3.20.37 \(\int \frac {(-4+x^5) \sqrt [4]{1-2 x^4+x^5}}{x^2 (1+x^5)} \, dx\) [1937]

Optimal. Leaf size=135 \[ \frac {4 \sqrt [4]{1-2 x^4+x^5}}{x}-2^{3/4} \text {ArcTan}\left (\frac {2^{3/4} x \sqrt [4]{1-2 x^4+x^5}}{\sqrt {2} x^2-\sqrt {1-2 x^4+x^5}}\right )-2^{3/4} \tanh ^{-1}\left (\frac {2 \sqrt [4]{2} x \sqrt [4]{1-2 x^4+x^5}}{2 x^2+\sqrt {2} \sqrt {1-2 x^4+x^5}}\right ) \]

[Out]

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

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Rubi [F]
time = 0.90, 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^5\right ) \sqrt [4]{1-2 x^4+x^5}}{x^2 \left (1+x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 1.07, size = 135, normalized size = 1.00 \begin {gather*} \frac {4 \sqrt [4]{1-2 x^4+x^5}}{x}-2^{3/4} \text {ArcTan}\left (\frac {2^{3/4} x \sqrt [4]{1-2 x^4+x^5}}{\sqrt {2} x^2-\sqrt {1-2 x^4+x^5}}\right )-2^{3/4} \tanh ^{-1}\left (\frac {2 \sqrt [4]{2} x \sqrt [4]{1-2 x^4+x^5}}{2 x^2+\sqrt {2} \sqrt {1-2 x^4+x^5}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 33.64, size = 334, normalized size = 2.47

method	result	size
trager	\(\frac {4 \left (x^{5}-2 x^{4}+1\right )^{\frac {1}{4}}}{x}-\RootOf \left (\textit {\_Z}^{4}+2\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+2\right )^{3} x^{5}-4 \RootOf \left (\textit {\_Z}^{4}+2\right )^{3} x^{4}-4 \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \left (x^{5}-2 x^{4}+1\right )^{\frac {1}{4}} x^{3}-4 \RootOf \left (\textit {\_Z}^{4}+2\right ) \sqrt {x^{5}-2 x^{4}+1}\, x^{2}-4 \left (x^{5}-2 x^{4}+1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{4}+2\right )^{3}}{\left (1+x \right ) \left (x^{4}-x^{3}+x^{2}-x +1\right )}\right )-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{5}-4 \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{4}-4 \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \left (x^{5}-2 x^{4}+1\right )^{\frac {1}{4}} x^{3}+4 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) \sqrt {x^{5}-2 x^{4}+1}\, x^{2}+4 \left (x^{5}-2 x^{4}+1\right )^{\frac {3}{4}} x +\RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right )}{\left (1+x \right ) \left (x^{4}-x^{3}+x^{2}-x +1\right )}\right )\)	\(334\)
risch	\(\text {Expression too large to display}\)	\(1652\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 809 vs. \(2 (113) = 226\).
time = 30.89, size = 809, normalized size = 5.99 \begin {gather*} -\frac {4 \cdot 2^{\frac {3}{4}} x \arctan \left (-\frac {2 \, x^{10} + 4 \, x^{5} + 4 \cdot 2^{\frac {3}{4}} {\left (x^{6} - 8 \, x^{5} + x\right )} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {3}{4}} + 8 \, \sqrt {2} {\left (x^{7} + x^{2}\right )} \sqrt {x^{5} - 2 \, x^{4} + 1} - \sqrt {2} {\left (32 \, \sqrt {2} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {3}{4}} x^{5} + 2^{\frac {3}{4}} {\left (x^{10} - 20 \, x^{9} + 32 \, x^{8} + 2 \, x^{5} - 20 \, x^{4} + 1\right )} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{7} - 8 \, x^{6} + x^{2}\right )} \sqrt {x^{5} - 2 \, x^{4} + 1} + 8 \, {\left (x^{8} + x^{3}\right )} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {4 \cdot 2^{\frac {3}{4}} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 8 \, \sqrt {x^{5} - 2 \, x^{4} + 1} x^{2} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (x^{5} + 1\right )}}{x^{5} + 1}} + 8 \cdot 2^{\frac {1}{4}} {\left (3 \, x^{8} - 8 \, x^{7} + 3 \, x^{3}\right )} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {1}{4}} + 2}{2 \, {\left (x^{10} - 32 \, x^{9} + 64 \, x^{8} + 2 \, x^{5} - 32 \, x^{4} + 1\right )}}\right ) - 4 \cdot 2^{\frac {3}{4}} x \arctan \left (-\frac {2 \, x^{10} + 4 \, x^{5} - 4 \cdot 2^{\frac {3}{4}} {\left (x^{6} - 8 \, x^{5} + x\right )} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {3}{4}} + 8 \, \sqrt {2} {\left (x^{7} + x^{2}\right )} \sqrt {x^{5} - 2 \, x^{4} + 1} - \sqrt {2} {\left (32 \, \sqrt {2} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {3}{4}} x^{5} - 2^{\frac {3}{4}} {\left (x^{10} - 20 \, x^{9} + 32 \, x^{8} + 2 \, x^{5} - 20 \, x^{4} + 1\right )} - 4 \cdot 2^{\frac {1}{4}} {\left (x^{7} - 8 \, x^{6} + x^{2}\right )} \sqrt {x^{5} - 2 \, x^{4} + 1} + 8 \, {\left (x^{8} + x^{3}\right )} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {1}{4}}\right )} \sqrt {-\frac {4 \cdot 2^{\frac {3}{4}} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 8 \, \sqrt {x^{5} - 2 \, x^{4} + 1} x^{2} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {3}{4}} x - \sqrt {2} {\left (x^{5} + 1\right )}}{x^{5} + 1}} - 8 \cdot 2^{\frac {1}{4}} {\left (3 \, x^{8} - 8 \, x^{7} + 3 \, x^{3}\right )} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {1}{4}} + 2}{2 \, {\left (x^{10} - 32 \, x^{9} + 64 \, x^{8} + 2 \, x^{5} - 32 \, x^{4} + 1\right )}}\right ) + 2^{\frac {3}{4}} x \log \left (\frac {2 \, {\left (4 \cdot 2^{\frac {3}{4}} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 8 \, \sqrt {x^{5} - 2 \, x^{4} + 1} x^{2} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (x^{5} + 1\right )}\right )}}{x^{5} + 1}\right ) - 2^{\frac {3}{4}} x \log \left (-\frac {2 \, {\left (4 \cdot 2^{\frac {3}{4}} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 8 \, \sqrt {x^{5} - 2 \, x^{4} + 1} x^{2} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {3}{4}} x - \sqrt {2} {\left (x^{5} + 1\right )}\right )}}{x^{5} + 1}\right ) - 16 \, {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {1}{4}}}{4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(((x - 1)*(x**4 - x**3 - x**2 - x - 1))**(1/4)*(x**5 - 4)/(x**2*(x + 1)*(x**4 - x**3 + x**2 - x + 1)),
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^5-4)*(x^5-2*x^4+1)^(1/4)/x^2/(x^5+1),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (x^5-4\right )\,{\left (x^5-2\,x^4+1\right )}^{1/4}}{x^2\,\left (x^5+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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