3.23.94 \(\int \frac {(-3+4 x) (-1+2 x+x^3)^{2/3}}{x^3 (2-4 x+x^3)} \, dx\)

Optimal. Leaf size=175 \[ \frac {1}{2} \left (\frac {3}{2}\right )^{2/3} \log \left (\sqrt [3]{2} 3^{2/3} \sqrt [3]{x^3+2 x-1}-3 x\right )-\frac {3 \sqrt [6]{3} \tan ^{-1}\left (\frac {3^{5/6} x}{2 \sqrt [3]{2} \sqrt [3]{x^3+2 x-1}+\sqrt [3]{3} x}\right )}{2\ 2^{2/3}}+\frac {3 \left (x^3+2 x-1\right )^{2/3}}{4 x^2}-\frac {1}{4} \left (\frac {3}{2}\right )^{2/3} \log \left (\sqrt [3]{2} 3^{2/3} \sqrt [3]{x^3+2 x-1} x+2^{2/3} \sqrt [3]{3} \left (x^3+2 x-1\right )^{2/3}+3 x^2\right ) \]

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Rubi [F]  time = 2.07, 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 {(-3+4 x) \left (-1+2 x+x^3\right )^{2/3}}{x^3 \left (2-4 x+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-27*2^(1/3)*(-1 + 2*x + x^3)^(2/3)*Defer[Int][(((4*(3/(9 + Sqrt[177]))^(1/3) - (2*(9 + Sqrt[177]))^(1/3))/6^(
2/3) + x)^(2/3)*((12 + 24*3^(1/3)*(2/(9 + Sqrt[177]))^(2/3) + 2^(1/3)*(3*(9 + Sqrt[177]))^(2/3))/18 - ((2*(6/(
9 + Sqrt[177]))^(1/3) - ((9 + Sqrt[177])/2)^(1/3))*x)/3^(2/3) + x^2)^(2/3))/x^3, x])/((6^(1/3)*(4*(3/(9 + Sqrt
[177]))^(1/3) - (2*(9 + Sqrt[177]))^(1/3)) + 6*x)^(2/3)*(12 + 24*3^(1/3)*(2/(9 + Sqrt[177]))^(2/3) + 2^(1/3)*(
3*(9 + Sqrt[177]))^(2/3) - 6*3^(1/3)*(2*(6/(9 + Sqrt[177]))^(1/3) - ((9 + Sqrt[177])/2)^(1/3))*x + 18*x^2)^(2/
3)) - (18*2^(1/3)*(-1 + 2*x + x^3)^(2/3)*Defer[Int][(((4*(3/(9 + Sqrt[177]))^(1/3) - (2*(9 + Sqrt[177]))^(1/3)
)/6^(2/3) + x)^(2/3)*((12 + 24*3^(1/3)*(2/(9 + Sqrt[177]))^(2/3) + 2^(1/3)*(3*(9 + Sqrt[177]))^(2/3))/18 - ((2
*(6/(9 + Sqrt[177]))^(1/3) - ((9 + Sqrt[177])/2)^(1/3))*x)/3^(2/3) + x^2)^(2/3))/x^2, x])/((6^(1/3)*(4*(3/(9 +
Sqrt[177]))^(1/3) - (2*(9 + Sqrt[177]))^(1/3)) + 6*x)^(2/3)*(12 + 24*3^(1/3)*(2/(9 + Sqrt[177]))^(2/3) + 2^(1
/3)*(3*(9 + Sqrt[177]))^(2/3) - 6*3^(1/3)*(2*(6/(9 + Sqrt[177]))^(1/3) - ((9 + Sqrt[177])/2)^(1/3))*x + 18*x^2
)^(2/3)) - (36*2^(1/3)*(-1 + 2*x + x^3)^(2/3)*Defer[Int][(((4*(3/(9 + Sqrt[177]))^(1/3) - (2*(9 + Sqrt[177]))^
(1/3))/6^(2/3) + x)^(2/3)*((12 + 24*3^(1/3)*(2/(9 + Sqrt[177]))^(2/3) + 2^(1/3)*(3*(9 + Sqrt[177]))^(2/3))/18
- ((2*(6/(9 + Sqrt[177]))^(1/3) - ((9 + Sqrt[177])/2)^(1/3))*x)/3^(2/3) + x^2)^(2/3))/x, x])/((6^(1/3)*(4*(3/(
9 + Sqrt[177]))^(1/3) - (2*(9 + Sqrt[177]))^(1/3)) + 6*x)^(2/3)*(12 + 24*3^(1/3)*(2/(9 + Sqrt[177]))^(2/3) + 2
^(1/3)*(3*(9 + Sqrt[177]))^(2/3) - 6*3^(1/3)*(2*(6/(9 + Sqrt[177]))^(1/3) - ((9 + Sqrt[177])/2)^(1/3))*x + 18*
x^2)^(2/3)) - (13*Defer[Int][(-1 + 2*x + x^3)^(2/3)/(2 - 4*x + x^3), x])/2 + Defer[Int][(x*(-1 + 2*x + x^3)^(2
/3))/(2 - 4*x + x^3), x] + 2*Defer[Int][(x^2*(-1 + 2*x + x^3)^(2/3))/(2 - 4*x + x^3), x]

Rubi steps

\begin {align*} \int \frac {(-3+4 x) \left (-1+2 x+x^3\right )^{2/3}}{x^3 \left (2-4 x+x^3\right )} \, dx &=\int \left (-\frac {3 \left (-1+2 x+x^3\right )^{2/3}}{2 x^3}-\frac {\left (-1+2 x+x^3\right )^{2/3}}{x^2}-\frac {2 \left (-1+2 x+x^3\right )^{2/3}}{x}+\frac {\left (-13+2 x+4 x^2\right ) \left (-1+2 x+x^3\right )^{2/3}}{2 \left (2-4 x+x^3\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {\left (-13+2 x+4 x^2\right ) \left (-1+2 x+x^3\right )^{2/3}}{2-4 x+x^3} \, dx-\frac {3}{2} \int \frac {\left (-1+2 x+x^3\right )^{2/3}}{x^3} \, dx-2 \int \frac {\left (-1+2 x+x^3\right )^{2/3}}{x} \, dx-\int \frac {\left (-1+2 x+x^3\right )^{2/3}}{x^2} \, dx\\ &=\frac {1}{2} \int \left (-\frac {13 \left (-1+2 x+x^3\right )^{2/3}}{2-4 x+x^3}+\frac {2 x \left (-1+2 x+x^3\right )^{2/3}}{2-4 x+x^3}+\frac {4 x^2 \left (-1+2 x+x^3\right )^{2/3}}{2-4 x+x^3}\right ) \, dx-\frac {\left (-1+2 x+x^3\right )^{2/3} \int \frac {\left (\frac {4 \sqrt [3]{\frac {3}{9+\sqrt {177}}}-\sqrt [3]{2 \left (9+\sqrt {177}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (12+24 \sqrt [3]{3} \left (\frac {2}{9+\sqrt {177}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (9+\sqrt {177}\right )\right )^{2/3}\right )-\frac {\left (2 \sqrt [3]{\frac {6}{9+\sqrt {177}}}-\sqrt [3]{\frac {1}{2} \left (9+\sqrt {177}\right )}\right ) x}{3^{2/3}}+x^2\right )^{2/3}}{x^2} \, dx}{\left (\frac {4 \sqrt [3]{\frac {3}{9+\sqrt {177}}}-\sqrt [3]{2 \left (9+\sqrt {177}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (12+24 \sqrt [3]{3} \left (\frac {2}{9+\sqrt {177}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (9+\sqrt {177}\right )\right )^{2/3}\right )-\frac {\left (2 \sqrt [3]{\frac {6}{9+\sqrt {177}}}-\sqrt [3]{\frac {1}{2} \left (9+\sqrt {177}\right )}\right ) x}{3^{2/3}}+x^2\right )^{2/3}}-\frac {\left (3 \left (-1+2 x+x^3\right )^{2/3}\right ) \int \frac {\left (\frac {4 \sqrt [3]{\frac {3}{9+\sqrt {177}}}-\sqrt [3]{2 \left (9+\sqrt {177}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (12+24 \sqrt [3]{3} \left (\frac {2}{9+\sqrt {177}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (9+\sqrt {177}\right )\right )^{2/3}\right )-\frac {\left (2 \sqrt [3]{\frac {6}{9+\sqrt {177}}}-\sqrt [3]{\frac {1}{2} \left (9+\sqrt {177}\right )}\right ) x}{3^{2/3}}+x^2\right )^{2/3}}{x^3} \, dx}{2 \left (\frac {4 \sqrt [3]{\frac {3}{9+\sqrt {177}}}-\sqrt [3]{2 \left (9+\sqrt {177}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (12+24 \sqrt [3]{3} \left (\frac {2}{9+\sqrt {177}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (9+\sqrt {177}\right )\right )^{2/3}\right )-\frac {\left (2 \sqrt [3]{\frac {6}{9+\sqrt {177}}}-\sqrt [3]{\frac {1}{2} \left (9+\sqrt {177}\right )}\right ) x}{3^{2/3}}+x^2\right )^{2/3}}-\frac {\left (2 \left (-1+2 x+x^3\right )^{2/3}\right ) \int \frac {\left (\frac {4 \sqrt [3]{\frac {3}{9+\sqrt {177}}}-\sqrt [3]{2 \left (9+\sqrt {177}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (12+24 \sqrt [3]{3} \left (\frac {2}{9+\sqrt {177}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (9+\sqrt {177}\right )\right )^{2/3}\right )-\frac {\left (2 \sqrt [3]{\frac {6}{9+\sqrt {177}}}-\sqrt [3]{\frac {1}{2} \left (9+\sqrt {177}\right )}\right ) x}{3^{2/3}}+x^2\right )^{2/3}}{x} \, dx}{\left (\frac {4 \sqrt [3]{\frac {3}{9+\sqrt {177}}}-\sqrt [3]{2 \left (9+\sqrt {177}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (12+24 \sqrt [3]{3} \left (\frac {2}{9+\sqrt {177}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (9+\sqrt {177}\right )\right )^{2/3}\right )-\frac {\left (2 \sqrt [3]{\frac {6}{9+\sqrt {177}}}-\sqrt [3]{\frac {1}{2} \left (9+\sqrt {177}\right )}\right ) x}{3^{2/3}}+x^2\right )^{2/3}}\\ &=2 \int \frac {x^2 \left (-1+2 x+x^3\right )^{2/3}}{2-4 x+x^3} \, dx-\frac {13}{2} \int \frac {\left (-1+2 x+x^3\right )^{2/3}}{2-4 x+x^3} \, dx-\frac {\left (-1+2 x+x^3\right )^{2/3} \int \frac {\left (\frac {4 \sqrt [3]{\frac {3}{9+\sqrt {177}}}-\sqrt [3]{2 \left (9+\sqrt {177}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (12+24 \sqrt [3]{3} \left (\frac {2}{9+\sqrt {177}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (9+\sqrt {177}\right )\right )^{2/3}\right )-\frac {\left (2 \sqrt [3]{\frac {6}{9+\sqrt {177}}}-\sqrt [3]{\frac {1}{2} \left (9+\sqrt {177}\right )}\right ) x}{3^{2/3}}+x^2\right )^{2/3}}{x^2} \, dx}{\left (\frac {4 \sqrt [3]{\frac {3}{9+\sqrt {177}}}-\sqrt [3]{2 \left (9+\sqrt {177}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (12+24 \sqrt [3]{3} \left (\frac {2}{9+\sqrt {177}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (9+\sqrt {177}\right )\right )^{2/3}\right )-\frac {\left (2 \sqrt [3]{\frac {6}{9+\sqrt {177}}}-\sqrt [3]{\frac {1}{2} \left (9+\sqrt {177}\right )}\right ) x}{3^{2/3}}+x^2\right )^{2/3}}-\frac {\left (3 \left (-1+2 x+x^3\right )^{2/3}\right ) \int \frac {\left (\frac {4 \sqrt [3]{\frac {3}{9+\sqrt {177}}}-\sqrt [3]{2 \left (9+\sqrt {177}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (12+24 \sqrt [3]{3} \left (\frac {2}{9+\sqrt {177}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (9+\sqrt {177}\right )\right )^{2/3}\right )-\frac {\left (2 \sqrt [3]{\frac {6}{9+\sqrt {177}}}-\sqrt [3]{\frac {1}{2} \left (9+\sqrt {177}\right )}\right ) x}{3^{2/3}}+x^2\right )^{2/3}}{x^3} \, dx}{2 \left (\frac {4 \sqrt [3]{\frac {3}{9+\sqrt {177}}}-\sqrt [3]{2 \left (9+\sqrt {177}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (12+24 \sqrt [3]{3} \left (\frac {2}{9+\sqrt {177}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (9+\sqrt {177}\right )\right )^{2/3}\right )-\frac {\left (2 \sqrt [3]{\frac {6}{9+\sqrt {177}}}-\sqrt [3]{\frac {1}{2} \left (9+\sqrt {177}\right )}\right ) x}{3^{2/3}}+x^2\right )^{2/3}}-\frac {\left (2 \left (-1+2 x+x^3\right )^{2/3}\right ) \int \frac {\left (\frac {4 \sqrt [3]{\frac {3}{9+\sqrt {177}}}-\sqrt [3]{2 \left (9+\sqrt {177}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (12+24 \sqrt [3]{3} \left (\frac {2}{9+\sqrt {177}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (9+\sqrt {177}\right )\right )^{2/3}\right )-\frac {\left (2 \sqrt [3]{\frac {6}{9+\sqrt {177}}}-\sqrt [3]{\frac {1}{2} \left (9+\sqrt {177}\right )}\right ) x}{3^{2/3}}+x^2\right )^{2/3}}{x} \, dx}{\left (\frac {4 \sqrt [3]{\frac {3}{9+\sqrt {177}}}-\sqrt [3]{2 \left (9+\sqrt {177}\right )}}{6^{2/3}}+x\right )^{2/3} \left (\frac {1}{18} \left (12+24 \sqrt [3]{3} \left (\frac {2}{9+\sqrt {177}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (9+\sqrt {177}\right )\right )^{2/3}\right )-\frac {\left (2 \sqrt [3]{\frac {6}{9+\sqrt {177}}}-\sqrt [3]{\frac {1}{2} \left (9+\sqrt {177}\right )}\right ) x}{3^{2/3}}+x^2\right )^{2/3}}+\int \frac {x \left (-1+2 x+x^3\right )^{2/3}}{2-4 x+x^3} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.33, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(-3+4 x) \left (-1+2 x+x^3\right )^{2/3}}{x^3 \left (2-4 x+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.61, size = 175, normalized size = 1.00 \begin {gather*} \frac {3 \left (-1+2 x+x^3\right )^{2/3}}{4 x^2}-\frac {3 \sqrt [6]{3} \tan ^{-1}\left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{2} \sqrt [3]{-1+2 x+x^3}}\right )}{2\ 2^{2/3}}+\frac {1}{2} \left (\frac {3}{2}\right )^{2/3} \log \left (-3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{-1+2 x+x^3}\right )-\frac {1}{4} \left (\frac {3}{2}\right )^{2/3} \log \left (3 x^2+\sqrt [3]{2} 3^{2/3} x \sqrt [3]{-1+2 x+x^3}+2^{2/3} \sqrt [3]{3} \left (-1+2 x+x^3\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(-1 + 2*x + x^3)^(2/3))/(4*x^2) - (3*3^(1/6)*ArcTan[(3^(5/6)*x)/(3^(1/3)*x + 2*2^(1/3)*(-1 + 2*x + x^3)^(1/
3))])/(2*2^(2/3)) + ((3/2)^(2/3)*Log[-3*x + 2^(1/3)*3^(2/3)*(-1 + 2*x + x^3)^(1/3)])/2 - ((3/2)^(2/3)*Log[3*x^
2 + 2^(1/3)*3^(2/3)*x*(-1 + 2*x + x^3)^(1/3) + 2^(2/3)*3^(1/3)*(-1 + 2*x + x^3)^(2/3)])/4

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fricas [B]  time = 17.29, size = 427, normalized size = 2.44 \begin {gather*} -\frac {4 \cdot 9^{\frac {1}{3}} 4^{\frac {1}{6}} \sqrt {3} x^{2} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (4 \cdot 9^{\frac {2}{3}} 4^{\frac {2}{3}} {\left (4 \, x^{7} - 14 \, x^{5} + 7 \, x^{4} - 8 \, x^{3} + 8 \, x^{2} - 2 \, x\right )} {\left (x^{3} + 2 \, x - 1\right )}^{\frac {2}{3}} - 12 \cdot 9^{\frac {1}{3}} {\left (55 \, x^{8} + 100 \, x^{6} - 50 \, x^{5} + 16 \, x^{4} - 16 \, x^{3} + 4 \, x^{2}\right )} {\left (x^{3} + 2 \, x - 1\right )}^{\frac {1}{3}} - 4^{\frac {1}{3}} {\left (377 \, x^{9} + 1200 \, x^{7} - 600 \, x^{6} + 816 \, x^{5} - 816 \, x^{4} + 268 \, x^{3} - 96 \, x^{2} + 48 \, x - 8\right )}\right )}}{6 \, {\left (487 \, x^{9} + 960 \, x^{7} - 480 \, x^{6} + 48 \, x^{5} - 48 \, x^{4} - 52 \, x^{3} + 96 \, x^{2} - 48 \, x + 8\right )}}\right ) - 2 \cdot 9^{\frac {1}{3}} 4^{\frac {2}{3}} x^{2} \log \left (-\frac {6 \cdot 9^{\frac {2}{3}} 4^{\frac {1}{3}} {\left (x^{3} + 2 \, x - 1\right )}^{\frac {1}{3}} x^{2} - 9^{\frac {1}{3}} 4^{\frac {2}{3}} {\left (x^{3} - 4 \, x + 2\right )} - 36 \, {\left (x^{3} + 2 \, x - 1\right )}^{\frac {2}{3}} x}{x^{3} - 4 \, x + 2}\right ) + 9^{\frac {1}{3}} 4^{\frac {2}{3}} x^{2} \log \left (\frac {18 \cdot 9^{\frac {1}{3}} 4^{\frac {2}{3}} {\left (4 \, x^{4} + 2 \, x^{2} - x\right )} {\left (x^{3} + 2 \, x - 1\right )}^{\frac {2}{3}} + 9^{\frac {2}{3}} 4^{\frac {1}{3}} {\left (55 \, x^{6} + 100 \, x^{4} - 50 \, x^{3} + 16 \, x^{2} - 16 \, x + 4\right )} + 54 \, {\left (7 \, x^{5} + 8 \, x^{3} - 4 \, x^{2}\right )} {\left (x^{3} + 2 \, x - 1\right )}^{\frac {1}{3}}}{x^{6} - 8 \, x^{4} + 4 \, x^{3} + 16 \, x^{2} - 16 \, x + 4}\right ) - 36 \, {\left (x^{3} + 2 \, x - 1\right )}^{\frac {2}{3}}}{48 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*(4*9^(1/3)*4^(1/6)*sqrt(3)*x^2*arctan(1/6*4^(1/6)*sqrt(3)*(4*9^(2/3)*4^(2/3)*(4*x^7 - 14*x^5 + 7*x^4 - 8
*x^3 + 8*x^2 - 2*x)*(x^3 + 2*x - 1)^(2/3) - 12*9^(1/3)*(55*x^8 + 100*x^6 - 50*x^5 + 16*x^4 - 16*x^3 + 4*x^2)*(
x^3 + 2*x - 1)^(1/3) - 4^(1/3)*(377*x^9 + 1200*x^7 - 600*x^6 + 816*x^5 - 816*x^4 + 268*x^3 - 96*x^2 + 48*x - 8
))/(487*x^9 + 960*x^7 - 480*x^6 + 48*x^5 - 48*x^4 - 52*x^3 + 96*x^2 - 48*x + 8)) - 2*9^(1/3)*4^(2/3)*x^2*log(-
(6*9^(2/3)*4^(1/3)*(x^3 + 2*x - 1)^(1/3)*x^2 - 9^(1/3)*4^(2/3)*(x^3 - 4*x + 2) - 36*(x^3 + 2*x - 1)^(2/3)*x)/(
x^3 - 4*x + 2)) + 9^(1/3)*4^(2/3)*x^2*log((18*9^(1/3)*4^(2/3)*(4*x^4 + 2*x^2 - x)*(x^3 + 2*x - 1)^(2/3) + 9^(2
/3)*4^(1/3)*(55*x^6 + 100*x^4 - 50*x^3 + 16*x^2 - 16*x + 4) + 54*(7*x^5 + 8*x^3 - 4*x^2)*(x^3 + 2*x - 1)^(1/3)
)/(x^6 - 8*x^4 + 4*x^3 + 16*x^2 - 16*x + 4)) - 36*(x^3 + 2*x - 1)^(2/3))/x^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 18.98, size = 1054, normalized size = 6.02

method	result	size
risch	\(\frac {3 \left (x^{3}+2 x -1\right )^{\frac {2}{3}}}{4 x^{2}}-\frac {\ln \left (-\frac {3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-18\right )^{3} x^{3}-12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-18\right )^{2} x^{3}+14 \left (x^{3}+2 x -1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-18\right )^{2} x -\left (x^{3}+2 x -1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-18\right )^{2} x^{2}+42 \left (x^{3}+2 x -1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-18\right ) x^{2}-6 \RootOf \left (\textit {\_Z}^{3}-18\right ) x^{3}+24 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+36 \textit {\_Z}^{2}\right ) x^{3}-6 \left (x^{3}+2 x -1\right )^{\frac {2}{3}} x -12 \RootOf \left (\textit {\_Z}^{3}-18\right ) x +48 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+36 \textit {\_Z}^{2}\right ) x +6 \RootOf \left (\textit {\_Z}^{3}-18\right )-24 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+36 \textit {\_Z}^{2}\right )}{x^{3}-4 x +2}\right ) \RootOf \left (\textit {\_Z}^{3}-18\right )}{4}-\frac {3 \ln \left (-\frac {3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-18\right )^{3} x^{3}-12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-18\right )^{2} x^{3}+14 \left (x^{3}+2 x -1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-18\right )^{2} x -\left (x^{3}+2 x -1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-18\right )^{2} x^{2}+42 \left (x^{3}+2 x -1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-18\right ) x^{2}-6 \RootOf \left (\textit {\_Z}^{3}-18\right ) x^{3}+24 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+36 \textit {\_Z}^{2}\right ) x^{3}-6 \left (x^{3}+2 x -1\right )^{\frac {2}{3}} x -12 \RootOf \left (\textit {\_Z}^{3}-18\right ) x +48 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+36 \textit {\_Z}^{2}\right ) x +6 \RootOf \left (\textit {\_Z}^{3}-18\right )-24 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+36 \textit {\_Z}^{2}\right )}{x^{3}-4 x +2}\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+36 \textit {\_Z}^{2}\right )}{2}+\frac {3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+36 \textit {\_Z}^{2}\right ) \ln \left (\frac {5 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-18\right )^{3} x^{3}+12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-18\right )^{2} x^{3}+14 \left (x^{3}+2 x -1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-18\right )^{2} x +8 \left (x^{3}+2 x -1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-18\right )^{2} x^{2}+42 \left (x^{3}+2 x -1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-18\right ) x^{2}+25 \RootOf \left (\textit {\_Z}^{3}-18\right ) x^{3}+60 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+36 \textit {\_Z}^{2}\right ) x^{3}+48 \left (x^{3}+2 x -1\right )^{\frac {2}{3}} x +20 \RootOf \left (\textit {\_Z}^{3}-18\right ) x +48 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+36 \textit {\_Z}^{2}\right ) x -10 \RootOf \left (\textit {\_Z}^{3}-18\right )-24 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-18\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-18\right )+36 \textit {\_Z}^{2}\right )}{x^{3}-4 x +2}\right )}{2}\)	\(1054\)
trager	\(\text {Expression too large to display}\)	\(1480\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*(x^3+2*x-1)^(2/3)/x^2-1/4*ln(-(3*RootOf(RootOf(_Z^3-18)^2+6*_Z*RootOf(_Z^3-18)+36*_Z^2)*RootOf(_Z^3-18)^3*
x^3-12*RootOf(RootOf(_Z^3-18)^2+6*_Z*RootOf(_Z^3-18)+36*_Z^2)^2*RootOf(_Z^3-18)^2*x^3+14*(x^3+2*x-1)^(2/3)*Roo
tOf(RootOf(_Z^3-18)^2+6*_Z*RootOf(_Z^3-18)+36*_Z^2)*RootOf(_Z^3-18)^2*x-(x^3+2*x-1)^(1/3)*RootOf(_Z^3-18)^2*x^
2+42*(x^3+2*x-1)^(1/3)*RootOf(RootOf(_Z^3-18)^2+6*_Z*RootOf(_Z^3-18)+36*_Z^2)*RootOf(_Z^3-18)*x^2-6*RootOf(_Z^
3-18)*x^3+24*RootOf(RootOf(_Z^3-18)^2+6*_Z*RootOf(_Z^3-18)+36*_Z^2)*x^3-6*(x^3+2*x-1)^(2/3)*x-12*RootOf(_Z^3-1
8)*x+48*RootOf(RootOf(_Z^3-18)^2+6*_Z*RootOf(_Z^3-18)+36*_Z^2)*x+6*RootOf(_Z^3-18)-24*RootOf(RootOf(_Z^3-18)^2
+6*_Z*RootOf(_Z^3-18)+36*_Z^2))/(x^3-4*x+2))*RootOf(_Z^3-18)-3/2*ln(-(3*RootOf(RootOf(_Z^3-18)^2+6*_Z*RootOf(_
Z^3-18)+36*_Z^2)*RootOf(_Z^3-18)^3*x^3-12*RootOf(RootOf(_Z^3-18)^2+6*_Z*RootOf(_Z^3-18)+36*_Z^2)^2*RootOf(_Z^3
-18)^2*x^3+14*(x^3+2*x-1)^(2/3)*RootOf(RootOf(_Z^3-18)^2+6*_Z*RootOf(_Z^3-18)+36*_Z^2)*RootOf(_Z^3-18)^2*x-(x^
3+2*x-1)^(1/3)*RootOf(_Z^3-18)^2*x^2+42*(x^3+2*x-1)^(1/3)*RootOf(RootOf(_Z^3-18)^2+6*_Z*RootOf(_Z^3-18)+36*_Z^
2)*RootOf(_Z^3-18)*x^2-6*RootOf(_Z^3-18)*x^3+24*RootOf(RootOf(_Z^3-18)^2+6*_Z*RootOf(_Z^3-18)+36*_Z^2)*x^3-6*(
x^3+2*x-1)^(2/3)*x-12*RootOf(_Z^3-18)*x+48*RootOf(RootOf(_Z^3-18)^2+6*_Z*RootOf(_Z^3-18)+36*_Z^2)*x+6*RootOf(_
Z^3-18)-24*RootOf(RootOf(_Z^3-18)^2+6*_Z*RootOf(_Z^3-18)+36*_Z^2))/(x^3-4*x+2))*RootOf(RootOf(_Z^3-18)^2+6*_Z*
RootOf(_Z^3-18)+36*_Z^2)+3/2*RootOf(RootOf(_Z^3-18)^2+6*_Z*RootOf(_Z^3-18)+36*_Z^2)*ln((5*RootOf(RootOf(_Z^3-1
8)^2+6*_Z*RootOf(_Z^3-18)+36*_Z^2)*RootOf(_Z^3-18)^3*x^3+12*RootOf(RootOf(_Z^3-18)^2+6*_Z*RootOf(_Z^3-18)+36*_
Z^2)^2*RootOf(_Z^3-18)^2*x^3+14*(x^3+2*x-1)^(2/3)*RootOf(RootOf(_Z^3-18)^2+6*_Z*RootOf(_Z^3-18)+36*_Z^2)*RootO
f(_Z^3-18)^2*x+8*(x^3+2*x-1)^(1/3)*RootOf(_Z^3-18)^2*x^2+42*(x^3+2*x-1)^(1/3)*RootOf(RootOf(_Z^3-18)^2+6*_Z*Ro
otOf(_Z^3-18)+36*_Z^2)*RootOf(_Z^3-18)*x^2+25*RootOf(_Z^3-18)*x^3+60*RootOf(RootOf(_Z^3-18)^2+6*_Z*RootOf(_Z^3
-18)+36*_Z^2)*x^3+48*(x^3+2*x-1)^(2/3)*x+20*RootOf(_Z^3-18)*x+48*RootOf(RootOf(_Z^3-18)^2+6*_Z*RootOf(_Z^3-18)
+36*_Z^2)*x-10*RootOf(_Z^3-18)-24*RootOf(RootOf(_Z^3-18)^2+6*_Z*RootOf(_Z^3-18)+36*_Z^2))/(x^3-4*x+2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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