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

Optimal. Leaf size=285 \[ 2 \text {RootSum}\left [\text {$\#$1}^3-\text {$\#$1}-1\& ,\frac {\text {$\#$1} \log \left (\sqrt [3]{x^3+x^2}-\text {$\#$1} x\right )-\text {$\#$1} \log (x)}{3 \text {$\#$1}^2-1}\& \right ]+2 \text {RootSum}\left [\text {$\#$1}^6+\text {$\#$1}^4-2 \text {$\#$1}^3+\text {$\#$1}^2-\text {$\#$1}+1\& ,\frac {2 \text {$\#$1}^4 \log \left (\sqrt [3]{x^3+x^2}-\text {$\#$1} x\right )-2 \text {$\#$1}^4 \log (x)+\text {$\#$1}^2 \log \left (\sqrt [3]{x^3+x^2}-\text {$\#$1} x\right )-\text {$\#$1}^2 \log (x)-2 \text {$\#$1} \log \left (\sqrt [3]{x^3+x^2}-\text {$\#$1} x\right )+2 \text {$\#$1} \log (x)}{6 \text {$\#$1}^5+4 \text {$\#$1}^3-6 \text {$\#$1}^2+2 \text {$\#$1}-1}\& \right ]-\log \left (\sqrt [3]{x^3+x^2}-x\right )+\frac {1}{2} \log \left (x^2+\sqrt [3]{x^3+x^2} x+\left (x^3+x^2\right )^{2/3}\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+x^2}+x}\right ) \]

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Rubi [F]  time = 0.61, 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 {1+x^2+x^3}{\left (-1+x^2+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(Sqrt[3]*x^(2/3)*(1 + x)^(1/3)*ArcTan[(1 + (2*x^(1/3))/(1 + x)^(1/3))/Sqrt[3]])/(x^2 + x^3)^(1/3) - (3*x^(2/3)
*(1 + x)^(1/3)*Log[x^(1/3) - (1 + x)^(1/3)])/(2*(x^2 + x^3)^(1/3)) + (6*x^(2/3)*(1 + x)^(1/3)*Defer[Subst][Def
er[Int][1/((1 + x^3)^(1/3)*(-1 + x^6 + x^9)), x], x, x^(1/3)])/(x^2 + x^3)^(1/3)

Rubi steps

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

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Mathematica [F]  time = 0.46, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x^2+x^3}{\left (-1+x^2+x^3\right ) \sqrt [3]{x^2+x^3}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.36, size = 285, normalized size = 1.00 \begin {gather*} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x^2+x^3}}\right )-\log \left (-x+\sqrt [3]{x^2+x^3}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{x^2+x^3}+\left (x^2+x^3\right )^{2/3}\right )+2 \text {RootSum}\left [-1-\text {$\#$1}+\text {$\#$1}^3\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [3]{x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}}{-1+3 \text {$\#$1}^2}\&\right ]+2 \text {RootSum}\left [1-\text {$\#$1}+\text {$\#$1}^2-2 \text {$\#$1}^3+\text {$\#$1}^4+\text {$\#$1}^6\&,\frac {2 \log (x) \text {$\#$1}-2 \log \left (\sqrt [3]{x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}-\log (x) \text {$\#$1}^2+\log \left (\sqrt [3]{x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^2-2 \log (x) \text {$\#$1}^4+2 \log \left (\sqrt [3]{x^2+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-1+2 \text {$\#$1}-6 \text {$\#$1}^2+4 \text {$\#$1}^3+6 \text {$\#$1}^5}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(x^2 + x^3)^(1/3))] - Log[-x + (x^2 + x^3)^(1/3)] + Log[x^2 + x*(x^2 + x^3)^
(1/3) + (x^2 + x^3)^(2/3)]/2 + 2*RootSum[-1 - #1 + #1^3 & , (-(Log[x]*#1) + Log[(x^2 + x^3)^(1/3) - x*#1]*#1)/
(-1 + 3*#1^2) & ] + 2*RootSum[1 - #1 + #1^2 - 2*#1^3 + #1^4 + #1^6 & , (2*Log[x]*#1 - 2*Log[(x^2 + x^3)^(1/3)
- x*#1]*#1 - Log[x]*#1^2 + Log[(x^2 + x^3)^(1/3) - x*#1]*#1^2 - 2*Log[x]*#1^4 + 2*Log[(x^2 + x^3)^(1/3) - x*#1
]*#1^4)/(-1 + 2*#1 - 6*#1^2 + 4*#1^3 + 6*#1^5) & ]

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fricas [B]  time = 11.80, size = 868, normalized size = 3.05

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/23*sqrt(23)*sqrt(1/69*(69*(100/4761*sqrt(69) + 4/23)^(1/3) - 4/(100/4761*sqrt(69) + 4/23)^(1/3))^2 + 16)*ar
ctan(1/4761000*(sqrt(1/23)*(sqrt(23)*x*(69*(100/4761*sqrt(69) + 4/23)^(1/3) - 4/(100/4761*sqrt(69) + 4/23)^(1/
3))^2 + 7176*sqrt(23)*x)*sqrt(1/69*(69*(100/4761*sqrt(69) + 4/23)^(1/3) - 4/(100/4761*sqrt(69) + 4/23)^(1/3))^
2 + 16)*sqrt(((2*x^2 + 9*(x^3 + x^2)^(1/3)*x)*(69*(100/4761*sqrt(69) + 4/23)^(1/3) - 4/(100/4761*sqrt(69) + 4/
23)^(1/3))^2 - 5796*x^2 + 138*(3*x^2 + (x^3 + x^2)^(1/3)*x)*(69*(100/4761*sqrt(69) + 4/23)^(1/3) - 4/(100/4761
*sqrt(69) + 4/23)^(1/3)) + 4968*(x^3 + x^2)^(1/3)*x + 20700*(x^3 + x^2)^(2/3))/x^2) - 15*((3*sqrt(23)*x + 2*sq
rt(23)*(x^3 + x^2)^(1/3))*(69*(100/4761*sqrt(69) + 4/23)^(1/3) - 4/(100/4761*sqrt(69) + 4/23)^(1/3))^2 + 92*sq
rt(23)*x*(69*(100/4761*sqrt(69) + 4/23)^(1/3) - 4/(100/4761*sqrt(69) + 4/23)^(1/3)) + 2484*sqrt(23)*x + 14352*
sqrt(23)*(x^3 + x^2)^(1/3))*sqrt(1/69*(69*(100/4761*sqrt(69) + 4/23)^(1/3) - 4/(100/4761*sqrt(69) + 4/23)^(1/3
))^2 + 16))/x) + 1/69*(69*(100/4761*sqrt(69) + 4/23)^(1/3) - 4/(100/4761*sqrt(69) + 4/23)^(1/3))*log(-1/69*(3*
x*(69*(100/4761*sqrt(69) + 4/23)^(1/3) - 4/(100/4761*sqrt(69) + 4/23)^(1/3))^2 + 46*x*(69*(100/4761*sqrt(69) +
4/23)^(1/3) - 4/(100/4761*sqrt(69) + 4/23)^(1/3)) + 1656*x - 6900*(x^3 + x^2)^(1/3))/x) + 1/46*(sqrt(23)*sqrt
(-0.4683433399583197? + 0.?e-33*I) + 34.26254145273487? + 0.?e-32*I)*log(-((71.2918670130777? + 0.?e-32*I)*sqr
t(23)*sqrt(-0.4683433399583197? + 0.?e-33*I)*x + (654.502264627727? + 0.?e-31*I)*x - 800*(x^3 + x^2)^(1/3))/x)
- 1/46*(sqrt(23)*sqrt(-0.4683433399583197? + 0.?e-33*I) - 34.26254145273487? + 0.?e-32*I)*log(-(-(71.29186701
30777? + 0.?e-32*I)*sqrt(23)*sqrt(-0.4683433399583197? + 0.?e-33*I)*x + (654.502264627727? + 0.?e-31*I)*x - 80
0*(x^3 + x^2)^(1/3))/x) - 1/138*(69*(100/4761*sqrt(69) + 4/23)^(1/3) - 4/(100/4761*sqrt(69) + 4/23)^(1/3))*log
(400/207*((2*x^2 + 9*(x^3 + x^2)^(1/3)*x)*(69*(100/4761*sqrt(69) + 4/23)^(1/3) - 4/(100/4761*sqrt(69) + 4/23)^
(1/3))^2 - 5796*x^2 + 138*(3*x^2 + (x^3 + x^2)^(1/3)*x)*(69*(100/4761*sqrt(69) + 4/23)^(1/3) - 4/(100/4761*sqr
t(69) + 4/23)^(1/3)) + 4968*(x^3 + x^2)^(1/3)*x + 20700*(x^3 + x^2)^(2/3))/x^2) - sqrt(3)*arctan(1/3*(sqrt(3)*
x + 2*sqrt(3)*(x^3 + x^2)^(1/3))/x) - (0.3106288296404671? + 0.5380249152329462?*I)*log(((264.9435914489492? -
458.8957615293512?*I)*x + 400*(x^3 + x^2)^(1/3))/x) - (0.4342090280276823? + 0.6093739760383123?*I)*log(((62.
30754086491403? - 341.9037831771358?*I)*x + 400*(x^3 + x^2)^(1/3))/x) - log(-(x - (x^3 + x^2)^(1/3))/x) - (0.4
342090280276823? - 0.6093739760383123?*I)*log(-(-(62.30754086491403? + 341.9037831771358?*I)*x - 400*(x^3 + x^
2)^(1/3))/x) - (0.3106288296404671? - 0.5380249152329462?*I)*log(-(-(264.9435914489492? + 458.8957615293512?*I
)*x - 400*(x^3 + x^2)^(1/3))/x) + 1/2*log((x^2 + (x^3 + x^2)^(1/3)*x + (x^3 + x^2)^(2/3))/x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 254.25, size = 290058, normalized size = 1017.75

method	result	size
trager	\(\text {Expression too large to display}\)	\(290058\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((x**3 + x**2 + 1)/((x**2*(x + 1))**(1/3)*(x**3 + x**2 - 1)), x)

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