3.9.40 \(\int \frac {\sqrt {-1+x^2+x^5} (2+3 x^5)}{1+x^4-2 x^5+x^{10}} \, dx\)

Optimal. Leaf size=63 \[ -\sqrt {1+i} \tan ^{-1}\left (\frac {\sqrt {-1-i} x}{\sqrt {x^5+x^2-1}}\right )-\sqrt {1-i} \tan ^{-1}\left (\frac {\sqrt {-1+i} x}{\sqrt {x^5+x^2-1}}\right ) \]

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Rubi [F]  time = 0.48, 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 {\sqrt {-1+x^2+x^5} \left (2+3 x^5\right )}{1+x^4-2 x^5+x^{10}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(Sqrt[-1 + x^2 + x^5]*(2 + 3*x^5))/(1 + x^4 - 2*x^5 + x^10),x]

[Out]

2*Defer[Int][Sqrt[-1 + x^2 + x^5]/(1 + x^4 - 2*x^5 + x^10), x] + 3*Defer[Int][(x^5*Sqrt[-1 + x^2 + x^5])/(1 +
x^4 - 2*x^5 + x^10), x]

Rubi steps

\begin {align*} \int \frac {\sqrt {-1+x^2+x^5} \left (2+3 x^5\right )}{1+x^4-2 x^5+x^{10}} \, dx &=\int \left (\frac {2 \sqrt {-1+x^2+x^5}}{1+x^4-2 x^5+x^{10}}+\frac {3 x^5 \sqrt {-1+x^2+x^5}}{1+x^4-2 x^5+x^{10}}\right ) \, dx\\ &=2 \int \frac {\sqrt {-1+x^2+x^5}}{1+x^4-2 x^5+x^{10}} \, dx+3 \int \frac {x^5 \sqrt {-1+x^2+x^5}}{1+x^4-2 x^5+x^{10}} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.14, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {-1+x^2+x^5} \left (2+3 x^5\right )}{1+x^4-2 x^5+x^{10}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(Sqrt[-1 + x^2 + x^5]*(2 + 3*x^5))/(1 + x^4 - 2*x^5 + x^10),x]

[Out]

Integrate[(Sqrt[-1 + x^2 + x^5]*(2 + 3*x^5))/(1 + x^4 - 2*x^5 + x^10), x]

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IntegrateAlgebraic [A]  time = 2.86, size = 63, normalized size = 1.00 \begin {gather*} -\sqrt {1+i} \tan ^{-1}\left (\frac {\sqrt {-1-i} x}{\sqrt {-1+x^2+x^5}}\right )-\sqrt {1-i} \tan ^{-1}\left (\frac {\sqrt {-1+i} x}{\sqrt {-1+x^2+x^5}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(Sqrt[-1 + x^2 + x^5]*(2 + 3*x^5))/(1 + x^4 - 2*x^5 + x^10),x]

[Out]

-(Sqrt[1 + I]*ArcTan[(Sqrt[-1 - I]*x)/Sqrt[-1 + x^2 + x^5]]) - Sqrt[1 - I]*ArcTan[(Sqrt[-1 + I]*x)/Sqrt[-1 + x
^2 + x^5]]

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fricas [B]  time = 1.53, size = 3547, normalized size = 56.30

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5+x^2-1)^(1/2)*(3*x^5+2)/(x^10-2*x^5+x^4+1),x, algorithm="fricas")

[Out]

1/16*2^(1/4)*sqrt(2*sqrt(2) + 4)*(sqrt(2) - 2)*log((x^10 + 4*x^7 - 2*x^5 + 5*x^4 + 2^(1/4)*sqrt(x^5 + x^2 - 1)
*(2*x^3 + sqrt(2)*(x^6 + x^3 - x))*sqrt(2*sqrt(2) + 4) - 4*x^2 + 4*sqrt(2)*(x^7 + x^4 - x^2) + 1)/(x^10 - 2*x^
5 + x^4 + 1)) - 1/16*2^(1/4)*sqrt(2*sqrt(2) + 4)*(sqrt(2) - 2)*log((x^10 + 4*x^7 - 2*x^5 + 5*x^4 - 2^(1/4)*sqr
t(x^5 + x^2 - 1)*(2*x^3 + sqrt(2)*(x^6 + x^3 - x))*sqrt(2*sqrt(2) + 4) - 4*x^2 + 4*sqrt(2)*(x^7 + x^4 - x^2) +
1)/(x^10 - 2*x^5 + x^4 + 1)) + 1/4*2^(3/4)*sqrt(2*sqrt(2) + 4)*arctan(1/4*(4*x^26 - 4*x^23 - 20*x^21 - 56*x^2
0 + 16*x^18 - 56*x^17 + 40*x^16 + 168*x^15 + 4*x^14 - 24*x^13 + 112*x^12 - 28*x^11 - 168*x^10 - 4*x^9 + 16*x^8
- 56*x^7 + 20*x^6 + 56*x^5 - 4*x^3 + sqrt(x^5 + x^2 - 1)*(2^(3/4)*(x^25 + 7*x^22 - 5*x^20 + 10*x^19 - 28*x^17
- 6*x^16 + 10*x^15 - 30*x^14 - 7*x^13 + 42*x^12 + 12*x^11 - 7*x^10 + 30*x^9 + 7*x^8 - 28*x^7 - 6*x^6 + 5*x^5
- 10*x^4 + 7*x^2 - sqrt(2)*(3*x^22 + 10*x^19 - 12*x^17 + 4*x^16 - 30*x^14 - 6*x^13 + 18*x^12 - 8*x^11 + x^10 +
30*x^9 + 6*x^8 - 12*x^7 + 4*x^6 - 10*x^4 + 3*x^2) - 1) + 2*2^(1/4)*(2*x^22 + 2*x^19 - 8*x^17 + 10*x^16 - 6*x^
14 + 18*x^13 + 12*x^12 - 20*x^11 + 8*x^10 + 6*x^9 - 18*x^8 - 8*x^7 + 10*x^6 - 2*x^4 + 2*x^2 + sqrt(2)*(x^22 -
4*x^19 - 4*x^17 - 12*x^16 + 12*x^14 - 12*x^13 + 6*x^12 + 24*x^11 - 5*x^10 - 12*x^9 + 12*x^8 - 4*x^7 - 12*x^6 +
4*x^4 + x^2)))*sqrt(2*sqrt(2) + 4) - 2*sqrt(2)*(2*x^26 + 6*x^23 - 10*x^21 + 4*x^20 - 24*x^18 + 4*x^17 + 20*x^
16 - 12*x^15 + 2*x^14 + 36*x^13 - 8*x^12 - 22*x^11 + 12*x^10 - 2*x^9 - 24*x^8 + 4*x^7 + 10*x^6 - 4*x^5 + 6*x^3
+ sqrt(2)*(x^26 - x^23 - 5*x^21 - 14*x^20 + 4*x^18 - 14*x^17 + 10*x^16 + 42*x^15 + x^14 - 6*x^13 + 28*x^12 -
7*x^11 - 42*x^10 - x^9 + 4*x^8 - 14*x^7 + 5*x^6 + 14*x^5 - x^3 - x) - 2*x) + 4*sqrt(2)*(x^26 + 3*x^23 - 5*x^21
+ 2*x^20 - 12*x^18 + 2*x^17 + 10*x^16 - 6*x^15 + x^14 + 18*x^13 - 4*x^12 - 11*x^11 + 6*x^10 - x^9 - 12*x^8 +
2*x^7 + 5*x^6 - 2*x^5 + 3*x^3 - x) + (24*x^23 + 16*x^20 - 96*x^18 - 48*x^17 - 48*x^15 - 48*x^14 + 144*x^13 + 9
6*x^12 - 8*x^11 + 48*x^10 + 48*x^9 - 96*x^8 - 48*x^7 - 16*x^5 + 24*x^3 - sqrt(x^5 + x^2 - 1)*(2^(3/4)*(x^25 +
7*x^22 - 5*x^20 + 2*x^19 - 28*x^17 - 14*x^16 + 10*x^15 - 6*x^14 - 15*x^13 + 42*x^12 + 28*x^11 - 15*x^10 + 6*x^
9 + 15*x^8 - 28*x^7 - 14*x^6 + 5*x^5 - 2*x^4 + 7*x^2 - sqrt(2)*(x^22 - 2*x^19 - 4*x^17 + 4*x^16 + 6*x^14 + 14*
x^13 + 6*x^12 - 8*x^11 + 3*x^10 - 6*x^9 - 14*x^8 - 4*x^7 + 4*x^6 + 2*x^4 + x^2) - 1) + 2*2^(1/4)*(2*x^22 + 6*x
^19 - 8*x^17 - 2*x^16 - 18*x^14 - 10*x^13 + 12*x^12 + 4*x^11 - 4*x^10 + 18*x^9 + 10*x^8 - 8*x^7 - 2*x^6 - 6*x^
4 + 2*x^2 + sqrt(2)*(x^22 - 4*x^17 - 8*x^16 - 8*x^13 + 6*x^12 + 16*x^11 - x^10 + 8*x^8 - 4*x^7 - 8*x^6 + x^2))
)*sqrt(2*sqrt(2) + 4) + 2*sqrt(2)*(2*x^26 + 10*x^23 - 10*x^21 + 4*x^20 - 40*x^18 - 28*x^17 + 20*x^16 - 12*x^15
- 30*x^14 + 60*x^13 + 56*x^12 - 26*x^11 + 12*x^10 + 30*x^9 - 40*x^8 - 28*x^7 + 10*x^6 - 4*x^5 + 10*x^3 + sqrt
(2)*(x^26 + 3*x^23 - 5*x^21 - 2*x^20 - 12*x^18 - 18*x^17 + 10*x^16 + 6*x^15 - 19*x^14 + 18*x^13 + 36*x^12 - 15
*x^11 - 6*x^10 + 19*x^9 - 12*x^8 - 18*x^7 + 5*x^6 + 2*x^5 + 3*x^3 - x) - 2*x) + 16*sqrt(2)*(x^23 - 4*x^18 - 4*
x^17 - 4*x^14 + 6*x^13 + 8*x^12 - x^11 + 4*x^9 - 4*x^8 - 4*x^7 + x^3))*sqrt((x^10 + 4*x^7 - 2*x^5 + 5*x^4 + 2^
(1/4)*sqrt(x^5 + x^2 - 1)*(2*x^3 + sqrt(2)*(x^6 + x^3 - x))*sqrt(2*sqrt(2) + 4) - 4*x^2 + 4*sqrt(2)*(x^7 + x^4
- x^2) + 1)/(x^10 - 2*x^5 + x^4 + 1)) - 4*x)/(x^26 + 9*x^23 - 5*x^21 + 2*x^20 - 36*x^18 - 30*x^17 + 10*x^16 -
6*x^15 - 31*x^14 + 54*x^13 + 60*x^12 - 17*x^11 + 6*x^10 + 31*x^9 - 36*x^8 - 30*x^7 + 5*x^6 - 2*x^5 + 9*x^3 -
x)) + 1/4*2^(3/4)*sqrt(2*sqrt(2) + 4)*arctan(-1/4*(4*x^26 - 4*x^23 - 20*x^21 - 56*x^20 + 16*x^18 - 56*x^17 + 4
0*x^16 + 168*x^15 + 4*x^14 - 24*x^13 + 112*x^12 - 28*x^11 - 168*x^10 - 4*x^9 + 16*x^8 - 56*x^7 + 20*x^6 + 56*x
^5 - 4*x^3 - sqrt(x^5 + x^2 - 1)*(2^(3/4)*(x^25 + 7*x^22 - 5*x^20 + 10*x^19 - 28*x^17 - 6*x^16 + 10*x^15 - 30*
x^14 - 7*x^13 + 42*x^12 + 12*x^11 - 7*x^10 + 30*x^9 + 7*x^8 - 28*x^7 - 6*x^6 + 5*x^5 - 10*x^4 + 7*x^2 - sqrt(2
)*(3*x^22 + 10*x^19 - 12*x^17 + 4*x^16 - 30*x^14 - 6*x^13 + 18*x^12 - 8*x^11 + x^10 + 30*x^9 + 6*x^8 - 12*x^7
+ 4*x^6 - 10*x^4 + 3*x^2) - 1) + 2*2^(1/4)*(2*x^22 + 2*x^19 - 8*x^17 + 10*x^16 - 6*x^14 + 18*x^13 + 12*x^12 -
20*x^11 + 8*x^10 + 6*x^9 - 18*x^8 - 8*x^7 + 10*x^6 - 2*x^4 + 2*x^2 + sqrt(2)*(x^22 - 4*x^19 - 4*x^17 - 12*x^16
+ 12*x^14 - 12*x^13 + 6*x^12 + 24*x^11 - 5*x^10 - 12*x^9 + 12*x^8 - 4*x^7 - 12*x^6 + 4*x^4 + x^2)))*sqrt(2*sq
rt(2) + 4) - 2*sqrt(2)*(2*x^26 + 6*x^23 - 10*x^21 + 4*x^20 - 24*x^18 + 4*x^17 + 20*x^16 - 12*x^15 + 2*x^14 + 3
6*x^13 - 8*x^12 - 22*x^11 + 12*x^10 - 2*x^9 - 24*x^8 + 4*x^7 + 10*x^6 - 4*x^5 + 6*x^3 + sqrt(2)*(x^26 - x^23 -
5*x^21 - 14*x^20 + 4*x^18 - 14*x^17 + 10*x^16 + 42*x^15 + x^14 - 6*x^13 + 28*x^12 - 7*x^11 - 42*x^10 - x^9 +
4*x^8 - 14*x^7 + 5*x^6 + 14*x^5 - x^3 - x) - 2*x) + 4*sqrt(2)*(x^26 + 3*x^23 - 5*x^21 + 2*x^20 - 12*x^18 + 2*x
^17 + 10*x^16 - 6*x^15 + x^14 + 18*x^13 - 4*x^12 - 11*x^11 + 6*x^10 - x^9 - 12*x^8 + 2*x^7 + 5*x^6 - 2*x^5 + 3
*x^3 - x) + (24*x^23 + 16*x^20 - 96*x^18 - 48*x^17 - 48*x^15 - 48*x^14 + 144*x^13 + 96*x^12 - 8*x^11 + 48*x^10
+ 48*x^9 - 96*x^8 - 48*x^7 - 16*x^5 + 24*x^3 + sqrt(x^5 + x^2 - 1)*(2^(3/4)*(x^25 + 7*x^22 - 5*x^20 + 2*x^19
- 28*x^17 - 14*x^16 + 10*x^15 - 6*x^14 - 15*x^13 + 42*x^12 + 28*x^11 - 15*x^10 + 6*x^9 + 15*x^8 - 28*x^7 - 14*
x^6 + 5*x^5 - 2*x^4 + 7*x^2 - sqrt(2)*(x^22 - 2*x^19 - 4*x^17 + 4*x^16 + 6*x^14 + 14*x^13 + 6*x^12 - 8*x^11 +
3*x^10 - 6*x^9 - 14*x^8 - 4*x^7 + 4*x^6 + 2*x^4 + x^2) - 1) + 2*2^(1/4)*(2*x^22 + 6*x^19 - 8*x^17 - 2*x^16 - 1
8*x^14 - 10*x^13 + 12*x^12 + 4*x^11 - 4*x^10 + 18*x^9 + 10*x^8 - 8*x^7 - 2*x^6 - 6*x^4 + 2*x^2 + sqrt(2)*(x^22
- 4*x^17 - 8*x^16 - 8*x^13 + 6*x^12 + 16*x^11 - x^10 + 8*x^8 - 4*x^7 - 8*x^6 + x^2)))*sqrt(2*sqrt(2) + 4) + 2
*sqrt(2)*(2*x^26 + 10*x^23 - 10*x^21 + 4*x^20 - 40*x^18 - 28*x^17 + 20*x^16 - 12*x^15 - 30*x^14 + 60*x^13 + 56
*x^12 - 26*x^11 + 12*x^10 + 30*x^9 - 40*x^8 - 28*x^7 + 10*x^6 - 4*x^5 + 10*x^3 + sqrt(2)*(x^26 + 3*x^23 - 5*x^
21 - 2*x^20 - 12*x^18 - 18*x^17 + 10*x^16 + 6*x^15 - 19*x^14 + 18*x^13 + 36*x^12 - 15*x^11 - 6*x^10 + 19*x^9 -
12*x^8 - 18*x^7 + 5*x^6 + 2*x^5 + 3*x^3 - x) - 2*x) + 16*sqrt(2)*(x^23 - 4*x^18 - 4*x^17 - 4*x^14 + 6*x^13 +
8*x^12 - x^11 + 4*x^9 - 4*x^8 - 4*x^7 + x^3))*sqrt((x^10 + 4*x^7 - 2*x^5 + 5*x^4 - 2^(1/4)*sqrt(x^5 + x^2 - 1)
*(2*x^3 + sqrt(2)*(x^6 + x^3 - x))*sqrt(2*sqrt(2) + 4) - 4*x^2 + 4*sqrt(2)*(x^7 + x^4 - x^2) + 1)/(x^10 - 2*x^
5 + x^4 + 1)) - 4*x)/(x^26 + 9*x^23 - 5*x^21 + 2*x^20 - 36*x^18 - 30*x^17 + 10*x^16 - 6*x^15 - 31*x^14 + 54*x^
13 + 60*x^12 - 17*x^11 + 6*x^10 + 31*x^9 - 36*x^8 - 30*x^7 + 5*x^6 - 2*x^5 + 9*x^3 - x))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (3 \, x^{5} + 2\right )} \sqrt {x^{5} + x^{2} - 1}}{x^{10} - 2 \, x^{5} + x^{4} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5+x^2-1)^(1/2)*(3*x^5+2)/(x^10-2*x^5+x^4+1),x, algorithm="giac")

[Out]

integrate((3*x^5 + 2)*sqrt(x^5 + x^2 - 1)/(x^10 - 2*x^5 + x^4 + 1), x)

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maple [C]  time = 3.75, size = 484, normalized size = 7.68

method	result	size
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2}+2\right ) \ln \left (\frac {-\RootOf \left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2} x^{5} \RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2}+2\right )-4 \RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2}+2\right ) \RootOf \left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{4} x^{2}-3 \RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2}+2\right ) \RootOf \left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2} x^{2}+4 \sqrt {x^{5}+x^{2}-1}\, \RootOf \left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2} x +\RootOf \left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2}+2\right )+\sqrt {x^{5}+x^{2}-1}\, x}{-x^{5}+4 \RootOf \left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2} x^{2}+x^{2}+1}\right )}{2}+\RootOf \left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-2 \RootOf \left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{3} x^{5}+8 \RootOf \left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{5} x^{2}-\RootOf \left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right ) x^{5}+2 \RootOf \left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{3} x^{2}+4 \sqrt {x^{5}+x^{2}-1}\, \RootOf \left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2} x +2 \RootOf \left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{3}-\RootOf \left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right ) x^{2}+\sqrt {x^{5}+x^{2}-1}\, x +\RootOf \left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )}{x^{5}+4 \RootOf \left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2} x^{2}+x^{2}-1}\right )\)	\(484\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5+x^2-1)^(1/2)*(3*x^5+2)/(x^10-2*x^5+x^4+1),x,method=_RETURNVERBOSE)

[Out]

1/2*RootOf(_Z^2+4*RootOf(8*_Z^4+4*_Z^2+1)^2+2)*ln((-RootOf(8*_Z^4+4*_Z^2+1)^2*x^5*RootOf(_Z^2+4*RootOf(8*_Z^4+
4*_Z^2+1)^2+2)-4*RootOf(_Z^2+4*RootOf(8*_Z^4+4*_Z^2+1)^2+2)*RootOf(8*_Z^4+4*_Z^2+1)^4*x^2-3*RootOf(_Z^2+4*Root
Of(8*_Z^4+4*_Z^2+1)^2+2)*RootOf(8*_Z^4+4*_Z^2+1)^2*x^2+4*(x^5+x^2-1)^(1/2)*RootOf(8*_Z^4+4*_Z^2+1)^2*x+RootOf(
8*_Z^4+4*_Z^2+1)^2*RootOf(_Z^2+4*RootOf(8*_Z^4+4*_Z^2+1)^2+2)+(x^5+x^2-1)^(1/2)*x)/(-x^5+4*RootOf(8*_Z^4+4*_Z^
2+1)^2*x^2+x^2+1))+RootOf(8*_Z^4+4*_Z^2+1)*ln(-(-2*RootOf(8*_Z^4+4*_Z^2+1)^3*x^5+8*RootOf(8*_Z^4+4*_Z^2+1)^5*x
^2-RootOf(8*_Z^4+4*_Z^2+1)*x^5+2*RootOf(8*_Z^4+4*_Z^2+1)^3*x^2+4*(x^5+x^2-1)^(1/2)*RootOf(8*_Z^4+4*_Z^2+1)^2*x
+2*RootOf(8*_Z^4+4*_Z^2+1)^3-RootOf(8*_Z^4+4*_Z^2+1)*x^2+(x^5+x^2-1)^(1/2)*x+RootOf(8*_Z^4+4*_Z^2+1))/(x^5+4*R
ootOf(8*_Z^4+4*_Z^2+1)^2*x^2+x^2-1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (3 \, x^{5} + 2\right )} \sqrt {x^{5} + x^{2} - 1}}{x^{10} - 2 \, x^{5} + x^{4} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5+x^2-1)^(1/2)*(3*x^5+2)/(x^10-2*x^5+x^4+1),x, algorithm="maxima")

[Out]

integrate((3*x^5 + 2)*sqrt(x^5 + x^2 - 1)/(x^10 - 2*x^5 + x^4 + 1), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\left (3\,x^5+2\right )\,\sqrt {x^5+x^2-1}}{x^{10}-2\,x^5+x^4+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x^5 + 2)*(x^2 + x^5 - 1)^(1/2))/(x^4 - 2*x^5 + x^10 + 1),x)

[Out]

int(((3*x^5 + 2)*(x^2 + x^5 - 1)^(1/2))/(x^4 - 2*x^5 + x^10 + 1), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (3 x^{5} + 2\right ) \sqrt {x^{5} + x^{2} - 1}}{x^{10} - 2 x^{5} + x^{4} + 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**5+x**2-1)**(1/2)*(3*x**5+2)/(x**10-2*x**5+x**4+1),x)

[Out]

Integral((3*x**5 + 2)*sqrt(x**5 + x**2 - 1)/(x**10 - 2*x**5 + x**4 + 1), x)

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