3.8.97 \(\int \frac {1}{(8+8 x-x^3+8 x^4)^{3/2}} \, dx\) [797]
Optimal. Leaf size=431 \[ -\frac {\left (66-\left (1+\frac {4}{x}\right )^2\right ) x^2}{1008 \sqrt {8+8 x-x^3+8 x^4}}+\frac {\left (216-7 \left (1+\frac {4}{x}\right )^2\right ) \left (1+\frac {4}{x}\right ) x^2}{12528 \sqrt {8+8 x-x^3+8 x^4}}+\frac {7 \left (261-6 \left (1+\frac {4}{x}\right )^2+\left (1+\frac {4}{x}\right )^4\right ) \left (1+\frac {4}{x}\right ) x^2}{432 \sqrt {29} \sqrt {8+8 x-x^3+8 x^4} \left (87+\frac {\sqrt {29} (4+x)^2}{x^2}\right )}-\frac {7 x^2 \sqrt {\frac {261-6 \left (1+\frac {4}{x}\right )^2+\left (1+\frac {4}{x}\right )^4}{\left (87+\frac {\sqrt {29} (4+x)^2}{x^2}\right )^2}} \left (87+\frac {\sqrt {29} (4+x)^2}{x^2}\right ) E\left (2 \tan ^{-1}\left (\frac {4+x}{\sqrt {3} \sqrt [4]{29} x}\right )|\frac {1}{58} \left (29+\sqrt {29}\right )\right )}{144 \sqrt {3} 29^{3/4} \sqrt {8+8 x-x^3+8 x^4}}+\frac {\left (14-5 \sqrt {29}\right ) x^2 \sqrt {\frac {261-6 \left (1+\frac {4}{x}\right )^2+\left (1+\frac {4}{x}\right )^4}{\left (87+\frac {\sqrt {29} (4+x)^2}{x^2}\right )^2}} \left (87+\frac {\sqrt {29} (4+x)^2}{x^2}\right ) F\left (2 \tan ^{-1}\left (\frac {4+x}{\sqrt {3} \sqrt [4]{29} x}\right )|\frac {1}{58} \left (29+\sqrt {29}\right )\right )}{576 \sqrt {3} 29^{3/4} \sqrt {8+8 x-x^3+8 x^4}} \]
[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)
________________________________________________________________________________________
Rubi [A]
time = 0.37, 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 = {2094, 12,
6851, 1687, 1692, 1211, 1117, 1209, 1261, 650} \begin {gather*} \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 \text {ArcTan}\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 \text {ArcTan}\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}}-\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 )} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(8 + 8*x - x^3 + 8*x^4)^(-3/2),x]
[Out]
-1/1008*((66 - (1 + 4/x)^2)*x^2)/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 650
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 1117
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)]/(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(
4*c))], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Rule 1209
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)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c))], 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 1211
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 1261
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 1687
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 1692
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 2094
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/(b - 4*a*x)^2)*(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, 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 6851
Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a*v^m*w^n)^FracPart[p]/(v^(m*Fr
acPart[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 \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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] Result contains higher order function than in optimal. Order 9 vs. order 4 in
optimal.
time = 16.04, size = 4865, normalized size = 11.29 \begin {gather*} \text {Result too large to show} \end {gather*}
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...
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.65, size = 4426, normalized size = 10.27
| | |
| method |
result |
size |
| | |
| default |
\(\text {Expression too large to display}\) |
\(4426\) |
| risch |
\(\text {Expression too large to display}\) |
\(4426\) |
| elliptic |
\(\text {Expression too large to display}\) |
\(4426\) |
| | |
|
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|
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(8*x^4-x^3+8*x+8)^(3/2),x,method=_RETURNVERBOSE)
[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=1)-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=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,in
dex=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)*Ellip
ticF(((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,ind
ex=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,in
dex=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=1)-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=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=1)-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=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,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-Roo
tOf(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)-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=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=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))*Ellipt
icPi(((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,ind
ex=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,in
dex=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=1)-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=2)-RootOf(8*_Z^4-_Z^3+8*_Z+8,in
dex=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-RootO
f(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))*((RootO
f(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))/(Root
Of(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,ind
ex=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,in
dex=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,i...
________________________________________________________________________________________
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(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)
________________________________________________________________________________________
Fricas [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(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)
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (8 x^{4} - x^{3} + 8 x + 8\right )^{\frac {3}{2}}}\, dx \end {gather*}
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)
________________________________________________________________________________________
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(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)
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (8\,x^4-x^3+8\,x+8\right )}^{3/2}} \,d x \end {gather*}
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)
________________________________________________________________________________________