3.8.15 \(\int \frac {(-2+x^3) \sqrt [3]{x+x^3+x^4}}{1+x^2+2 x^3+x^4+x^5+x^6} \, dx\) [715]

Optimal. Leaf size=55 \[ \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\& ,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [3]{x+x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{-1+2 \text {$\#$1}^3}\& \right ] \]

[Out]

Unintegrable

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Rubi [F]
time = 0.89, 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 [3]{x+x^3+x^4}}{1+x^2+2 x^3+x^4+x^5+x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-2 + x^3)*(x + x^3 + x^4)^(1/3))/(1 + x^2 + 2*x^3 + x^4 + x^5 + x^6),x]

[Out]

(-6*(x + x^3 + x^4)^(1/3)*Defer[Subst][Defer[Int][(x^3*(1 + x^6 + x^9)^(1/3))/(1 + x^6 + 2*x^9 + x^12 + x^15 +
x^18), x], x, x^(1/3)])/(x^(1/3)*(1 + x^2 + x^3)^(1/3)) + (3*(x + x^3 + x^4)^(1/3)*Defer[Subst][Defer[Int][(x
^12*(1 + x^6 + x^9)^(1/3))/(1 + x^6 + 2*x^9 + x^12 + x^15 + x^18), x], x, x^(1/3)])/(x^(1/3)*(1 + x^2 + x^3)^(
1/3))

Rubi steps

\begin {align*} \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^3+x^4}}{1+x^2+2 x^3+x^4+x^5+x^6} \, dx &=\frac {\sqrt [3]{x+x^3+x^4} \int \frac {\sqrt [3]{x} \left (-2+x^3\right ) \sqrt [3]{1+x^2+x^3}}{1+x^2+2 x^3+x^4+x^5+x^6} \, dx}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}\\ &=\frac {\left (3 \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^3 \left (-2+x^9\right ) \sqrt [3]{1+x^6+x^9}}{1+x^6+2 x^9+x^{12}+x^{15}+x^{18}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}\\ &=\frac {\left (3 \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \left (-\frac {2 x^3 \sqrt [3]{1+x^6+x^9}}{1+x^6+2 x^9+x^{12}+x^{15}+x^{18}}+\frac {x^{12} \sqrt [3]{1+x^6+x^9}}{1+x^6+2 x^9+x^{12}+x^{15}+x^{18}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}\\ &=\frac {\left (3 \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^{12} \sqrt [3]{1+x^6+x^9}}{1+x^6+2 x^9+x^{12}+x^{15}+x^{18}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}-\frac {\left (6 \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^3 \sqrt [3]{1+x^6+x^9}}{1+x^6+2 x^9+x^{12}+x^{15}+x^{18}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}\\ \end {align*}

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Mathematica [A]
time = 2.70, size = 93, normalized size = 1.69 \begin {gather*} \frac {\sqrt [3]{x+x^3+x^4} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-2 \log \left (\sqrt [3]{x}\right ) \text {$\#$1}+\log \left (\sqrt [3]{1+x^2+x^3}-x^{2/3} \text {$\#$1}\right ) \text {$\#$1}}{-1+2 \text {$\#$1}^3}\&\right ]}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-2 + x^3)*(x + x^3 + x^4)^(1/3))/(1 + x^2 + 2*x^3 + x^4 + x^5 + x^6),x]

[Out]

((x + x^3 + x^4)^(1/3)*RootSum[1 - #1^3 + #1^6 & , (-2*Log[x^(1/3)]*#1 + Log[(1 + x^2 + x^3)^(1/3) - x^(2/3)*#
1]*#1)/(-1 + 2*#1^3) & ])/(x^(1/3)*(1 + x^2 + x^3)^(1/3))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 1.
time = 10.11, size = 2597, normalized size = 47.22

method	result	size
trager	\(\text {Expression too large to display}\)	\(2597\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3-2)*(x^4+x^3+x)^(1/3)/(x^6+x^5+x^4+2*x^3+x^2+1),x,method=_RETURNVERBOSE)

[Out]

18*RootOf(27*_Z^6+9*_Z^3+1)^5*ln((243*RootOf(27*_Z^6+9*_Z^3+1)^7*x^3-486*RootOf(27*_Z^6+9*_Z^3+1)^7*x^2-324*Ro
otOf(27*_Z^6+9*_Z^3+1)^5*(x^4+x^3+x)^(1/3)*x+243*RootOf(27*_Z^6+9*_Z^3+1)^7+180*RootOf(27*_Z^6+9*_Z^3+1)^4*x^3
-171*RootOf(27*_Z^6+9*_Z^3+1)^4*x^2+99*RootOf(27*_Z^6+9*_Z^3+1)^3*(x^4+x^3+x)^(2/3)-39*RootOf(27*_Z^6+9*_Z^3+1
)^2*(x^4+x^3+x)^(1/3)*x+180*RootOf(27*_Z^6+9*_Z^3+1)^4+32*RootOf(27*_Z^6+9*_Z^3+1)*x^3-8*RootOf(27*_Z^6+9*_Z^3
+1)*x^2+23*(x^4+x^3+x)^(2/3)+32*RootOf(27*_Z^6+9*_Z^3+1))/(9*RootOf(27*_Z^6+9*_Z^3+1)^3*x^3-18*RootOf(27*_Z^6+
9*_Z^3+1)^3*x^2+9*RootOf(27*_Z^6+9*_Z^3+1)^3-x^3-5*x^2-1))-9*RootOf(27*_Z^6+9*_Z^3+1)^5*ln((-16659*RootOf(27*_
Z^6+9*_Z^3+1)^5*(x^4+x^3+x)^(1/3)*x-2880*RootOf(27*_Z^6+9*_Z^3+1)^4*x^3-5760*RootOf(27*_Z^6+9*_Z^3+1)^4*x^2+26
73*RootOf(27*_Z^6+9*_Z^3+1)^3*(x^4+x^3+x)^(2/3)-2811*RootOf(27*_Z^6+9*_Z^3+1)^2*(x^4+x^3+x)^(1/3)*x-2880*RootO
f(27*_Z^6+9*_Z^3+1)^4-23*RootOf(27*_Z^6+9*_Z^3+1)*x^3-46*RootOf(27*_Z^6+9*_Z^3+1)*x^2+914*(x^4+x^3+x)^(2/3)-23
*RootOf(27*_Z^6+9*_Z^3+1))/(9*RootOf(27*_Z^6+9*_Z^3+1)^3*x^3-18*RootOf(27*_Z^6+9*_Z^3+1)^3*x^2+9*RootOf(27*_Z^
6+9*_Z^3+1)^3-x^3-5*x^2-1))+9*RootOf(27*_Z^6+9*_Z^3+1)^4*ln(-(-8226*RootOf(27*_Z^6+9*_Z^3+1)^5*x^3-16452*RootO
f(27*_Z^6+9*_Z^3+1)^5*x^2-8433*(x^4+x^3+x)^(1/3)*RootOf(27*_Z^6+9*_Z^3+1)^4*x+2673*RootOf(27*_Z^6+9*_Z^3+1)^3*
(x^4+x^3+x)^(2/3)-8226*RootOf(27*_Z^6+9*_Z^3+1)^5-891*RootOf(27*_Z^6+9*_Z^3+1)^2*x^3-1782*RootOf(27*_Z^6+9*_Z^
3+1)^2*x^2-1851*(x^4+x^3+x)^(1/3)*RootOf(27*_Z^6+9*_Z^3+1)*x-23*(x^4+x^3+x)^(2/3)-891*RootOf(27*_Z^6+9*_Z^3+1)
^2)/(9*RootOf(27*_Z^6+9*_Z^3+1)^3*x^3-18*RootOf(27*_Z^6+9*_Z^3+1)^3*x^2+9*RootOf(27*_Z^6+9*_Z^3+1)^3+4*x^3-x^2
+4))-9*RootOf(27*_Z^6+9*_Z^3+1)^4*ln(-(187515*RootOf(27*_Z^6+9*_Z^3+1)^8*x^3-375030*RootOf(27*_Z^6+9*_Z^3+1)^8
*x^2+187515*RootOf(27*_Z^6+9*_Z^3+1)^8+22932*RootOf(27*_Z^6+9*_Z^3+1)^5*x^3-191709*RootOf(27*_Z^6+9*_Z^3+1)^5*
x^2-39573*(x^4+x^3+x)^(1/3)*RootOf(27*_Z^6+9*_Z^3+1)^4*x+31275*RootOf(27*_Z^6+9*_Z^3+1)^3*(x^4+x^3+x)^(2/3)+22
932*RootOf(27*_Z^6+9*_Z^3+1)^5-4863*RootOf(27*_Z^6+9*_Z^3+1)^2*x^3-24315*RootOf(27*_Z^6+9*_Z^3+1)^2*x^2-11808*
(x^4+x^3+x)^(1/3)*RootOf(27*_Z^6+9*_Z^3+1)*x+3014*(x^4+x^3+x)^(2/3)-4863*RootOf(27*_Z^6+9*_Z^3+1)^2)/(9*RootOf
(27*_Z^6+9*_Z^3+1)^3*x^3-18*RootOf(27*_Z^6+9*_Z^3+1)^3*x^2+9*RootOf(27*_Z^6+9*_Z^3+1)^3+4*x^3-x^2+4))+3*RootOf
(27*_Z^6+9*_Z^3+1)^2*ln((243*RootOf(27*_Z^6+9*_Z^3+1)^7*x^3-486*RootOf(27*_Z^6+9*_Z^3+1)^7*x^2-324*RootOf(27*_
Z^6+9*_Z^3+1)^5*(x^4+x^3+x)^(1/3)*x+243*RootOf(27*_Z^6+9*_Z^3+1)^7+180*RootOf(27*_Z^6+9*_Z^3+1)^4*x^3-171*Root
Of(27*_Z^6+9*_Z^3+1)^4*x^2+99*RootOf(27*_Z^6+9*_Z^3+1)^3*(x^4+x^3+x)^(2/3)-39*RootOf(27*_Z^6+9*_Z^3+1)^2*(x^4+
x^3+x)^(1/3)*x+180*RootOf(27*_Z^6+9*_Z^3+1)^4+32*RootOf(27*_Z^6+9*_Z^3+1)*x^3-8*RootOf(27*_Z^6+9*_Z^3+1)*x^2+2
3*(x^4+x^3+x)^(2/3)+32*RootOf(27*_Z^6+9*_Z^3+1))/(9*RootOf(27*_Z^6+9*_Z^3+1)^3*x^3-18*RootOf(27*_Z^6+9*_Z^3+1)
^3*x^2+9*RootOf(27*_Z^6+9*_Z^3+1)^3-x^3-5*x^2-1))-3*RootOf(27*_Z^6+9*_Z^3+1)^2*ln((-16659*RootOf(27*_Z^6+9*_Z^
3+1)^5*(x^4+x^3+x)^(1/3)*x-2880*RootOf(27*_Z^6+9*_Z^3+1)^4*x^3-5760*RootOf(27*_Z^6+9*_Z^3+1)^4*x^2+2673*RootOf
(27*_Z^6+9*_Z^3+1)^3*(x^4+x^3+x)^(2/3)-2811*RootOf(27*_Z^6+9*_Z^3+1)^2*(x^4+x^3+x)^(1/3)*x-2880*RootOf(27*_Z^6
+9*_Z^3+1)^4-23*RootOf(27*_Z^6+9*_Z^3+1)*x^3-46*RootOf(27*_Z^6+9*_Z^3+1)*x^2+914*(x^4+x^3+x)^(2/3)-23*RootOf(2
7*_Z^6+9*_Z^3+1))/(9*RootOf(27*_Z^6+9*_Z^3+1)^3*x^3-18*RootOf(27*_Z^6+9*_Z^3+1)^3*x^2+9*RootOf(27*_Z^6+9*_Z^3+
1)^3-x^3-5*x^2-1))+RootOf(27*_Z^6+9*_Z^3+1)*ln(-(-8226*RootOf(27*_Z^6+9*_Z^3+1)^5*x^3-16452*RootOf(27*_Z^6+9*_
Z^3+1)^5*x^2-8433*(x^4+x^3+x)^(1/3)*RootOf(27*_Z^6+9*_Z^3+1)^4*x+2673*RootOf(27*_Z^6+9*_Z^3+1)^3*(x^4+x^3+x)^(
2/3)-8226*RootOf(27*_Z^6+9*_Z^3+1)^5-891*RootOf(27*_Z^6+9*_Z^3+1)^2*x^3-1782*RootOf(27*_Z^6+9*_Z^3+1)^2*x^2-18
51*(x^4+x^3+x)^(1/3)*RootOf(27*_Z^6+9*_Z^3+1)*x-23*(x^4+x^3+x)^(2/3)-891*RootOf(27*_Z^6+9*_Z^3+1)^2)/(9*RootOf
(27*_Z^6+9*_Z^3+1)^3*x^3-18*RootOf(27*_Z^6+9*_Z^3+1)^3*x^2+9*RootOf(27*_Z^6+9*_Z^3+1)^3+4*x^3-x^2+4))-2*RootOf
(27*_Z^6+9*_Z^3+1)*ln(-(187515*RootOf(27*_Z^6+9*_Z^3+1)^8*x^3-375030*RootOf(27*_Z^6+9*_Z^3+1)^8*x^2+187515*Roo
tOf(27*_Z^6+9*_Z^3+1)^8+22932*RootOf(27*_Z^6+9*_Z^3+1)^5*x^3-191709*RootOf(27*_Z^6+9*_Z^3+1)^5*x^2-39573*(x^4+
x^3+x)^(1/3)*RootOf(27*_Z^6+9*_Z^3+1)^4*x+31275*RootOf(27*_Z^6+9*_Z^3+1)^3*(x^4+x^3+x)^(2/3)+22932*RootOf(27*_
Z^6+9*_Z^3+1)^5-4863*RootOf(27*_Z^6+9*_Z^3+1)^2*x^3-24315*RootOf(27*_Z^6+9*_Z^3+1)^2*x^2-11808*(x^4+x^3+x)^(1/
3)*RootOf(27*_Z^6+9*_Z^3+1)*x+3014*(x^4+x^3+x)^(2/3)-4863*RootOf(27*_Z^6+9*_Z^3+1)^2)/(9*RootOf(27*_Z^6+9*_Z^3
+1)^3*x^3-18*RootOf(27*_Z^6+9*_Z^3+1)^3*x^2+9*RootOf(27*_Z^6+9*_Z^3+1)^3+4*x^3-x^2+4))

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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^4+x^3+x)^(1/3)/(x^6+x^5+x^4+2*x^3+x^2+1),x, algorithm="maxima")

[Out]

integrate((x^4 + x^3 + x)^(1/3)*(x^3 - 2)/(x^6 + x^5 + x^4 + 2*x^3 + x^2 + 1), x)

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-2)*(x^4+x^3+x)^(1/3)/(x^6+x^5+x^4+2*x^3+x^2+1),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (tr
ace 0)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{x \left (x^{3} + x^{2} + 1\right )} \left (x^{3} - 2\right )}{x^{6} + x^{5} + x^{4} + 2 x^{3} + x^{2} + 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3-2)*(x**4+x**3+x)**(1/3)/(x**6+x**5+x**4+2*x**3+x**2+1),x)

[Out]

Integral((x*(x**3 + x**2 + 1))**(1/3)*(x**3 - 2)/(x**6 + x**5 + x**4 + 2*x**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^4+x^3+x)^(1/3)/(x^6+x^5+x^4+2*x^3+x^2+1),x, algorithm="giac")

[Out]

integrate((x^4 + x^3 + x)^(1/3)*(x^3 - 2)/(x^6 + x^5 + x^4 + 2*x^3 + x^2 + 1), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\left (x^3-2\right )\,{\left (x^4+x^3+x\right )}^{1/3}}{x^6+x^5+x^4+2\,x^3+x^2+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3 - 2)*(x + x^3 + x^4)^(1/3))/(x^2 + 2*x^3 + x^4 + x^5 + x^6 + 1),x)

[Out]

int(((x^3 - 2)*(x + x^3 + x^4)^(1/3))/(x^2 + 2*x^3 + x^4 + x^5 + x^6 + 1), x)

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