2*x^2)^(1/3)])/(4*((1 - 2*x^2)/
(1 + 2*x^2))^(2/3)*(1 + 2*x^2)^(2/3)) - (22*(1 - 2*x^2)^(2/3)*Defer[Int][1/((1 - 2*x^2)^(2/3)*(1 + 2*x^2)^(1/3
)*(3 - 7*x + 7*x^2 - 6*x^3 + 2*x^4)), x])/(((1 - 2*x^2)/(1 + 2*x^2))^(2/3)*(1 + 2*x^2)^(2/3)) + (51*(1 - 2*x^2
)^(2/3)*Defer[Int][x/((1 - 2*x^2)^(2/3)*(1 + 2*x^2)^(1/3)*(3 - 7*x + 7*x^2 - 6*x^3 + 2*x^4)), x])/(((1 - 2*x^2
)/(1 + 2*x^2))^(2/3)*(1 + 2*x^2)^(2/3)) - (44*(1 - 2*x^2)^(2/3)*Defer[Int][x^2/((1 - 2*x^2)^(2/3)*(1 + 2*x^2)^
(1/3)*(3 - 7*x + 7*x^2 - 6*x^3 + 2*x^4)), x])/(((1 - 2*x^2)/(1 + 2*x^2))^(2/3)*(1 + 2*x^2)^(2/3)) + (22*(1 - 2
*x^2)^(2/3)*Defer[Int][x^3/((1 - 2*x^2)^(2/3)*(1 + 2*x^2)^(1/3)*(3 - 7*x + 7*x^2 - 6*x^3 + 2*x^4)), x])/(((1 -
2*x^2)/(1 + 2*x^2))^(2/3)*(1 + 2*x^2)^(2/3))

Rubi steps

\begin {align*} \int \frac {(-1+x) \left (-3+8 x-8 x^2+12 x^4\right )}{x \left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3} \left (1+2 x^2\right ) \left (3-7 x+7 x^2-6 x^3+2 x^4\right )} \, dx &=\frac {\left (1-2 x^2\right )^{2/3} \int \frac {(-1+x) \left (-3+8 x-8 x^2+12 x^4\right )}{x \left (1-2 x^2\right )^{2/3} \sqrt [3]{1+2 x^2} \left (3-7 x+7 x^2-6 x^3+2 x^4\right )} \, dx}{\left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3} \left (1+2 x^2\right )^{2/3}}\\ &=\frac {\left (1-2 x^2\right )^{2/3} \int \left (\frac {6}{\left (1-2 x^2\right )^{2/3} \sqrt [3]{1+2 x^2}}+\frac {1}{x \left (1-2 x^2\right )^{2/3} \sqrt [3]{1+2 x^2}}+\frac {-22+51 x-44 x^2+22 x^3}{\left (1-2 x^2\right )^{2/3} \sqrt [3]{1+2 x^2} \left (3-7 x+7 x^2-6 x^3+2 x^4\right )}\right ) \, dx}{\left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3} \left (1+2 x^2\right )^{2/3}}\\ &=\frac {\left (1-2 x^2\right )^{2/3} \int \frac {1}{x \left (1-2 x^2\right )^{2/3} \sqrt [3]{1+2 x^2}} \, dx}{\left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3} \left (1+2 x^2\right )^{2/3}}+\frac {\left (1-2 x^2\right )^{2/3} \int \frac {-22+51 x-44 x^2+22 x^3}{\left (1-2 x^2\right )^{2/3} \sqrt [3]{1+2 x^2} \left (3-7 x+7 x^2-6 x^3+2 x^4\right )} \, dx}{\left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3} \left (1+2 x^2\right )^{2/3}}+\frac {\left (6 \left (1-2 x^2\right )^{2/3}\right ) \int \frac {1}{\left (1-2 x^2\right )^{2/3} \sqrt [3]{1+2 x^2}} \, dx}{\left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3} \left (1+2 x^2\right )^{2/3}}\\ &=\frac {6 x \left (1-2 x^2\right )^{2/3} F_1\left (\frac {1}{2};\frac {2}{3},\frac {1}{3};\frac {3}{2};2 x^2,-2 x^2\right )}{\left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3} \left (1+2 x^2\right )^{2/3}}+\frac {\left (1-2 x^2\right )^{2/3} \operatorname {Subst}\left (\int \frac {1}{(1-2 x)^{2/3} x \sqrt [3]{1+2 x}} \, dx,x,x^2\right )}{2 \left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3} \left (1+2 x^2\right )^{2/3}}+\frac {\left (1-2 x^2\right )^{2/3} \int \left (-\frac {22}{\left (1-2 x^2\right )^{2/3} \sqrt [3]{1+2 x^2} \left (3-7 x+7 x^2-6 x^3+2 x^4\right )}+\frac {51 x}{\left (1-2 x^2\right )^{2/3} \sqrt [3]{1+2 x^2} \left (3-7 x+7 x^2-6 x^3+2 x^4\right )}-\frac {44 x^2}{\left (1-2 x^2\right )^{2/3} \sqrt [3]{1+2 x^2} \left (3-7 x+7 x^2-6 x^3+2 x^4\right )}+\frac {22 x^3}{\left (1-2 x^2\right )^{2/3} \sqrt [3]{1+2 x^2} \left (3-7 x+7 x^2-6 x^3+2 x^4\right )}\right ) \, dx}{\left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3} \left (1+2 x^2\right )^{2/3}}\\ &=\frac {6 x \left (1-2 x^2\right )^{2/3} F_1\left (\frac {1}{2};\frac {2}{3},\frac {1}{3};\frac {3}{2};2 x^2,-2 x^2\right )}{\left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3} \left (1+2 x^2\right )^{2/3}}+\frac {\sqrt {3} \left (1-2 x^2\right )^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+2 x^2}}{\sqrt {3} \sqrt [3]{1-2 x^2}}\right )}{2 \left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3} \left (1+2 x^2\right )^{2/3}}-\frac {\left (1-2 x^2\right )^{2/3} \log (x)}{2 \left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3} \left (1+2 x^2\right )^{2/3}}+\frac {3 \left (1-2 x^2\right )^{2/3} \log \left (\sqrt [3]{1-2 x^2}-\sqrt [3]{1+2 x^2}\right )}{4 \left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3} \left (1+2 x^2\right )^{2/3}}-\frac {\left (22 \left (1-2 x^2\right )^{2/3}\right ) \int \frac {1}{\left (1-2 x^2\right )^{2/3} \sqrt [3]{1+2 x^2} \left (3-7 x+7 x^2-6 x^3+2 x^4\right )} \, dx}{\left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3} \left (1+2 x^2\right )^{2/3}}+\frac {\left (22 \left (1-2 x^2\right )^{2/3}\right ) \int \frac {x^3}{\left (1-2 x^2\right )^{2/3} \sqrt [3]{1+2 x^2} \left (3-7 x+7 x^2-6 x^3+2 x^4\right )} \, dx}{\left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3} \left (1+2 x^2\right )^{2/3}}-\frac {\left (44 \left (1-2 x^2\right )^{2/3}\right ) \int \frac {x^2}{\left (1-2 x^2\right )^{2/3} \sqrt [3]{1+2 x^2} \left (3-7 x+7 x^2-6 x^3+2 x^4\right )} \, dx}{\left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3} \left (1+2 x^2\right )^{2/3}}+\frac {\left (51 \left (1-2 x^2\right )^{2/3}\right ) \int \frac {x}{\left (1-2 x^2\right )^{2/3} \sqrt [3]{1+2 x^2} \left (3-7 x+7 x^2-6 x^3+2 x^4\right )} \, dx}{\left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3} \left (1+2 x^2\right )^{2/3}}\\ \end {align*}

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Mathematica [F]  time = 0.62, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(-1+x) \left (-3+8 x-8 x^2+12 x^4\right )}{x \left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3} \left (1+2 x^2\right ) \left (3-7 x+7 x^2-6 x^3+2 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((-1 + x)*(-3 + 8*x - 8*x^2 + 12*x^4))/(x*((1 - 2*x^2)/(1 + 2*x^2))^(2/3)*(1 + 2*x^2)*(3 - 7*x + 7*x
^2 - 6*x^3 + 2*x^4)),x]

[Out]

Integrate[((-1 + x)*(-3 + 8*x - 8*x^2 + 12*x^4))/(x*((1 - 2*x^2)/(1 + 2*x^2))^(2/3)*(1 + 2*x^2)*(3 - 7*x + 7*x
^2 - 6*x^3 + 2*x^4)), x]

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IntegrateAlgebraic [A]  time = 3.09, size = 138, normalized size = 1.00 \begin {gather*} \sqrt {3} \tan ^{-1}\left (\frac {-\sqrt {3}+\sqrt {3} x}{-1+x-2 \sqrt [3]{\frac {1-2 x^2}{1+2 x^2}}}\right )+\log \left (-1+x+\sqrt [3]{\frac {1-2 x^2}{1+2 x^2}}\right )-\frac {1}{2} \log \left (1-2 x+x^2+(1-x) \sqrt [3]{\frac {1-2 x^2}{1+2 x^2}}+\left (\frac {1-2 x^2}{1+2 x^2}\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 + x)*(-3 + 8*x - 8*x^2 + 12*x^4))/(x*((1 - 2*x^2)/(1 + 2*x^2))^(2/3)*(1 + 2*x^2)*(3 -
7*x + 7*x^2 - 6*x^3 + 2*x^4)),x]

[Out]

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

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fricas [B]  time = 3.61, size = 278, normalized size = 2.01 \begin {gather*} \sqrt {3} \arctan \left (\frac {434 \, \sqrt {3} {\left (2 \, x^{3} - 2 \, x^{2} + x - 1\right )} \left (-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}\right )^{\frac {2}{3}} + 682 \, \sqrt {3} {\left (2 \, x^{4} - 4 \, x^{3} + 3 \, x^{2} - 2 \, x + 1\right )} \left (-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}\right )^{\frac {1}{3}} + \sqrt {3} {\left (242 \, x^{5} - 726 \, x^{4} + 847 \, x^{3} - 1095 \, x^{2} + 363 \, x + 124\right )}}{2662 \, x^{5} - 7986 \, x^{4} + 9317 \, x^{3} - 5969 \, x^{2} + 3993 \, x - 1674}\right ) + \frac {1}{2} \, \log \left (\frac {2 \, x^{5} - 6 \, x^{4} + 7 \, x^{3} - 7 \, x^{2} + 3 \, {\left (2 \, x^{3} - 2 \, x^{2} + x - 1\right )} \left (-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}\right )^{\frac {2}{3}} + 3 \, {\left (2 \, x^{4} - 4 \, x^{3} + 3 \, x^{2} - 2 \, x + 1\right )} \left (-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}\right )^{\frac {1}{3}} + 3 \, x}{2 \, x^{5} - 6 \, x^{4} + 7 \, x^{3} - 7 \, x^{2} + 3 \, x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)*(12*x^4-8*x^2+8*x-3)/x/((-2*x^2+1)/(2*x^2+1))^(2/3)/(2*x^2+1)/(2*x^4-6*x^3+7*x^2-7*x+3),x, al
gorithm="fricas")

[Out]

sqrt(3)*arctan((434*sqrt(3)*(2*x^3 - 2*x^2 + x - 1)*(-(2*x^2 - 1)/(2*x^2 + 1))^(2/3) + 682*sqrt(3)*(2*x^4 - 4*
x^3 + 3*x^2 - 2*x + 1)*(-(2*x^2 - 1)/(2*x^2 + 1))^(1/3) + sqrt(3)*(242*x^5 - 726*x^4 + 847*x^3 - 1095*x^2 + 36
3*x + 124))/(2662*x^5 - 7986*x^4 + 9317*x^3 - 5969*x^2 + 3993*x - 1674)) + 1/2*log((2*x^5 - 6*x^4 + 7*x^3 - 7*
x^2 + 3*(2*x^3 - 2*x^2 + x - 1)*(-(2*x^2 - 1)/(2*x^2 + 1))^(2/3) + 3*(2*x^4 - 4*x^3 + 3*x^2 - 2*x + 1)*(-(2*x^
2 - 1)/(2*x^2 + 1))^(1/3) + 3*x)/(2*x^5 - 6*x^4 + 7*x^3 - 7*x^2 + 3*x))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (12 \, x^{4} - 8 \, x^{2} + 8 \, x - 3\right )} {\left (x - 1\right )}}{{\left (2 \, x^{4} - 6 \, x^{3} + 7 \, x^{2} - 7 \, x + 3\right )} {\left (2 \, x^{2} + 1\right )} x \left (-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}\right )^{\frac {2}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)*(12*x^4-8*x^2+8*x-3)/x/((-2*x^2+1)/(2*x^2+1))^(2/3)/(2*x^2+1)/(2*x^4-6*x^3+7*x^2-7*x+3),x, al
gorithm="giac")

[Out]

integrate((12*x^4 - 8*x^2 + 8*x - 3)*(x - 1)/((2*x^4 - 6*x^3 + 7*x^2 - 7*x + 3)*(2*x^2 + 1)*x*(-(2*x^2 - 1)/(2
*x^2 + 1))^(2/3)), x)

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maple [C]  time = 5.30, size = 1426, normalized size = 10.33

method	result	size
trager	\(\ln \left (\frac {14-21 x -15 x \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {2}{3}}+15 \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {2}{3}}-18 \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}}+36 x \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}}-14 x^{5}+21 x^{2}-49 x^{3}+42 x^{4}+58 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+44 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}+12 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {2}{3}} x^{3}-12 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {2}{3}} x^{2}-60 x^{4} \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}}+120 x^{3} \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}}-90 x^{2} \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}}+60 x \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}}+6 x \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {2}{3}}-88 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{2}-44 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{5}+132 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{4}-154 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{3}+38 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}-66 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x -30 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}}-6 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {2}{3}}-54 \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}} x^{2}+30 \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {2}{3}} x^{2}-36 \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}} x^{4}+72 \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}} x^{3}-30 \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {2}{3}} x^{3}}{x \left (2 x^{4}-6 x^{3}+7 x^{2}-7 x +3\right )}\right )+2 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \ln \left (-\frac {7-21 x +18 x \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {2}{3}}-18 \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {2}{3}}+15 \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}}-30 x \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}}-14 x^{5}+35 x^{2}-49 x^{3}+42 x^{4}-22 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+16 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}+12 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {2}{3}} x^{3}-12 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {2}{3}} x^{2}+72 x^{4} \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}}-144 x^{3} \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}}+108 x^{2} \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}}-72 x \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}}+6 x \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {2}{3}}-32 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{2}+16 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{5}-48 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{4}+56 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{3}-12 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{2}+24 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x +36 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}}-6 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {2}{3}}+45 \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}} x^{2}-36 \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {2}{3}} x^{2}+30 \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}} x^{4}-60 \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {1}{3}} x^{3}+36 \left (-\frac {2 x^{2}-1}{2 x^{2}+1}\right )^{\frac {2}{3}} x^{3}}{x \left (2 x^{4}-6 x^{3}+7 x^{2}-7 x +3\right )}\right )\)	\(1426\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-1+x)*(12*x^4-8*x^2+8*x-3)/x/((-2*x^2+1)/(2*x^2+1))^(2/3)/(2*x^2+1)/(2*x^4-6*x^3+7*x^2-7*x+3),x,method=_R
ETURNVERBOSE)

[Out]

ln((14-21*x-88*RootOf(4*_Z^2+2*_Z+1)^2*x^2-44*RootOf(4*_Z^2+2*_Z+1)*x^5-15*x*(-(2*x^2-1)/(2*x^2+1))^(2/3)+132*
RootOf(4*_Z^2+2*_Z+1)*x^4-154*RootOf(4*_Z^2+2*_Z+1)*x^3+38*RootOf(4*_Z^2+2*_Z+1)*x^2-66*RootOf(4*_Z^2+2*_Z+1)*
x+15*(-(2*x^2-1)/(2*x^2+1))^(2/3)-18*(-(2*x^2-1)/(2*x^2+1))^(1/3)+36*x*(-(2*x^2-1)/(2*x^2+1))^(1/3)-14*x^5+21*
x^2-49*x^3+42*x^4+44*RootOf(4*_Z^2+2*_Z+1)^2+58*RootOf(4*_Z^2+2*_Z+1)+12*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*
x^2+1))^(2/3)*x^3-12*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^(2/3)*x^2-60*x^4*RootOf(4*_Z^2+2*_Z+1)*(-(2*
x^2-1)/(2*x^2+1))^(1/3)+120*x^3*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^(1/3)-90*x^2*RootOf(4*_Z^2+2*_Z+1
)*(-(2*x^2-1)/(2*x^2+1))^(1/3)+60*x*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^(1/3)+6*x*RootOf(4*_Z^2+2*_Z+
1)*(-(2*x^2-1)/(2*x^2+1))^(2/3)-54*(-(2*x^2-1)/(2*x^2+1))^(1/3)*x^2+30*(-(2*x^2-1)/(2*x^2+1))^(2/3)*x^2-36*(-(
2*x^2-1)/(2*x^2+1))^(1/3)*x^4+72*(-(2*x^2-1)/(2*x^2+1))^(1/3)*x^3-30*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+
1))^(1/3)-30*(-(2*x^2-1)/(2*x^2+1))^(2/3)*x^3-6*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^(2/3))/x/(2*x^4-6
*x^3+7*x^2-7*x+3))+2*RootOf(4*_Z^2+2*_Z+1)*ln(-(7-21*x-32*RootOf(4*_Z^2+2*_Z+1)^2*x^2+16*RootOf(4*_Z^2+2*_Z+1)
*x^5+18*x*(-(2*x^2-1)/(2*x^2+1))^(2/3)-48*RootOf(4*_Z^2+2*_Z+1)*x^4+56*RootOf(4*_Z^2+2*_Z+1)*x^3-12*RootOf(4*_
Z^2+2*_Z+1)*x^2+24*RootOf(4*_Z^2+2*_Z+1)*x-18*(-(2*x^2-1)/(2*x^2+1))^(2/3)+15*(-(2*x^2-1)/(2*x^2+1))^(1/3)-30*
x*(-(2*x^2-1)/(2*x^2+1))^(1/3)-14*x^5+35*x^2-49*x^3+42*x^4+16*RootOf(4*_Z^2+2*_Z+1)^2-22*RootOf(4*_Z^2+2*_Z+1)
+12*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^(2/3)*x^3-12*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^(2/
3)*x^2+72*x^4*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^(1/3)-144*x^3*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*
x^2+1))^(1/3)+108*x^2*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^(1/3)-72*x*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1
)/(2*x^2+1))^(1/3)+6*x*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^(2/3)+45*(-(2*x^2-1)/(2*x^2+1))^(1/3)*x^2-
36*(-(2*x^2-1)/(2*x^2+1))^(2/3)*x^2+30*(-(2*x^2-1)/(2*x^2+1))^(1/3)*x^4-60*(-(2*x^2-1)/(2*x^2+1))^(1/3)*x^3+36
*RootOf(4*_Z^2+2*_Z+1)*(-(2*x^2-1)/(2*x^2+1))^(1/3)+36*(-(2*x^2-1)/(2*x^2+1))^(2/3)*x^3-6*RootOf(4*_Z^2+2*_Z+1
)*(-(2*x^2-1)/(2*x^2+1))^(2/3))/x/(2*x^4-6*x^3+7*x^2-7*x+3))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (12 \, x^{4} - 8 \, x^{2} + 8 \, x - 3\right )} {\left (x - 1\right )}}{{\left (2 \, x^{4} - 6 \, x^{3} + 7 \, x^{2} - 7 \, x + 3\right )} {\left (2 \, x^{2} + 1\right )} x \left (-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}\right )^{\frac {2}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)*(12*x^4-8*x^2+8*x-3)/x/((-2*x^2+1)/(2*x^2+1))^(2/3)/(2*x^2+1)/(2*x^4-6*x^3+7*x^2-7*x+3),x, al
gorithm="maxima")

[Out]

integrate((12*x^4 - 8*x^2 + 8*x - 3)*(x - 1)/((2*x^4 - 6*x^3 + 7*x^2 - 7*x + 3)*(2*x^2 + 1)*x*(-(2*x^2 - 1)/(2
*x^2 + 1))^(2/3)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (x-1\right )\,\left (12\,x^4-8\,x^2+8\,x-3\right )}{x\,\left (2\,x^2+1\right )\,{\left (-\frac {2\,x^2-1}{2\,x^2+1}\right )}^{2/3}\,\left (2\,x^4-6\,x^3+7\,x^2-7\,x+3\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x - 1)*(8*x - 8*x^2 + 12*x^4 - 3))/(x*(2*x^2 + 1)*(-(2*x^2 - 1)/(2*x^2 + 1))^(2/3)*(7*x^2 - 7*x - 6*x^3
+ 2*x^4 + 3)),x)

[Out]

int(((x - 1)*(8*x - 8*x^2 + 12*x^4 - 3))/(x*(2*x^2 + 1)*(-(2*x^2 - 1)/(2*x^2 + 1))^(2/3)*(7*x^2 - 7*x - 6*x^3
+ 2*x^4 + 3)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)*(12*x**4-8*x**2+8*x-3)/x/((-2*x**2+1)/(2*x**2+1))**(2/3)/(2*x**2+1)/(2*x**4-6*x**3+7*x**2-7*x
+3),x)

[Out]

Timed out

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