3.7.65 \(\int \frac {(-2+x^3) \sqrt {1+x^3} (2-x^2+2 x^3)}{x^4 (1-3 x^2+x^3)} \, dx\) [665]

Optimal. Leaf size=53 \[ \frac {2 \sqrt {1+x^3} \left (2+15 x^2+2 x^3\right )}{3 x^3}-10 \sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {1+x^3}}\right ) \]

[Out]

2/3*(x^3+1)^(1/2)*(2*x^3+15*x^2+2)/x^3-10*3^(1/2)*arctanh(3^(1/2)*x/(x^3+1)^(1/2))

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Rubi [F]
time = 0.64, 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 (-2+x^3\right ) \sqrt {1+x^3} \left (2-x^2+2 x^3\right )}{x^4 \left (1-3 x^2+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(4*Sqrt[1 + x^3])/3 + (4*Sqrt[1 + x^3])/(3*x^3) + (10*Sqrt[1 + x^3])/x - (30*Sqrt[1 + x^3])/(1 + Sqrt[3] + x)
+ (15*3^(1/4)*Sqrt[2 - Sqrt[3]]*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*EllipticE[ArcSin[(1 - Sqrt[3]
+ x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3]) - (10*Sqrt[2]*3^(3
/4)*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*EllipticF[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7
- 4*Sqrt[3]])/(Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3]) - 30*Defer[Int][Sqrt[1 + x^3]/(1 - 3*x^2 + x^3
), x] + 15*Defer[Int][(x*Sqrt[1 + x^3])/(1 - 3*x^2 + x^3), x]

Rubi steps

\begin {align*} \int \frac {\left (-2+x^3\right ) \sqrt {1+x^3} \left (2-x^2+2 x^3\right )}{x^4 \left (1-3 x^2+x^3\right )} \, dx &=\int \left (-\frac {4 \sqrt {1+x^3}}{x^4}-\frac {10 \sqrt {1+x^3}}{x^2}+\frac {2 \sqrt {1+x^3}}{x}+\frac {15 (-2+x) \sqrt {1+x^3}}{1-3 x^2+x^3}\right ) \, dx\\ &=2 \int \frac {\sqrt {1+x^3}}{x} \, dx-4 \int \frac {\sqrt {1+x^3}}{x^4} \, dx-10 \int \frac {\sqrt {1+x^3}}{x^2} \, dx+15 \int \frac {(-2+x) \sqrt {1+x^3}}{1-3 x^2+x^3} \, dx\\ &=\frac {10 \sqrt {1+x^3}}{x}+\frac {2}{3} \text {Subst}\left (\int \frac {\sqrt {1+x}}{x} \, dx,x,x^3\right )-\frac {4}{3} \text {Subst}\left (\int \frac {\sqrt {1+x}}{x^2} \, dx,x,x^3\right )-15 \int \frac {x}{\sqrt {1+x^3}} \, dx+15 \int \left (-\frac {2 \sqrt {1+x^3}}{1-3 x^2+x^3}+\frac {x \sqrt {1+x^3}}{1-3 x^2+x^3}\right ) \, dx\\ &=\frac {4 \sqrt {1+x^3}}{3}+\frac {4 \sqrt {1+x^3}}{3 x^3}+\frac {10 \sqrt {1+x^3}}{x}-15 \int \frac {1-\sqrt {3}+x}{\sqrt {1+x^3}} \, dx+15 \int \frac {x \sqrt {1+x^3}}{1-3 x^2+x^3} \, dx-30 \int \frac {\sqrt {1+x^3}}{1-3 x^2+x^3} \, dx-\left (15 \sqrt {2 \left (2-\sqrt {3}\right )}\right ) \int \frac {1}{\sqrt {1+x^3}} \, dx\\ &=\frac {4 \sqrt {1+x^3}}{3}+\frac {4 \sqrt {1+x^3}}{3 x^3}+\frac {10 \sqrt {1+x^3}}{x}-\frac {30 \sqrt {1+x^3}}{1+\sqrt {3}+x}+\frac {15 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} E\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{\sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {10 \sqrt {2} 3^{3/4} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{\sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+15 \int \frac {x \sqrt {1+x^3}}{1-3 x^2+x^3} \, dx-30 \int \frac {\sqrt {1+x^3}}{1-3 x^2+x^3} \, dx\\ \end {align*}

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Mathematica [A]
time = 0.53, size = 53, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {1+x^3} \left (2+15 x^2+2 x^3\right )}{3 x^3}-10 \sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {1+x^3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[1 + x^3]*(2 + 15*x^2 + 2*x^3))/(3*x^3) - 10*Sqrt[3]*ArcTanh[(Sqrt[3]*x)/Sqrt[1 + x^3]]

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 1.71, size = 330, normalized size = 6.23

method	result	size
trager	\(\frac {2 \sqrt {x^{3}+1}\, \left (2 x^{3}+15 x^{2}+2\right )}{3 x^{3}}-5 \RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) x^{3}+3 \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{2}+6 x \sqrt {x^{3}+1}+\RootOf \left (\textit {\_Z}^{2}-3\right )}{x^{3}-3 x^{2}+1}\right )\)	\(87\)
default	\(\frac {30 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}+5 \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}-3 \textit {\_Z}^{2}+1\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \left (-\underline {\hspace {1.25 ex}}\alpha ^{2}+4 \underline {\hspace {1.25 ex}}\alpha -4\right ) \left (3-i \sqrt {3}\right ) \sqrt {\frac {1+x}{3-i \sqrt {3}}}\, \sqrt {\frac {-1+2 x -i \sqrt {3}}{-3-i \sqrt {3}}}\, \sqrt {\frac {-1+2 x +i \sqrt {3}}{-3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{2}}{2}-2 \underline {\hspace {1.25 ex}}\alpha +2-\frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{6}+\frac {2 i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha }{3}-\frac {2 i \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}\right )+\frac {10 \sqrt {x^{3}+1}}{x}+\frac {4 \sqrt {x^{3}+1}}{3}+\frac {4 \sqrt {x^{3}+1}}{3 x^{3}}\)	\(330\)
elliptic	\(\frac {30 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}+5 \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}-3 \textit {\_Z}^{2}+1\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \left (-\underline {\hspace {1.25 ex}}\alpha ^{2}+4 \underline {\hspace {1.25 ex}}\alpha -4\right ) \left (3-i \sqrt {3}\right ) \sqrt {\frac {1+x}{3-i \sqrt {3}}}\, \sqrt {\frac {-1+2 x -i \sqrt {3}}{-3-i \sqrt {3}}}\, \sqrt {\frac {-1+2 x +i \sqrt {3}}{-3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{2}}{2}-2 \underline {\hspace {1.25 ex}}\alpha +2-\frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{6}+\frac {2 i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha }{3}-\frac {2 i \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}\right )+\frac {10 \sqrt {x^{3}+1}}{x}+\frac {4 \sqrt {x^{3}+1}}{3}+\frac {4 \sqrt {x^{3}+1}}{3 x^{3}}\)	\(330\)
risch	\(\frac {\frac {8}{3} x^{3}+\frac {4}{3}+10 x^{5}+10 x^{2}+\frac {4}{3} x^{6}}{\sqrt {x^{3}+1}\, x^{3}}+\frac {30 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}+5 \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}-3 \textit {\_Z}^{2}+1\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \left (-\underline {\hspace {1.25 ex}}\alpha ^{2}+4 \underline {\hspace {1.25 ex}}\alpha -4\right ) \left (3-i \sqrt {3}\right ) \sqrt {\frac {1+x}{3-i \sqrt {3}}}\, \sqrt {\frac {-1+2 x -i \sqrt {3}}{-3-i \sqrt {3}}}\, \sqrt {\frac {-1+2 x +i \sqrt {3}}{-3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{2}}{2}-2 \underline {\hspace {1.25 ex}}\alpha +2-\frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{6}+\frac {2 i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha }{3}-\frac {2 i \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}\right )\)	\(331\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

30*(3/2-1/2*I*3^(1/2))*((1+x)/(3/2-1/2*I*3^(1/2)))^(1/2)*((x-1/2-1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2)*((
x-1/2+1/2*I*3^(1/2))/(-3/2+1/2*I*3^(1/2)))^(1/2)/(x^3+1)^(1/2)*EllipticF(((1+x)/(3/2-1/2*I*3^(1/2)))^(1/2),((-
3/2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2))+5*2^(1/2)*sum(_alpha*(-_alpha^2+4*_alpha-4)*(3-I*3^(1/2))*((1+
x)/(3-I*3^(1/2)))^(1/2)*((-1+2*x-I*3^(1/2))/(-3-I*3^(1/2)))^(1/2)*((-1+2*x+I*3^(1/2))/(-3+I*3^(1/2)))^(1/2)/(x
^3+1)^(1/2)*EllipticPi(((1+x)/(3/2-1/2*I*3^(1/2)))^(1/2),1/2*_alpha^2-2*_alpha+2-1/6*I*3^(1/2)*_alpha^2+2/3*I*
3^(1/2)*_alpha-2/3*I*3^(1/2),((-3/2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2)))^(1/2)),_alpha=RootOf(_Z^3-3*_Z^2+1))+
10*(x^3+1)^(1/2)/x+4/3*(x^3+1)^(1/2)+4/3*(x^3+1)^(1/2)/x^3

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (43) = 86\).
time = 0.44, size = 112, normalized size = 2.11 \begin {gather*} \frac {15 \, \sqrt {3} x^{3} \log \left (-\frac {x^{6} + 18 \, x^{5} + 9 \, x^{4} + 2 \, x^{3} - 4 \, \sqrt {3} {\left (x^{4} + 3 \, x^{3} + x\right )} \sqrt {x^{3} + 1} + 18 \, x^{2} + 1}{x^{6} - 6 \, x^{5} + 9 \, x^{4} + 2 \, x^{3} - 6 \, x^{2} + 1}\right ) + 4 \, {\left (2 \, x^{3} + 15 \, x^{2} + 2\right )} \sqrt {x^{3} + 1}}{6 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(15*sqrt(3)*x^3*log(-(x^6 + 18*x^5 + 9*x^4 + 2*x^3 - 4*sqrt(3)*(x^4 + 3*x^3 + x)*sqrt(x^3 + 1) + 18*x^2 +
1)/(x^6 - 6*x^5 + 9*x^4 + 2*x^3 - 6*x^2 + 1)) + 4*(2*x^3 + 15*x^2 + 2)*sqrt(x^3 + 1))/x^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [B]
time = 1.13, size = 76, normalized size = 1.43 \begin {gather*} 5\,\sqrt {3}\,\ln \left (\frac {3\,x^2+x^3-2\,\sqrt {3}\,x\,\sqrt {x^3+1}+1}{x^3-3\,x^2+1}\right )+\frac {4\,\sqrt {x^3+1}}{3}+\frac {10\,\sqrt {x^3+1}}{x}+\frac {4\,\sqrt {x^3+1}}{3\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*3^(1/2)*log((3*x^2 + x^3 - 2*3^(1/2)*x*(x^3 + 1)^(1/2) + 1)/(x^3 - 3*x^2 + 1)) + (4*(x^3 + 1)^(1/2))/3 + (10
*(x^3 + 1)^(1/2))/x + (4*(x^3 + 1)^(1/2))/(3*x^3)

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