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

Optimal. Leaf size=44 \[ -\frac {\sqrt {x^3-x^2+1} x}{x^3+1}-\tan ^{-1}\left (\frac {x}{\sqrt {x^3-x^2+1}}\right ) \]

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Rubi [F]  time = 180.00, 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*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

$Aborted

Rubi steps

\begin {align*} \int \frac {\left (-2+x^3\right ) \sqrt {1-x^2+x^3}}{\left (1+x^3\right )^2} \, dx &=\int \left (-\frac {3 \sqrt {1-x^2+x^3}}{\left (1+x^3\right )^2}+\frac {\sqrt {1-x^2+x^3}}{1+x^3}\right ) \, dx\\ &=-\left (3 \int \frac {\sqrt {1-x^2+x^3}}{\left (1+x^3\right )^2} \, dx\right )+\int \frac {\sqrt {1-x^2+x^3}}{1+x^3} \, dx\\ &=-\left (3 \int \left (\frac {\sqrt {1-x^2+x^3}}{9 (1+x)^2}+\frac {2 \sqrt {1-x^2+x^3}}{9 (1+x)}+\frac {(1-x) \sqrt {1-x^2+x^3}}{3 \left (1-x+x^2\right )^2}+\frac {(3-2 x) \sqrt {1-x^2+x^3}}{9 \left (1-x+x^2\right )}\right ) \, dx\right )+\int \left (\frac {\sqrt {1-x^2+x^3}}{3 (1+x)}+\frac {(2-x) \sqrt {1-x^2+x^3}}{3 \left (1-x+x^2\right )}\right ) \, dx\\ &=-\left (\frac {1}{3} \int \frac {\sqrt {1-x^2+x^3}}{(1+x)^2} \, dx\right )+\frac {1}{3} \int \frac {\sqrt {1-x^2+x^3}}{1+x} \, dx-\frac {1}{3} \int \frac {(3-2 x) \sqrt {1-x^2+x^3}}{1-x+x^2} \, dx+\frac {1}{3} \int \frac {(2-x) \sqrt {1-x^2+x^3}}{1-x+x^2} \, dx-\frac {2}{3} \int \frac {\sqrt {1-x^2+x^3}}{1+x} \, dx-\int \frac {(1-x) \sqrt {1-x^2+x^3}}{\left (1-x+x^2\right )^2} \, dx\\ \end {align*}

rest of steps removed due to Latex formating problem.

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Mathematica [C]  time = 3.13, size = 1609, normalized size = 36.57

result too large to display

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-((x - x^3 + x^4)/(1 + x^3)) + (EllipticF[ArcSin[Sqrt[(-x + Root[1 - #1^2 + #1^3 & , 3, 0])/(-Root[1 - #1^2 +
#1^3 & , 2, 0] + Root[1 - #1^2 + #1^3 & , 3, 0])]], (Root[1 - #1^2 + #1^3 & , 2, 0] - Root[1 - #1^2 + #1^3 &
, 3, 0])/(Root[1 - #1^2 + #1^3 & , 1, 0] - Root[1 - #1^2 + #1^3 & , 3, 0])]*(x - Root[1 - #1^2 + #1^3 & , 3, 0
])*Sqrt[(-x + Root[1 - #1^2 + #1^3 & , 1, 0])/(Root[1 - #1^2 + #1^3 & , 1, 0] - Root[1 - #1^2 + #1^3 & , 3, 0]
)]*Sqrt[(-x + Root[1 - #1^2 + #1^3 & , 2, 0])/(Root[1 - #1^2 + #1^3 & , 2, 0] - Root[1 - #1^2 + #1^3 & , 3, 0]
)])/Sqrt[(x - Root[1 - #1^2 + #1^3 & , 3, 0])/(Root[1 - #1^2 + #1^3 & , 2, 0] - Root[1 - #1^2 + #1^3 & , 3, 0]
)] + (3*(-1)^(2/3)*EllipticPi[(Root[1 - #1^2 + #1^3 & , 2, 0] - Root[1 - #1^2 + #1^3 & , 3, 0])/((-1)^(1/3) -
Root[1 - #1^2 + #1^3 & , 3, 0]), ArcSin[Sqrt[(-x + Root[1 - #1^2 + #1^3 & , 3, 0])/(-Root[1 - #1^2 + #1^3 & ,
2, 0] + Root[1 - #1^2 + #1^3 & , 3, 0])]], (Root[1 - #1^2 + #1^3 & , 2, 0] - Root[1 - #1^2 + #1^3 & , 3, 0])/(
Root[1 - #1^2 + #1^3 & , 1, 0] - Root[1 - #1^2 + #1^3 & , 3, 0])]*Sqrt[(-x + Root[1 - #1^2 + #1^3 & , 1, 0])/(
Root[1 - #1^2 + #1^3 & , 1, 0] - Root[1 - #1^2 + #1^3 & , 3, 0])]*Sqrt[-(((x - Root[1 - #1^2 + #1^3 & , 2, 0])
*(x - Root[1 - #1^2 + #1^3 & , 3, 0]))/(Root[1 - #1^2 + #1^3 & , 2, 0] - Root[1 - #1^2 + #1^3 & , 3, 0])^2)]*(
-Root[1 - #1^2 + #1^3 & , 2, 0] + Root[1 - #1^2 + #1^3 & , 3, 0]))/((1 + (-1)^(1/3))^2*((-1)^(1/3) - Root[1 -
#1^2 + #1^3 & , 3, 0])) + (EllipticPi[(-Root[1 - #1^2 + #1^3 & , 2, 0] + Root[1 - #1^2 + #1^3 & , 3, 0])/(1 +
Root[1 - #1^2 + #1^3 & , 3, 0]), ArcSin[Sqrt[(-x + Root[1 - #1^2 + #1^3 & , 3, 0])/(-Root[1 - #1^2 + #1^3 & ,
2, 0] + Root[1 - #1^2 + #1^3 & , 3, 0])]], (Root[1 - #1^2 + #1^3 & , 2, 0] - Root[1 - #1^2 + #1^3 & , 3, 0])/(
Root[1 - #1^2 + #1^3 & , 1, 0] - Root[1 - #1^2 + #1^3 & , 3, 0])]*Sqrt[(-x + Root[1 - #1^2 + #1^3 & , 1, 0])/(
Root[1 - #1^2 + #1^3 & , 1, 0] - Root[1 - #1^2 + #1^3 & , 3, 0])]*Sqrt[-(((x - Root[1 - #1^2 + #1^3 & , 2, 0])
*(x - Root[1 - #1^2 + #1^3 & , 3, 0]))/(Root[1 - #1^2 + #1^3 & , 2, 0] - Root[1 - #1^2 + #1^3 & , 3, 0])^2)]*(
-Root[1 - #1^2 + #1^3 & , 2, 0] + Root[1 - #1^2 + #1^3 & , 3, 0]))/(1 + Root[1 - #1^2 + #1^3 & , 3, 0]) + (2*(
-1)^(2/3)*EllipticPi[(-Root[1 - #1^2 + #1^3 & , 2, 0] + Root[1 - #1^2 + #1^3 & , 3, 0])/((-1)^(2/3) + Root[1 -
#1^2 + #1^3 & , 3, 0]), ArcSin[Sqrt[(-x + Root[1 - #1^2 + #1^3 & , 3, 0])/(-Root[1 - #1^2 + #1^3 & , 2, 0] +
Root[1 - #1^2 + #1^3 & , 3, 0])]], (Root[1 - #1^2 + #1^3 & , 2, 0] - Root[1 - #1^2 + #1^3 & , 3, 0])/(Root[1 -
#1^2 + #1^3 & , 1, 0] - Root[1 - #1^2 + #1^3 & , 3, 0])]*Sqrt[(-x + Root[1 - #1^2 + #1^3 & , 1, 0])/(Root[1 -
#1^2 + #1^3 & , 1, 0] - Root[1 - #1^2 + #1^3 & , 3, 0])]*Sqrt[-(((x - Root[1 - #1^2 + #1^3 & , 2, 0])*(x - Ro
ot[1 - #1^2 + #1^3 & , 3, 0]))/(Root[1 - #1^2 + #1^3 & , 2, 0] - Root[1 - #1^2 + #1^3 & , 3, 0])^2)]*(-Root[1
- #1^2 + #1^3 & , 2, 0] + Root[1 - #1^2 + #1^3 & , 3, 0]))/(-1 + I*Sqrt[3] + 2*Root[1 - #1^2 + #1^3 & , 3, 0])
)/Sqrt[1 - x^2 + x^3]

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IntegrateAlgebraic [A]  time = 0.29, size = 44, normalized size = 1.00 \begin {gather*} -\frac {x \sqrt {1-x^2+x^3}}{1+x^3}-\tan ^{-1}\left (\frac {x}{\sqrt {1-x^2+x^3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((x*Sqrt[1 - x^2 + x^3])/(1 + x^3)) - ArcTan[x/Sqrt[1 - x^2 + x^3]]

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fricas [A]  time = 0.53, size = 68, normalized size = 1.55 \begin {gather*} \frac {{\left (x^{3} + 1\right )} \arctan \left (\frac {\sqrt {x^{3} - x^{2} + 1} {\left (x^{3} - 2 \, x^{2} + 1\right )}}{2 \, {\left (x^{4} - x^{3} + x\right )}}\right ) - 2 \, \sqrt {x^{3} - x^{2} + 1} x}{2 \, {\left (x^{3} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.97, size = 96, normalized size = 2.18

method	result	size
trager	\(-\frac {x \sqrt {x^{3}-x^{2}+1}}{x^{3}+1}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \sqrt {x^{3}-x^{2}+1}\, x -\RootOf \left (\textit {\_Z}^{2}+1\right )}{\left (1+x \right ) \left (x^{2}-x +1\right )}\right )}{2}\)	\(96\)
risch	\(\text {Expression too large to display}\)	\(3383\)
default	\(\text {Expression too large to display}\)	\(5687\)
elliptic	\(\text {Expression too large to display}\)	\(50736\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.13, size = 61, normalized size = 1.39 \begin {gather*} -\frac {x\,\sqrt {x^3-x^2+1}}{x^3+1}+\frac {\ln \left (\frac {x^3-2\,x^2+1+x\,\sqrt {x^3-x^2+1}\,2{}\mathrm {i}}{x^3+1}\right )\,1{}\mathrm {i}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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