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

Optimal. Leaf size=85 \[ -\log \left (\sqrt [3]{x^5+x^2}+x\right )+\frac {1}{2} \log \left (x^2-\sqrt [3]{x^5+x^2} x+\left (x^5+x^2\right )^{2/3}\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^5+x^2}-x}\right ) \]

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Rubi [F]  time = 1.00, 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+2 x^3}{\left (1+x+x^3\right ) \sqrt [3]{x^2+x^5}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(6*x*(1 + x^3)^(1/3)*Hypergeometric2F1[1/9, 1/3, 10/9, -x^3])/(x^2 + x^5)^(1/3) - (9*x^(2/3)*(1 + x^3)^(1/3)*D
efer[Subst][Defer[Int][1/((1 + x^9)^(1/3)*(1 + x^3 + x^9)), x], x, x^(1/3)])/(x^2 + x^5)^(1/3) - (6*x^(2/3)*(1
+ x^3)^(1/3)*Defer[Subst][Defer[Int][x^3/((1 + x^9)^(1/3)*(1 + x^3 + x^9)), x], x, x^(1/3)])/(x^2 + x^5)^(1/3
)

Rubi steps

\begin {align*} \int \frac {-1+2 x^3}{\left (1+x+x^3\right ) \sqrt [3]{x^2+x^5}} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{1+x^3}\right ) \int \frac {-1+2 x^3}{x^{2/3} \sqrt [3]{1+x^3} \left (1+x+x^3\right )} \, dx}{\sqrt [3]{x^2+x^5}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{1+x^3}\right ) \operatorname {Subst}\left (\int \frac {-1+2 x^9}{\sqrt [3]{1+x^9} \left (1+x^3+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^5}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{1+x^3}\right ) \operatorname {Subst}\left (\int \left (\frac {2}{\sqrt [3]{1+x^9}}-\frac {3+2 x^3}{\sqrt [3]{1+x^9} \left (1+x^3+x^9\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^5}}\\ &=-\frac {\left (3 x^{2/3} \sqrt [3]{1+x^3}\right ) \operatorname {Subst}\left (\int \frac {3+2 x^3}{\sqrt [3]{1+x^9} \left (1+x^3+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^5}}+\frac {\left (6 x^{2/3} \sqrt [3]{1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^9}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^5}}\\ &=\frac {6 x \sqrt [3]{1+x^3} \, _2F_1\left (\frac {1}{9},\frac {1}{3};\frac {10}{9};-x^3\right )}{\sqrt [3]{x^2+x^5}}-\frac {\left (3 x^{2/3} \sqrt [3]{1+x^3}\right ) \operatorname {Subst}\left (\int \left (\frac {3}{\sqrt [3]{1+x^9} \left (1+x^3+x^9\right )}+\frac {2 x^3}{\sqrt [3]{1+x^9} \left (1+x^3+x^9\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^5}}\\ &=\frac {6 x \sqrt [3]{1+x^3} \, _2F_1\left (\frac {1}{9},\frac {1}{3};\frac {10}{9};-x^3\right )}{\sqrt [3]{x^2+x^5}}-\frac {\left (6 x^{2/3} \sqrt [3]{1+x^3}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt [3]{1+x^9} \left (1+x^3+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^5}}-\frac {\left (9 x^{2/3} \sqrt [3]{1+x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^9} \left (1+x^3+x^9\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2+x^5}}\\ \end {align*}

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Mathematica [F]  time = 0.20, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-1+2 x^3}{\left (1+x+x^3\right ) \sqrt [3]{x^2+x^5}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.93, size = 85, normalized size = 1.00 \begin {gather*} -\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{x^2+x^5}}\right )-\log \left (x+\sqrt [3]{x^2+x^5}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{x^2+x^5}+\left (x^2+x^5\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2*(x^2 + x^5)^(1/3))]) - Log[x + (x^2 + x^5)^(1/3)] + Log[x^2 - x*(x^2 + x^
5)^(1/3) + (x^2 + x^5)^(2/3)]/2

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fricas [A]  time = 2.18, size = 108, normalized size = 1.27 \begin {gather*} -\sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} {\left (x^{5} + x^{2}\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (x^{4} + x^{2} + x\right )} + 2 \, \sqrt {3} {\left (x^{5} + x^{2}\right )}^{\frac {2}{3}}}{3 \, {\left (x^{4} - x^{2} + x\right )}}\right ) - \frac {1}{2} \, \log \left (\frac {x^{4} + x^{2} + 3 \, {\left (x^{5} + x^{2}\right )}^{\frac {1}{3}} x + x + 3 \, {\left (x^{5} + x^{2}\right )}^{\frac {2}{3}}}{x^{4} + x^{2} + x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(3)*arctan(1/3*(2*sqrt(3)*(x^5 + x^2)^(1/3)*x + sqrt(3)*(x^4 + x^2 + x) + 2*sqrt(3)*(x^5 + x^2)^(2/3))/(x
^4 - x^2 + x)) - 1/2*log((x^4 + x^2 + 3*(x^5 + x^2)^(1/3)*x + x + 3*(x^5 + x^2)^(2/3))/(x^4 + x^2 + x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 3.42, size = 368, normalized size = 4.33

method	result	size
trager	\(-\ln \left (\frac {-7 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{4}+22 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{4}+14 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}-16 x^{4}+21 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{5}+x^{2}\right )^{\frac {2}{3}}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{5}+x^{2}\right )^{\frac {1}{3}} x -7 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x -23 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-24 \left (x^{5}+x^{2}\right )^{\frac {2}{3}}-21 x \left (x^{5}+x^{2}\right )^{\frac {1}{3}}+22 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +8 x^{2}-16 x}{x \left (x^{3}+x +1\right )}\right )+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\frac {99 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{4}+257 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{4}-198 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{2}+158 x^{4}+474 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{5}+x^{2}\right )^{\frac {2}{3}}-297 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{5}+x^{2}\right )^{\frac {1}{3}} x +99 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x -217 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-771 \left (x^{5}+x^{2}\right )^{\frac {2}{3}}-474 x \left (x^{5}+x^{2}\right )^{\frac {1}{3}}+257 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x +158 x^{2}+158 x}{x \left (x^{3}+x +1\right )}\right )\)	\(368\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-ln((-7*RootOf(_Z^2-_Z+1)^2*x^4+22*RootOf(_Z^2-_Z+1)*x^4+14*RootOf(_Z^2-_Z+1)^2*x^2-16*x^4+21*RootOf(_Z^2-_Z+1
)*(x^5+x^2)^(2/3)-3*RootOf(_Z^2-_Z+1)*(x^5+x^2)^(1/3)*x-7*RootOf(_Z^2-_Z+1)^2*x-23*RootOf(_Z^2-_Z+1)*x^2-24*(x
^5+x^2)^(2/3)-21*x*(x^5+x^2)^(1/3)+22*RootOf(_Z^2-_Z+1)*x+8*x^2-16*x)/x/(x^3+x+1))+RootOf(_Z^2-_Z+1)*ln((99*Ro
otOf(_Z^2-_Z+1)^2*x^4+257*RootOf(_Z^2-_Z+1)*x^4-198*RootOf(_Z^2-_Z+1)^2*x^2+158*x^4+474*RootOf(_Z^2-_Z+1)*(x^5
+x^2)^(2/3)-297*RootOf(_Z^2-_Z+1)*(x^5+x^2)^(1/3)*x+99*RootOf(_Z^2-_Z+1)^2*x-217*RootOf(_Z^2-_Z+1)*x^2-771*(x^
5+x^2)^(2/3)-474*x*(x^5+x^2)^(1/3)+257*RootOf(_Z^2-_Z+1)*x+158*x^2+158*x)/x/(x^3+x+1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {2\,x^3-1}{{\left (x^5+x^2\right )}^{1/3}\,\left (x^3+x+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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