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

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 = 2.18, size = 111, normalized size = 1.00 \begin {gather*} \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 ) \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 = 5.32, 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 = 1.99, size = 1119, normalized size = 10.08

result too large to display

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)

[Out]

3/2*(x^5+x^3-2)^(1/3)/x+(1/2*RootOf(_Z^2+_Z+1)*ln(-(2*x^10*RootOf(_Z^2+_Z+1)-2*RootOf(_Z^2+_Z+1)^2*x^8+x^10+3*
RootOf(_Z^2+_Z+1)*x^8+3*RootOf(_Z^2+_Z+1)*(x^10+2*x^8+x^6-4*x^5-4*x^3+4)^(1/3)*x^6-2*RootOf(_Z^2+_Z+1)^2*x^6+2
*x^8+3*(x^10+2*x^8+x^6-4*x^5-4*x^3+4)^(1/3)*x^6+RootOf(_Z^2+_Z+1)*x^6+3*RootOf(_Z^2+_Z+1)*(x^10+2*x^8+x^6-4*x^
5-4*x^3+4)^(1/3)*x^4-8*RootOf(_Z^2+_Z+1)*x^5+x^6+3*RootOf(_Z^2+_Z+1)*(x^10+2*x^8+x^6-4*x^5-4*x^3+4)^(2/3)*x^2+
3*(x^10+2*x^8+x^6-4*x^5-4*x^3+4)^(1/3)*x^4+4*RootOf(_Z^2+_Z+1)^2*x^3-4*x^5+3*(x^10+2*x^8+x^6-4*x^5-4*x^3+4)^(2
/3)*x^2-6*RootOf(_Z^2+_Z+1)*x^3-6*RootOf(_Z^2+_Z+1)*(x^10+2*x^8+x^6-4*x^5-4*x^3+4)^(1/3)*x-4*x^3-6*(x^10+2*x^8
+x^6-4*x^5-4*x^3+4)^(1/3)*x+8*RootOf(_Z^2+_Z+1)+4)/(x^5-2)/(x^4+x^3+2*x^2+2*x+2)/(-1+x))-1/2*ln((2*x^10*RootOf
(_Z^2+_Z+1)+2*RootOf(_Z^2+_Z+1)^2*x^8+x^10+7*RootOf(_Z^2+_Z+1)*x^8+3*RootOf(_Z^2+_Z+1)*(x^10+2*x^8+x^6-4*x^5-4
*x^3+4)^(1/3)*x^6+2*RootOf(_Z^2+_Z+1)^2*x^6+3*x^8+5*RootOf(_Z^2+_Z+1)*x^6+3*RootOf(_Z^2+_Z+1)*(x^10+2*x^8+x^6-
4*x^5-4*x^3+4)^(1/3)*x^4-8*RootOf(_Z^2+_Z+1)*x^5+2*x^6+3*RootOf(_Z^2+_Z+1)*(x^10+2*x^8+x^6-4*x^5-4*x^3+4)^(2/3
)*x^2-4*RootOf(_Z^2+_Z+1)^2*x^3-4*x^5-14*RootOf(_Z^2+_Z+1)*x^3-6*RootOf(_Z^2+_Z+1)*(x^10+2*x^8+x^6-4*x^5-4*x^3
+4)^(1/3)*x-6*x^3+8*RootOf(_Z^2+_Z+1)+4)/(x^5-2)/(x^4+x^3+2*x^2+2*x+2)/(-1+x))*RootOf(_Z^2+_Z+1)-1/2*ln((2*x^1
0*RootOf(_Z^2+_Z+1)+2*RootOf(_Z^2+_Z+1)^2*x^8+x^10+7*RootOf(_Z^2+_Z+1)*x^8+3*RootOf(_Z^2+_Z+1)*(x^10+2*x^8+x^6
-4*x^5-4*x^3+4)^(1/3)*x^6+2*RootOf(_Z^2+_Z+1)^2*x^6+3*x^8+5*RootOf(_Z^2+_Z+1)*x^6+3*RootOf(_Z^2+_Z+1)*(x^10+2*
x^8+x^6-4*x^5-4*x^3+4)^(1/3)*x^4-8*RootOf(_Z^2+_Z+1)*x^5+2*x^6+3*RootOf(_Z^2+_Z+1)*(x^10+2*x^8+x^6-4*x^5-4*x^3
+4)^(2/3)*x^2-4*RootOf(_Z^2+_Z+1)^2*x^3-4*x^5-14*RootOf(_Z^2+_Z+1)*x^3-6*RootOf(_Z^2+_Z+1)*(x^10+2*x^8+x^6-4*x
^5-4*x^3+4)^(1/3)*x-6*x^3+8*RootOf(_Z^2+_Z+1)+4)/(x^5-2)/(x^4+x^3+2*x^2+2*x+2)/(-1+x)))/(x^5+x^3-2)^(2/3)*((x^
5+x^3-2)^2)^(1/3)

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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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