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

Optimal. Leaf size=59 \[ -\text {RootSum}\left [\text {$\#$1}^6-\text {$\#$1}^3+1\& ,\frac {\text {$\#$1} \log \left (\sqrt [3]{-x^4+x^3+x}-\text {$\#$1} x\right )-\text {$\#$1} \log (x)}{2 \text {$\#$1}^3-1}\& \right ] \]

________________________________________________________________________________________

Rubi [F]  time = 1.58, 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} \sqrt [3]{1+x^2-x^3} \left (2+x^3\right )}{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 \sqrt [3]{1+x^6-x^9} \left (2+x^9\right )}{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*}

________________________________________________________________________________________

Mathematica [F]  time = 0.47, 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]

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.13, size = 59, 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) & ]

________________________________________________________________________________________

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)

________________________________________________________________________________________

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)

________________________________________________________________________________________

maple [B]  time = 13.59, size = 1808, normalized size = 30.64

method	result	size
trager	\(\text {Expression too large to display}\)	\(1808\)

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((567*RootOf(27*_Z^6-9*_Z^3+1)^7*x^3-567*RootOf(27*_Z^6-9*_Z^3+1)^7+3132*(-x^4
+x^3+x)^(1/3)*RootOf(27*_Z^6-9*_Z^3+1)^5*x+855*RootOf(27*_Z^6-9*_Z^3+1)^4*x^3+63*RootOf(27*_Z^6-9*_Z^3+1)^4*x^
2-1107*RootOf(27*_Z^6-9*_Z^3+1)^3*(-x^4+x^3+x)^(2/3)-855*RootOf(27*_Z^6-9*_Z^3+1)^4-717*(-x^4+x^3+x)^(1/3)*Roo
tOf(27*_Z^6-9*_Z^3+1)^2*x-218*RootOf(27*_Z^6-9*_Z^3+1)*x^3+109*RootOf(27*_Z^6-9*_Z^3+1)*x^2+130*(-x^4+x^3+x)^(
2/3)+218*RootOf(27*_Z^6-9*_Z^3+1))/(9*RootOf(27*_Z^6-9*_Z^3+1)^3*x^3-9*RootOf(27*_Z^6-9*_Z^3+1)^3-x^3-x^2+1))+
18*ln(-(5751*(-x^4+x^3+x)^(1/3)*RootOf(27*_Z^6-9*_Z^3+1)^5*x-819*RootOf(27*_Z^6-9*_Z^3+1)^4*x^3+1638*RootOf(27
*_Z^6-9*_Z^3+1)^4*x^2-2736*RootOf(27*_Z^6-9*_Z^3+1)^3*(-x^4+x^3+x)^(2/3)+819*RootOf(27*_Z^6-9*_Z^3+1)^4-1551*(
-x^4+x^3+x)^(1/3)*RootOf(27*_Z^6-9*_Z^3+1)^2*x-122*RootOf(27*_Z^6-9*_Z^3+1)*x^3+244*RootOf(27*_Z^6-9*_Z^3+1)*x
^2+395*(-x^4+x^3+x)^(2/3)+122*RootOf(27*_Z^6-9*_Z^3+1))/(9*RootOf(27*_Z^6-9*_Z^3+1)^3*x^3-9*RootOf(27*_Z^6-9*_
Z^3+1)^3-x^3-x^2+1))*RootOf(27*_Z^6-9*_Z^3+1)^5-9*ln((8829*RootOf(27*_Z^6-9*_Z^3+1)^8*x^3-8829*RootOf(27*_Z^6-
9*_Z^3+1)^8-792*RootOf(27*_Z^6-9*_Z^3+1)^5*x^3-981*RootOf(27*_Z^6-9*_Z^3+1)^5*x^2-1170*RootOf(27*_Z^6-9*_Z^3+1
)^4*(-x^4+x^3+x)^(1/3)*x+792*RootOf(27*_Z^6-9*_Z^3+1)^5+1107*RootOf(27*_Z^6-9*_Z^3+1)^3*(-x^4+x^3+x)^(2/3)-21*
RootOf(27*_Z^6-9*_Z^3+1)^2*x^3-21*RootOf(27*_Z^6-9*_Z^3+1)^2*x^2+369*RootOf(27*_Z^6-9*_Z^3+1)*(-x^4+x^3+x)^(1/
3)*x+21*RootOf(27*_Z^6-9*_Z^3+1)^2-239*(-x^4+x^3+x)^(2/3))/(9*RootOf(27*_Z^6-9*_Z^3+1)^3*x^3-9*RootOf(27*_Z^6-
9*_Z^3+1)^3-2*x^3+x^2+2))*RootOf(27*_Z^6-9*_Z^3+1)^4-3*ln(-(5751*(-x^4+x^3+x)^(1/3)*RootOf(27*_Z^6-9*_Z^3+1)^5
*x-819*RootOf(27*_Z^6-9*_Z^3+1)^4*x^3+1638*RootOf(27*_Z^6-9*_Z^3+1)^4*x^2-2736*RootOf(27*_Z^6-9*_Z^3+1)^3*(-x^
4+x^3+x)^(2/3)+819*RootOf(27*_Z^6-9*_Z^3+1)^4-1551*(-x^4+x^3+x)^(1/3)*RootOf(27*_Z^6-9*_Z^3+1)^2*x-122*RootOf(
27*_Z^6-9*_Z^3+1)*x^3+244*RootOf(27*_Z^6-9*_Z^3+1)*x^2+395*(-x^4+x^3+x)^(2/3)+122*RootOf(27*_Z^6-9*_Z^3+1))/(9
*RootOf(27*_Z^6-9*_Z^3+1)^3*x^3-9*RootOf(27*_Z^6-9*_Z^3+1)^3-x^3-x^2+1))*RootOf(27*_Z^6-9*_Z^3+1)^2+ln((8829*R
ootOf(27*_Z^6-9*_Z^3+1)^8*x^3-8829*RootOf(27*_Z^6-9*_Z^3+1)^8-792*RootOf(27*_Z^6-9*_Z^3+1)^5*x^3-981*RootOf(27
*_Z^6-9*_Z^3+1)^5*x^2-1170*RootOf(27*_Z^6-9*_Z^3+1)^4*(-x^4+x^3+x)^(1/3)*x+792*RootOf(27*_Z^6-9*_Z^3+1)^5+1107
*RootOf(27*_Z^6-9*_Z^3+1)^3*(-x^4+x^3+x)^(2/3)-21*RootOf(27*_Z^6-9*_Z^3+1)^2*x^3-21*RootOf(27*_Z^6-9*_Z^3+1)^2
*x^2+369*RootOf(27*_Z^6-9*_Z^3+1)*(-x^4+x^3+x)^(1/3)*x+21*RootOf(27*_Z^6-9*_Z^3+1)^2-239*(-x^4+x^3+x)^(2/3))/(
9*RootOf(27*_Z^6-9*_Z^3+1)^3*x^3-9*RootOf(27*_Z^6-9*_Z^3+1)^3-2*x^3+x^2+2))*RootOf(27*_Z^6-9*_Z^3+1)+RootOf(27
*_Z^6-9*_Z^3+1)*ln(-(1098*RootOf(27*_Z^6-9*_Z^3+1)^5*x^3-2196*RootOf(27*_Z^6-9*_Z^3+1)^5*x^2-3555*RootOf(27*_Z
^6-9*_Z^3+1)^4*(-x^4+x^3+x)^(1/3)*x-1098*RootOf(27*_Z^6-9*_Z^3+1)^5+2736*RootOf(27*_Z^6-9*_Z^3+1)^3*(-x^4+x^3+
x)^(2/3)+273*RootOf(27*_Z^6-9*_Z^3+1)^2*x^3-546*RootOf(27*_Z^6-9*_Z^3+1)^2*x^2+912*RootOf(27*_Z^6-9*_Z^3+1)*(-
x^4+x^3+x)^(1/3)*x-273*RootOf(27*_Z^6-9*_Z^3+1)^2-517*(-x^4+x^3+x)^(2/3))/(9*RootOf(27*_Z^6-9*_Z^3+1)^3*x^3-9*
RootOf(27*_Z^6-9*_Z^3+1)^3-2*x^3+x^2+2))

________________________________________________________________________________________

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)

________________________________________________________________________________________

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)

________________________________________________________________________________________

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)

________________________________________________________________________________________