3.22.83 \(\int \frac {(-8+x^5) (2+x^5) \sqrt [4]{2-3 x^4+x^5}}{x^6 (4-3 x^4+2 x^5)} \, dx\)

Optimal. Leaf size=162 \[ \frac {\sqrt [4]{x^5-3 x^4+2} \left (2 x^5+9 x^4+4\right )}{5 x^5}+\frac {3 \sqrt [4]{3} \tan ^{-1}\left (\frac {6^{3/4} x \sqrt [4]{x^5-3 x^4+2}}{\sqrt {6} \sqrt {x^5-3 x^4+2}-3 x^2}\right )}{2\ 2^{3/4}}-\frac {3 \sqrt [4]{3} \tanh ^{-1}\left (\frac {6^{3/4} x \sqrt [4]{x^5-3 x^4+2}}{3 x^2+\sqrt {6} \sqrt {x^5-3 x^4+2}}\right )}{2\ 2^{3/4}} \]

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Rubi [F]  time = 1.53, 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 (-8+x^5\right ) \left (2+x^5\right ) \sqrt [4]{2-3 x^4+x^5}}{x^6 \left (4-3 x^4+2 x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-8 + x^5)*(2 + x^5)*(2 - 3*x^4 + x^5)^(1/4))/(x^6*(4 - 3*x^4 + 2*x^5)),x]

[Out]

-4*Defer[Int][(2 - 3*x^4 + x^5)^(1/4)/x^6, x] - 3*Defer[Int][(2 - 3*x^4 + x^5)^(1/4)/x^2, x] + Defer[Int][(2 -
3*x^4 + x^5)^(1/4)/x, x]/2 - 9*Defer[Int][(x^2*(2 - 3*x^4 + x^5)^(1/4))/(4 - 3*x^4 + 2*x^5), x] + (15*Defer[I
nt][(x^3*(2 - 3*x^4 + x^5)^(1/4))/(4 - 3*x^4 + 2*x^5), x])/2

Rubi steps

\begin {align*} \int \frac {\left (-8+x^5\right ) \left (2+x^5\right ) \sqrt [4]{2-3 x^4+x^5}}{x^6 \left (4-3 x^4+2 x^5\right )} \, dx &=\int \left (-\frac {4 \sqrt [4]{2-3 x^4+x^5}}{x^6}-\frac {3 \sqrt [4]{2-3 x^4+x^5}}{x^2}+\frac {\sqrt [4]{2-3 x^4+x^5}}{2 x}+\frac {3 x^2 (-6+5 x) \sqrt [4]{2-3 x^4+x^5}}{2 \left (4-3 x^4+2 x^5\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {\sqrt [4]{2-3 x^4+x^5}}{x} \, dx+\frac {3}{2} \int \frac {x^2 (-6+5 x) \sqrt [4]{2-3 x^4+x^5}}{4-3 x^4+2 x^5} \, dx-3 \int \frac {\sqrt [4]{2-3 x^4+x^5}}{x^2} \, dx-4 \int \frac {\sqrt [4]{2-3 x^4+x^5}}{x^6} \, dx\\ &=\frac {1}{2} \int \frac {\sqrt [4]{2-3 x^4+x^5}}{x} \, dx+\frac {3}{2} \int \left (-\frac {6 x^2 \sqrt [4]{2-3 x^4+x^5}}{4-3 x^4+2 x^5}+\frac {5 x^3 \sqrt [4]{2-3 x^4+x^5}}{4-3 x^4+2 x^5}\right ) \, dx-3 \int \frac {\sqrt [4]{2-3 x^4+x^5}}{x^2} \, dx-4 \int \frac {\sqrt [4]{2-3 x^4+x^5}}{x^6} \, dx\\ &=\frac {1}{2} \int \frac {\sqrt [4]{2-3 x^4+x^5}}{x} \, dx-3 \int \frac {\sqrt [4]{2-3 x^4+x^5}}{x^2} \, dx-4 \int \frac {\sqrt [4]{2-3 x^4+x^5}}{x^6} \, dx+\frac {15}{2} \int \frac {x^3 \sqrt [4]{2-3 x^4+x^5}}{4-3 x^4+2 x^5} \, dx-9 \int \frac {x^2 \sqrt [4]{2-3 x^4+x^5}}{4-3 x^4+2 x^5} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.51, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-8+x^5\right ) \left (2+x^5\right ) \sqrt [4]{2-3 x^4+x^5}}{x^6 \left (4-3 x^4+2 x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((-8 + x^5)*(2 + x^5)*(2 - 3*x^4 + x^5)^(1/4))/(x^6*(4 - 3*x^4 + 2*x^5)),x]

[Out]

Integrate[((-8 + x^5)*(2 + x^5)*(2 - 3*x^4 + x^5)^(1/4))/(x^6*(4 - 3*x^4 + 2*x^5)), x]

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IntegrateAlgebraic [A]  time = 2.98, size = 167, normalized size = 1.03 \begin {gather*} \frac {\sqrt [4]{2-3 x^4+x^5} \left (4+9 x^4+2 x^5\right )}{5 x^5}-\frac {3 \sqrt [4]{3} \tan ^{-1}\left (\frac {-\frac {\sqrt [4]{3} x^2}{2^{3/4}}+\frac {\sqrt {2-3 x^4+x^5}}{\sqrt [4]{6}}}{x \sqrt [4]{2-3 x^4+x^5}}\right )}{2\ 2^{3/4}}-\frac {3 \sqrt [4]{3} \tanh ^{-1}\left (\frac {6^{3/4} x \sqrt [4]{2-3 x^4+x^5}}{3 x^2+\sqrt {6} \sqrt {2-3 x^4+x^5}}\right )}{2\ 2^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-8 + x^5)*(2 + x^5)*(2 - 3*x^4 + x^5)^(1/4))/(x^6*(4 - 3*x^4 + 2*x^5)),x]

[Out]

((2 - 3*x^4 + x^5)^(1/4)*(4 + 9*x^4 + 2*x^5))/(5*x^5) - (3*3^(1/4)*ArcTan[(-((3^(1/4)*x^2)/2^(3/4)) + Sqrt[2 -
3*x^4 + x^5]/6^(1/4))/(x*(2 - 3*x^4 + x^5)^(1/4))])/(2*2^(3/4)) - (3*3^(1/4)*ArcTanh[(6^(3/4)*x*(2 - 3*x^4 +
x^5)^(1/4))/(3*x^2 + Sqrt[6]*Sqrt[2 - 3*x^4 + x^5])])/(2*2^(3/4))

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fricas [B]  time = 91.25, size = 1060, normalized size = 6.54

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5-8)*(x^5+2)*(x^5-3*x^4+2)^(1/4)/x^6/(2*x^5-3*x^4+4),x, algorithm="fricas")

[Out]

-1/80*(60*3^(1/4)*2^(1/4)*x^5*arctan(-1/3*(12*x^10 - 36*x^9 + 27*x^8 + 48*x^5 - 72*x^4 + 18*3^(3/4)*2^(3/4)*(2
*x^8 - 7*x^7 + 4*x^3)*(x^5 - 3*x^4 + 2)^(1/4) + 12*sqrt(3)*sqrt(2)*(2*x^7 - 3*x^6 + 4*x^2)*sqrt(x^5 - 3*x^4 +
2) + 12*3^(1/4)*2^(1/4)*(2*x^6 - 15*x^5 + 4*x)*(x^5 - 3*x^4 + 2)^(3/4) - sqrt(3)*(48*sqrt(3)*sqrt(2)*(x^5 - 3*
x^4 + 2)^(3/4)*x^5 + 2*3^(3/4)*2^(3/4)*(2*x^7 - 15*x^6 + 4*x^2)*sqrt(x^5 - 3*x^4 + 2) + 3^(1/4)*2^(1/4)*(4*x^1
0 - 72*x^9 + 171*x^8 + 16*x^5 - 144*x^4 + 16) + 12*(2*x^8 - 3*x^7 + 4*x^3)*(x^5 - 3*x^4 + 2)^(1/4))*sqrt((12*3
^(1/4)*2^(1/4)*(x^5 - 3*x^4 + 2)^(1/4)*x^3 + 4*3^(3/4)*2^(3/4)*(x^5 - 3*x^4 + 2)^(3/4)*x + 24*sqrt(x^5 - 3*x^4
+ 2)*x^2 + sqrt(3)*sqrt(2)*(2*x^5 - 3*x^4 + 4))/(2*x^5 - 3*x^4 + 4)) + 48)/(4*x^10 - 108*x^9 + 297*x^8 + 16*x
^5 - 216*x^4 + 16)) - 60*3^(1/4)*2^(1/4)*x^5*arctan(-1/3*(12*x^10 - 36*x^9 + 27*x^8 + 48*x^5 - 72*x^4 - 18*3^(
3/4)*2^(3/4)*(2*x^8 - 7*x^7 + 4*x^3)*(x^5 - 3*x^4 + 2)^(1/4) + 12*sqrt(3)*sqrt(2)*(2*x^7 - 3*x^6 + 4*x^2)*sqrt
(x^5 - 3*x^4 + 2) - 12*3^(1/4)*2^(1/4)*(2*x^6 - 15*x^5 + 4*x)*(x^5 - 3*x^4 + 2)^(3/4) - sqrt(3)*(48*sqrt(3)*sq
rt(2)*(x^5 - 3*x^4 + 2)^(3/4)*x^5 - 2*3^(3/4)*2^(3/4)*(2*x^7 - 15*x^6 + 4*x^2)*sqrt(x^5 - 3*x^4 + 2) - 3^(1/4)
*2^(1/4)*(4*x^10 - 72*x^9 + 171*x^8 + 16*x^5 - 144*x^4 + 16) + 12*(2*x^8 - 3*x^7 + 4*x^3)*(x^5 - 3*x^4 + 2)^(1
/4))*sqrt(-(12*3^(1/4)*2^(1/4)*(x^5 - 3*x^4 + 2)^(1/4)*x^3 + 4*3^(3/4)*2^(3/4)*(x^5 - 3*x^4 + 2)^(3/4)*x - 24*
sqrt(x^5 - 3*x^4 + 2)*x^2 - sqrt(3)*sqrt(2)*(2*x^5 - 3*x^4 + 4))/(2*x^5 - 3*x^4 + 4)) + 48)/(4*x^10 - 108*x^9
+ 297*x^8 + 16*x^5 - 216*x^4 + 16)) + 15*3^(1/4)*2^(1/4)*x^5*log(3*(12*3^(1/4)*2^(1/4)*(x^5 - 3*x^4 + 2)^(1/4)
*x^3 + 4*3^(3/4)*2^(3/4)*(x^5 - 3*x^4 + 2)^(3/4)*x + 24*sqrt(x^5 - 3*x^4 + 2)*x^2 + sqrt(3)*sqrt(2)*(2*x^5 - 3
*x^4 + 4))/(2*x^5 - 3*x^4 + 4)) - 15*3^(1/4)*2^(1/4)*x^5*log(-3*(12*3^(1/4)*2^(1/4)*(x^5 - 3*x^4 + 2)^(1/4)*x^
3 + 4*3^(3/4)*2^(3/4)*(x^5 - 3*x^4 + 2)^(3/4)*x - 24*sqrt(x^5 - 3*x^4 + 2)*x^2 - sqrt(3)*sqrt(2)*(2*x^5 - 3*x^
4 + 4))/(2*x^5 - 3*x^4 + 4)) - 16*(2*x^5 + 9*x^4 + 4)*(x^5 - 3*x^4 + 2)^(1/4))/x^5

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{5} - 3 \, x^{4} + 2\right )}^{\frac {1}{4}} {\left (x^{5} + 2\right )} {\left (x^{5} - 8\right )}}{{\left (2 \, x^{5} - 3 \, x^{4} + 4\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5-8)*(x^5+2)*(x^5-3*x^4+2)^(1/4)/x^6/(2*x^5-3*x^4+4),x, algorithm="giac")

[Out]

integrate((x^5 - 3*x^4 + 2)^(1/4)*(x^5 + 2)*(x^5 - 8)/((2*x^5 - 3*x^4 + 4)*x^6), x)

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{5}-8\right ) \left (x^{5}+2\right ) \left (x^{5}-3 x^{4}+2\right )^{\frac {1}{4}}}{x^{6} \left (2 x^{5}-3 x^{4}+4\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5-8)*(x^5+2)*(x^5-3*x^4+2)^(1/4)/x^6/(2*x^5-3*x^4+4),x)

[Out]

int((x^5-8)*(x^5+2)*(x^5-3*x^4+2)^(1/4)/x^6/(2*x^5-3*x^4+4),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{5} - 3 \, x^{4} + 2\right )}^{\frac {1}{4}} {\left (x^{5} + 2\right )} {\left (x^{5} - 8\right )}}{{\left (2 \, x^{5} - 3 \, x^{4} + 4\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5-8)*(x^5+2)*(x^5-3*x^4+2)^(1/4)/x^6/(2*x^5-3*x^4+4),x, algorithm="maxima")

[Out]

integrate((x^5 - 3*x^4 + 2)^(1/4)*(x^5 + 2)*(x^5 - 8)/((2*x^5 - 3*x^4 + 4)*x^6), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (x^5+2\right )\,\left (x^5-8\right )\,{\left (x^5-3\,x^4+2\right )}^{1/4}}{x^6\,\left (2\,x^5-3\,x^4+4\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^5 + 2)*(x^5 - 8)*(x^5 - 3*x^4 + 2)^(1/4))/(x^6*(2*x^5 - 3*x^4 + 4)),x)

[Out]

int(((x^5 + 2)*(x^5 - 8)*(x^5 - 3*x^4 + 2)^(1/4))/(x^6*(2*x^5 - 3*x^4 + 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} - 2 x^{3} - 2 x^{2} - 2 x - 2\right )} \left (x^{5} - 8\right ) \left (x^{5} + 2\right )}{x^{6} \left (2 x^{5} - 3 x^{4} + 4\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**5-8)*(x**5+2)*(x**5-3*x**4+2)**(1/4)/x**6/(2*x**5-3*x**4+4),x)

[Out]

Integral(((x - 1)*(x**4 - 2*x**3 - 2*x**2 - 2*x - 2))**(1/4)*(x**5 - 8)*(x**5 + 2)/(x**6*(2*x**5 - 3*x**4 + 4)
), x)

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