∫ (−2+x3) 3√______x+x3 +x4 __________________________________

3.8.16 $\int \frac {(-2+x^3) \sqrt [3]{x+x^3+x^4}}{1+x^2+2 x^3+x^4+x^5+x^6} \, dx$

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

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Rubi [F] time = 1.03, 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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 [F] time = 0.11, size = 0, normalized size = 0.00 \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]

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]

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

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IntegrateAlgebraic [A] time = 0.00, size = 55, normalized size = 1.00 \begin {gather*} \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 ] \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

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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maple [B] time = 14.84, size = 2189, normalized size = 39.80


method	result	size

trager	$\text {Expression too large to display}$	$2189$

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]

-9*RootOf(27*_Z^6+9*_Z^3+1)^5*ln((3*(x^4+x^3+x)^(1/3)*RootOf(27*_Z^6+9*_Z^3+1)^2*x-RootOf(27*_Z^6+9*_Z^3+1)*x^

3-2*RootOf(27*_Z^6+9*_Z^3+1)*x^2+(x^4+x^3+x)^(2/3)-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*ln(-(131301*RootOf(27*_Z^6+9*_

Z^3+1)^7*x^3-262602*RootOf(27*_Z^6+9*_Z^3+1)^7*x^2+131301*RootOf(27*_Z^6+9*_Z^3+1)^7-93825*(x^4+x^3+x)^(1/3)*R

ootOf(27*_Z^6+9*_Z^3+1)^5*x+79191*RootOf(27*_Z^6+9*_Z^3+1)^4*x^3-56259*RootOf(27*_Z^6+9*_Z^3+1)^4*x^2-31275*(x

^4+x^3+x)^(2/3)*RootOf(27*_Z^6+9*_Z^3+1)^3+79191*RootOf(27*_Z^6+9*_Z^3+1)^4-22233*(x^4+x^3+x)^(1/3)*RootOf(27*

_Z^6+9*_Z^3+1)^2*x+9260*RootOf(27*_Z^6+9*_Z^3+1)*x^3-2315*RootOf(27*_Z^6+9*_Z^3+1)*x^2-7411*(x^4+x^3+x)^(2/3)+

9260*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))*RootOf(27*_Z^6+9*_Z^3+1)^5-9*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+22932*RootOf(27*_Z^6+9*_Z^

3+1)^5+31275*(x^4+x^3+x)^(2/3)*RootOf(27*_Z^6+9*_Z^3+1)^3-4863*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-4863*RootOf(27*_Z^6+9*_Z^3+1)^2+3014*(

x^4+x^3+x)^(2/3))/(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))*RootOf(27*_Z^6+9*_Z^3+1)^4+9*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+8226*RootOf(27*_Z^6+9*_Z^3+1)^5-2673*(x^4+x

^3+x)^(2/3)*RootOf(27*_Z^6+9*_Z^3+1)^3+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+891*RootOf(27*_Z^6+9*_Z^3+1)^2+23*(x^4+x^3+x)^(2/3))/(9*Root

Of(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))*RootOf

(27*_Z^6+9*_Z^3+1)^4-3*ln(-(131301*RootOf(27*_Z^6+9*_Z^3+1)^7*x^3-262602*RootOf(27*_Z^6+9*_Z^3+1)^7*x^2+131301

*RootOf(27*_Z^6+9*_Z^3+1)^7-93825*(x^4+x^3+x)^(1/3)*RootOf(27*_Z^6+9*_Z^3+1)^5*x+79191*RootOf(27*_Z^6+9*_Z^3+1

)^4*x^3-56259*RootOf(27*_Z^6+9*_Z^3+1)^4*x^2-31275*(x^4+x^3+x)^(2/3)*RootOf(27*_Z^6+9*_Z^3+1)^3+79191*RootOf(2

7*_Z^6+9*_Z^3+1)^4-22233*(x^4+x^3+x)^(1/3)*RootOf(27*_Z^6+9*_Z^3+1)^2*x+9260*RootOf(27*_Z^6+9*_Z^3+1)*x^3-2315

*RootOf(27*_Z^6+9*_Z^3+1)*x^2-7411*(x^4+x^3+x)^(2/3)+9260*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))*RootOf(27*_Z^6+9*_Z^3+1)^

2-2*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)*Ro

otOf(27*_Z^6+9*_Z^3+1)^4*x+22932*RootOf(27*_Z^6+9*_Z^3+1)^5+31275*(x^4+x^3+x)^(2/3)*RootOf(27*_Z^6+9*_Z^3+1)^3

-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-4863*RootOf(27*_Z^6+9*_Z^3+1)^2+3014*(x^4+x^3+x)^(2/3))/(9*RootOf(27*_Z^6+9*_Z^3+1)^3*x^3-18*Ro

otOf(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))*RootOf(27*_Z^6+9*_Z^3+1)+ln((8226*Root

Of(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+8226*RootOf(27*_Z^6+9*_Z^3+1)^5-2673*(x^4+x^3+x)^(2/3)*RootOf(27*_Z^6+9*_Z^3+1)^3+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+891*RootOf(

27*_Z^6+9*_Z^3+1)^2+23*(x^4+x^3+x)^(2/3))/(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))*RootOf(27*_Z^6+9*_Z^3+1)

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

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