3.28.36 \(\int \frac {1}{x \sqrt [5]{1-4 x+6 x^2-4 x^3+x^4}} \, dx\)

Optimal. Leaf size=253 \[ \frac {(x-1)^{4/5} \left (\log \left (\sqrt [5]{x-1}+1\right )+\frac {1}{4} \left (-1-\sqrt {5}\right ) \log \left (2 (x-1)^{2/5}+\left (-1-\sqrt {5}\right ) \sqrt [5]{x-1}+2\right )+\frac {1}{4} \left (\sqrt {5}-1\right ) \log \left (2 (x-1)^{2/5}+\left (\sqrt {5}-1\right ) \sqrt [5]{x-1}+2\right )-\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tan ^{-1}\left (-\frac {4 \sqrt [5]{x-1}}{\sqrt {10-2 \sqrt {5}}}+\frac {1}{\sqrt {10-2 \sqrt {5}}}+\sqrt {\frac {5}{10-2 \sqrt {5}}}\right )+\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\frac {4 \sqrt [5]{x-1}}{\sqrt {10+2 \sqrt {5}}}-\frac {1}{\sqrt {10+2 \sqrt {5}}}+\sqrt {\frac {5}{10+2 \sqrt {5}}}\right )\right )}{\sqrt [5]{(x-1)^4}} \]

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Rubi [F]  time = 0.08, 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 {1}{x \sqrt [5]{1-4 x+6 x^2-4 x^3+x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[1/(x*(1 - 4*x + 6*x^2 - 4*x^3 + x^4)^(1/5)),x]

[Out]

Defer[Int][1/(x*(1 - 4*x + 6*x^2 - 4*x^3 + x^4)^(1/5)), x]

Rubi steps

\begin {align*} \int \frac {1}{x \sqrt [5]{1-4 x+6 x^2-4 x^3+x^4}} \, dx &=\int \frac {1}{x \sqrt [5]{1-4 x+6 x^2-4 x^3+x^4}} \, dx\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 27, normalized size = 0.11 \begin {gather*} \frac {5 (x-1) \, _2F_1\left (\frac {1}{5},1;\frac {6}{5};1-x\right )}{\sqrt [5]{(x-1)^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(1 - 4*x + 6*x^2 - 4*x^3 + x^4)^(1/5)),x]

[Out]

(5*(-1 + x)*Hypergeometric2F1[1/5, 1, 6/5, 1 - x])/((-1 + x)^4)^(1/5)

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IntegrateAlgebraic [A]  time = 1.19, size = 411, normalized size = 1.62 \begin {gather*} -\sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {10+2 \sqrt {5}} \sqrt [5]{1-4 x+6 x^2-4 x^3+x^4}}{-4+4 x+\left (-1+\sqrt {5}\right ) \sqrt [5]{1-4 x+6 x^2-4 x^3+x^4}}\right )+\sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {10-2 \sqrt {5}} \sqrt [5]{1-4 x+6 x^2-4 x^3+x^4}}{4-4 x+\left (1+\sqrt {5}\right ) \sqrt [5]{1-4 x+6 x^2-4 x^3+x^4}}\right )-\frac {4}{5} \log (-1+x)+\frac {1}{5} \log \left (1-4 x+6 x^2-4 x^3+x^4\right )+\log \left (-1+x+\sqrt [5]{1-4 x+6 x^2-4 x^3+x^4}\right )+\frac {1}{4} \left (-1-\sqrt {5}\right ) \log \left (2-4 x+2 x^2+\left (1+\sqrt {5} (1-x)-x\right ) \sqrt [5]{1-4 x+6 x^2-4 x^3+x^4}+2 \left (1-4 x+6 x^2-4 x^3+x^4\right )^{2/5}\right )+\frac {1}{4} \left (-1+\sqrt {5}\right ) \log \left (2-4 x+2 x^2+\left (1+\sqrt {5} (-1+x)-x\right ) \sqrt [5]{1-4 x+6 x^2-4 x^3+x^4}+2 \left (1-4 x+6 x^2-4 x^3+x^4\right )^{2/5}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x*(1 - 4*x + 6*x^2 - 4*x^3 + x^4)^(1/5)),x]

[Out]

-(Sqrt[(5 + Sqrt[5])/2]*ArcTan[(Sqrt[10 + 2*Sqrt[5]]*(1 - 4*x + 6*x^2 - 4*x^3 + x^4)^(1/5))/(-4 + 4*x + (-1 +
Sqrt[5])*(1 - 4*x + 6*x^2 - 4*x^3 + x^4)^(1/5))]) + Sqrt[(5 - Sqrt[5])/2]*ArcTan[(Sqrt[10 - 2*Sqrt[5]]*(1 - 4*
x + 6*x^2 - 4*x^3 + x^4)^(1/5))/(4 - 4*x + (1 + Sqrt[5])*(1 - 4*x + 6*x^2 - 4*x^3 + x^4)^(1/5))] - (4*Log[-1 +
x])/5 + Log[1 - 4*x + 6*x^2 - 4*x^3 + x^4]/5 + Log[-1 + x + (1 - 4*x + 6*x^2 - 4*x^3 + x^4)^(1/5)] + ((-1 - S
qrt[5])*Log[2 - 4*x + 2*x^2 + (1 + Sqrt[5]*(1 - x) - x)*(1 - 4*x + 6*x^2 - 4*x^3 + x^4)^(1/5) + 2*(1 - 4*x + 6
*x^2 - 4*x^3 + x^4)^(2/5)])/4 + ((-1 + Sqrt[5])*Log[2 - 4*x + 2*x^2 + (1 + Sqrt[5]*(-1 + x) - x)*(1 - 4*x + 6*
x^2 - 4*x^3 + x^4)^(1/5) + 2*(1 - 4*x + 6*x^2 - 4*x^3 + x^4)^(2/5)])/4

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fricas [B]  time = 1.27, size = 1084, normalized size = 4.28

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x^4-4*x^3+6*x^2-4*x+1)^(1/5),x, algorithm="fricas")

[Out]

-1/4*(sqrt(5) - 2*sqrt(1/2*sqrt(5) - 5/2) + 1)*log(-1/64*((x - 1)*(sqrt(5) + 2*sqrt(1/2*sqrt(5) - 5/2) + 1)^3
+ (x - 1)*(sqrt(5) + 2*sqrt(1/2*sqrt(5) - 5/2) + 1)*(sqrt(5) - 2*sqrt(1/2*sqrt(5) - 5/2) + 1)^2 - 4*(x - 1)*(s
qrt(5) + 2*sqrt(1/2*sqrt(5) - 5/2) + 1)^2 + 16*(x - 1)*(sqrt(5) + 2*sqrt(1/2*sqrt(5) - 5/2) + 1) + ((x - 1)*(s
qrt(5) + 2*sqrt(1/2*sqrt(5) - 5/2) + 1)^2 - 4*(x - 1)*(sqrt(5) + 2*sqrt(1/2*sqrt(5) - 5/2) + 1))*(sqrt(5) - 2*
sqrt(1/2*sqrt(5) - 5/2) + 1) - 64*(x^4 - 4*x^3 + 6*x^2 - 4*x + 1)^(1/5))/(x - 1)) - 1/4*(sqrt(5) + 2*sqrt(1/2*
sqrt(5) - 5/2) + 1)*log(1/64*((x - 1)*(sqrt(5) + 2*sqrt(1/2*sqrt(5) - 5/2) + 1)^3 - 4*(x - 1)*(sqrt(5) + 2*sqr
t(1/2*sqrt(5) - 5/2) + 1)^2 + 16*(x - 1)*(sqrt(5) + 2*sqrt(1/2*sqrt(5) - 5/2) + 1) - 64*x + 64*(x^4 - 4*x^3 +
6*x^2 - 4*x + 1)^(1/5) + 64)/(x - 1)) + 1/4*(sqrt(5) - 2*sqrt(-3/16*(sqrt(5) + 2*sqrt(1/2*sqrt(5) - 5/2) + 1)^
2 - 1/8*(sqrt(5) + 2*sqrt(1/2*sqrt(5) - 5/2) - 3)*(sqrt(5) - 2*sqrt(1/2*sqrt(5) - 5/2) + 1) - 3/16*(sqrt(5) -
2*sqrt(1/2*sqrt(5) - 5/2) + 1)^2 + 1/2*sqrt(5) + sqrt(1/2*sqrt(5) - 5/2) - 5/2) - 1)*log(1/64*((x - 1)*(sqrt(5
) + 2*sqrt(1/2*sqrt(5) - 5/2) + 1)*(sqrt(5) - 2*sqrt(1/2*sqrt(5) - 5/2) + 1)^2 + 4*sqrt(-3/16*(sqrt(5) + 2*sqr
t(1/2*sqrt(5) - 5/2) + 1)^2 - 1/8*(sqrt(5) + 2*sqrt(1/2*sqrt(5) - 5/2) - 3)*(sqrt(5) - 2*sqrt(1/2*sqrt(5) - 5/
2) + 1) - 3/16*(sqrt(5) - 2*sqrt(1/2*sqrt(5) - 5/2) + 1)^2 + 1/2*sqrt(5) + sqrt(1/2*sqrt(5) - 5/2) - 5/2)*(x -
1)*(sqrt(5) + 2*sqrt(1/2*sqrt(5) - 5/2) + 1)*(sqrt(5) - 2*sqrt(1/2*sqrt(5) - 5/2) + 1) + ((x - 1)*(sqrt(5) +
2*sqrt(1/2*sqrt(5) - 5/2) + 1)^2 - 4*(x - 1)*(sqrt(5) + 2*sqrt(1/2*sqrt(5) - 5/2) + 1))*(sqrt(5) - 2*sqrt(1/2*
sqrt(5) - 5/2) + 1) + 128*(x^4 - 4*x^3 + 6*x^2 - 4*x + 1)^(1/5))/(x - 1)) + 1/4*(sqrt(5) + 2*sqrt(-3/16*(sqrt(
5) + 2*sqrt(1/2*sqrt(5) - 5/2) + 1)^2 - 1/8*(sqrt(5) + 2*sqrt(1/2*sqrt(5) - 5/2) - 3)*(sqrt(5) - 2*sqrt(1/2*sq
rt(5) - 5/2) + 1) - 3/16*(sqrt(5) - 2*sqrt(1/2*sqrt(5) - 5/2) + 1)^2 + 1/2*sqrt(5) + sqrt(1/2*sqrt(5) - 5/2) -
5/2) - 1)*log(1/64*((x - 1)*(sqrt(5) + 2*sqrt(1/2*sqrt(5) - 5/2) + 1)*(sqrt(5) - 2*sqrt(1/2*sqrt(5) - 5/2) +
1)^2 - 4*sqrt(-3/16*(sqrt(5) + 2*sqrt(1/2*sqrt(5) - 5/2) + 1)^2 - 1/8*(sqrt(5) + 2*sqrt(1/2*sqrt(5) - 5/2) - 3
)*(sqrt(5) - 2*sqrt(1/2*sqrt(5) - 5/2) + 1) - 3/16*(sqrt(5) - 2*sqrt(1/2*sqrt(5) - 5/2) + 1)^2 + 1/2*sqrt(5) +
sqrt(1/2*sqrt(5) - 5/2) - 5/2)*(x - 1)*(sqrt(5) + 2*sqrt(1/2*sqrt(5) - 5/2) + 1)*(sqrt(5) - 2*sqrt(1/2*sqrt(5
) - 5/2) + 1) + ((x - 1)*(sqrt(5) + 2*sqrt(1/2*sqrt(5) - 5/2) + 1)^2 - 4*(x - 1)*(sqrt(5) + 2*sqrt(1/2*sqrt(5)
- 5/2) + 1))*(sqrt(5) - 2*sqrt(1/2*sqrt(5) - 5/2) + 1) + 128*(x^4 - 4*x^3 + 6*x^2 - 4*x + 1)^(1/5))/(x - 1))
+ log((x + (x^4 - 4*x^3 + 6*x^2 - 4*x + 1)^(1/5) - 1)/(x - 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}^{\frac {1}{5}} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x^4-4*x^3+6*x^2-4*x+1)^(1/5),x, algorithm="giac")

[Out]

integrate(1/((x^4 - 4*x^3 + 6*x^2 - 4*x + 1)^(1/5)*x), x)

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maple [C]  time = 15.04, size = 10683, normalized size = 42.23

method	result	size
trager	\(\text {Expression too large to display}\)	\(10683\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(x^4-4*x^3+6*x^2-4*x+1)^(1/5),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}^{\frac {1}{5}} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x^4-4*x^3+6*x^2-4*x+1)^(1/5),x, algorithm="maxima")

[Out]

integrate(1/((x^4 - 4*x^3 + 6*x^2 - 4*x + 1)^(1/5)*x), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{x\,{\left (x^4-4\,x^3+6\,x^2-4\,x+1\right )}^{1/5}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(6*x^2 - 4*x - 4*x^3 + x^4 + 1)^(1/5)),x)

[Out]

int(1/(x*(6*x^2 - 4*x - 4*x^3 + x^4 + 1)^(1/5)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x**4-4*x**3+6*x**2-4*x+1)**(1/5),x)

[Out]

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

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