3.5.13 \(\int \frac {(3+x^4) \sqrt {x-x^5}}{1-2 x^4-x^6+x^8} \, dx\)

Optimal. Leaf size=33 \[ \tan ^{-1}\left (\frac {x^2}{\sqrt {x-x^5}}\right )+\tanh ^{-1}\left (\frac {x^2}{\sqrt {x-x^5}}\right ) \]

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Rubi [F]  time = 2.09, 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 {\left (3+x^4\right ) \sqrt {x-x^5}}{1-2 x^4-x^6+x^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((3 + x^4)*Sqrt[x - x^5])/(1 - 2*x^4 - x^6 + x^8),x]

[Out]

(-3*Sqrt[x - x^5]*Defer[Subst][Defer[Int][(x^2*Sqrt[1 - x^8])/(-1 - x^6 + x^8), x], x, Sqrt[x]])/(Sqrt[x]*Sqrt
[1 - x^4]) + (4*Sqrt[x - x^5]*Defer[Subst][Defer[Int][(x^4*Sqrt[1 - x^8])/(-1 - x^6 + x^8), x], x, Sqrt[x]])/(
Sqrt[x]*Sqrt[1 - x^4]) - (3*Sqrt[x - x^5]*Defer[Subst][Defer[Int][(x^2*Sqrt[1 - x^8])/(-1 + x^6 + x^8), x], x,
Sqrt[x]])/(Sqrt[x]*Sqrt[1 - x^4]) - (4*Sqrt[x - x^5]*Defer[Subst][Defer[Int][(x^4*Sqrt[1 - x^8])/(-1 + x^6 +
x^8), x], x, Sqrt[x]])/(Sqrt[x]*Sqrt[1 - x^4])

Rubi steps

\begin {align*} \int \frac {\left (3+x^4\right ) \sqrt {x-x^5}}{1-2 x^4-x^6+x^8} \, dx &=\frac {\sqrt {x-x^5} \int \frac {\sqrt {x} \sqrt {1-x^4} \left (3+x^4\right )}{1-2 x^4-x^6+x^8} \, dx}{\sqrt {x} \sqrt {1-x^4}}\\ &=\frac {\left (2 \sqrt {x-x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {1-x^8} \left (3+x^8\right )}{1-2 x^8-x^{12}+x^{16}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1-x^4}}\\ &=\frac {\left (2 \sqrt {x-x^5}\right ) \operatorname {Subst}\left (\int \left (\frac {x^2 \left (3-4 x^2\right ) \sqrt {1-x^8}}{2 \left (1+x^6-x^8\right )}-\frac {x^2 \left (3+4 x^2\right ) \sqrt {1-x^8}}{2 \left (-1+x^6+x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1-x^4}}\\ &=\frac {\sqrt {x-x^5} \operatorname {Subst}\left (\int \frac {x^2 \left (3-4 x^2\right ) \sqrt {1-x^8}}{1+x^6-x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1-x^4}}-\frac {\sqrt {x-x^5} \operatorname {Subst}\left (\int \frac {x^2 \left (3+4 x^2\right ) \sqrt {1-x^8}}{-1+x^6+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1-x^4}}\\ &=\frac {\sqrt {x-x^5} \operatorname {Subst}\left (\int \left (-\frac {3 x^2 \sqrt {1-x^8}}{-1-x^6+x^8}+\frac {4 x^4 \sqrt {1-x^8}}{-1-x^6+x^8}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1-x^4}}-\frac {\sqrt {x-x^5} \operatorname {Subst}\left (\int \left (\frac {3 x^2 \sqrt {1-x^8}}{-1+x^6+x^8}+\frac {4 x^4 \sqrt {1-x^8}}{-1+x^6+x^8}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1-x^4}}\\ &=-\frac {\left (3 \sqrt {x-x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {1-x^8}}{-1-x^6+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1-x^4}}-\frac {\left (3 \sqrt {x-x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {1-x^8}}{-1+x^6+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1-x^4}}+\frac {\left (4 \sqrt {x-x^5}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt {1-x^8}}{-1-x^6+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1-x^4}}-\frac {\left (4 \sqrt {x-x^5}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt {1-x^8}}{-1+x^6+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1-x^4}}\\ \end {align*}

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Mathematica [F]  time = 0.56, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (3+x^4\right ) \sqrt {x-x^5}}{1-2 x^4-x^6+x^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((3 + x^4)*Sqrt[x - x^5])/(1 - 2*x^4 - x^6 + x^8),x]

[Out]

Integrate[((3 + x^4)*Sqrt[x - x^5])/(1 - 2*x^4 - x^6 + x^8), x]

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IntegrateAlgebraic [A]  time = 0.76, size = 33, normalized size = 1.00 \begin {gather*} \tan ^{-1}\left (\frac {x^2}{\sqrt {x-x^5}}\right )+\tanh ^{-1}\left (\frac {x^2}{\sqrt {x-x^5}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((3 + x^4)*Sqrt[x - x^5])/(1 - 2*x^4 - x^6 + x^8),x]

[Out]

ArcTan[x^2/Sqrt[x - x^5]] + ArcTanh[x^2/Sqrt[x - x^5]]

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fricas [B]  time = 0.67, size = 71, normalized size = 2.15 \begin {gather*} -\frac {1}{2} \, \arctan \left (\frac {\sqrt {-x^{5} + x} {\left (x^{4} + x^{3} - 1\right )}}{2 \, {\left (x^{6} - x^{2}\right )}}\right ) + \frac {1}{2} \, \log \left (-\frac {x^{4} - x^{3} - 2 \, \sqrt {-x^{5} + x} x - 1}{x^{4} + x^{3} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+3)*(-x^5+x)^(1/2)/(x^8-x^6-2*x^4+1),x, algorithm="fricas")

[Out]

-1/2*arctan(1/2*sqrt(-x^5 + x)*(x^4 + x^3 - 1)/(x^6 - x^2)) + 1/2*log(-(x^4 - x^3 - 2*sqrt(-x^5 + x)*x - 1)/(x
^4 + x^3 - 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {-x^{5} + x} {\left (x^{4} + 3\right )}}{x^{8} - x^{6} - 2 \, x^{4} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+3)*(-x^5+x)^(1/2)/(x^8-x^6-2*x^4+1),x, algorithm="giac")

[Out]

integrate(sqrt(-x^5 + x)*(x^4 + 3)/(x^8 - x^6 - 2*x^4 + 1), x)

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maple [C]  time = 0.64, size = 101, normalized size = 3.06

method	result	size
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \sqrt {-x^{5}+x}\, x +\RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{4}-x^{3}-1}\right )}{2}-\frac {\ln \left (\frac {x^{4}-x^{3}+2 \sqrt {-x^{5}+x}\, x -1}{x^{4}+x^{3}-1}\right )}{2}\)	\(101\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4+3)*(-x^5+x)^(1/2)/(x^8-x^6-2*x^4+1),x,method=_RETURNVERBOSE)

[Out]

1/2*RootOf(_Z^2+1)*ln((-RootOf(_Z^2+1)*x^4-RootOf(_Z^2+1)*x^3+2*(-x^5+x)^(1/2)*x+RootOf(_Z^2+1))/(x^4-x^3-1))-
1/2*ln((x^4-x^3+2*(-x^5+x)^(1/2)*x-1)/(x^4+x^3-1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {-x^{5} + x} {\left (x^{4} + 3\right )}}{x^{8} - x^{6} - 2 \, x^{4} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+3)*(-x^5+x)^(1/2)/(x^8-x^6-2*x^4+1),x, algorithm="maxima")

[Out]

integrate(sqrt(-x^5 + x)*(x^4 + 3)/(x^8 - x^6 - 2*x^4 + 1), x)

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mupad [B]  time = 3.24, size = 75, normalized size = 2.27 \begin {gather*} \frac {\ln \left (\frac {2\,x\,\sqrt {x-x^5}+x^3-x^4+1}{x^4+x^3-1}\right )}{2}+\frac {\ln \left (\frac {x^3+x^4-1+x\,\sqrt {x-x^5}\,2{}\mathrm {i}}{-x^4+x^3+1}\right )\,1{}\mathrm {i}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((x - x^5)^(1/2)*(x^4 + 3))/(2*x^4 + x^6 - x^8 - 1),x)

[Out]

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

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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**4+3)*(-x**5+x)**(1/2)/(x**8-x**6-2*x**4+1),x)

[Out]

Timed out

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