3.6.14 \(\int \frac {(2+x^3) \sqrt {-1+x^2+x^3}}{(-1+x^3)^2} \, dx\) [514]

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

[Out]

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

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Rubi [F]
time = 180.01, 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.

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Mathematica [A]
time = 0.24, 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]

Integrate[((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]]

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 2.60, size = 5693, normalized size = 142.32

method	result	size
trager	\(-\frac {x \sqrt {x^{3}+x^{2}-1}}{x^{3}-1}-\frac {\ln \left (-\frac {x^{3}+2 \sqrt {x^{3}+x^{2}-1}\, x +2 x^{2}-1}{\left (-1+x \right ) \left (x^{2}+x +1\right )}\right )}{2}\)	\(63\)
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]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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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Fricas [A]
time = 0.41, 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)

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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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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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)

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