3.4.38 \(\int \frac {x^2 (-2+x^3) \sqrt {1+x^3}}{1+3 x^3+x^9} \, dx\)

Optimal. Leaf size=28 \[ -\frac {2 \tanh ^{-1}\left (\frac {\left (x^3+1\right )^{3/2}}{\sqrt {3} x^3}\right )}{3 \sqrt {3}} \]

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Rubi [A]  time = 0.38, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6715, 2094, 207} \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\left (x^3+1\right )^{3/2}}{\sqrt {3} x^3}\right )}{3 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^2*(-2 + x^3)*Sqrt[1 + x^3])/(1 + 3*x^3 + x^9),x]

[Out]

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

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 2094

Int[((x_)^(m_.)*((A_) + (B_.)*(x_)^(n_.)))/((a_) + (b_.)*(x_)^(k_.) + (c_.)*(x_)^(n_.) + (d_.)*(x_)^(n2_)), x_
Symbol] :> Dist[(A^2*(m - n + 1))/(m + 1), Subst[Int[1/(a + A^2*b*(m - n + 1)^2*x^2), x], x, x^(m + 1)/(A*(m -
n + 1) + B*(m + 1)*x^n)], x] /; FreeQ[{a, b, c, d, A, B, m, n}, x] && EqQ[n2, 2*n] && EqQ[k, 2*(m + 1)] && Eq
Q[a*B^2*(m + 1)^2 - A^2*d*(m - n + 1)^2, 0] && EqQ[B*c*(m + 1) - 2*A*d*(m - n + 1), 0]

Rule 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rubi steps

\begin {align*} \int \frac {x^2 \left (-2+x^3\right ) \sqrt {1+x^3}}{1+3 x^3+x^9} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {(-2+x) \sqrt {1+x}}{1+3 x+x^3} \, dx,x,x^3\right )\\ &=\frac {2}{3} \operatorname {Subst}\left (\int \frac {x^2 \left (-3+x^2\right )}{-3+6 x^2-3 x^4+x^6} \, dx,x,\sqrt {1+x^3}\right )\\ &=2 \operatorname {Subst}\left (\int \frac {1}{-3+9 x^2} \, dx,x,\frac {\left (1+x^3\right )^{3/2}}{3 x^3}\right )\\ &=-\frac {2 \tanh ^{-1}\left (\frac {\left (1+x^3\right )^{3/2}}{\sqrt {3} x^3}\right )}{3 \sqrt {3}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 28, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\left (x^3+1\right )^{3/2}}{\sqrt {3} x^3}\right )}{3 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^2*(-2 + x^3)*Sqrt[1 + x^3])/(1 + 3*x^3 + x^9),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.07, size = 28, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {3} x^3}{\left (1+x^3\right )^{3/2}}\right )}{3 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^2*(-2 + x^3)*Sqrt[1 + x^3])/(1 + 3*x^3 + x^9),x]

[Out]

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

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fricas [B]  time = 0.48, size = 53, normalized size = 1.89 \begin {gather*} \frac {1}{9} \, \sqrt {3} \log \left (\frac {x^{9} + 6 \, x^{6} + 3 \, x^{3} - 2 \, \sqrt {3} {\left (x^{6} + x^{3}\right )} \sqrt {x^{3} + 1} + 1}{x^{9} + 3 \, x^{3} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(x^3-2)*(x^3+1)^(1/2)/(x^9+3*x^3+1),x, algorithm="fricas")

[Out]

1/9*sqrt(3)*log((x^9 + 6*x^6 + 3*x^3 - 2*sqrt(3)*(x^6 + x^3)*sqrt(x^3 + 1) + 1)/(x^9 + 3*x^3 + 1))

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giac [B]  time = 0.29, size = 58, normalized size = 2.07 \begin {gather*} -\frac {1}{9} \, \sqrt {3} \log \left ({\left | \sqrt {3} {\left (x^{3} + 1\right )} + {\left (x^{3} + 1\right )}^{\frac {3}{2}} - \sqrt {3} \right |}\right ) + \frac {1}{9} \, \sqrt {3} \log \left ({\left | -\sqrt {3} {\left (x^{3} + 1\right )} + {\left (x^{3} + 1\right )}^{\frac {3}{2}} + \sqrt {3} \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(x^3-2)*(x^3+1)^(1/2)/(x^9+3*x^3+1),x, algorithm="giac")

[Out]

-1/9*sqrt(3)*log(abs(sqrt(3)*(x^3 + 1) + (x^3 + 1)^(3/2) - sqrt(3))) + 1/9*sqrt(3)*log(abs(-sqrt(3)*(x^3 + 1)
+ (x^3 + 1)^(3/2) + sqrt(3)))

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maple [C]  time = 0.38, size = 87, normalized size = 3.11

method	result	size
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) x^{9}+6 \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{6}-6 \sqrt {x^{3}+1}\, x^{6}+3 \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{3}-6 \sqrt {x^{3}+1}\, x^{3}+\RootOf \left (\textit {\_Z}^{2}-3\right )}{x^{9}+3 x^{3}+1}\right )}{9}\)	\(87\)
default	\(\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{9}+3 \textit {\_Z}^{3}+1\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{6}+\underline {\hspace {1.25 ex}}\alpha ^{3}-1\right ) \left (-\underline {\hspace {1.25 ex}}\alpha ^{8}+\underline {\hspace {1.25 ex}}\alpha ^{7}-\underline {\hspace {1.25 ex}}\alpha ^{6}+\underline {\hspace {1.25 ex}}\alpha ^{5}-\underline {\hspace {1.25 ex}}\alpha ^{4}+\underline {\hspace {1.25 ex}}\alpha ^{3}-4 \underline {\hspace {1.25 ex}}\alpha ^{2}+4 \underline {\hspace {1.25 ex}}\alpha -4\right ) \left (3-i \sqrt {3}\right ) \sqrt {\frac {1+x}{3-i \sqrt {3}}}\, \sqrt {\frac {-1+2 x -i \sqrt {3}}{-3-i \sqrt {3}}}\, \sqrt {\frac {-1+2 x +i \sqrt {3}}{-3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{8}}{2}-\frac {\underline {\hspace {1.25 ex}}\alpha ^{7}}{2}+\frac {\underline {\hspace {1.25 ex}}\alpha ^{6}}{2}-\frac {\underline {\hspace {1.25 ex}}\alpha ^{5}}{2}+\frac {\underline {\hspace {1.25 ex}}\alpha ^{4}}{2}-\frac {\underline {\hspace {1.25 ex}}\alpha ^{3}}{2}+2 \underline {\hspace {1.25 ex}}\alpha ^{2}-2 \underline {\hspace {1.25 ex}}\alpha +2+\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{7}}{6}-\frac {i \underline {\hspace {1.25 ex}}\alpha ^{4} \sqrt {3}}{6}-\frac {2 i \sqrt {3}}{3}+\frac {i \underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {3}}{6}-\frac {2 i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{3}-\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{6}}{6}-\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{8}}{6}+\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{5}}{6}+\frac {2 i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha }{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}\right )}{27}\)	\(298\)
elliptic	\(\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{9}+3 \textit {\_Z}^{3}+1\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{6}+\underline {\hspace {1.25 ex}}\alpha ^{3}-1\right ) \left (-\underline {\hspace {1.25 ex}}\alpha ^{8}+\underline {\hspace {1.25 ex}}\alpha ^{7}-\underline {\hspace {1.25 ex}}\alpha ^{6}+\underline {\hspace {1.25 ex}}\alpha ^{5}-\underline {\hspace {1.25 ex}}\alpha ^{4}+\underline {\hspace {1.25 ex}}\alpha ^{3}-4 \underline {\hspace {1.25 ex}}\alpha ^{2}+4 \underline {\hspace {1.25 ex}}\alpha -4\right ) \left (3-i \sqrt {3}\right ) \sqrt {\frac {1+x}{3-i \sqrt {3}}}\, \sqrt {\frac {-1+2 x -i \sqrt {3}}{-3-i \sqrt {3}}}\, \sqrt {\frac {-1+2 x +i \sqrt {3}}{-3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{8}}{2}-\frac {\underline {\hspace {1.25 ex}}\alpha ^{7}}{2}+\frac {\underline {\hspace {1.25 ex}}\alpha ^{6}}{2}-\frac {\underline {\hspace {1.25 ex}}\alpha ^{5}}{2}+\frac {\underline {\hspace {1.25 ex}}\alpha ^{4}}{2}-\frac {\underline {\hspace {1.25 ex}}\alpha ^{3}}{2}+2 \underline {\hspace {1.25 ex}}\alpha ^{2}-2 \underline {\hspace {1.25 ex}}\alpha +2+\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{7}}{6}-\frac {i \underline {\hspace {1.25 ex}}\alpha ^{4} \sqrt {3}}{6}-\frac {2 i \sqrt {3}}{3}+\frac {i \underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {3}}{6}-\frac {2 i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{3}-\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{6}}{6}-\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{8}}{6}+\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{5}}{6}+\frac {2 i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha }{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}\right )}{27}\)	\(298\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(x^3-2)*(x^3+1)^(1/2)/(x^9+3*x^3+1),x,method=_RETURNVERBOSE)

[Out]

1/9*RootOf(_Z^2-3)*ln(-(RootOf(_Z^2-3)*x^9+6*RootOf(_Z^2-3)*x^6-6*(x^3+1)^(1/2)*x^6+3*RootOf(_Z^2-3)*x^3-6*(x^
3+1)^(1/2)*x^3+RootOf(_Z^2-3))/(x^9+3*x^3+1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{3} + 1} {\left (x^{3} - 2\right )} x^{2}}{x^{9} + 3 \, x^{3} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(x^3-2)*(x^3+1)^(1/2)/(x^9+3*x^3+1),x, algorithm="maxima")

[Out]

integrate(sqrt(x^3 + 1)*(x^3 - 2)*x^2/(x^9 + 3*x^3 + 1), x)

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mupad [B]  time = 1.49, size = 236, normalized size = 8.43 \begin {gather*} \sum _{k=1}^9\frac {\sqrt {6}\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {-\left (-3+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (x+1\right )}\,\Pi \left (\frac {3+\sqrt {3}\,1{}\mathrm {i}}{2\,\left (\mathrm {root}\left (z^9+3\,z^3+1,z,k\right )+1\right )};\mathrm {asin}\left (\frac {\sqrt {6}\,\sqrt {-\left (-3+\sqrt {3}\,1{}\mathrm {i}\right )\,\left (x+1\right )}}{6}\right )\middle |\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-{\mathrm {root}\left (z^9+3\,z^3+1,z,k\right )}^6+{\mathrm {root}\left (z^9+3\,z^3+1,z,k\right )}^3+2\right )\,{\mathrm {root}\left (z^9+3\,z^3+1,z,k\right )}^2\,\sqrt {3-3\,x+\sqrt {3}\,x\,1{}\mathrm {i}+\sqrt {3}\,1{}\mathrm {i}}\,\sqrt {3-3\,x-\sqrt {3}\,x\,1{}\mathrm {i}-\sqrt {3}\,1{}\mathrm {i}}}{162\,\left (\mathrm {root}\left (z^9+3\,z^3+1,z,k\right )+1\right )\,\sqrt {x^3+1}\,\left ({\mathrm {root}\left (z^9+3\,z^3+1,z,k\right )}^8+{\mathrm {root}\left (z^9+3\,z^3+1,z,k\right )}^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*(x^3 + 1)^(1/2)*(x^3 - 2))/(3*x^3 + x^9 + 1),x)

[Out]

symsum((6^(1/2)*((3^(1/2)*1i)/2 + 3/2)*(-(3^(1/2)*1i - 3)*(x + 1))^(1/2)*ellipticPi((3^(1/2)*1i + 3)/(2*(root(
z^9 + 3*z^3 + 1, z, k) + 1)), asin((6^(1/2)*(-(3^(1/2)*1i - 3)*(x + 1))^(1/2))/6), (3^(1/2)*1i)/2 + 1/2)*(root
(z^9 + 3*z^3 + 1, z, k)^3 - root(z^9 + 3*z^3 + 1, z, k)^6 + 2)*root(z^9 + 3*z^3 + 1, z, k)^2*(3^(1/2)*x*1i - 3
*x + 3^(1/2)*1i + 3)^(1/2)*(3 - 3^(1/2)*x*1i - 3^(1/2)*1i - 3*x)^(1/2))/(162*(root(z^9 + 3*z^3 + 1, z, k) + 1)
*(x^3 + 1)^(1/2)*(root(z^9 + 3*z^3 + 1, z, k)^2 + root(z^9 + 3*z^3 + 1, z, k)^8)), k, 1, 9)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(x**3-2)*(x**3+1)**(1/2)/(x**9+3*x**3+1),x)

[Out]

Timed out

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