3.9.4 \(\int \frac {1}{(9-6 x-44 x^2+15 x^3+3 x^4)^{3/2}} \, dx\) [804]
Optimal. Leaf size=444 \[ -\frac {\left (176-23 \left (1-\frac {6}{x}\right )^2\right ) x^2}{51759 \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}+\frac {\left (45401-3722 \left (1-\frac {6}{x}\right )^2\right ) \left (1-\frac {6}{x}\right ) x^2}{31728267 \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}+\frac {3722 \left (613-182 \left (1-\frac {6}{x}\right )^2+\left (-1+\frac {6}{x}\right )^4\right ) \left (1-\frac {6}{x}\right ) x^2}{31728267 \left (\sqrt {613}+\frac {(6-x)^2}{x^2}\right ) \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}+\frac {3722 \sqrt {\frac {613-182 \left (1-\frac {6}{x}\right )^2+\left (-1+\frac {6}{x}\right )^4}{\left (\sqrt {613}+\frac {(6-x)^2}{x^2}\right )^2}} \left (\sqrt {613}+\frac {(6-x)^2}{x^2}\right ) x^2 E\left (2 \tan ^{-1}\left (\frac {6-x}{\sqrt [4]{613} x}\right )|\frac {613+91 \sqrt {613}}{1226}\right )}{51759\ 613^{3/4} \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}-\frac {\left (7444-145 \sqrt {613}\right ) \sqrt {\frac {613-182 \left (1-\frac {6}{x}\right )^2+\left (-1+\frac {6}{x}\right )^4}{\left (\sqrt {613}+\frac {(6-x)^2}{x^2}\right )^2}} \left (\sqrt {613}+\frac {(6-x)^2}{x^2}\right ) x^2 F\left (2 \tan ^{-1}\left (\frac {6-x}{\sqrt [4]{613} x}\right )|\frac {613+91 \sqrt {613}}{1226}\right )}{207036\ 613^{3/4} \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}} \]
[Out]
-1/51759*(176-23*(1-6/x)^2)*x^2/(3*x^4+15*x^3-44*x^2-6*x+9)^(1/2)+1/31728267*(45401-3722*(1-6/x)^2)*(1-6/x)*x^
2/(3*x^4+15*x^3-44*x^2-6*x+9)^(1/2)+3722/31728267*(613-182*(1-6/x)^2+(-1+6/x)^4)*(1-6/x)*x^2/((6-x)^2/x^2+613^
(1/2))/(3*x^4+15*x^3-44*x^2-6*x+9)^(1/2)+3722/31728267*x^2*(cos(2*arctan(1/613*(6-x)*613^(3/4)/x))^2)^(1/2)/co
s(2*arctan(1/613*(6-x)*613^(3/4)/x))*EllipticE(sin(2*arctan(1/613*(6-x)*613^(3/4)/x)),1/1226*(751538+111566*61
3^(1/2))^(1/2))*((6-x)^2/x^2+613^(1/2))*((613-182*(1-6/x)^2+(-1+6/x)^4)/((6-x)^2/x^2+613^(1/2))^2)^(1/2)*613^(
1/4)/(3*x^4+15*x^3-44*x^2-6*x+9)^(1/2)-1/126913068*x^2*(cos(2*arctan(1/613*(6-x)*613^(3/4)/x))^2)^(1/2)/cos(2*
arctan(1/613*(6-x)*613^(3/4)/x))*EllipticF(sin(2*arctan(1/613*(6-x)*613^(3/4)/x)),1/1226*(751538+111566*613^(1
/2))^(1/2))*(7444-145*613^(1/2))*((6-x)^2/x^2+613^(1/2))*((613-182*(1-6/x)^2+(-1+6/x)^4)/((6-x)^2/x^2+613^(1/2
))^2)^(1/2)*613^(1/4)/(3*x^4+15*x^3-44*x^2-6*x+9)^(1/2)
________________________________________________________________________________________
Rubi [A]
time = 0.32, antiderivative size = 444, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2094, 12,
6851, 1687, 1692, 1197, 1110, 1196, 1261, 650} \begin {gather*} -\frac {\left (7444-145 \sqrt {613}\right ) \sqrt {\frac {\left (\frac {6}{x}-1\right )^4-182 \left (1-\frac {6}{x}\right )^2+613}{\left (\frac {(6-x)^2}{x^2}+\sqrt {613}\right )^2}} \left (\frac {(6-x)^2}{x^2}+\sqrt {613}\right ) x^2 F\left (2 \text {ArcTan}\left (\frac {6-x}{\sqrt [4]{613} x}\right )|\frac {613+91 \sqrt {613}}{1226}\right )}{207036\ 613^{3/4} \sqrt {3 x^4+15 x^3-44 x^2-6 x+9}}+\frac {3722 \sqrt {\frac {\left (\frac {6}{x}-1\right )^4-182 \left (1-\frac {6}{x}\right )^2+613}{\left (\frac {(6-x)^2}{x^2}+\sqrt {613}\right )^2}} \left (\frac {(6-x)^2}{x^2}+\sqrt {613}\right ) x^2 E\left (2 \text {ArcTan}\left (\frac {6-x}{\sqrt [4]{613} x}\right )|\frac {613+91 \sqrt {613}}{1226}\right )}{51759\ 613^{3/4} \sqrt {3 x^4+15 x^3-44 x^2-6 x+9}}-\frac {\left (176-23 \left (1-\frac {6}{x}\right )^2\right ) x^2}{51759 \sqrt {3 x^4+15 x^3-44 x^2-6 x+9}}+\frac {\left (45401-3722 \left (1-\frac {6}{x}\right )^2\right ) \left (1-\frac {6}{x}\right ) x^2}{31728267 \sqrt {3 x^4+15 x^3-44 x^2-6 x+9}}+\frac {3722 \left (\left (\frac {6}{x}-1\right )^4-182 \left (1-\frac {6}{x}\right )^2+613\right ) \left (1-\frac {6}{x}\right ) x^2}{31728267 \left (\frac {(6-x)^2}{x^2}+\sqrt {613}\right ) \sqrt {3 x^4+15 x^3-44 x^2-6 x+9}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4)^(-3/2),x]
[Out]
-1/51759*((176 - 23*(1 - 6/x)^2)*x^2)/Sqrt[9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4] + ((45401 - 3722*(1 - 6/x)^2)*(1
- 6/x)*x^2)/(31728267*Sqrt[9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4]) + (3722*(613 - 182*(1 - 6/x)^2 + (-1 + 6/x)^4)
*(1 - 6/x)*x^2)/(31728267*(Sqrt[613] + (6 - x)^2/x^2)*Sqrt[9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4]) + (3722*Sqrt[(6
13 - 182*(1 - 6/x)^2 + (-1 + 6/x)^4)/(Sqrt[613] + (6 - x)^2/x^2)^2]*(Sqrt[613] + (6 - x)^2/x^2)*x^2*EllipticE[
2*ArcTan[(6 - x)/(613^(1/4)*x)], (613 + 91*Sqrt[613])/1226])/(51759*613^(3/4)*Sqrt[9 - 6*x - 44*x^2 + 15*x^3 +
3*x^4]) - ((7444 - 145*Sqrt[613])*Sqrt[(613 - 182*(1 - 6/x)^2 + (-1 + 6/x)^4)/(Sqrt[613] + (6 - x)^2/x^2)^2]*
(Sqrt[613] + (6 - x)^2/x^2)*x^2*EllipticF[2*ArcTan[(6 - x)/(613^(1/4)*x)], (613 + 91*Sqrt[613])/1226])/(207036
*613^(3/4)*Sqrt[9 - 6*x - 44*x^2 + 15*x^3 + 3*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 1110
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] && GtQ[b^2 - 4*a*c, 0] && GtQ[c/a, 0] && LtQ[b/a, 0]
Rule 1196
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] && GtQ[b^2 - 4*a*c, 0] && GtQ[c/a, 0] && LtQ[b/a, 0]
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] && GtQ[b^2 - 4*a*c, 0] && GtQ[c/a, 0] && LtQ[b/a, 0]
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 (9-6 x-44 x^2+15 x^3+3 x^4\right )^{3/2}} \, dx &=-\left (1296 \text {Subst}\left (\int \frac {1}{27 (-6-36 x)^2 \left (\frac {794448-8491392 x^2+1679616 x^4}{(-6-36 x)^4}\right )^{3/2}} \, dx,x,-\frac {1}{6}+\frac {1}{x}\right )\right )\\ &=-\left (48 \text {Subst}\left (\int \frac {1}{(-6-36 x)^2 \left (\frac {794448-8491392 x^2+1679616 x^4}{(-6-36 x)^4}\right )^{3/2}} \, dx,x,-\frac {1}{6}+\frac {1}{x}\right )\right )\\ &=-\frac {\left (\sqrt {794448-8491392 \left (-\frac {1}{6}+\frac {1}{x}\right )^2+1679616 \left (-\frac {1}{6}+\frac {1}{x}\right )^4} x^2\right ) \text {Subst}\left (\int \frac {(-6-36 x)^4}{\left (794448-8491392 x^2+1679616 x^4\right )^{3/2}} \, dx,x,-\frac {1}{6}+\frac {1}{x}\right )}{9 \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}\\ &=-\frac {\left (\sqrt {794448-8491392 \left (-\frac {1}{6}+\frac {1}{x}\right )^2+1679616 \left (-\frac {1}{6}+\frac {1}{x}\right )^4} x^2\right ) \text {Subst}\left (\int \frac {x \left (31104+1119744 x^2\right )}{\left (794448-8491392 x^2+1679616 x^4\right )^{3/2}} \, dx,x,-\frac {1}{6}+\frac {1}{x}\right )}{9 \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}-\frac {\left (\sqrt {794448-8491392 \left (-\frac {1}{6}+\frac {1}{x}\right )^2+1679616 \left (-\frac {1}{6}+\frac {1}{x}\right )^4} x^2\right ) \text {Subst}\left (\int \frac {1296+279936 x^2+1679616 x^4}{\left (794448-8491392 x^2+1679616 x^4\right )^{3/2}} \, dx,x,-\frac {1}{6}+\frac {1}{x}\right )}{9 \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}\\ &=\frac {\left (45401-3722 \left (1-\frac {6}{x}\right )^2\right ) \left (1-\frac {6}{x}\right ) x^2}{31728267 \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}+\frac {\left (\sqrt {794448-8491392 \left (-\frac {1}{6}+\frac {1}{x}\right )^2+1679616 \left (-\frac {1}{6}+\frac {1}{x}\right )^4} x^2\right ) \text {Subst}\left (\int \frac {4012069665987624960-12096197079035019264 x^2}{\sqrt {794448-8491392 x^2+1679616 x^4}} \, dx,x,-\frac {1}{6}+\frac {1}{x}\right )}{477380951360582713344 \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}-\frac {\left (\sqrt {794448-8491392 \left (-\frac {1}{6}+\frac {1}{x}\right )^2+1679616 \left (-\frac {1}{6}+\frac {1}{x}\right )^4} x^2\right ) \text {Subst}\left (\int \frac {31104+1119744 x}{\left (794448-8491392 x+1679616 x^2\right )^{3/2}} \, dx,x,\left (-\frac {1}{6}+\frac {1}{x}\right )^2\right )}{18 \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}\\ &=-\frac {\left (176-23 \left (1-\frac {6}{x}\right )^2\right ) x^2}{51759 \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}+\frac {\left (45401-3722 \left (1-\frac {6}{x}\right )^2\right ) \left (1-\frac {6}{x}\right ) x^2}{31728267 \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}+\frac {\left (7444 \sqrt {794448-8491392 \left (-\frac {1}{6}+\frac {1}{x}\right )^2+1679616 \left (-\frac {1}{6}+\frac {1}{x}\right )^4} x^2\right ) \text {Subst}\left (\int \frac {1-\frac {36 x^2}{\sqrt {613}}}{\sqrt {794448-8491392 x^2+1679616 x^4}} \, dx,x,-\frac {1}{6}+\frac {1}{x}\right )}{17253 \sqrt {613} \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}+\frac {\left (\left (88885-7444 \sqrt {613}\right ) \sqrt {794448-8491392 \left (-\frac {1}{6}+\frac {1}{x}\right )^2+1679616 \left (-\frac {1}{6}+\frac {1}{x}\right )^4} x^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {794448-8491392 x^2+1679616 x^4}} \, dx,x,-\frac {1}{6}+\frac {1}{x}\right )}{10576089 \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}\\ &=-\frac {\left (176-23 \left (1-\frac {6}{x}\right )^2\right ) x^2}{51759 \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}+\frac {\left (45401-3722 \left (1-\frac {6}{x}\right )^2\right ) \left (1-\frac {6}{x}\right ) x^2}{31728267 \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}+\frac {3722 \left (613-182 \left (1-\frac {6}{x}\right )^2+\left (-1+\frac {6}{x}\right )^4\right ) \left (1-\frac {6}{x}\right ) x^2}{31728267 \left (\sqrt {613}+\frac {(6-x)^2}{x^2}\right ) \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}+\frac {3722 \sqrt {\frac {613-182 \left (1-\frac {6}{x}\right )^2+\left (-1+\frac {6}{x}\right )^4}{\left (\sqrt {613}+\frac {(6-x)^2}{x^2}\right )^2}} \left (\sqrt {613}+\frac {(6-x)^2}{x^2}\right ) x^2 E\left (2 \tan ^{-1}\left (\frac {6-x}{\sqrt [4]{613} x}\right )|\frac {613+91 \sqrt {613}}{1226}\right )}{51759\ 613^{3/4} \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}-\frac {\left (7444-145 \sqrt {613}\right ) \sqrt {\frac {613-182 \left (1-\frac {6}{x}\right )^2+\left (-1+\frac {6}{x}\right )^4}{\left (\sqrt {613}+\frac {(6-x)^2}{x^2}\right )^2}} \left (\sqrt {613}+\frac {(6-x)^2}{x^2}\right ) x^2 F\left (2 \tan ^{-1}\left (\frac {6-x}{\sqrt [4]{613} x}\right )|\frac {613+91 \sqrt {613}}{1226}\right )}{207036\ 613^{3/4} \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 4 in
optimal.
time = 15.79, size = 4974, normalized size = 11.20 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[(9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4)^(-3/2),x]
[Out]
(2*(106926 + 592639*x - 232005*x^2 - 44664*x^3 + 81441*EllipticF[ArcSin[Sqrt[((x - Root[9 - 6*#1 - 44*#1^2 + 1
5*#1^3 + 3*#1^4 & , 1, 0])*(Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0] - Root[9 - 6*#1 - 44*#1^2 + 1
5*#1^3 + 3*#1^4 & , 4, 0]))/((x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0])*(Root[9 - 6*#1 - 44*#1
^2 + 15*#1^3 + 3*#1^4 & , 1, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 4, 0]))]], ((Root[9 - 6*#1 -
44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 3, 0])*(Root[9 - 6*#1 -
44*#1^2 + 15*#1^3 + 3*#1^4 & , 1, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 4, 0]))/((Root[9 - 6*#1
- 44*#1^2 + 15*#1^3 + 3*#1^4 & , 1, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 3, 0])*(Root[9 - 6*#1
- 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 4, 0]))]*Sqrt[(x - Roo
t[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 1, 0])/(x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0])]
*(x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0])^2*Sqrt[(x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*
#1^4 & , 3, 0])/(x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0])]*Sqrt[(x - Root[9 - 6*#1 - 44*#1^2
+ 15*#1^3 + 3*#1^4 & , 4, 0])/((x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0])*(Root[9 - 6*#1 - 44*
#1^2 + 15*#1^3 + 3*#1^4 & , 1, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 3, 0])*(Root[9 - 6*#1 - 44*
#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 4, 0]))] - (223320*(x - Roo
t[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0])^2*(EllipticPi[(Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 &
, 1, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 4, 0])/(Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 &
, 2, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 4, 0]), ArcSin[Sqrt[((x - Root[9 - 6*#1 - 44*#1^2 +
15*#1^3 + 3*#1^4 & , 1, 0])*(Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0] - Root[9 - 6*#1 - 44*#1^2 +
15*#1^3 + 3*#1^4 & , 4, 0]))/((x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0])*(Root[9 - 6*#1 - 44*#
1^2 + 15*#1^3 + 3*#1^4 & , 1, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 4, 0]))]], -(((Root[9 - 6*#1
- 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 3, 0])*(Root[9 - 6*#1
- 44*#1^2 + 15*#1^3 + 3*#1^4 & , 1, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 4, 0]))/((-Root[9 - 6
*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 1, 0] + Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 3, 0])*(Root[9 - 6
*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 4, 0])))]*(Root[9
- 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 1, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0]) + Ellipt
icF[ArcSin[Sqrt[((x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 1, 0])*(Root[9 - 6*#1 - 44*#1^2 + 15*#1^3
+ 3*#1^4 & , 2, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 4, 0]))/((x - Root[9 - 6*#1 - 44*#1^2 + 1
5*#1^3 + 3*#1^4 & , 2, 0])*(Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 1, 0] - Root[9 - 6*#1 - 44*#1^2 + 1
5*#1^3 + 3*#1^4 & , 4, 0]))]], -(((Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0] - Root[9 - 6*#1 - 44*#
1^2 + 15*#1^3 + 3*#1^4 & , 3, 0])*(Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 1, 0] - Root[9 - 6*#1 - 44*#
1^2 + 15*#1^3 + 3*#1^4 & , 4, 0]))/((-Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 1, 0] + Root[9 - 6*#1 - 4
4*#1^2 + 15*#1^3 + 3*#1^4 & , 3, 0])*(Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0] - Root[9 - 6*#1 - 4
4*#1^2 + 15*#1^3 + 3*#1^4 & , 4, 0])))]*Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0])*Sqrt[(x - Root[9
- 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 3, 0])/(x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0])]*Sq
rt[(x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 4, 0])/((x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4
& , 2, 0])*(Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 1, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4
& , 3, 0])*(Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 1, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4
& , 4, 0]))]*Sqrt[((x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 1, 0])*(Root[9 - 6*#1 - 44*#1^2 + 15*#
1^3 + 3*#1^4 & , 2, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 4, 0]))/((x - Root[9 - 6*#1 - 44*#1^2
+ 15*#1^3 + 3*#1^4 & , 2, 0])*(Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 1, 0] - Root[9 - 6*#1 - 44*#1^2
+ 15*#1^3 + 3*#1^4 & , 4, 0]))]*(-Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 1, 0] + Root[9 - 6*#1 - 44*#1
^2 + 15*#1^3 + 3*#1^4 & , 4, 0]))/(-Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0] + Root[9 - 6*#1 - 44*
#1^2 + 15*#1^3 + 3*#1^4 & , 4, 0]) + 44664*((x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 1, 0])*(x - Ro
ot[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 3, 0])*(x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 4, 0])
- ((x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0])^2*Sqrt[(x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 +
3*#1^4 & , 3, 0])/(x - Root[9 - 6*#1 - 44*#1^2...
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.13, size = 5427, normalized size = 12.22
| | |
| method |
result |
size |
| | |
| default |
\(\text {Expression too large to display}\) |
\(5427\) |
| risch |
\(\text {Expression too large to display}\) |
\(5427\) |
| elliptic |
\(\text {Expression too large to display}\) |
\(5427\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(3*x^4+15*x^3-44*x^2-6*x+9)^(3/2),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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(3*x^4+15*x^3-44*x^2-6*x+9)^(3/2),x, algorithm="maxima")
[Out]
integrate((3*x^4 + 15*x^3 - 44*x^2 - 6*x + 9)^(-3/2), x)
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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/(3*x^4+15*x^3-44*x^2-6*x+9)^(3/2),x, algorithm="fricas")
[Out]
integral(sqrt(3*x^4 + 15*x^3 - 44*x^2 - 6*x + 9)/(9*x^8 + 90*x^7 - 39*x^6 - 1356*x^5 + 1810*x^4 + 798*x^3 - 75
6*x^2 - 108*x + 81), x)
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (3 x^{4} + 15 x^{3} - 44 x^{2} - 6 x + 9\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(3*x**4+15*x**3-44*x**2-6*x+9)**(3/2),x)
[Out]
Integral((3*x**4 + 15*x**3 - 44*x**2 - 6*x + 9)**(-3/2), 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(1/(3*x^4+15*x^3-44*x^2-6*x+9)^(3/2),x, algorithm="giac")
[Out]
integrate((3*x^4 + 15*x^3 - 44*x^2 - 6*x + 9)^(-3/2), x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (3\,x^4+15\,x^3-44\,x^2-6\,x+9\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(15*x^3 - 44*x^2 - 6*x + 3*x^4 + 9)^(3/2),x)
[Out]
int(1/(15*x^3 - 44*x^2 - 6*x + 3*x^4 + 9)^(3/2), x)
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