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

Optimal. Leaf size=71 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {2} x \sqrt {x^5+1}}{x^5-x^2+1}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {x^5+1}}{x^5+x^2+1}\right )}{\sqrt {2}} \]

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Rubi [F]  time = 0.60, 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^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^5]*(-2 + 3*x^5))/(1 + x^4 + 2*x^5 + x^10),x]

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.13, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1+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^5]*(-2 + 3*x^5))/(1 + x^4 + 2*x^5 + x^10),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 5.64, size = 82, normalized size = 1.15 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {2} x \sqrt {1+x^5}}{1-x^2+x^5}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {\frac {1}{\sqrt {2}}+\frac {x^2}{\sqrt {2}}+\frac {x^5}{\sqrt {2}}}{x \sqrt {1+x^5}}\right )}{\sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.51, size = 387, normalized size = 5.45 \begin {gather*} -\frac {1}{2} \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (x^{5} - x^{2} + 1\right )} \sqrt {x^{5} + 1} + {\left (2 \, x^{6} - \sqrt {2} {\left (x^{5} + x^{2} + 1\right )} \sqrt {x^{5} + 1} + 2 \, x\right )} \sqrt {\frac {x^{10} + 4 \, x^{7} + 2 \, x^{5} + x^{4} + 2 \, \sqrt {2} {\left (x^{6} + x^{3} + x\right )} \sqrt {x^{5} + 1} + 4 \, x^{2} + 1}{x^{10} + 2 \, x^{5} + x^{4} + 1}}}{2 \, {\left (x^{6} + x\right )}}\right ) - \frac {1}{2} \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (x^{5} - x^{2} + 1\right )} \sqrt {x^{5} + 1} - {\left (2 \, x^{6} + \sqrt {2} {\left (x^{5} + x^{2} + 1\right )} \sqrt {x^{5} + 1} + 2 \, x\right )} \sqrt {\frac {x^{10} + 4 \, x^{7} + 2 \, x^{5} + x^{4} - 2 \, \sqrt {2} {\left (x^{6} + x^{3} + x\right )} \sqrt {x^{5} + 1} + 4 \, x^{2} + 1}{x^{10} + 2 \, x^{5} + x^{4} + 1}}}{2 \, {\left (x^{6} + x\right )}}\right ) - \frac {1}{8} \, \sqrt {2} \log \left (\frac {x^{10} + 4 \, x^{7} + 2 \, x^{5} + x^{4} + 2 \, \sqrt {2} {\left (x^{6} + x^{3} + x\right )} \sqrt {x^{5} + 1} + 4 \, x^{2} + 1}{x^{10} + 2 \, x^{5} + x^{4} + 1}\right ) + \frac {1}{8} \, \sqrt {2} \log \left (\frac {x^{10} + 4 \, x^{7} + 2 \, x^{5} + x^{4} - 2 \, \sqrt {2} {\left (x^{6} + x^{3} + x\right )} \sqrt {x^{5} + 1} + 4 \, x^{2} + 1}{x^{10} + 2 \, x^{5} + x^{4} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(2)*arctan(-1/2*(sqrt(2)*(x^5 - x^2 + 1)*sqrt(x^5 + 1) + (2*x^6 - sqrt(2)*(x^5 + x^2 + 1)*sqrt(x^5 +
1) + 2*x)*sqrt((x^10 + 4*x^7 + 2*x^5 + x^4 + 2*sqrt(2)*(x^6 + x^3 + x)*sqrt(x^5 + 1) + 4*x^2 + 1)/(x^10 + 2*x^
5 + x^4 + 1)))/(x^6 + x)) - 1/2*sqrt(2)*arctan(-1/2*(sqrt(2)*(x^5 - x^2 + 1)*sqrt(x^5 + 1) - (2*x^6 + sqrt(2)*
(x^5 + x^2 + 1)*sqrt(x^5 + 1) + 2*x)*sqrt((x^10 + 4*x^7 + 2*x^5 + x^4 - 2*sqrt(2)*(x^6 + x^3 + x)*sqrt(x^5 + 1
) + 4*x^2 + 1)/(x^10 + 2*x^5 + x^4 + 1)))/(x^6 + x)) - 1/8*sqrt(2)*log((x^10 + 4*x^7 + 2*x^5 + x^4 + 2*sqrt(2)
*(x^6 + x^3 + x)*sqrt(x^5 + 1) + 4*x^2 + 1)/(x^10 + 2*x^5 + x^4 + 1)) + 1/8*sqrt(2)*log((x^10 + 4*x^7 + 2*x^5
+ x^4 - 2*sqrt(2)*(x^6 + x^3 + x)*sqrt(x^5 + 1) + 4*x^2 + 1)/(x^10 + 2*x^5 + x^4 + 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} + 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+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 + 1)/(x^10 + 2*x^5 + x^4 + 1), x)

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maple [C]  time = 1.53, size = 153, normalized size = 2.15

method	result	size
trager	\(\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{5}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{5} x^{2}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3}+2 \sqrt {x^{5}+1}\, x}{-x^{5}+\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-1}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{5}+\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{2}+2 \sqrt {x^{5}+1}\, x -\RootOf \left (\textit {\_Z}^{4}+1\right )}{x^{5}+\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+1}\right )}{2}\)	\(153\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*RootOf(_Z^4+1)^3*ln((-RootOf(_Z^4+1)^3*x^5-RootOf(_Z^4+1)^5*x^2-RootOf(_Z^4+1)^3+2*(x^5+1)^(1/2)*x)/(-x^5+
RootOf(_Z^4+1)^2*x^2-1))+1/2*RootOf(_Z^4+1)*ln((-RootOf(_Z^4+1)*x^5+RootOf(_Z^4+1)^3*x^2+2*(x^5+1)^(1/2)*x-Roo
tOf(_Z^4+1))/(x^5+RootOf(_Z^4+1)^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} + 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+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 + 1)/(x^10 + 2*x^5 + x^4 + 1), x)

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mupad [B]  time = 12.79, size = 173, normalized size = 2.44 \begin {gather*} \frac {{\left (-1\right )}^{1/4}\,\ln \left (2\,x^5-x^4+x^{10}+1-x^2\,2{}\mathrm {i}-x^7\,2{}\mathrm {i}+\sqrt {2}\,x^3\,\sqrt {x^5+1}\,\left (1+1{}\mathrm {i}\right )+\sqrt {2}\,x\,{\left (x^5+1\right )}^{3/2}\,\left (-1+1{}\mathrm {i}\right )\right )}{2}-\frac {{\left (-1\right )}^{1/4}\,\ln \left (x^{10}+2\,x^5+x^4+1\right )}{2}+\sqrt {2}\,\ln \left (2\,x^5-x^4+x^{10}+1+x^2\,2{}\mathrm {i}+x^7\,2{}\mathrm {i}+\sqrt {2}\,x^3\,\sqrt {x^5+1}\,\left (1-\mathrm {i}\right )+\sqrt {2}\,x\,{\left (x^5+1\right )}^{3/2}\,\left (-1-\mathrm {i}\right )\right )\,\left (\frac {1}{4}-\frac {1}{4}{}\mathrm {i}\right )+\sqrt {2}\,\ln \left (x^{10}+2\,x^5+x^4+1\right )\,\left (-\frac {1}{4}+\frac {1}{4}{}\mathrm {i}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2^(1/2)*log(x^2*2i - x^4 + 2*x^5 + x^7*2i + x^10 + 2^(1/2)*x^3*(x^5 + 1)^(1/2)*(1 - 1i) - 2^(1/2)*x*(x^5 + 1)^
(3/2)*(1 + 1i) + 1)*(1/4 - 1i/4) + ((-1)^(1/4)*log(2*x^5 - x^4 - x^2*2i - x^7*2i + x^10 + 2^(1/2)*x^3*(x^5 + 1
)^(1/2)*(1 + 1i) - 2^(1/2)*x*(x^5 + 1)^(3/2)*(1 - 1i) + 1))/2 - 2^(1/2)*log(x^4 + 2*x^5 + x^10 + 1)*(1/4 - 1i/
4) - ((-1)^(1/4)*log(x^4 + 2*x^5 + x^10 + 1))/2

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

[Out]

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

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