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

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

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Rubi [F]  time = 1.12, 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 ) \sqrt [3]{-2+x^3+x^5}}{x^2 \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)^(1/3))/(x^2*(-2 + x^5)),x]

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.55, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (3+x^5\right ) \sqrt [3]{-2+x^3+x^5}}{x^2 \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)^(1/3))/(x^2*(-2 + x^5)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 2.33, size = 111, normalized size = 1.00 \begin {gather*} \frac {3 \sqrt [3]{-2+x^3+x^5}}{2 x}+\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)^(1/3))/(x^2*(-2 + x^5)),x]

[Out]

(3*(-2 + x^3 + x^5)^(1/3))/(2*x) + (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 = 5.69, size = 136, normalized size = 1.23 \begin {gather*} \frac {2 \, \sqrt {3} x \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 \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 ) + 6 \, {\left (x^{5} + x^{3} - 2\right )}^{\frac {1}{3}}}{4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*sqrt(3)*x*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*log((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)) + 6*(x^5 + x^3 - 2)^(1/3))/x

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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 {1}{3}} {\left (x^{5} + 3\right )}}{{\left (x^{5} - 2\right )} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 9.85, size = 423, normalized size = 3.81

method	result	size
trager	\(\frac {3 \left (x^{5}+x^{3}-2\right )^{\frac {1}{3}}}{2 x}+\frac {\ln \left (\frac {-3687964213260096 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{5}-1951741557610164 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{5}+13829865799725360 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{3}-88450351904093 x^{5}+2398485078249264 \RootOf \left (144 \textit {\_Z}^{2}+12 \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 (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{2}-1382259724622424 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{3}-11355288584745 \left (x^{5}+x^{3}-2\right )^{\frac {2}{3}} x [phone] \left (x^{5}+x^{3}-2\right )^{\frac {1}{3}} x^{2}-103833021800457 x^{3}+7375928426520192 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}+3903483115220328 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )[phone]}{x^{5}-2}\right )}{2}+6 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \ln \left (-\frac {-1472859748183680 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{5}[phone] \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{5}+5523224055688800 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{3}-73067682007729 x^{5}+2398485078249264 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{5}+x^{3}-2\right )^{\frac {2}{3}} x -2534748541266204 \left (x^{5}+x^{3}-2\right )^{\frac {1}{3}} \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{2}[phone] \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{3}[phone] \left (x^{5}+x^{3}-2\right )^{\frac {2}{3}} x -11355288584745 \left (x^{5}+x^{3}-2\right )^{\frac {1}{3}} x^{2}-161518033911822 x^{3}+2945719496367360 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}-1535198044824816 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )[phone]}{x^{5}-2}\right )\)	\(423\)
risch	\(\frac {3 \left (x^{5}+x^{3}-2\right )^{\frac {1}{3}}}{2 x}+\frac {\left (\frac {\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{10}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{8}-x^{10}-3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{8}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}-3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{10}+2 x^{8}+x^{6}-4 x^{5}-4 x^{3}+4\right )^{\frac {1}{3}} x^{6}-2 x^{8}-x^{6} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-3 \left (x^{10}+2 x^{8}+x^{6}-4 x^{5}-4 x^{3}+4\right )^{\frac {1}{3}} x^{6}-3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{10}+2 x^{8}+x^{6}-4 x^{5}-4 x^{3}+4\right )^{\frac {1}{3}} x^{4}+8 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{5}-x^{6}-4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{10}+2 x^{8}+x^{6}-4 x^{5}-4 x^{3}+4\right )^{\frac {2}{3}} x^{2}-3 \left (x^{10}+2 x^{8}+x^{6}-4 x^{5}-4 x^{3}+4\right )^{\frac {1}{3}} x^{4}+4 x^{5}+6 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-3 \left (x^{10}+2 x^{8}+x^{6}-4 x^{5}-4 x^{3}+4\right )^{\frac {2}{3}} x^{2}+6 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{10}+2 x^{8}+x^{6}-4 x^{5}-4 x^{3}+4\right )^{\frac {1}{3}} x +4 x^{3}+6 \left (x^{10}+2 x^{8}+x^{6}-4 x^{5}-4 x^{3}+4\right )^{\frac {1}{3}} x -8 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-4}{\left (-1+x \right ) \left (x^{5}-2\right ) \left (x^{4}+x^{3}+2 x^{2}+2 x +2\right )}\right )}{2}-\frac {\ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{10}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{8}+x^{10}+7 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{8}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{10}+2 x^{8}+x^{6}-4 x^{5}-4 x^{3}+4\right )^{\frac {1}{3}} x^{6}+3 x^{8}+5 x^{6} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{10}+2 x^{8}+x^{6}-4 x^{5}-4 x^{3}+4\right )^{\frac {1}{3}} x^{4}-8 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{5}+2 x^{6}-4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{10}+2 x^{8}+x^{6}-4 x^{5}-4 x^{3}+4\right )^{\frac {2}{3}} x^{2}-4 x^{5}-14 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-6 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{10}+2 x^{8}+x^{6}-4 x^{5}-4 x^{3}+4\right )^{\frac {1}{3}} x -6 x^{3}+8 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+4}{\left (-1+x \right ) \left (x^{5}-2\right ) \left (x^{4}+x^{3}+2 x^{2}+2 x +2\right )}\right ) \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{2}-\frac {\ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{10}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{8}+x^{10}+7 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{8}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{10}+2 x^{8}+x^{6}-4 x^{5}-4 x^{3}+4\right )^{\frac {1}{3}} x^{6}+3 x^{8}+5 x^{6} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{10}+2 x^{8}+x^{6}-4 x^{5}-4 x^{3}+4\right )^{\frac {1}{3}} x^{4}-8 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{5}+2 x^{6}-4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{10}+2 x^{8}+x^{6}-4 x^{5}-4 x^{3}+4\right )^{\frac {2}{3}} x^{2}-4 x^{5}-14 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-6 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{10}+2 x^{8}+x^{6}-4 x^{5}-4 x^{3}+4\right )^{\frac {1}{3}} x -6 x^{3}+8 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+4}{\left (-1+x \right ) \left (x^{5}-2\right ) \left (x^{4}+x^{3}+2 x^{2}+2 x +2\right )}\right )}{2}\right ) \left (\left (x^{5}+x^{3}-2\right )^{2}\right )^{\frac {1}{3}}}{\left (x^{5}+x^{3}-2\right )^{\frac {2}{3}}}\)	\(1123\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*(x^5+x^3-2)^(1/3)/x+1/2*ln((-3687964213260096*RootOf(144*_Z^2+12*_Z+1)^2*x^5-1951741557610164*RootOf(144*_
Z^2+12*_Z+1)*x^5+13829865799725360*RootOf(144*_Z^2+12*_Z+1)^2*x^3-88450351904093*x^5+2398485078249264*RootOf(1
44*_Z^2+12*_Z+1)*(x^5+x^3-2)^(2/3)*x+136263463016940*(x^5+x^3-2)^(1/3)*RootOf(144*_Z^2+12*_Z+1)*x^2-1382259724
622424*RootOf(144*_Z^2+12*_Z+1)*x^3-11355288584745*(x^5+x^3-2)^(2/3)*x+211229045105517*(x^5+x^3-2)^(1/3)*x^2-1
03833021800457*x^3+7375928426520192*RootOf(144*_Z^2+12*_Z+1)^2+3903483115220328*RootOf(144*_Z^2+12*_Z+1)+17690
0703808186)/(x^5-2))+6*RootOf(144*_Z^2+12*_Z+1)*ln(-(-1472859748183680*RootOf(144*_Z^2+12*_Z+1)^2*x^5+76759902
2412408*RootOf(144*_Z^2+12*_Z+1)*x^5+5523224055688800*RootOf(144*_Z^2+12*_Z+1)^2*x^3-73067682007729*x^5+239848
5078249264*RootOf(144*_Z^2+12*_Z+1)*(x^5+x^3-2)^(2/3)*x-2534748541266204*(x^5+x^3-2)^(1/3)*RootOf(144*_Z^2+12*
_Z+1)*x^2+596532134324340*RootOf(144*_Z^2+12*_Z+1)*x^3+211229045105517*(x^5+x^3-2)^(2/3)*x-11355288584745*(x^5
+x^3-2)^(1/3)*x^2-161518033911822*x^3+2945719496367360*RootOf(144*_Z^2+12*_Z+1)^2-1535198044824816*RootOf(144*
_Z^2+12*_Z+1)[phone])/(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 {1}{3}} {\left (x^{5} + 3\right )}}{{\left (x^{5} - 2\right )} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

Timed out

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