∫ 1_______ (8+8x−x3+8x4)3∕2 dx

3.797 \(\int \frac {1}{(8+8 x-x^3+8 x^4)^{3/2}} \, dx\)

Optimal. Leaf size=431 \[ -\frac {\left (66-\left (\frac {4}{x}+1\right )^2\right ) x^2}{1008 \sqrt {8 x^4-x^3+8 x+8}}+\frac {\left (216-7 \left (\frac {4}{x}+1\right )^2\right ) \left (\frac {4}{x}+1\right ) x^2}{12528 \sqrt {8 x^4-x^3+8 x+8}}+\frac {7 \left (\left (\frac {4}{x}+1\right )^4-6 \left (\frac {4}{x}+1\right )^2+261\right ) \left (\frac {4}{x}+1\right ) x^2}{432 \sqrt {29} \sqrt {8 x^4-x^3+8 x+8} \left (\frac {\sqrt {29} (x+4)^2}{x^2}+87\right )}+\frac {\left (14-5 \sqrt {29}\right ) \sqrt {\frac {\left (\frac {4}{x}+1\right )^4-6 \left (\frac {4}{x}+1\right )^2+261}{\left (\frac {\sqrt {29} (x+4)^2}{x^2}+87\right )^2}} \left (\frac {\sqrt {29} (x+4)^2}{x^2}+87\right ) x^2 F\left (2 \tan ^{-1}\left (\frac {x+4}{\sqrt {3} \sqrt [4]{29} x}\right )|\frac {1}{58} \left (29+\sqrt {29}\right )\right )}{576 \sqrt {3} 29^{3/4} \sqrt {8 x^4-x^3+8 x+8}}-\frac {7 \sqrt {\frac {\left (\frac {4}{x}+1\right )^4-6 \left (\frac {4}{x}+1\right )^2+261}{\left (\frac {\sqrt {29} (x+4)^2}{x^2}+87\right )^2}} \left (\frac {\sqrt {29} (x+4)^2}{x^2}+87\right ) x^2 E\left (2 \tan ^{-1}\left (\frac {x+4}{\sqrt {3} \sqrt [4]{29} x}\right )|\frac {1}{58} \left (29+\sqrt {29}\right )\right )}{144 \sqrt {3} 29^{3/4} \sqrt {8 x^4-x^3+8 x+8}} \]

[Out]

-1/1008*(66-(1+4/x)^2)*x^2/(8*x^4-x^3+8*x+8)^(1/2)+1/12528*(216-7*(1+4/x)^2)*(1+4/x)*x^2/(8*x^4-x^3+8*x+8)^(1/

2)+7/12528*(261-6*(1+4/x)^2+(1+4/x)^4)*(1+4/x)*x^2*29^(1/2)/(87+(4+x)^2*29^(1/2)/x^2)/(8*x^4-x^3+8*x+8)^(1/2)-

7/12528*x^2*(cos(2*arctan(1/87*(4+x)*29^(3/4)/x*3^(1/2)))^2)^(1/2)/cos(2*arctan(1/87*(4+x)*29^(3/4)/x*3^(1/2))

)*EllipticE(sin(2*arctan(1/87*(4+x)*29^(3/4)/x*3^(1/2))),1/58*(1682+58*29^(1/2))^(1/2))*(87+(4+x)^2*29^(1/2)/x

^2)*((261-6*(1+4/x)^2+(1+4/x)^4)/(87+(4+x)^2*29^(1/2)/x^2)^2)^(1/2)*29^(1/4)*3^(1/2)/(8*x^4-x^3+8*x+8)^(1/2)+1

/50112*x^2*(cos(2*arctan(1/87*(4+x)*29^(3/4)/x*3^(1/2)))^2)^(1/2)/cos(2*arctan(1/87*(4+x)*29^(3/4)/x*3^(1/2)))

*EllipticF(sin(2*arctan(1/87*(4+x)*29^(3/4)/x*3^(1/2))),1/58*(1682+58*29^(1/2))^(1/2))*(14-5*29^(1/2))*(87+(4+

x)^2*29^(1/2)/x^2)*((261-6*(1+4/x)^2+(1+4/x)^4)/(87+(4+x)^2*29^(1/2)/x^2)^2)^(1/2)*29^(1/4)*3^(1/2)/(8*x^4-x^3

+8*x+8)^(1/2)

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Rubi [A] time = 0.53, antiderivative size = 431, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {2069, 12, 6719, 1673, 1678, 1197, 1103, 1195, 1247, 636} \[ -\frac {\left (66-\left (\frac {4}{x}+1\right )^2\right ) x^2}{1008 \sqrt {8 x^4-x^3+8 x+8}}+\frac {\left (216-7 \left (\frac {4}{x}+1\right )^2\right ) \left (\frac {4}{x}+1\right ) x^2}{12528 \sqrt {8 x^4-x^3+8 x+8}}+\frac {7 \left (\left (\frac {4}{x}+1\right )^4-6 \left (\frac {4}{x}+1\right )^2+261\right ) \left (\frac {4}{x}+1\right ) x^2}{432 \sqrt {29} \sqrt {8 x^4-x^3+8 x+8} \left (\frac {\sqrt {29} (x+4)^2}{x^2}+87\right )}+\frac {\left (14-5 \sqrt {29}\right ) \sqrt {\frac {\left (\frac {4}{x}+1\right )^4-6 \left (\frac {4}{x}+1\right )^2+261}{\left (\frac {\sqrt {29} (x+4)^2}{x^2}+87\right )^2}} \left (\frac {\sqrt {29} (x+4)^2}{x^2}+87\right ) x^2 F\left (2 \tan ^{-1}\left (\frac {x+4}{\sqrt {3} \sqrt [4]{29} x}\right )|\frac {1}{58} \left (29+\sqrt {29}\right )\right )}{576 \sqrt {3} 29^{3/4} \sqrt {8 x^4-x^3+8 x+8}}-\frac {7 \sqrt {\frac {\left (\frac {4}{x}+1\right )^4-6 \left (\frac {4}{x}+1\right )^2+261}{\left (\frac {\sqrt {29} (x+4)^2}{x^2}+87\right )^2}} \left (\frac {\sqrt {29} (x+4)^2}{x^2}+87\right ) x^2 E\left (2 \tan ^{-1}\left (\frac {x+4}{\sqrt {3} \sqrt [4]{29} x}\right )|\frac {1}{58} \left (29+\sqrt {29}\right )\right )}{144 \sqrt {3} 29^{3/4} \sqrt {8 x^4-x^3+8 x+8}} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-((66 - (1 + 4/x)^2)*x^2)/(1008*Sqrt[8 + 8*x - x^3 + 8*x^4]) + ((216 - 7*(1 + 4/x)^2)*(1 + 4/x)*x^2)/(12528*Sq

rt[8 + 8*x - x^3 + 8*x^4]) + (7*(261 - 6*(1 + 4/x)^2 + (1 + 4/x)^4)*(1 + 4/x)*x^2)/(432*Sqrt[29]*Sqrt[8 + 8*x

- x^3 + 8*x^4]*(87 + (Sqrt[29]*(4 + x)^2)/x^2)) - (7*x^2*Sqrt[(261 - 6*(1 + 4/x)^2 + (1 + 4/x)^4)/(87 + (Sqrt[

29]*(4 + x)^2)/x^2)^2]*(87 + (Sqrt[29]*(4 + x)^2)/x^2)*EllipticE[2*ArcTan[(4 + x)/(Sqrt[3]*29^(1/4)*x)], (29 +

Sqrt[29])/58])/(144*Sqrt[3]*29^(3/4)*Sqrt[8 + 8*x - x^3 + 8*x^4]) + ((14 - 5*Sqrt[29])*x^2*Sqrt[(261 - 6*(1 +

4/x)^2 + (1 + 4/x)^4)/(87 + (Sqrt[29]*(4 + x)^2)/x^2)^2]*(87 + (Sqrt[29]*(4 + x)^2)/x^2)*EllipticF[2*ArcTan[(

4 + x)/(Sqrt[3]*29^(1/4)*x)], (29 + Sqrt[29])/58])/(576*Sqrt[3]*29^(3/4)*Sqrt[8 + 8*x - x^3 + 8*x^4])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 636

Int[((d_.) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(-2*(b*d - 2*a*e + (2*c*

d - b*e)*x))/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] &&

NeQ[b^2 - 4*a*c, 0]

Rule 1103

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(

a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2 - (b*q^2)/(4*c)])/(2*q*Sqrt[a + b*x^2 + c

*x^4]), x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 1195

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, -Simp[

(d*x*Sqrt[a + b*x^2 + c*x^4])/(a*(1 + q^2*x^2)), x] + Simp[(d*(1 + q^2*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q

^2*x^2)^2)]*EllipticE[2*ArcTan[q*x], 1/2 - (b*q^2)/(4*c)])/(q*Sqrt[a + b*x^2 + c*x^4]), x] /; EqQ[e + d*q^2, 0

]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 1197

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 2]}, Dist[(

e + d*q)/q, Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] - Dist[e/q, Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x]

/; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 1247

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[

Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rule 1673

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], k}, Int[Sum[Coeff[

Pq, x, 2*k]*x^(2*k), {k, 0, q/2}]*(a + b*x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pq, x, 2*k + 1]*x^(2*k), {k, 0,

(q - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && !PolyQ[Pq, x^2]

Rule 1678

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainder[Pq, a +

b*x^2 + c*x^4, x], x, 0], e = Coeff[PolynomialRemainder[Pq, a + b*x^2 + c*x^4, x], x, 2]}, Simp[(x*(a + b*x^2

+ c*x^4)^(p + 1)*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*

a*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x^2 + c*x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuot

ient[Pq, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4*p + 7)*(b*d - 2*a*e)*x^2,

x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && Expon[Pq, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1

]

Rule 2069

Int[(P4_)^(p_), x_Symbol] :> With[{a = Coeff[P4, x, 0], b = Coeff[P4, x, 1], c = Coeff[P4, x, 2], d = Coeff[P4

, x, 3], e = Coeff[P4, x, 4]}, Dist[-16*a^2, Subst[Int[(1*((a*(-3*b^4 + 16*a*b^2*c - 64*a^2*b*d + 256*a^3*e -

32*a^2*(3*b^2 - 8*a*c)*x^2 + 256*a^4*x^4))/(b - 4*a*x)^4)^p)/(b - 4*a*x)^2, x], x, b/(4*a) + 1/x], x] /; NeQ[a

, 0] && NeQ[b, 0] && EqQ[b^3 - 4*a*b*c + 8*a^2*d, 0]] /; FreeQ[p, x] && PolyQ[P4, x, 4] && IntegerQ[2*p] && !

IGtQ[p, 0]

Rule 6719

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n)^FracPart[p])/(v^(m*F

racPart[p])*w^(n*FracPart[p])), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] && !IntegerQ[p] && !

FreeQ[v, x] && !FreeQ[w, x]

Rubi steps

\begin {align*} \int \frac {1}{\left (8+8 x-x^3+8 x^4\right )^{3/2}} \, dx &=-\left (1024 \operatorname {Subst}\left (\int \frac {1}{16 \sqrt {2} (8-32 x)^2 \left (\frac {1069056-393216 x^2+1048576 x^4}{(8-32 x)^4}\right )^{3/2}} \, dx,x,\frac {1}{4}+\frac {1}{x}\right )\right )\\ &=-\left (\left (32 \sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{(8-32 x)^2 \left (\frac {1069056-393216 x^2+1048576 x^4}{(8-32 x)^4}\right )^{3/2}} \, dx,x,\frac {1}{4}+\frac {1}{x}\right )\right )\\ &=-\frac {\left (\sqrt {1069056-393216 \left (\frac {1}{4}+\frac {1}{x}\right )^2+1048576 \left (\frac {1}{4}+\frac {1}{x}\right )^4} x^2\right ) \operatorname {Subst}\left (\int \frac {(8-32 x)^4}{\left (1069056-393216 x^2+1048576 x^4\right )^{3/2}} \, dx,x,\frac {1}{4}+\frac {1}{x}\right )}{8 \sqrt {8+8 x-x^3+8 x^4}}\\ &=-\frac {\left (\sqrt {1069056-393216 \left (\frac {1}{4}+\frac {1}{x}\right )^2+1048576 \left (\frac {1}{4}+\frac {1}{x}\right )^4} x^2\right ) \operatorname {Subst}\left (\int \frac {x \left (-65536-1048576 x^2\right )}{\left (1069056-393216 x^2+1048576 x^4\right )^{3/2}} \, dx,x,\frac {1}{4}+\frac {1}{x}\right )}{8 \sqrt {8+8 x-x^3+8 x^4}}-\frac {\left (\sqrt {1069056-393216 \left (\frac {1}{4}+\frac {1}{x}\right )^2+1048576 \left (\frac {1}{4}+\frac {1}{x}\right )^4} x^2\right ) \operatorname {Subst}\left (\int \frac {4096+393216 x^2+1048576 x^4}{\left (1069056-393216 x^2+1048576 x^4\right )^{3/2}} \, dx,x,\frac {1}{4}+\frac {1}{x}\right )}{8 \sqrt {8+8 x-x^3+8 x^4}}\\ &=\frac {\left (216-7 \left (1+\frac {4}{x}\right )^2\right ) \left (1+\frac {4}{x}\right ) x^2}{12528 \sqrt {8+8 x-x^3+8 x^4}}-\frac {\left (\sqrt {1069056-393216 \left (\frac {1}{4}+\frac {1}{x}\right )^2+1048576 \left (\frac {1}{4}+\frac {1}{x}\right )^4} x^2\right ) \operatorname {Subst}\left (\int \frac {2571273912251842560-1324058290446925824 x^2}{\sqrt {1069056-393216 x^2+1048576 x^4}} \, dx,x,\frac {1}{4}+\frac {1}{x}\right )}{37026344336426532864 \sqrt {8+8 x-x^3+8 x^4}}-\frac {\left (\sqrt {1069056-393216 \left (\frac {1}{4}+\frac {1}{x}\right )^2+1048576 \left (\frac {1}{4}+\frac {1}{x}\right )^4} x^2\right ) \operatorname {Subst}\left (\int \frac {-65536-1048576 x}{\left (1069056-393216 x+1048576 x^2\right )^{3/2}} \, dx,x,\left (\frac {1}{4}+\frac {1}{x}\right )^2\right )}{16 \sqrt {8+8 x-x^3+8 x^4}}\\ &=-\frac {\left (66-\left (1+\frac {4}{x}\right )^2\right ) x^2}{1008 \sqrt {8+8 x-x^3+8 x^4}}+\frac {\left (216-7 \left (1+\frac {4}{x}\right )^2\right ) \left (1+\frac {4}{x}\right ) x^2}{12528 \sqrt {8+8 x-x^3+8 x^4}}-\frac {\left (7 \sqrt {1069056-393216 \left (\frac {1}{4}+\frac {1}{x}\right )^2+1048576 \left (\frac {1}{4}+\frac {1}{x}\right )^4} x^2\right ) \operatorname {Subst}\left (\int \frac {1-\frac {16 x^2}{3 \sqrt {29}}}{\sqrt {1069056-393216 x^2+1048576 x^4}} \, dx,x,\frac {1}{4}+\frac {1}{x}\right )}{36 \sqrt {29} \sqrt {8+8 x-x^3+8 x^4}}-\frac {\left (\left (145-14 \sqrt {29}\right ) \sqrt {1069056-393216 \left (\frac {1}{4}+\frac {1}{x}\right )^2+1048576 \left (\frac {1}{4}+\frac {1}{x}\right )^4} x^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1069056-393216 x^2+1048576 x^4}} \, dx,x,\frac {1}{4}+\frac {1}{x}\right )}{2088 \sqrt {8+8 x-x^3+8 x^4}}\\ &=-\frac {\left (66-\left (1+\frac {4}{x}\right )^2\right ) x^2}{1008 \sqrt {8+8 x-x^3+8 x^4}}+\frac {\left (216-7 \left (1+\frac {4}{x}\right )^2\right ) \left (1+\frac {4}{x}\right ) x^2}{12528 \sqrt {8+8 x-x^3+8 x^4}}+\frac {7 \left (261-6 \left (1+\frac {4}{x}\right )^2+\left (1+\frac {4}{x}\right )^4\right ) \left (1+\frac {4}{x}\right ) x^2}{432 \sqrt {29} \sqrt {8+8 x-x^3+8 x^4} \left (87+\frac {\sqrt {29} (4+x)^2}{x^2}\right )}-\frac {7 x^2 \sqrt {\frac {261-6 \left (1+\frac {4}{x}\right )^2+\left (1+\frac {4}{x}\right )^4}{\left (87+\frac {\sqrt {29} (4+x)^2}{x^2}\right )^2}} \left (87+\frac {\sqrt {29} (4+x)^2}{x^2}\right ) E\left (2 \tan ^{-1}\left (\frac {4+x}{\sqrt {3} \sqrt [4]{29} x}\right )|\frac {1}{58} \left (29+\sqrt {29}\right )\right )}{144 \sqrt {3} 29^{3/4} \sqrt {8+8 x-x^3+8 x^4}}+\frac {\left (14-5 \sqrt {29}\right ) x^2 \sqrt {\frac {261-6 \left (1+\frac {4}{x}\right )^2+\left (1+\frac {4}{x}\right )^4}{\left (87+\frac {\sqrt {29} (4+x)^2}{x^2}\right )^2}} \left (87+\frac {\sqrt {29} (4+x)^2}{x^2}\right ) F\left (2 \tan ^{-1}\left (\frac {4+x}{\sqrt {3} \sqrt [4]{29} x}\right )|\frac {1}{58} \left (29+\sqrt {29}\right )\right )}{576 \sqrt {3} 29^{3/4} \sqrt {8+8 x-x^3+8 x^4}}\\ \end {align*}

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Mathematica [C] time = 6.05, size = 4865, normalized size = 11.29 \[ \text {Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(544 + 1539*x - 1146*x^2 + 784*x^3)/(21924*Sqrt[8 + 8*x - x^3 + 8*x^4]) + ((28*(x - Root[8 + 8*#1 - #1^3 + 8*#

1^4 & , 2, 0])^2*(-(EllipticF[ArcSin[Sqrt[((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0])*(Root[8 + 8*#1 - #1^3

+ 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])

*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))]], -(((Root[8 + 8*#1 - #

1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - R

oot[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8

*#1^4 & , 3, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0])))]*Root[8

+ 8*#1 - #1^3 + 8*#1^4 & , 2, 0]) + EllipticPi[(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1

^3 + 8*#1^4 & , 4, 0])/(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]), A

rcSin[Sqrt[((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 +

8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4

& , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))]], -(((Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8

+ 8*#1 - #1^3 + 8*#1^4 & , 3, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 &

, 4, 0]))/((-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0])*(Root[8 + 8*#1

- #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0])))]*(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1

, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0]))*Sqrt[((-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*

#1 - #1^3 + 8*#1^4 & , 2, 0])*(x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0]))/((x - Root[8 + 8*#1 - #1^3 + 8*#1

^4 & , 2, 0])*(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0]))]*(Root[8 +

8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0])*Sqrt[((x - Root[8 + 8*#1 - #1^3 + 8

*#1^4 & , 1, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((x - Ro

ot[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^

4 & , 4, 0]))]*Sqrt[((-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(x -

Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(-Root[8 + 8*#1 - #1

^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))])/(Sqrt[8 + 8*x - x^3 + 8*x^4]*(-Root[8 + 8*

#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2,

0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0])) + (842*EllipticF[ArcSin[Sqrt[((x - Root[8 + 8*#1 - #1^3 + 8*#1^

4 & , 1, 0])*(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((x - Root[

8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4

& , 4, 0]))]], ((Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0])*(Root[8 +

8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((Root[8 + 8*#1 - #1^3 + 8*#1^4 & ,

1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #

1^3 + 8*#1^4 & , 4, 0]))]*(x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])^2*Sqrt[((-Root[8 + 8*#1 - #1^3 + 8*#1^

4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0]))/((x - Ro

ot[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1

^4 & , 3, 0]))]*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0])*Sqrt[((-Ro

ot[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(x - Root[8 + 8*#1 - #1^3 + 8

*#1^4 & , 4, 0]))/((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + R

oot[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))]*Sqrt[((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0])*(-Root[8 + 8*#1

- #1^3 + 8*#1^4 & , 2, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & ,

2, 0])*(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))])/(Sqrt[8 + 8*x -

x^3 + 8*x^4]*(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(-Root[8 +

8*#1 - #1^3 + 8*#1^4 & , 2, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0])) - (224*((x - Root[8 + 8*#1 - #1^3 +

8*#1^4 & , 1, 0])*(x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0])*(x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0])

+ (x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])^2*Sqrt[((-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 +

8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0]))/((x - Root[8 + 8*#1 - #1^3 + 8*

#1^4 & , 2, 0])*(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0]))]*Sqrt[((

x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 +

8*#1^4 & , 4, 0]))/((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] -

Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))]*Sqrt[((-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #

1^3 + 8*#1^4 & , 2, 0])*(x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & ,

2, 0])*(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))]*(-Root[8 + 8*#1

- #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0])*((EllipticE[ArcSin[Sqrt[((x - Root[8 + 8

*#1 - #1^3 + 8*#1^4 & , 1, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4,

0]))/((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1

- #1^3 + 8*#1^4 & , 4, 0]))]], -(((Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & ,

3, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((-Root[8 + 8*#1

- #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0]

- Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0])))]*(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3

+ 8*#1^4 & , 3, 0]))/(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0]) + (

EllipticF[ArcSin[Sqrt[((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0]

- Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(Root[8 + 8*#1 - #1

^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))]], -(((Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2,

0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3

+ 8*#1^4 & , 4, 0]))/((-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0])*(Ro

ot[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0])))]*(Root[8 + 8*#1 - #1^3 + 8*

#1^4 & , 2, 0]*(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]) - Root[8 +

8*#1 - #1^3 + 8*#1^4 & , 1, 0]*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4

, 0])))/((-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(-Root[8 + 8*#1

- #1^3 + 8*#1^4 & , 2, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0])) - (EllipticPi[(-Root[8 + 8*#1 - #1^3 + 8

*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0])/(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] + Root[8

+ 8*#1 - #1^3 + 8*#1^4 & , 4, 0]), ArcSin[Sqrt[((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0])*(Root[8 + 8*#1 -

#1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2

, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))]], -(((Root[8 + 8*#

1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0

] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^

3 + 8*#1^4 & , 3, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0])))]*(

-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*

#1^4 & , 3, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] + Root[8

+ 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))))/Sqrt[8 + 8*x - x^3 + 8*x^4])/6264

________________________________________________________________________________________

fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {8 \, x^{4} - x^{3} + 8 \, x + 8}}{64 \, x^{8} - 16 \, x^{7} + x^{6} + 128 \, x^{5} + 112 \, x^{4} - 16 \, x^{3} + 64 \, x^{2} + 128 \, x + 64}, x\right ) \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(8*x^4 - x^3 + 8*x + 8)/(64*x^8 - 16*x^7 + x^6 + 128*x^5 + 112*x^4 - 16*x^3 + 64*x^2 + 128*x + 64

), x)

________________________________________________________________________________________

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [C] time = 0.12, size = 4426, normalized size = 10.27 \[ \text {output too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-16*(-17/10962-57/12992*x+191/58464*x^2-7/3132*x^3)/(8*x^4-x^3+8*x+8)^(1/2)+421/12528*(-RootOf(8*_Z^4-_Z^3+8*_

Z+8,index=4)+RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))*((RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8*_Z^4-_Z^3+8*_Z+

8,index=2))*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8*_Z^4-_Z^3+8*_Z

+8,index=1))/(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)))^(1/2)*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2))^2*((RootOf(8

*_Z^4-_Z^3+8*_Z+8,index=2)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=3))/(RootOf(

8*_Z^4-_Z^3+8*_Z+8,index=3)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)))^(1/2)*

((RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)

)/(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2

)))^(1/2)/(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2))/(RootOf(8*_Z^4-_Z^3+8*_Z+8,i

ndex=2)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))*2^(1/2)/((x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))*(x-RootOf(8*_Z^4-_

Z^3+8*_Z+8,index=2))*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=3))*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)))^(1/2)*Elli

pticF(((RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2))*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,in

dex=1))/(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,i

ndex=2)))^(1/2),((RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=3))*(-RootOf(8*_Z^4-_Z^3+

8*_Z+8,index=4)+RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=3)+RootOf(8*_Z^4-_Z^3+8*

_Z+8,index=1))/(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)))^(1/2))+7/6264*(-RootOf

(8*_Z^4-_Z^3+8*_Z+8,index=4)+RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))*((RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8

*_Z^4-_Z^3+8*_Z+8,index=2))*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(

8*_Z^4-_Z^3+8*_Z+8,index=1))/(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)))^(1/2)*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=

2))^2*((RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,in

dex=3))/(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=3)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,i

ndex=2)))^(1/2)*((RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))*(x-RootOf(8*_Z^4-_Z^3

+8*_Z+8,index=4))/(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(x-RootOf(8*_Z^4-_Z^

3+8*_Z+8,index=2)))^(1/2)/(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2))/(RootOf(8*_Z

^4-_Z^3+8*_Z+8,index=2)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))*2^(1/2)/((x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))*(x

-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2))*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=3))*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index

=4)))^(1/2)*(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)*EllipticF(((RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8*_Z^4-_

Z^3+8*_Z+8,index=2))*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8*_Z^4-

_Z^3+8*_Z+8,index=1))/(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)))^(1/2),((RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)-RootO

f(8*_Z^4-_Z^3+8*_Z+8,index=3))*(-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)+RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(-Root

Of(8*_Z^4-_Z^3+8*_Z+8,index=3)+RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)-RootOf(

8*_Z^4-_Z^3+8*_Z+8,index=4)))^(1/2))+(-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)+RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))*

EllipticPi(((RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2))*(x-RootOf(8*_Z^4-_Z^3+8*_Z

+8,index=1))/(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(x-RootOf(8*_Z^4-_Z^3+8*_

Z+8,index=2)))^(1/2),(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(RootOf(8*_Z^4-_Z

^3+8*_Z+8,index=4)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)),((RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)-RootOf(8*_Z^4-_Z^3

+8*_Z+8,index=3))*(-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)+RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(-RootOf(8*_Z^4-_Z^

3+8*_Z+8,index=3)+RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)-RootOf(8*_Z^4-_Z^3+8

*_Z+8,index=4)))^(1/2)))-7/783*((x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=3))*

(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4))+(-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)+RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1)

)*((RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2))*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=

1))/(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index

=2)))^(1/2)*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2))^2*((RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)-RootOf(8*_Z^4-_Z^3+8

*_Z+8,index=1))*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=3))/(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=3)-RootOf(8*_Z^4-_Z^3+

8*_Z+8,index=1))/(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)))^(1/2)*((RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)-RootOf(8*_

Z^4-_Z^3+8*_Z+8,index=1))*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4))/(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8*

_Z^4-_Z^3+8*_Z+8,index=1))/(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)))^(1/2)*((RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)*

RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1)*RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)+RootO

f(8*_Z^4-_Z^3+8*_Z+8,index=2)*RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)+RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)^2)/(RootOf

(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2))/(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)-RootOf(8*

_Z^4-_Z^3+8*_Z+8,index=1))*EllipticF(((RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2))*

(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))

/(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)))^(1/2),((RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)-RootOf(8*_Z^4-_Z^3+8*_Z+8,

index=3))*(-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)+RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(-RootOf(8*_Z^4-_Z^3+8*_Z+8

,index=3)+RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)-RootOf(8*_Z^4-_Z^3+8*_Z+8,in

dex=4)))^(1/2))+(-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=3)+RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))*EllipticE(((RootOf(8*

_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2))*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(RootOf(8

*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)))^(1/2),(

(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=3))*(-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)+R

ootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=3)+RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(R

ootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)))^(1/2))/(RootOf(8*_Z^4-_Z^3+8*_Z+8,index

=2)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))-1/8/(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8*_Z^4-_Z^3+8*_Z+8,inde

x=2))*EllipticPi(((RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2))*(x-RootOf(8*_Z^4-_Z^

3+8*_Z+8,index=1))/(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(x-RootOf(8*_Z^4-_Z

^3+8*_Z+8,index=2)))^(1/2),(-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)+RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(RootOf(8*

_Z^4-_Z^3+8*_Z+8,index=2)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)),((RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)-RootOf(8*_Z

^4-_Z^3+8*_Z+8,index=3))*(-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)+RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(-RootOf(8*_

Z^4-_Z^3+8*_Z+8,index=3)+RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)-RootOf(8*_Z^4

-_Z^3+8*_Z+8,index=4)))^(1/2))))*2^(1/2)/((x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,

index=2))*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=3))*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)))^(1/2)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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