3.27.99 \(\int \frac {x^6 (4+x^5)}{(-1+x^5)^{3/4} (1-2 x^5+x^8+x^{10})} \, dx\)

Optimal. Leaf size=245 \[ -\frac {1}{2} \sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{x^5-1}}{\sqrt {x^5-1}-x^2}\right )+\frac {1}{2} \sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\left (\sqrt {\frac {2}{2-\sqrt {2}}}-\frac {2}{\sqrt {2-\sqrt {2}}}\right ) x \sqrt [4]{x^5-1}}{\sqrt {x^5-1}-x^2}\right )+\frac {1}{2} \sqrt {2-\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{x^5-1}}{\sqrt {x^5-1}+x^2}\right )+\frac {1}{2} \sqrt {2+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{x^5-1}}{\sqrt {x^5-1}+x^2}\right ) \]

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Rubi [F]  time = 1.14, 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 {x^6 \left (4+x^5\right )}{\left (-1+x^5\right )^{3/4} \left (1-2 x^5+x^8+x^{10}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^6*(4 + x^5))/((-1 + x^5)^(3/4)*(1 - 2*x^5 + x^8 + x^10)),x]

[Out]

(x^2*(1 - x^5)^(3/4)*Hypergeometric2F1[2/5, 3/4, 7/5, x^5])/(2*(-1 + x^5)^(3/4)) - Defer[Int][x/((-1 + x^5)^(3
/4)*(1 - 2*x^5 + x^8 + x^10)), x] + 6*Defer[Int][x^6/((-1 + x^5)^(3/4)*(1 - 2*x^5 + x^8 + x^10)), x] - Defer[I
nt][x^9/((-1 + x^5)^(3/4)*(1 - 2*x^5 + x^8 + x^10)), x]

Rubi steps

\begin {align*} \int \frac {x^6 \left (4+x^5\right )}{\left (-1+x^5\right )^{3/4} \left (1-2 x^5+x^8+x^{10}\right )} \, dx &=\int \left (\frac {x}{\left (-1+x^5\right )^{3/4}}+\frac {x \left (-1+6 x^5-x^8\right )}{\left (-1+x^5\right )^{3/4} \left (1-2 x^5+x^8+x^{10}\right )}\right ) \, dx\\ &=\int \frac {x}{\left (-1+x^5\right )^{3/4}} \, dx+\int \frac {x \left (-1+6 x^5-x^8\right )}{\left (-1+x^5\right )^{3/4} \left (1-2 x^5+x^8+x^{10}\right )} \, dx\\ &=\frac {\left (1-x^5\right )^{3/4} \int \frac {x}{\left (1-x^5\right )^{3/4}} \, dx}{\left (-1+x^5\right )^{3/4}}+\int \left (-\frac {x}{\left (-1+x^5\right )^{3/4} \left (1-2 x^5+x^8+x^{10}\right )}+\frac {6 x^6}{\left (-1+x^5\right )^{3/4} \left (1-2 x^5+x^8+x^{10}\right )}-\frac {x^9}{\left (-1+x^5\right )^{3/4} \left (1-2 x^5+x^8+x^{10}\right )}\right ) \, dx\\ &=\frac {x^2 \left (1-x^5\right )^{3/4} \, _2F_1\left (\frac {2}{5},\frac {3}{4};\frac {7}{5};x^5\right )}{2 \left (-1+x^5\right )^{3/4}}+6 \int \frac {x^6}{\left (-1+x^5\right )^{3/4} \left (1-2 x^5+x^8+x^{10}\right )} \, dx-\int \frac {x}{\left (-1+x^5\right )^{3/4} \left (1-2 x^5+x^8+x^{10}\right )} \, dx-\int \frac {x^9}{\left (-1+x^5\right )^{3/4} \left (1-2 x^5+x^8+x^{10}\right )} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.28, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^6 \left (4+x^5\right )}{\left (-1+x^5\right )^{3/4} \left (1-2 x^5+x^8+x^{10}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(x^6*(4 + x^5))/((-1 + x^5)^(3/4)*(1 - 2*x^5 + x^8 + x^10)),x]

[Out]

Integrate[(x^6*(4 + x^5))/((-1 + x^5)^(3/4)*(1 - 2*x^5 + x^8 + x^10)), x]

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IntegrateAlgebraic [A]  time = 13.32, size = 225, normalized size = 0.92 \begin {gather*} -\frac {1}{2} \sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{-1+x^5}}{-x^2+\sqrt {-1+x^5}}\right )-\frac {1}{2} \sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^5}}{-x^2+\sqrt {-1+x^5}}\right )+\frac {1}{2} \sqrt {2-\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} x \sqrt [4]{-1+x^5}}{x^2+\sqrt {-1+x^5}}\right )+\frac {1}{2} \sqrt {2+\sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} x \sqrt [4]{-1+x^5}}{x^2+\sqrt {-1+x^5}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^6*(4 + x^5))/((-1 + x^5)^(3/4)*(1 - 2*x^5 + x^8 + x^10)),x]

[Out]

-1/2*(Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[2 - Sqrt[2]]*x*(-1 + x^5)^(1/4))/(-x^2 + Sqrt[-1 + x^5])]) - (Sqrt[2 + Sq
rt[2]]*ArcTan[(Sqrt[2 + Sqrt[2]]*x*(-1 + x^5)^(1/4))/(-x^2 + Sqrt[-1 + x^5])])/2 + (Sqrt[2 - Sqrt[2]]*ArcTanh[
(Sqrt[2 - Sqrt[2]]*x*(-1 + x^5)^(1/4))/(x^2 + Sqrt[-1 + x^5])])/2 + (Sqrt[2 + Sqrt[2]]*ArcTanh[(Sqrt[2 + Sqrt[
2]]*x*(-1 + x^5)^(1/4))/(x^2 + Sqrt[-1 + x^5])])/2

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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*(x^5+4)/(x^5-1)^(3/4)/(x^10+x^8-2*x^5+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 {{\left (x^{5} + 4\right )} x^{6}}{{\left (x^{10} + x^{8} - 2 \, x^{5} + 1\right )} {\left (x^{5} - 1\right )}^{\frac {3}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(x^5+4)/(x^5-1)^(3/4)/(x^10+x^8-2*x^5+1),x, algorithm="giac")

[Out]

integrate((x^5 + 4)*x^6/((x^10 + x^8 - 2*x^5 + 1)*(x^5 - 1)^(3/4)), x)

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maple [C]  time = 12.58, size = 462, normalized size = 1.89

method	result	size
trager	\(\frac {\RootOf \left (\textit {\_Z}^{8}+1\right )^{3} \ln \left (\frac {\RootOf \left (\textit {\_Z}^{8}+1\right )^{9} x^{4}+2 \left (x^{5}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{8}+1\right )^{6} x^{3}-\RootOf \left (\textit {\_Z}^{8}+1\right )^{5} x^{5}+2 \sqrt {x^{5}-1}\, \RootOf \left (\textit {\_Z}^{8}+1\right )^{3} x^{2}+\RootOf \left (\textit {\_Z}^{8}+1\right )^{5}+2 \left (x^{5}-1\right )^{\frac {3}{4}} x}{\RootOf \left (\textit {\_Z}^{8}+1\right )^{4} x^{4}+x^{5}-1}\right )}{2}-\frac {\RootOf \left (\textit {\_Z}^{8}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{8}+1\right )^{11} x^{4}+\RootOf \left (\textit {\_Z}^{8}+1\right )^{7} x^{5}-\RootOf \left (\textit {\_Z}^{8}+1\right )^{7}+2 \left (x^{5}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{8}+1\right )^{2} x^{3}-2 \sqrt {x^{5}-1}\, \RootOf \left (\textit {\_Z}^{8}+1\right ) x^{2}+2 \left (x^{5}-1\right )^{\frac {3}{4}} x}{\RootOf \left (\textit {\_Z}^{8}+1\right )^{4} x^{4}-x^{5}+1}\right )}{2}-\frac {\RootOf \left (\textit {\_Z}^{8}+1\right )^{5} \ln \left (\frac {\RootOf \left (\textit {\_Z}^{8}+1\right )^{7} x^{4}-2 \sqrt {x^{5}-1}\, \RootOf \left (\textit {\_Z}^{8}+1\right )^{5} x^{2}+\RootOf \left (\textit {\_Z}^{8}+1\right )^{3} x^{5}-2 \left (x^{5}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{8}+1\right )^{2} x^{3}+2 \left (x^{5}-1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{8}+1\right )^{3}}{\RootOf \left (\textit {\_Z}^{8}+1\right )^{4} x^{4}-x^{5}+1}\right )}{2}-\frac {\RootOf \left (\textit {\_Z}^{8}+1\right )^{7} \ln \left (\frac {-2 \sqrt {x^{5}-1}\, \RootOf \left (\textit {\_Z}^{8}+1\right )^{7} x^{2}-2 \left (x^{5}-1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{8}+1\right )^{6} x^{3}-\RootOf \left (\textit {\_Z}^{8}+1\right )^{5} x^{4}+\RootOf \left (\textit {\_Z}^{8}+1\right ) x^{5}+2 \left (x^{5}-1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{8}+1\right )}{\RootOf \left (\textit {\_Z}^{8}+1\right )^{4} x^{4}+x^{5}-1}\right )}{2}\)	\(462\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(x^5+4)/(x^5-1)^(3/4)/(x^10+x^8-2*x^5+1),x,method=_RETURNVERBOSE)

[Out]

1/2*RootOf(_Z^8+1)^3*ln((RootOf(_Z^8+1)^9*x^4+2*(x^5-1)^(1/4)*RootOf(_Z^8+1)^6*x^3-RootOf(_Z^8+1)^5*x^5+2*(x^5
-1)^(1/2)*RootOf(_Z^8+1)^3*x^2+RootOf(_Z^8+1)^5+2*(x^5-1)^(3/4)*x)/(RootOf(_Z^8+1)^4*x^4+x^5-1))-1/2*RootOf(_Z
^8+1)*ln((RootOf(_Z^8+1)^11*x^4+RootOf(_Z^8+1)^7*x^5-RootOf(_Z^8+1)^7+2*(x^5-1)^(1/4)*RootOf(_Z^8+1)^2*x^3-2*(
x^5-1)^(1/2)*RootOf(_Z^8+1)*x^2+2*(x^5-1)^(3/4)*x)/(RootOf(_Z^8+1)^4*x^4-x^5+1))-1/2*RootOf(_Z^8+1)^5*ln((Root
Of(_Z^8+1)^7*x^4-2*(x^5-1)^(1/2)*RootOf(_Z^8+1)^5*x^2+RootOf(_Z^8+1)^3*x^5-2*(x^5-1)^(1/4)*RootOf(_Z^8+1)^2*x^
3+2*(x^5-1)^(3/4)*x-RootOf(_Z^8+1)^3)/(RootOf(_Z^8+1)^4*x^4-x^5+1))-1/2*RootOf(_Z^8+1)^7*ln((-2*(x^5-1)^(1/2)*
RootOf(_Z^8+1)^7*x^2-2*(x^5-1)^(1/4)*RootOf(_Z^8+1)^6*x^3-RootOf(_Z^8+1)^5*x^4+RootOf(_Z^8+1)*x^5+2*(x^5-1)^(3
/4)*x-RootOf(_Z^8+1))/(RootOf(_Z^8+1)^4*x^4+x^5-1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{5} + 4\right )} x^{6}}{{\left (x^{10} + x^{8} - 2 \, x^{5} + 1\right )} {\left (x^{5} - 1\right )}^{\frac {3}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(x^5+4)/(x^5-1)^(3/4)/(x^10+x^8-2*x^5+1),x, algorithm="maxima")

[Out]

integrate((x^5 + 4)*x^6/((x^10 + x^8 - 2*x^5 + 1)*(x^5 - 1)^(3/4)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^6\,\left (x^5+4\right )}{{\left (x^5-1\right )}^{3/4}\,\left (x^{10}+x^8-2\,x^5+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6*(x^5 + 4))/((x^5 - 1)^(3/4)*(x^8 - 2*x^5 + x^10 + 1)),x)

[Out]

int((x^6*(x^5 + 4))/((x^5 - 1)^(3/4)*(x^8 - 2*x^5 + x^10 + 1)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{6} \left (x^{5} + 4\right )}{\left (\left (x - 1\right ) \left (x^{4} + x^{3} + x^{2} + x + 1\right )\right )^{\frac {3}{4}} \left (x^{10} + x^{8} - 2 x^{5} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(x**5+4)/(x**5-1)**(3/4)/(x**10+x**8-2*x**5+1),x)

[Out]

Integral(x**6*(x**5 + 4)/(((x - 1)*(x**4 + x**3 + x**2 + x + 1))**(3/4)*(x**10 + x**8 - 2*x**5 + 1)), x)

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