3.10.85 \(\int \frac {1}{\sqrt [4]{-1+3 x-3 x^2+x^3} (-1-2 x+x^2+3 x^3)} \, dx\) [985]
Optimal. Leaf size=75 \[ \frac {\left ((-1+x)^3\right )^{3/4} \text {RootSum}\left [1+9 \text {$\#$1}^4+10 \text {$\#$1}^8+3 \text {$\#$1}^{12}\& ,\frac {\log \left (\sqrt [4]{-1+x}-\text {$\#$1}\right )}{9 \text {$\#$1}^3+20 \text {$\#$1}^7+9 \text {$\#$1}^{11}}\& \right ]}{(-1+x)^{9/4}} \]
[Out]
Unintegrable
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Rubi [F]
time = 0.07, 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 {1}{\sqrt [4]{-1+3 x-3 x^2+x^3} \left (-1-2 x+x^2+3 x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[1/((-1 + 3*x - 3*x^2 + x^3)^(1/4)*(-1 - 2*x + x^2 + 3*x^3)),x]
[Out]
Defer[Int][1/((-1 + 3*x - 3*x^2 + x^3)^(1/4)*(-1 - 2*x + x^2 + 3*x^3)), x]
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [4]{-1+3 x-3 x^2+x^3} \left (-1-2 x+x^2+3 x^3\right )} \, dx &=\int \frac {1}{\sqrt [4]{-1+3 x-3 x^2+x^3} \left (-1-2 x+x^2+3 x^3\right )} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 75, normalized size = 1.00 \begin {gather*} \frac {(-1+x)^{3/4} \text {RootSum}\left [1+9 \text {$\#$1}^4+10 \text {$\#$1}^8+3 \text {$\#$1}^{12}\&,\frac {\log \left (\sqrt [4]{-1+x}-\text {$\#$1}\right )}{9 \text {$\#$1}^3+20 \text {$\#$1}^7+9 \text {$\#$1}^{11}}\&\right ]}{\sqrt [4]{(-1+x)^3}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/((-1 + 3*x - 3*x^2 + x^3)^(1/4)*(-1 - 2*x + x^2 + 3*x^3)),x]
[Out]
((-1 + x)^(3/4)*RootSum[1 + 9*#1^4 + 10*#1^8 + 3*#1^12 & , Log[(-1 + x)^(1/4) - #1]/(9*#1^3 + 20*#1^7 + 9*#1^1
1) & ])/((-1 + x)^3)^(1/4)
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (x^{3}-3 x^{2}+3 x -1\right )^{\frac {1}{4}} \left (3 x^{3}+x^{2}-2 x -1\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(x^3-3*x^2+3*x-1)^(1/4)/(3*x^3+x^2-2*x-1),x)
[Out]
int(1/(x^3-3*x^2+3*x-1)^(1/4)/(3*x^3+x^2-2*x-1),x)
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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(1/(x^3-3*x^2+3*x-1)^(1/4)/(3*x^3+x^2-2*x-1),x, algorithm="maxima")
[Out]
integrate(1/((3*x^3 + x^2 - 2*x - 1)*(x^3 - 3*x^2 + 3*x - 1)^(1/4)), x)
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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order
1.
time = 1.52, size = 6190, normalized size = 82.53 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(x^3-3*x^2+3*x-1)^(1/4)/(3*x^3+x^2-2*x-1),x, algorithm="fricas")
[Out]
2/31*sqrt(31)*sqrt(2)*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I
*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) - 2*sqrt(31)*sqrt(-3/124*((1/18)^(1/3)*(621503*sqrt(93) - 6210
333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260)^2 +
390/31*(1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 14135940/31*(1/18)^(2/3)*(I*sqrt(3) +
1)/(621503*sqrt(93) - 6210333)^(1/3) + 74864/31) - 520)^(1/4)*arctan(1/31037243945331168*sqrt(2)*(14386551444
*sqrt(31)*(x^3 - 3*x^2 + 3*x - 1)^(1/4)*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 362
46*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260)^2 - sqrt(1864509)*sqrt(1/31)*(643*sqr
t(31)*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(6
21503*sqrt(93) - 6210333)^(1/3) + 260)^2*(x - 1) - 389166*sqrt(31)*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(
1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260)*(x - 1) +
62*sqrt(-3/124*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3
) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260)^2 + 390/31*(1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*
sqrt(3) + 1) + 14135940/31*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 74864/31)*(643*((1
/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqr
t(93) - 6210333)^(1/3) + 260)*(x - 1) - 112374*x + 112374) + 46529388*sqrt(31)*(x - 1))*sqrt(((3239*(x^2 - 2*x
+ 1)*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(6
21503*sqrt(93) - 6210333)^(1/3) + 260)^2 - 551100*(x^2 - 2*x + 1)*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1
/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260) - 159338016
*x^2 + 2*(3239*sqrt(31)*(x^2 - 2*x + 1)*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 362
46*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260) - 1975320*sqrt(31)*(x^2 - 2*x + 1))*s
qrt(-3/124*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) +
1)/(621503*sqrt(93) - 6210333)^(1/3) + 260)^2 + 390/31*(1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt
(3) + 1) + 14135940/31*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 74864/31) + 318676032*
x - 159338016)*sqrt((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sq
rt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) - 2*sqrt(31)*sqrt(-3/124*((1/18)^(1/3)*(621503*sqrt(93) - 6210333
)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260)^2 + 390
/31*(1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 14135940/31*(1/18)^(2/3)*(I*sqrt(3) + 1)
/(621503*sqrt(93) - 6210333)^(1/3) + 74864/31) - 520) + 8323168176*sqrt(x^3 - 3*x^2 + 3*x - 1))/(x^2 - 2*x + 1
)) - 8707242113928*sqrt(31)*(x^3 - 3*x^2 + 3*x - 1)^(1/4)*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*
sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260) + 1387194696*sqrt(-
3/124*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(6
21503*sqrt(93) - 6210333)^(1/3) + 260)^2 + 390/31*(1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) +
1) + 14135940/31*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 74864/31)*(643*(x^3 - 3*x^2
+ 3*x - 1)^(1/4)*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqr
t(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260) - 112374*(x^3 - 3*x^2 + 3*x - 1)^(1/4)) + 1041053552285904*
sqrt(31)*(x^3 - 3*x^2 + 3*x - 1)^(1/4))*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 362
46*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) - 2*sqrt(31)*sqrt(-3/124*((1/18)^(1/3)*(6215
03*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)
^(1/3) + 260)^2 + 390/31*(1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 14135940/31*(1/18)^
(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 74864/31) - 520)^(1/4)/(x - 1)) - 2/31*sqrt(31)*sqrt
(2)*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621
503*sqrt(93) - 6210333)^(1/3) + 2*sqrt(31)*sqrt(-3/124*((1/18)^(1/3)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqr
t(3) + 1) + 36246*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(93) - 6210333)^(1/3) + 260)^2 + 390/31*(1/18)^(1/3
)*(621503*sqrt(93) - 6210333)^(1/3)*(-I*sqrt(3) + 1) + 14135940/31*(1/18)^(2/3)*(I*sqrt(3) + 1)/(621503*sqrt(9
3) - 6210333)^(1/3) + 74864/31) - 520)^(1/4)*arctan(1/31037243945331168*(sqrt(1864509)*sqrt(2)*sqrt(1/31)*(643
*sqrt(31)*((1/18)^(1/3)*(621503*sqrt(93) - 6210...
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (3 x^{3} + x^{2} - 2 x - 1\right ) \sqrt [4]{\left (x - 1\right )^{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(x**3-3*x**2+3*x-1)**(1/4)/(3*x**3+x**2-2*x-1),x)
[Out]
Integral(1/((3*x**3 + x**2 - 2*x - 1)*((x - 1)**3)**(1/4)), 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(1/(x^3-3*x^2+3*x-1)^(1/4)/(3*x^3+x^2-2*x-1),x, algorithm="giac")
[Out]
integrate(1/((3*x^3 + x^2 - 2*x - 1)*(x^3 - 3*x^2 + 3*x - 1)^(1/4)), x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {1}{{\left (x^3-3\,x^2+3\,x-1\right )}^{1/4}\,\left (-3\,x^3-x^2+2\,x+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-1/((3*x - 3*x^2 + x^3 - 1)^(1/4)*(2*x - x^2 - 3*x^3 + 1)),x)
[Out]
-int(1/((3*x - 3*x^2 + x^3 - 1)^(1/4)*(2*x - x^2 - 3*x^3 + 1)), x)
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