3.23.54 \(\int \frac {(4+x^2+x^5) \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{x^2 (-2-x^2+2 x^5)} \, dx\) [2254]

Optimal. Leaf size=170 \[ \frac {2 \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{x}-\frac {\text {ArcTan}\left (\frac {2^{3/4} x \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{\sqrt {2} x^2-\sqrt {-2-x^2-2 x^4+2 x^5}}\right )}{\sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {2 \sqrt [4]{2} x \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{2 x^2+\sqrt {2} \sqrt {-2-x^2-2 x^4+2 x^5}}\right )}{\sqrt [4]{2}} \]

[Out]

2*(2*x^5-2*x^4-x^2-2)^(1/4)/x-1/2*arctan(2^(3/4)*x*(2*x^5-2*x^4-x^2-2)^(1/4)/(x^2*2^(1/2)-(2*x^5-2*x^4-x^2-2)^
(1/2)))*2^(3/4)-1/2*arctanh(2*2^(1/4)*x*(2*x^5-2*x^4-x^2-2)^(1/4)/(2*x^2+2^(1/2)*(2*x^5-2*x^4-x^2-2)^(1/2)))*2
^(3/4)

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Rubi [F]
time = 0.71, 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 (4+x^2+x^5\right ) \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{x^2 \left (-2-x^2+2 x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.31, size = 159, normalized size = 0.94 \begin {gather*} \frac {2 \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{x}-\frac {\text {ArcTan}\left (\frac {2^{3/4} x \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{\sqrt {2} x^2-\sqrt {-2-x^2-2 x^4+2 x^5}}\right )}{\sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {2 x \sqrt [4]{-4-2 x^2-4 x^4+4 x^5}}{2 x^2+\sqrt {-4-2 x^2-4 x^4+4 x^5}}\right )}{\sqrt [4]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{5}+x^{2}+4\right ) \left (2 x^{5}-2 x^{4}-x^{2}-2\right )^{\frac {1}{4}}}{x^{2} \left (2 x^{5}-x^{2}-2\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5+x^2+4)*(2*x^5-2*x^4-x^2-2)^(1/4)/x^2/(2*x^5-x^2-2),x)

[Out]

int((x^5+x^2+4)*(2*x^5-2*x^4-x^2-2)^(1/4)/x^2/(2*x^5-x^2-2),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((2*x^5 - 2*x^4 - x^2 - 2)^(1/4)*(x^5 + x^2 + 4)/((2*x^5 - x^2 - 2)*x^2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1274 vs. \(2 (148) = 296\).
time = 31.78, size = 1274, normalized size = 7.49 \begin {gather*} \frac {4 \cdot 8^{\frac {3}{4}} \sqrt {2} x \arctan \left (\frac {32 \, x^{10} - 32 \, x^{7} - 64 \, x^{5} + 8 \, x^{4} + 4 \cdot 8^{\frac {3}{4}} \sqrt {2} {\left (2 \, x^{6} - 8 \, x^{5} - x^{3} - 2 \, x\right )} {\left (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {3}{4}} + 16 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (6 \, x^{8} - 8 \, x^{7} - 3 \, x^{5} - 6 \, x^{3}\right )} {\left (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {1}{4}} + 32 \, \sqrt {2} {\left (2 \, x^{7} - x^{4} - 2 \, x^{2}\right )} \sqrt {2 \, x^{5} - 2 \, x^{4} - x^{2} - 2} + 32 \, x^{2} + \sqrt {2} {\left (128 \, \sqrt {2} {\left (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {3}{4}} x^{5} + 8^{\frac {3}{4}} \sqrt {2} {\left (4 \, x^{10} - 40 \, x^{9} + 32 \, x^{8} - 4 \, x^{7} + 20 \, x^{6} - 8 \, x^{5} + 41 \, x^{4} + 4 \, x^{2} + 4\right )} + 8 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (2 \, x^{7} - 8 \, x^{6} - x^{4} - 2 \, x^{2}\right )} \sqrt {2 \, x^{5} - 2 \, x^{4} - x^{2} - 2} + 32 \, {\left (2 \, x^{8} - x^{5} - 2 \, x^{3}\right )} {\left (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {8^{\frac {3}{4}} \sqrt {2} {\left (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {1}{4}} x^{3} + 2 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {3}{4}} x + 8 \, \sqrt {2 \, x^{5} - 2 \, x^{4} - x^{2} - 2} x^{2} + \sqrt {2} {\left (2 \, x^{5} - x^{2} - 2\right )}}{2 \, x^{5} - x^{2} - 2}} + 32}{8 \, {\left (4 \, x^{10} - 64 \, x^{9} + 64 \, x^{8} - 4 \, x^{7} + 32 \, x^{6} - 8 \, x^{5} + 65 \, x^{4} + 4 \, x^{2} + 4\right )}}\right ) - 4 \cdot 8^{\frac {3}{4}} \sqrt {2} x \arctan \left (\frac {32 \, x^{10} - 32 \, x^{7} - 64 \, x^{5} + 8 \, x^{4} - 4 \cdot 8^{\frac {3}{4}} \sqrt {2} {\left (2 \, x^{6} - 8 \, x^{5} - x^{3} - 2 \, x\right )} {\left (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {3}{4}} - 16 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (6 \, x^{8} - 8 \, x^{7} - 3 \, x^{5} - 6 \, x^{3}\right )} {\left (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {1}{4}} + 32 \, \sqrt {2} {\left (2 \, x^{7} - x^{4} - 2 \, x^{2}\right )} \sqrt {2 \, x^{5} - 2 \, x^{4} - x^{2} - 2} + 32 \, x^{2} + \sqrt {2} {\left (128 \, \sqrt {2} {\left (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {3}{4}} x^{5} - 8^{\frac {3}{4}} \sqrt {2} {\left (4 \, x^{10} - 40 \, x^{9} + 32 \, x^{8} - 4 \, x^{7} + 20 \, x^{6} - 8 \, x^{5} + 41 \, x^{4} + 4 \, x^{2} + 4\right )} - 8 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (2 \, x^{7} - 8 \, x^{6} - x^{4} - 2 \, x^{2}\right )} \sqrt {2 \, x^{5} - 2 \, x^{4} - x^{2} - 2} + 32 \, {\left (2 \, x^{8} - x^{5} - 2 \, x^{3}\right )} {\left (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {1}{4}}\right )} \sqrt {-\frac {8^{\frac {3}{4}} \sqrt {2} {\left (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {1}{4}} x^{3} + 2 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {3}{4}} x - 8 \, \sqrt {2 \, x^{5} - 2 \, x^{4} - x^{2} - 2} x^{2} - \sqrt {2} {\left (2 \, x^{5} - x^{2} - 2\right )}}{2 \, x^{5} - x^{2} - 2}} + 32}{8 \, {\left (4 \, x^{10} - 64 \, x^{9} + 64 \, x^{8} - 4 \, x^{7} + 32 \, x^{6} - 8 \, x^{5} + 65 \, x^{4} + 4 \, x^{2} + 4\right )}}\right ) - 8^{\frac {3}{4}} \sqrt {2} x \log \left (\frac {8 \, {\left (8^{\frac {3}{4}} \sqrt {2} {\left (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {1}{4}} x^{3} + 2 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {3}{4}} x + 8 \, \sqrt {2 \, x^{5} - 2 \, x^{4} - x^{2} - 2} x^{2} + \sqrt {2} {\left (2 \, x^{5} - x^{2} - 2\right )}\right )}}{2 \, x^{5} - x^{2} - 2}\right ) + 8^{\frac {3}{4}} \sqrt {2} x \log \left (-\frac {8 \, {\left (8^{\frac {3}{4}} \sqrt {2} {\left (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {1}{4}} x^{3} + 2 \cdot 8^{\frac {1}{4}} \sqrt {2} {\left (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {3}{4}} x - 8 \, \sqrt {2 \, x^{5} - 2 \, x^{4} - x^{2} - 2} x^{2} - \sqrt {2} {\left (2 \, x^{5} - x^{2} - 2\right )}\right )}}{2 \, x^{5} - x^{2} - 2}\right ) + 64 \, {\left (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {1}{4}}}{32 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5+x^2+4)*(2*x^5-2*x^4-x^2-2)^(1/4)/x^2/(2*x^5-x^2-2),x, algorithm="fricas")

[Out]

1/32*(4*8^(3/4)*sqrt(2)*x*arctan(1/8*(32*x^10 - 32*x^7 - 64*x^5 + 8*x^4 + 4*8^(3/4)*sqrt(2)*(2*x^6 - 8*x^5 - x
^3 - 2*x)*(2*x^5 - 2*x^4 - x^2 - 2)^(3/4) + 16*8^(1/4)*sqrt(2)*(6*x^8 - 8*x^7 - 3*x^5 - 6*x^3)*(2*x^5 - 2*x^4
- x^2 - 2)^(1/4) + 32*sqrt(2)*(2*x^7 - x^4 - 2*x^2)*sqrt(2*x^5 - 2*x^4 - x^2 - 2) + 32*x^2 + sqrt(2)*(128*sqrt
(2)*(2*x^5 - 2*x^4 - x^2 - 2)^(3/4)*x^5 + 8^(3/4)*sqrt(2)*(4*x^10 - 40*x^9 + 32*x^8 - 4*x^7 + 20*x^6 - 8*x^5 +
41*x^4 + 4*x^2 + 4) + 8*8^(1/4)*sqrt(2)*(2*x^7 - 8*x^6 - x^4 - 2*x^2)*sqrt(2*x^5 - 2*x^4 - x^2 - 2) + 32*(2*x
^8 - x^5 - 2*x^3)*(2*x^5 - 2*x^4 - x^2 - 2)^(1/4))*sqrt((8^(3/4)*sqrt(2)*(2*x^5 - 2*x^4 - x^2 - 2)^(1/4)*x^3 +
2*8^(1/4)*sqrt(2)*(2*x^5 - 2*x^4 - x^2 - 2)^(3/4)*x + 8*sqrt(2*x^5 - 2*x^4 - x^2 - 2)*x^2 + sqrt(2)*(2*x^5 -
x^2 - 2))/(2*x^5 - x^2 - 2)) + 32)/(4*x^10 - 64*x^9 + 64*x^8 - 4*x^7 + 32*x^6 - 8*x^5 + 65*x^4 + 4*x^2 + 4)) -
4*8^(3/4)*sqrt(2)*x*arctan(1/8*(32*x^10 - 32*x^7 - 64*x^5 + 8*x^4 - 4*8^(3/4)*sqrt(2)*(2*x^6 - 8*x^5 - x^3 -
2*x)*(2*x^5 - 2*x^4 - x^2 - 2)^(3/4) - 16*8^(1/4)*sqrt(2)*(6*x^8 - 8*x^7 - 3*x^5 - 6*x^3)*(2*x^5 - 2*x^4 - x^2
- 2)^(1/4) + 32*sqrt(2)*(2*x^7 - x^4 - 2*x^2)*sqrt(2*x^5 - 2*x^4 - x^2 - 2) + 32*x^2 + sqrt(2)*(128*sqrt(2)*(
2*x^5 - 2*x^4 - x^2 - 2)^(3/4)*x^5 - 8^(3/4)*sqrt(2)*(4*x^10 - 40*x^9 + 32*x^8 - 4*x^7 + 20*x^6 - 8*x^5 + 41*x
^4 + 4*x^2 + 4) - 8*8^(1/4)*sqrt(2)*(2*x^7 - 8*x^6 - x^4 - 2*x^2)*sqrt(2*x^5 - 2*x^4 - x^2 - 2) + 32*(2*x^8 -
x^5 - 2*x^3)*(2*x^5 - 2*x^4 - x^2 - 2)^(1/4))*sqrt(-(8^(3/4)*sqrt(2)*(2*x^5 - 2*x^4 - x^2 - 2)^(1/4)*x^3 + 2*8
^(1/4)*sqrt(2)*(2*x^5 - 2*x^4 - x^2 - 2)^(3/4)*x - 8*sqrt(2*x^5 - 2*x^4 - x^2 - 2)*x^2 - sqrt(2)*(2*x^5 - x^2
- 2))/(2*x^5 - x^2 - 2)) + 32)/(4*x^10 - 64*x^9 + 64*x^8 - 4*x^7 + 32*x^6 - 8*x^5 + 65*x^4 + 4*x^2 + 4)) - 8^(
3/4)*sqrt(2)*x*log(8*(8^(3/4)*sqrt(2)*(2*x^5 - 2*x^4 - x^2 - 2)^(1/4)*x^3 + 2*8^(1/4)*sqrt(2)*(2*x^5 - 2*x^4 -
x^2 - 2)^(3/4)*x + 8*sqrt(2*x^5 - 2*x^4 - x^2 - 2)*x^2 + sqrt(2)*(2*x^5 - x^2 - 2))/(2*x^5 - x^2 - 2)) + 8^(3
/4)*sqrt(2)*x*log(-8*(8^(3/4)*sqrt(2)*(2*x^5 - 2*x^4 - x^2 - 2)^(1/4)*x^3 + 2*8^(1/4)*sqrt(2)*(2*x^5 - 2*x^4 -
x^2 - 2)^(3/4)*x - 8*sqrt(2*x^5 - 2*x^4 - x^2 - 2)*x^2 - sqrt(2)*(2*x^5 - x^2 - 2))/(2*x^5 - x^2 - 2)) + 64*(
2*x^5 - 2*x^4 - x^2 - 2)^(1/4))/x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((2*x^5 - 2*x^4 - x^2 - 2)^(1/4)*(x^5 + x^2 + 4)/((2*x^5 - x^2 - 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+x^2+4\right )\,{\left (2\,x^5-2\,x^4-x^2-2\right )}^{1/4}}{x^2\,\left (-2\,x^5+x^2+2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((x^2 + x^5 + 4)*(2*x^5 - 2*x^4 - x^2 - 2)^(1/4))/(x^2*(x^2 - 2*x^5 + 2)),x)

[Out]

int(-((x^2 + x^5 + 4)*(2*x^5 - 2*x^4 - x^2 - 2)^(1/4))/(x^2*(x^2 - 2*x^5 + 2)), x)

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