3.23.73 \(\int \frac {1}{(1+x) (-2+3 x-2 x^2+3 x^3-2 x^4)^{3/2}} \, dx\)

Optimal. Leaf size=173 \[ \frac {i (x-1) \sqrt {2 x^2+x+2} \left (-\frac {i \tanh ^{-1}\left (-\frac {\sqrt {2 x^2+x+2}}{\sqrt {3}}+\sqrt {\frac {2}{3}} x+\sqrt {\frac {2}{3}}\right )}{12 \sqrt {3}}-\frac {91 i \tanh ^{-1}\left (\frac {\sqrt {2 x^2+x+2}}{\sqrt {5}}-\sqrt {\frac {2}{5}} x+\sqrt {\frac {2}{5}}\right )}{200 \sqrt {5}}+\frac {i \left (22 x^3+73 x^2-107 x-18\right )}{600 (x-1)^2 \sqrt {2 x^2+x+2}}\right )}{\sqrt {-(x-1)^2 \left (2 x^2+x+2\right )}} \]

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Rubi [F]  time = 0.14, 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}{(1+x) \left (-2+3 x-2 x^2+3 x^3-2 x^4\right )^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.15, size = 146, normalized size = 0.84 \begin {gather*} \frac {-250 \sqrt {3} \sqrt {2 x^2+x+2} (x-1)^2 \tanh ^{-1}\left (\frac {\sqrt {3} (1-x)}{2 \sqrt {2 x^2+x+2}}\right )+819 \sqrt {5} \sqrt {2 x^2+x+2} (x-1)^2 \tanh ^{-1}\left (\frac {\sqrt {5} (x+1)}{2 \sqrt {2 x^2+x+2}}\right )-30 \left (22 x^3+73 x^2-107 x-18\right )}{18000 (x-1) \sqrt {-(x-1)^2 \left (2 x^2+x+2\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-30*(-18 - 107*x + 73*x^2 + 22*x^3) - 250*Sqrt[3]*(-1 + x)^2*Sqrt[2 + x + 2*x^2]*ArcTanh[(Sqrt[3]*(1 - x))/(2
*Sqrt[2 + x + 2*x^2])] + 819*Sqrt[5]*(-1 + x)^2*Sqrt[2 + x + 2*x^2]*ArcTanh[(Sqrt[5]*(1 + x))/(2*Sqrt[2 + x +
2*x^2])])/(18000*(-1 + x)*Sqrt[-((-1 + x)^2*(2 + x + 2*x^2))])

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IntegrateAlgebraic [A]  time = 10.31, size = 173, normalized size = 1.00 \begin {gather*} \frac {i (-1+x) \sqrt {2+x+2 x^2} \left (\frac {i \left (-18-107 x+73 x^2+22 x^3\right )}{600 (-1+x)^2 \sqrt {2+x+2 x^2}}-\frac {i \tanh ^{-1}\left (\sqrt {\frac {2}{3}}+\sqrt {\frac {2}{3}} x-\frac {\sqrt {2+x+2 x^2}}{\sqrt {3}}\right )}{12 \sqrt {3}}-\frac {91 i \tanh ^{-1}\left (\sqrt {\frac {2}{5}}-\sqrt {\frac {2}{5}} x+\frac {\sqrt {2+x+2 x^2}}{\sqrt {5}}\right )}{200 \sqrt {5}}\right )}{\sqrt {-(-1+x)^2 \left (2+x+2 x^2\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(I*(-1 + x)*Sqrt[2 + x + 2*x^2]*(((I/600)*(-18 - 107*x + 73*x^2 + 22*x^3))/((-1 + x)^2*Sqrt[2 + x + 2*x^2]) -
((I/12)*ArcTanh[Sqrt[2/3] + Sqrt[2/3]*x - Sqrt[2 + x + 2*x^2]/Sqrt[3]])/Sqrt[3] - (((91*I)/200)*ArcTanh[Sqrt[2
/5] - Sqrt[2/5]*x + Sqrt[2 + x + 2*x^2]/Sqrt[5]])/Sqrt[5]))/Sqrt[-((-1 + x)^2*(2 + x + 2*x^2))]

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fricas [A]  time = 0.64, size = 213, normalized size = 1.23 \begin {gather*} -\frac {819 \, \sqrt {5} {\left (2 \, x^{5} - 5 \, x^{4} + 5 \, x^{3} - 5 \, x^{2} + 5 \, x - 2\right )} \arctan \left (\frac {\sqrt {5} \sqrt {-2 \, x^{4} + 3 \, x^{3} - 2 \, x^{2} + 3 \, x - 2} {\left (x + 1\right )}}{2 \, {\left (2 \, x^{3} - x^{2} + x - 2\right )}}\right ) + 250 \, \sqrt {3} {\left (2 \, x^{5} - 5 \, x^{4} + 5 \, x^{3} - 5 \, x^{2} + 5 \, x - 2\right )} \arctan \left (\frac {\sqrt {3} \sqrt {-2 \, x^{4} + 3 \, x^{3} - 2 \, x^{2} + 3 \, x - 2}}{2 \, {\left (2 \, x^{2} + x + 2\right )}}\right ) - 30 \, \sqrt {-2 \, x^{4} + 3 \, x^{3} - 2 \, x^{2} + 3 \, x - 2} {\left (22 \, x^{3} + 73 \, x^{2} - 107 \, x - 18\right )}}{18000 \, {\left (2 \, x^{5} - 5 \, x^{4} + 5 \, x^{3} - 5 \, x^{2} + 5 \, x - 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/18000*(819*sqrt(5)*(2*x^5 - 5*x^4 + 5*x^3 - 5*x^2 + 5*x - 2)*arctan(1/2*sqrt(5)*sqrt(-2*x^4 + 3*x^3 - 2*x^2
+ 3*x - 2)*(x + 1)/(2*x^3 - x^2 + x - 2)) + 250*sqrt(3)*(2*x^5 - 5*x^4 + 5*x^3 - 5*x^2 + 5*x - 2)*arctan(1/2*
sqrt(3)*sqrt(-2*x^4 + 3*x^3 - 2*x^2 + 3*x - 2)/(2*x^2 + x + 2)) - 30*sqrt(-2*x^4 + 3*x^3 - 2*x^2 + 3*x - 2)*(2
2*x^3 + 73*x^2 - 107*x - 18))/(2*x^5 - 5*x^4 + 5*x^3 - 5*x^2 + 5*x - 2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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maple [A]  time = 0.35, size = 134, normalized size = 0.77

method	result	size
risch	\(-\frac {22 x^{3}+73 x^{2}-107 x -18}{600 \left (-1+x \right ) \sqrt {-\left (-1+x \right )^{2} \left (2 x^{2}+x +2\right )}}-\frac {\left (\frac {91 \sqrt {5}\, \arctan \left (\frac {\left (-5-5 x \right ) \sqrt {5}}{10 \sqrt {-2 \left (-1+x \right )^{2}-5 x}}\right )}{2000}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (-3+3 x \right ) \sqrt {3}}{6 \sqrt {-2 \left (1+x \right )^{2}+3 x}}\right )}{72}\right ) \left (-1+x \right ) \sqrt {-2 x^{2}-x -2}}{\sqrt {-\left (-1+x \right )^{2} \left (2 x^{2}+x +2\right )}}\)	\(134\)
trager	\(\frac {\left (22 x^{3}+73 x^{2}-107 x -18\right ) \sqrt {-2 x^{4}+3 x^{3}-2 x^{2}+3 x -2}}{600 \left (-1+x \right )^{3} \left (2 x^{2}+x +2\right )}+\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) x^{2}-2 \RootOf \left (\textit {\_Z}^{2}+3\right ) x +\RootOf \left (\textit {\_Z}^{2}+3\right )-2 \sqrt {-2 x^{4}+3 x^{3}-2 x^{2}+3 x -2}}{\left (-1+x \right ) \left (1+x \right )}\right )}{72}-\frac {91 \RootOf \left (\textit {\_Z}^{2}+5\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+5\right ) x^{2}-\RootOf \left (\textit {\_Z}^{2}+5\right )+2 \sqrt {-2 x^{4}+3 x^{3}-2 x^{2}+3 x -2}}{\left (-1+x \right )^{2}}\right )}{2000}\)	\(184\)
default	\(-\frac {\left (250 \sqrt {3}\, \arctan \left (\frac {\left (-1+x \right ) \sqrt {3}}{2 \sqrt {-2 x^{2}-x -2}}\right ) \sqrt {-2 x^{2}-x -2}\, x^{2}+819 \sqrt {5}\, \arctan \left (\frac {\left (1+x \right ) \sqrt {5}}{2 \sqrt {-2 x^{2}-x -2}}\right ) \sqrt {-2 x^{2}-x -2}\, x^{2}-500 \sqrt {3}\, \arctan \left (\frac {\left (-1+x \right ) \sqrt {3}}{2 \sqrt {-2 x^{2}-x -2}}\right ) \sqrt {-2 x^{2}-x -2}\, x -1638 \sqrt {5}\, \arctan \left (\frac {\left (1+x \right ) \sqrt {5}}{2 \sqrt {-2 x^{2}-x -2}}\right ) \sqrt {-2 x^{2}-x -2}\, x +250 \sqrt {3}\, \arctan \left (\frac {\left (-1+x \right ) \sqrt {3}}{2 \sqrt {-2 x^{2}-x -2}}\right ) \sqrt {-2 x^{2}-x -2}+819 \sqrt {5}\, \arctan \left (\frac {\left (1+x \right ) \sqrt {5}}{2 \sqrt {-2 x^{2}-x -2}}\right ) \sqrt {-2 x^{2}-x -2}-660 x^{3}-2190 x^{2}+3210 x +540\right ) \left (-1+x \right ) \left (2 x^{2}+x +2\right )}{18000 \left (-2 x^{4}+3 x^{3}-2 x^{2}+3 x -2\right )^{\frac {3}{2}}}\)	\(287\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/600*(22*x^3+73*x^2-107*x-18)/(-1+x)/(-(-1+x)^2*(2*x^2+x+2))^(1/2)-(91/2000*5^(1/2)*arctan(1/10*(-5-5*x)*5^(
1/2)/(-2*(-1+x)^2-5*x)^(1/2))-1/72*3^(1/2)*arctan(1/6*(-3+3*x)*3^(1/2)/(-2*(1+x)^2+3*x)^(1/2)))*(-1+x)*(-2*x^2
-x-2)^(1/2)/(-(-1+x)^2*(2*x^2+x+2))^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\left (x+1\right )\,{\left (-2\,x^4+3\,x^3-2\,x^2+3\,x-2\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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