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

Optimal. Leaf size=45 \[ 2 \tanh ^{-1}\left (\frac {x}{\sqrt {x^5+x^2+1}}\right )-2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^5+x^2+1}}\right ) \]

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Rubi [F]  time = 1.64, 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 )}{\left (1+x^5\right ) \left (1-x^2+x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.32, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1+x^2+x^5} \left (-2+3 x^5\right )}{\left (1+x^5\right ) \left (1-x^2+x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 2.65, size = 45, normalized size = 1.00 \begin {gather*} 2 \tanh ^{-1}\left (\frac {x}{\sqrt {1+x^2+x^5}}\right )-2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} 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^5)*(1 - x^2 + x^5)),x]

[Out]

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

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fricas [B]  time = 0.51, size = 115, normalized size = 2.56 \begin {gather*} \frac {1}{2} \, \sqrt {2} \log \left (\frac {x^{10} + 14 \, x^{7} + 2 \, x^{5} + 17 \, x^{4} - 4 \, \sqrt {2} {\left (x^{6} + 3 \, x^{3} + x\right )} \sqrt {x^{5} + x^{2} + 1} + 14 \, x^{2} + 1}{x^{10} - 2 \, x^{7} + 2 \, x^{5} + x^{4} - 2 \, x^{2} + 1}\right ) + \log \left (\frac {x^{5} + 2 \, x^{2} + 2 \, \sqrt {x^{5} + x^{2} + 1} x + 1}{x^{5} + 1}\right ) \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^5+1)/(x^5-x^2+1),x, algorithm="fricas")

[Out]

1/2*sqrt(2)*log((x^10 + 14*x^7 + 2*x^5 + 17*x^4 - 4*sqrt(2)*(x^6 + 3*x^3 + x)*sqrt(x^5 + x^2 + 1) + 14*x^2 + 1
)/(x^10 - 2*x^7 + 2*x^5 + x^4 - 2*x^2 + 1)) + log((x^5 + 2*x^2 + 2*sqrt(x^5 + x^2 + 1)*x + 1)/(x^5 + 1))

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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}}{{\left (x^{5} - x^{2} + 1\right )} {\left (x^{5} + 1\right )}}\,{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^5+1)/(x^5-x^2+1),x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.44, size = 113, normalized size = 2.51

method	result	size
trager	\(\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) x^{5}+3 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{2}-4 \sqrt {x^{5}+x^{2}+1}\, x +\RootOf \left (\textit {\_Z}^{2}-2\right )}{x^{5}-x^{2}+1}\right )+\ln \left (-\frac {x^{5}+2 \sqrt {x^{5}+x^{2}+1}\, x +2 x^{2}+1}{\left (1+x \right ) \left (x^{4}-x^{3}+x^{2}-x +1\right )}\right )\)	\(113\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootOf(_Z^2-2)*ln((RootOf(_Z^2-2)*x^5+3*RootOf(_Z^2-2)*x^2-4*(x^5+x^2+1)^(1/2)*x+RootOf(_Z^2-2))/(x^5-x^2+1))+
ln(-(x^5+2*(x^5+x^2+1)^(1/2)*x+2*x^2+1)/(1+x)/(x^4-x^3+x^2-x+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}}{{\left (x^{5} - x^{2} + 1\right )} {\left (x^{5} + 1\right )}}\,{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^5+1)/(x^5-x^2+1),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 3.55, size = 77, normalized size = 1.71 \begin {gather*} \ln \left (\frac {2\,x\,\sqrt {x^5+x^2+1}+2\,x^2+x^5+1}{x^5+1}\right )+\sqrt {2}\,\ln \left (\frac {3\,x^2+x^5-2\,\sqrt {2}\,x\,\sqrt {x^5+x^2+1}+1}{x^5-x^2+1}\right ) \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^5 + 1)*(x^5 - x^2 + 1)),x)

[Out]

log((2*x*(x^2 + x^5 + 1)^(1/2) + 2*x^2 + x^5 + 1)/(x^5 + 1)) + 2^(1/2)*log((3*x^2 + x^5 - 2*2^(1/2)*x*(x^2 + x
^5 + 1)^(1/2) + 1)/(x^5 - x^2 + 1))

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

[Out]

Timed out

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