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

Optimal. Leaf size=49 \[ -\frac {1}{3} \text {RootSum}\left [\text {$\#$1}^{12}+3 \text {$\#$1}^4-1\& ,\frac {\log \left (\sqrt [4]{x^5-x^3}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ] \]

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Rubi [F]  time = 180.00, 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*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

$Aborted

Rubi steps

\begin {align*} \int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1+x^3-x^6\right )} \, dx &=\frac {\left (x^{3/4} \sqrt [4]{-1+x^2}\right ) \int \frac {1+x^6}{x^{3/4} \sqrt [4]{-1+x^2} \left (1+x^3-x^6\right )} \, dx}{\sqrt [4]{-x^3+x^5}}\\ &=\frac {\left (4 x^{3/4} \sqrt [4]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1+x^{24}}{\sqrt [4]{-1+x^8} \left (1+x^{12}-x^{24}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x^3+x^5}}\\ &=\frac {\left (4 x^{3/4} \sqrt [4]{-1+x^2}\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{\sqrt [4]{-1+x^8}}+\frac {2+x^{12}}{\sqrt [4]{-1+x^8} \left (1+x^{12}-x^{24}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x^3+x^5}}\\ &=-\frac {\left (4 x^{3/4} \sqrt [4]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1+x^8}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x^3+x^5}}+\frac {\left (4 x^{3/4} \sqrt [4]{-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {2+x^{12}}{\sqrt [4]{-1+x^8} \left (1+x^{12}-x^{24}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x^3+x^5}}\\ &=-\frac {\left (4 x^{3/4} \sqrt [4]{1-x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1-x^8}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x^3+x^5}}+\frac {\left (4 x^{3/4} \sqrt [4]{-1+x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1-\sqrt {5}}{\sqrt [4]{-1+x^8} \left (1-\sqrt {5}-2 x^{12}\right )}+\frac {1+\sqrt {5}}{\sqrt [4]{-1+x^8} \left (1+\sqrt {5}-2 x^{12}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x^3+x^5}} \end {align*}

rest of steps removed due to Latex formating problem.

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Mathematica [F]  time = 0.81, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x^6}{\sqrt [4]{-x^3+x^5} \left (1+x^3-x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.00, size = 49, normalized size = 1.00 \begin {gather*} -\frac {1}{3} \text {RootSum}\left [-1+3 \text {$\#$1}^4+\text {$\#$1}^{12}\&,\frac {-\log (x)+\log \left (\sqrt [4]{-x^3+x^5}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*RootSum[-1 + 3*#1^4 + #1^12 & , (-Log[x] + Log[(-x^3 + x^5)^(1/4) - x*#1])/#1 & ]

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F(-1)]  time = 180.00, size = 0, normalized size = 0.00 hanged

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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