3.23.94 \(\int \frac {(-3+4 x) (-1+2 x+x^3)^{2/3}}{x^3 (2-4 x+x^3)} \, dx\) [2294]

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

[Out]

3/4*(x^3+2*x-1)^(2/3)/x^2-3/4*3^(1/6)*arctan(3^(5/6)*x/(3^(1/3)*x+2*2^(1/3)*(x^3+2*x-1)^(1/3)))*2^(1/3)+1/4*2^
(1/3)*3^(2/3)*ln(-3*x+2^(1/3)*3^(2/3)*(x^3+2*x-1)^(1/3))-1/8*2^(1/3)*3^(2/3)*ln(3*x^2+2^(1/3)*3^(2/3)*x*(x^3+2
*x-1)^(1/3)+2^(2/3)*3^(1/3)*(x^3+2*x-1)^(2/3))

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Rubi [F]
time = 1.39, 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 [A]
time = 0.45, 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} \text {ArcTan}\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]

Integrate[((-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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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 11.12, size = 942, normalized size = 5.38

method	result	size
risch	\(\text {Expression too large to display}\)	\(942\)
trager	\(\text {Expression too large to display}\)	\(1463\)

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((12*RootOf(RootOf(_Z^3-18)^2+6*_Z*RootOf(_Z^3-18)+36*_Z^2)^2*RootOf(_Z^3-18)^
2*x^3+RootOf(RootOf(_Z^3-18)^2+6*_Z*RootOf(_Z^3-18)+36*_Z^2)*RootOf(_Z^3-18)^3*x^3-18*(x^3+2*x-1)^(1/3)*RootOf
(_Z^3-18)*RootOf(RootOf(_Z^3-18)^2+6*_Z*RootOf(_Z^3-18)+36*_Z^2)*x^2-3*RootOf(_Z^3-18)^2*(x^3+2*x-1)^(1/3)*x^2
+60*RootOf(RootOf(_Z^3-18)^2+6*_Z*RootOf(_Z^3-18)+36*_Z^2)*x^3+5*RootOf(_Z^3-18)*x^3+18*(x^3+2*x-1)^(2/3)*x+48
*RootOf(RootOf(_Z^3-18)^2+6*_Z*RootOf(_Z^3-18)+36*_Z^2)*x+4*RootOf(_Z^3-18)*x-24*RootOf(RootOf(_Z^3-18)^2+6*_Z
*RootOf(_Z^3-18)+36*_Z^2)-2*RootOf(_Z^3-18))/(x^3-4*x+2))*RootOf(_Z^3-18)-3/2*ln((12*RootOf(RootOf(_Z^3-18)^2+
6*_Z*RootOf(_Z^3-18)+36*_Z^2)^2*RootOf(_Z^3-18)^2*x^3+RootOf(RootOf(_Z^3-18)^2+6*_Z*RootOf(_Z^3-18)+36*_Z^2)*R
ootOf(_Z^3-18)^3*x^3-18*(x^3+2*x-1)^(1/3)*RootOf(_Z^3-18)*RootOf(RootOf(_Z^3-18)^2+6*_Z*RootOf(_Z^3-18)+36*_Z^
2)*x^2-3*RootOf(_Z^3-18)^2*(x^3+2*x-1)^(1/3)*x^2+60*RootOf(RootOf(_Z^3-18)^2+6*_Z*RootOf(_Z^3-18)+36*_Z^2)*x^3
+5*RootOf(_Z^3-18)*x^3+18*(x^3+2*x-1)^(2/3)*x+48*RootOf(RootOf(_Z^3-18)^2+6*_Z*RootOf(_Z^3-18)+36*_Z^2)*x+4*Ro
otOf(_Z^3-18)*x-24*RootOf(RootOf(_Z^3-18)^2+6*_Z*RootOf(_Z^3-18)+36*_Z^2)-2*RootOf(_Z^3-18))/(x^3-4*x+2))*Root
Of(RootOf(_Z^3-18)^2+6*_Z*RootOf(_Z^3-18)+36*_Z^2)+1/4*RootOf(_Z^3-18)*ln(-(15*RootOf(RootOf(_Z^3-18)^2+6*_Z*R
ootOf(_Z^3-18)+36*_Z^2)^2*RootOf(_Z^3-18)^2*x^3+RootOf(RootOf(_Z^3-18)^2+6*_Z*RootOf(_Z^3-18)+36*_Z^2)*RootOf(
_Z^3-18)^3*x^3+7*(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+
3*(x^3+2*x-1)^(1/3)*RootOf(_Z^3-18)*RootOf(RootOf(_Z^3-18)^2+6*_Z*RootOf(_Z^3-18)+36*_Z^2)*x^2+4*RootOf(_Z^3-1
8)^2*(x^3+2*x-1)^(1/3)*x^2-30*RootOf(RootOf(_Z^3-18)^2+6*_Z*RootOf(_Z^3-18)+36*_Z^2)*x^3-2*RootOf(_Z^3-18)*x^3
-3*(x^3+2*x-1)^(2/3)*x-60*RootOf(RootOf(_Z^3-18)^2+6*_Z*RootOf(_Z^3-18)+36*_Z^2)*x-4*RootOf(_Z^3-18)*x+30*Root
Of(RootOf(_Z^3-18)^2+6*_Z*RootOf(_Z^3-18)+36*_Z^2)+2*RootOf(_Z^3-18))/(x^3-4*x+2))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (133) = 266\).
time = 9.12, 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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (4 x - 3\right ) \left (x^{3} + 2 x - 1\right )^{\frac {2}{3}}}{x^{3} \left (x^{3} - 4 x + 2\right )}\, dx \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]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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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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