3.17.44 \(\int \frac {(-3+x^5) (2+x^3+x^5)^{2/3}}{x^3 (2+x^5)} \, dx\)

Optimal. Leaf size=111 \[ \frac {1}{2} \log \left (\sqrt [3]{x^5+x^3+2}-x\right )-\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^5+x^3+2}+x}\right )+\frac {3 \left (x^5+x^3+2\right )^{2/3}}{4 x^2}-\frac {1}{4} \log \left (x^2+\sqrt [3]{x^5+x^3+2} x+\left (x^5+x^3+2\right )^{2/3}\right ) \]

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Rubi [F]  time = 1.18, 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 (-3+x^5\right ) \left (2+x^3+x^5\right )^{2/3}}{x^3 \left (2+x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-3*Defer[Int][(2 + x^3 + x^5)^(2/3)/x^3, x])/2 + Defer[Int][(2 + x^3 + x^5)^(2/3)/(2^(1/5) + x), x]/(2*2^(2/5
)) - ((-1)^(3/5)*Defer[Int][(2 + x^3 + x^5)^(2/3)/(2^(1/5) - (-1)^(1/5)*x), x])/(2*2^(2/5)) - ((-1)^(1/5)*Defe
r[Int][(2 + x^3 + x^5)^(2/3)/(2^(1/5) + (-1)^(2/5)*x), x])/(2*2^(2/5)) + ((-1)^(4/5)*Defer[Int][(2 + x^3 + x^5
)^(2/3)/(2^(1/5) - (-1)^(3/5)*x), x])/(2*2^(2/5)) + ((-1/2)^(2/5)*Defer[Int][(2 + x^3 + x^5)^(2/3)/(2^(1/5) +
(-1)^(4/5)*x), x])/2

Rubi steps

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

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Mathematica [F]  time = 0.53, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-3+x^5\right ) \left (2+x^3+x^5\right )^{2/3}}{x^3 \left (2+x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 2.67, size = 111, normalized size = 1.00 \begin {gather*} \frac {3 \left (2+x^3+x^5\right )^{2/3}}{4 x^2}-\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2+x^3+x^5}}\right )+\frac {1}{2} \log \left (-x+\sqrt [3]{2+x^3+x^5}\right )-\frac {1}{4} \log \left (x^2+x \sqrt [3]{2+x^3+x^5}+\left (2+x^3+x^5\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(2 + x^3 + x^5)^(2/3))/(4*x^2) - (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(2 + x^3 + x^5)^(1/3))])/2 + Log[-x + (
2 + x^3 + x^5)^(1/3)]/2 - Log[x^2 + x*(2 + x^3 + x^5)^(1/3) + (2 + x^3 + x^5)^(2/3)]/4

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fricas [A]  time = 7.72, size = 141, normalized size = 1.27 \begin {gather*} -\frac {2 \, \sqrt {3} x^{2} \arctan \left (-\frac {240779826 \, \sqrt {3} {\left (x^{5} + x^{3} + 2\right )}^{\frac {1}{3}} x^{2} - 64389332 \, \sqrt {3} {\left (x^{5} + x^{3} + 2\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (18550880 \, x^{5} + 88195247 \, x^{3} + 37101760\right )}}{3 \, {\left (2863288 \, x^{5} + 152584579 \, x^{3} + 5726576\right )}}\right ) - x^{2} \log \left (\frac {x^{5} + 3 \, {\left (x^{5} + x^{3} + 2\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{5} + x^{3} + 2\right )}^{\frac {2}{3}} x + 2}{x^{5} + 2}\right ) - 3 \, {\left (x^{5} + x^{3} + 2\right )}^{\frac {2}{3}}}{4 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*sqrt(3)*x^2*arctan(-1/3*(240779826*sqrt(3)*(x^5 + x^3 + 2)^(1/3)*x^2 - 64389332*sqrt(3)*(x^5 + x^3 + 2
)^(2/3)*x + sqrt(3)*(18550880*x^5 + 88195247*x^3 + 37101760))/(2863288*x^5 + 152584579*x^3 + 5726576)) - x^2*l
og((x^5 + 3*(x^5 + x^3 + 2)^(1/3)*x^2 - 3*(x^5 + x^3 + 2)^(2/3)*x + 2)/(x^5 + 2)) - 3*(x^5 + x^3 + 2)^(2/3))/x
^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 10.09, size = 381, normalized size = 3.43

method	result	size
risch	\(\frac {3 \left (x^{5}+x^{3}+2\right )^{\frac {2}{3}}}{4 x^{2}}-\frac {\ln \left (-\frac {2 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{5}-4 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{3}+2 x^{5}-2 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{3}+3 \left (x^{5}+x^{3}+2\right )^{\frac {2}{3}} x +3 \left (x^{5}+x^{3}+2\right )^{\frac {1}{3}} x^{2}+2 x^{3}+4 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+4}{x^{5}+2}\right )}{2}-\ln \left (-\frac {2 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{5}-4 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{3}+2 x^{5}-2 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{3}+3 \left (x^{5}+x^{3}+2\right )^{\frac {2}{3}} x +3 \left (x^{5}+x^{3}+2\right )^{\frac {1}{3}} x^{2}+2 x^{3}+4 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+4}{x^{5}+2}\right ) \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+\RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \ln \left (-\frac {-2 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{5}-4 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{3}+x^{5}-2 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{3}+3 \left (x^{5}+x^{3}+2\right )^{\frac {2}{3}} x +3 \left (x^{5}+x^{3}+2\right )^{\frac {1}{3}} x^{2}+2 x^{3}-4 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+2}{x^{5}+2}\right )\)	\(381\)
trager	\(\frac {3 \left (x^{5}+x^{3}+2\right )^{\frac {2}{3}}}{4 x^{2}}+\frac {\ln \left (-\frac {-1221827760 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{5}-12434247822 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{5}[phone] \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{3}-9513381685 x^{5}[phone] \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \left (x^{5}+x^{3}+2\right )^{\frac {2}{3}} x -101522694438 \left (x^{5}+x^{3}+2\right )^{\frac {1}{3}} \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{2}[phone] \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{3}[phone] \left (x^{5}+x^{3}+2\right )^{\frac {2}{3}} x -5402572071 \left (x^{5}+x^{3}+2\right )^{\frac {1}{3}} x^{2}-11416058022 x^{3}-2443655520 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2}-24868495644 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )-19026763370}{x^{5}+2}\right )}{2}-\frac {\ln \left (\frac {67274520372 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{5}-33433622226 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{5}-201823561116 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{3}-7610705348 x^{5}[phone] \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \left (x^{5}+x^{3}+2\right )^{\frac {2}{3}} x [phone] \left (x^{5}+x^{3}+2\right )^{\frac {1}{3}} \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{2}-135159954624 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{3}-5402572071 \left (x^{5}+x^{3}+2\right )^{\frac {2}{3}} x [phone] \left (x^{5}+x^{3}+2\right )^{\frac {1}{3}} x^{2}-17124087033 x^{3}[phone] \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2}-66867244452 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )-15221410696}{x^{5}+2}\right )}{2}-3 \ln \left (\frac {67274520372 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{5}-33433622226 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{5}-201823561116 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{3}-7610705348 x^{5}[phone] \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \left (x^{5}+x^{3}+2\right )^{\frac {2}{3}} x [phone] \left (x^{5}+x^{3}+2\right )^{\frac {1}{3}} \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{2}-135159954624 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{3}-5402572071 \left (x^{5}+x^{3}+2\right )^{\frac {2}{3}} x [phone] \left (x^{5}+x^{3}+2\right )^{\frac {1}{3}} x^{2}-17124087033 x^{3}[phone] \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2}-66867244452 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )-15221410696}{x^{5}+2}\right ) \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )\)	\(620\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*(x^5+x^3+2)^(2/3)/x^2-1/2*ln(-(2*RootOf(4*_Z^2+2*_Z+1)*x^5-4*RootOf(4*_Z^2+2*_Z+1)^2*x^3+2*x^5-2*RootOf(4*
_Z^2+2*_Z+1)*x^3+3*(x^5+x^3+2)^(2/3)*x+3*(x^5+x^3+2)^(1/3)*x^2+2*x^3+4*RootOf(4*_Z^2+2*_Z+1)+4)/(x^5+2))-ln(-(
2*RootOf(4*_Z^2+2*_Z+1)*x^5-4*RootOf(4*_Z^2+2*_Z+1)^2*x^3+2*x^5-2*RootOf(4*_Z^2+2*_Z+1)*x^3+3*(x^5+x^3+2)^(2/3
)*x+3*(x^5+x^3+2)^(1/3)*x^2+2*x^3+4*RootOf(4*_Z^2+2*_Z+1)+4)/(x^5+2))*RootOf(4*_Z^2+2*_Z+1)+RootOf(4*_Z^2+2*_Z
+1)*ln(-(-2*RootOf(4*_Z^2+2*_Z+1)*x^5-4*RootOf(4*_Z^2+2*_Z+1)^2*x^3+x^5-2*RootOf(4*_Z^2+2*_Z+1)*x^3+3*(x^5+x^3
+2)^(2/3)*x+3*(x^5+x^3+2)^(1/3)*x^2+2*x^3-4*RootOf(4*_Z^2+2*_Z+1)+2)/(x^5+2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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