3.17.5 \(\int \frac {\sqrt [3]{1+x^5} (-3+2 x^5)}{x^2 (2-x^3+2 x^5)} \, dx\)

Optimal. Leaf size=133 \[ \frac {3 \sqrt [3]{x^5+1}}{2 x}+\frac {\log \left (\sqrt [3]{2} \sqrt [3]{x^5+1}-x\right )}{2 \sqrt [3]{2}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{2} \sqrt [3]{x^5+1}+x}\right )}{2 \sqrt [3]{2}}-\frac {\log \left (\sqrt [3]{2} \sqrt [3]{x^5+1} x+2^{2/3} \left (x^5+1\right )^{2/3}+x^2\right )}{4 \sqrt [3]{2}} \]

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Rubi [F]  time = 0.85, 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 {\sqrt [3]{1+x^5} \left (-3+2 x^5\right )}{x^2 \left (2-x^3+2 x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(3*Hypergeometric2F1[-1/3, -1/5, 4/5, -x^5])/(2*x) - (3*Defer[Int][(x*(1 + x^5)^(1/3))/(2 - x^3 + 2*x^5), x])/
2 + 5*Defer[Int][(x^3*(1 + x^5)^(1/3))/(2 - x^3 + 2*x^5), x]

Rubi steps

\begin {align*} \int \frac {\sqrt [3]{1+x^5} \left (-3+2 x^5\right )}{x^2 \left (2-x^3+2 x^5\right )} \, dx &=\int \left (-\frac {3 \sqrt [3]{1+x^5}}{2 x^2}+\frac {x \left (-3+10 x^2\right ) \sqrt [3]{1+x^5}}{2 \left (2-x^3+2 x^5\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {x \left (-3+10 x^2\right ) \sqrt [3]{1+x^5}}{2-x^3+2 x^5} \, dx-\frac {3}{2} \int \frac {\sqrt [3]{1+x^5}}{x^2} \, dx\\ &=\frac {3 \, _2F_1\left (-\frac {1}{3},-\frac {1}{5};\frac {4}{5};-x^5\right )}{2 x}+\frac {1}{2} \int \left (-\frac {3 x \sqrt [3]{1+x^5}}{2-x^3+2 x^5}+\frac {10 x^3 \sqrt [3]{1+x^5}}{2-x^3+2 x^5}\right ) \, dx\\ &=\frac {3 \, _2F_1\left (-\frac {1}{3},-\frac {1}{5};\frac {4}{5};-x^5\right )}{2 x}-\frac {3}{2} \int \frac {x \sqrt [3]{1+x^5}}{2-x^3+2 x^5} \, dx+5 \int \frac {x^3 \sqrt [3]{1+x^5}}{2-x^3+2 x^5} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.36, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{1+x^5} \left (-3+2 x^5\right )}{x^2 \left (2-x^3+2 x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 2.68, size = 133, normalized size = 1.00 \begin {gather*} \frac {3 \sqrt [3]{x^5+1}}{2 x}+\frac {\log \left (\sqrt [3]{2} \sqrt [3]{x^5+1}-x\right )}{2 \sqrt [3]{2}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{2} \sqrt [3]{x^5+1}+x}\right )}{2 \sqrt [3]{2}}-\frac {\log \left (\sqrt [3]{2} \sqrt [3]{x^5+1} x+2^{2/3} \left (x^5+1\right )^{2/3}+x^2\right )}{4 \sqrt [3]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 100.24, size = 384, normalized size = 2.89 \begin {gather*} \frac {2 \, \sqrt {3} 2^{\frac {2}{3}} x \arctan \left (\frac {\sqrt {3} 2^{\frac {1}{6}} {\left (24 \, \sqrt {2} {\left (2 \, x^{11} + x^{9} - x^{7} + 4 \, x^{6} + x^{4} + 2 \, x\right )} {\left (x^{5} + 1\right )}^{\frac {2}{3}} + 2^{\frac {5}{6}} {\left (8 \, x^{15} + 60 \, x^{13} + 24 \, x^{11} + 24 \, x^{10} - x^{9} + 120 \, x^{8} + 24 \, x^{6} + 24 \, x^{5} + 60 \, x^{3} + 8\right )} + 12 \cdot 2^{\frac {1}{6}} {\left (4 \, x^{12} + 14 \, x^{10} + x^{8} + 8 \, x^{7} + 14 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{5} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (8 \, x^{15} - 12 \, x^{13} - 48 \, x^{11} + 24 \, x^{10} - x^{9} - 24 \, x^{8} - 48 \, x^{6} + 24 \, x^{5} - 12 \, x^{3} + 8\right )}}\right ) + 2 \cdot 2^{\frac {2}{3}} x \log \left (\frac {3 \cdot 2^{\frac {2}{3}} {\left (x^{5} + 1\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (x^{5} + 1\right )}^{\frac {2}{3}} x + 2^{\frac {1}{3}} {\left (2 \, x^{5} - x^{3} + 2\right )}}{2 \, x^{5} - x^{3} + 2}\right ) - 2^{\frac {2}{3}} x \log \left (\frac {12 \cdot 2^{\frac {1}{3}} {\left (x^{6} + x^{4} + x\right )} {\left (x^{5} + 1\right )}^{\frac {2}{3}} + 2^{\frac {2}{3}} {\left (4 \, x^{10} + 14 \, x^{8} + x^{6} + 8 \, x^{5} + 14 \, x^{3} + 4\right )} + 6 \, {\left (4 \, x^{7} + x^{5} + 4 \, x^{2}\right )} {\left (x^{5} + 1\right )}^{\frac {1}{3}}}{4 \, x^{10} - 4 \, x^{8} + x^{6} + 8 \, x^{5} - 4 \, x^{3} + 4}\right ) + 36 \, {\left (x^{5} + 1\right )}^{\frac {1}{3}}}{24 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(2*sqrt(3)*2^(2/3)*x*arctan(1/6*sqrt(3)*2^(1/6)*(24*sqrt(2)*(2*x^11 + x^9 - x^7 + 4*x^6 + x^4 + 2*x)*(x^5
+ 1)^(2/3) + 2^(5/6)*(8*x^15 + 60*x^13 + 24*x^11 + 24*x^10 - x^9 + 120*x^8 + 24*x^6 + 24*x^5 + 60*x^3 + 8) +
12*2^(1/6)*(4*x^12 + 14*x^10 + x^8 + 8*x^7 + 14*x^5 + 4*x^2)*(x^5 + 1)^(1/3))/(8*x^15 - 12*x^13 - 48*x^11 + 24
*x^10 - x^9 - 24*x^8 - 48*x^6 + 24*x^5 - 12*x^3 + 8)) + 2*2^(2/3)*x*log((3*2^(2/3)*(x^5 + 1)^(1/3)*x^2 - 6*(x^
5 + 1)^(2/3)*x + 2^(1/3)*(2*x^5 - x^3 + 2))/(2*x^5 - x^3 + 2)) - 2^(2/3)*x*log((12*2^(1/3)*(x^6 + x^4 + x)*(x^
5 + 1)^(2/3) + 2^(2/3)*(4*x^10 + 14*x^8 + x^6 + 8*x^5 + 14*x^3 + 4) + 6*(4*x^7 + x^5 + 4*x^2)*(x^5 + 1)^(1/3))
/(4*x^10 - 4*x^8 + x^6 + 8*x^5 - 4*x^3 + 4)) + 36*(x^5 + 1)^(1/3))/x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 8.29, size = 1656, normalized size = 12.45 \begin {gather*} \text {Expression too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*(x^5+1)^(1/3)/x+(-1/4*ln(-(RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)^3*x^8+4*Root
Of(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)^2*RootOf(_Z^3-4)^2*x^8+4*RootOf(_Z^3-4)*x^10+16*RootOf(RootOf(
_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x^10+12*(x^10+2*x^5+1)^(1/3)*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)
+4*_Z^2)*RootOf(_Z^3-4)*x^6+2*RootOf(_Z^3-4)*x^8+8*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x^8+6*(
x^10+2*x^5+1)^(2/3)*RootOf(_Z^3-4)^2*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x^2+RootOf(RootOf(_Z^
3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)^3*x^3+4*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)^
2*RootOf(_Z^3-4)^2*x^3+8*RootOf(_Z^3-4)*x^5+32*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x^5+12*(x^1
0+2*x^5+1)^(1/3)*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)*x+2*RootOf(_Z^3-4)*x^3+8*R
ootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x^3+4*RootOf(_Z^3-4)+16*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf
(_Z^3-4)+4*_Z^2))/(2*x^5-x^3+2)/(1+x)/(x^4-x^3+x^2-x+1))*RootOf(_Z^3-4)-1/2*ln(-(RootOf(RootOf(_Z^3-4)^2+2*_Z*
RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)^3*x^8+4*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)^2*RootOf(_Z^
3-4)^2*x^8+4*RootOf(_Z^3-4)*x^10+16*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x^10+12*(x^10+2*x^5+1)
^(1/3)*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)*x^6+2*RootOf(_Z^3-4)*x^8+8*RootOf(Ro
otOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x^8+6*(x^10+2*x^5+1)^(2/3)*RootOf(_Z^3-4)^2*RootOf(RootOf(_Z^3-4)^2
+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x^2+RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)^3*x^3+4*Ro
otOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)^2*RootOf(_Z^3-4)^2*x^3+8*RootOf(_Z^3-4)*x^5+32*RootOf(RootOf
(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x^5+12*(x^10+2*x^5+1)^(1/3)*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)
+4*_Z^2)*RootOf(_Z^3-4)*x+2*RootOf(_Z^3-4)*x^3+8*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x^3+4*Roo
tOf(_Z^3-4)+16*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2))/(2*x^5-x^3+2)/(1+x)/(x^4-x^3+x^2-x+1))*Roo
tOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)+1/2*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*ln((-
RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)^3*x^8-4*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf
(_Z^3-4)+4*_Z^2)^2*RootOf(_Z^3-4)^2*x^8+4*RootOf(_Z^3-4)*x^10+16*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4
*_Z^2)*x^10+6*RootOf(_Z^3-4)^2*(x^10+2*x^5+1)^(1/3)*x^6+12*(x^10+2*x^5+1)^(1/3)*RootOf(RootOf(_Z^3-4)^2+2*_Z*R
ootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)*x^6+6*(x^10+2*x^5+1)^(2/3)*RootOf(_Z^3-4)^2*RootOf(RootOf(_Z^3-4)^2+2*_Z*
RootOf(_Z^3-4)+4*_Z^2)*x^2-RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)^3*x^3-4*RootOf(R
ootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)^2*RootOf(_Z^3-4)^2*x^3+8*RootOf(_Z^3-4)*x^5+32*RootOf(RootOf(_Z^3-
4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*x^5+12*(x^10+2*x^5+1)^(2/3)*x^2+6*RootOf(_Z^3-4)^2*(x^10+2*x^5+1)^(1/3)*x+12*
(x^10+2*x^5+1)^(1/3)*RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2)*RootOf(_Z^3-4)*x+4*RootOf(_Z^3-4)+16*
RootOf(RootOf(_Z^3-4)^2+2*_Z*RootOf(_Z^3-4)+4*_Z^2))/(2*x^5-x^3+2)/(1+x)/(x^4-x^3+x^2-x+1)))/(x^5+1)^(2/3)*((x
^5+1)^2)^(1/3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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