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

Optimal. Leaf size=40 \[ -\frac {\sqrt {x^3+x^2-1} x}{x^3-1}-\tanh ^{-1}\left (\frac {x}{\sqrt {x^3+x^2-1}}\right ) \]

________________________________________________________________________________________

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\\ &=3 \int \frac {\sqrt {-1+x^2+x^3}}{\left (-1+x^3\right )^2} \, dx+\int \frac {\sqrt {-1+x^2+x^3}}{-1+x^3} \, dx\\ &=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+\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\\ &=\frac {1}{3} \int \frac {\sqrt {-1+x^2+x^3}}{(-1+x)^2} \, dx+\frac {1}{3} \int \frac {\sqrt {-1+x^2+x^3}}{-1+x} \, dx+\frac {1}{3} \int \frac {(-2-x) \sqrt {-1+x^2+x^3}}{1+x+x^2} \, dx+\frac {1}{3} \int \frac {(3+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.

________________________________________________________________________________________

Mathematica [C]  time = 2.95, size = 1451, normalized size = 36.28

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*(-1 + x^2 + x^3))/(-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])] - (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]) + (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 - Ro
ot[-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])) - (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])]], (Roo
t[-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] + R
oot[-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]

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.29, size = 40, normalized size = 1.00 \begin {gather*} -\frac {x \sqrt {-1+x^2+x^3}}{-1+x^3}-\tanh ^{-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)) - ArcTanh[x/Sqrt[-1 + x^2 + x^3]]

________________________________________________________________________________________

fricas [A]  time = 0.46, size = 61, normalized size = 1.52 \begin {gather*} \frac {{\left (x^{3} - 1\right )} \log \left (\frac {x^{3} + 2 \, x^{2} - 2 \, \sqrt {x^{3} + x^{2} - 1} x - 1}{x^{3} - 1}\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)*log((x^3 + 2*x^2 - 2*sqrt(x^3 + x^2 - 1)*x - 1)/(x^3 - 1)) - 2*sqrt(x^3 + x^2 - 1)*x)/(x^3 - 1)

________________________________________________________________________________________

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)

________________________________________________________________________________________

maple [A]  time = 1.57, size = 65, normalized size = 1.62

method	result	size
trager	\(-\frac {x \sqrt {x^{3}+x^{2}-1}}{x^{3}-1}+\frac {\ln \left (-\frac {-x^{3}+2 \sqrt {x^{3}+x^{2}-1}\, x -2 x^{2}+1}{\left (-1+x \right ) \left (x^{2}+x +1\right )}\right )}{2}\)	\(65\)
risch	\(\text {Expression too large to display}\)	\(3389\)
default	\(\text {Expression too large to display}\)	\(5693\)
elliptic	\(\text {Expression too large to display}\)	\(51070\)

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

________________________________________________________________________________________

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)

________________________________________________________________________________________

mupad [B]  time = 0.70, size = 57, normalized size = 1.42 \begin {gather*} \frac {\ln \left (\frac {2\,x\,\sqrt {x^3+x^2-1}-2\,x^2-x^3+1}{x^3-1}\right )}{2}-\frac {x\,\sqrt {x^3+x^2-1}}{x^3-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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)

________________________________________________________________________________________