3.804 \(\int \frac{1}{(9-6 x-44 x^2+15 x^3+3 x^4)^{3/2}} \, dx\)

Optimal. Leaf size=444 \[ -\frac{\left (176-23 \left (1-\frac{6}{x}\right )^2\right ) x^2}{51759 \sqrt{3 x^4+15 x^3-44 x^2-6 x+9}}+\frac{\left (45401-3722 \left (1-\frac{6}{x}\right )^2\right ) \left (1-\frac{6}{x}\right ) x^2}{31728267 \sqrt{3 x^4+15 x^3-44 x^2-6 x+9}}+\frac{3722 \left (\left (\frac{6}{x}-1\right )^4-182 \left (1-\frac{6}{x}\right )^2+613\right ) \left (1-\frac{6}{x}\right ) x^2}{31728267 \left (\frac{(6-x)^2}{x^2}+\sqrt{613}\right ) \sqrt{3 x^4+15 x^3-44 x^2-6 x+9}}-\frac{\left (7444-145 \sqrt{613}\right ) \sqrt{\frac{\left (\frac{6}{x}-1\right )^4-182 \left (1-\frac{6}{x}\right )^2+613}{\left (\frac{(6-x)^2}{x^2}+\sqrt{613}\right )^2}} \left (\frac{(6-x)^2}{x^2}+\sqrt{613}\right ) x^2 F\left (2 \tan ^{-1}\left (\frac{6-x}{\sqrt [4]{613} x}\right )|\frac{613+91 \sqrt{613}}{1226}\right )}{207036\ 613^{3/4} \sqrt{3 x^4+15 x^3-44 x^2-6 x+9}}+\frac{3722 \sqrt{\frac{\left (\frac{6}{x}-1\right )^4-182 \left (1-\frac{6}{x}\right )^2+613}{\left (\frac{(6-x)^2}{x^2}+\sqrt{613}\right )^2}} \left (\frac{(6-x)^2}{x^2}+\sqrt{613}\right ) x^2 E\left (2 \tan ^{-1}\left (\frac{6-x}{\sqrt [4]{613} x}\right )|\frac{613+91 \sqrt{613}}{1226}\right )}{51759\ 613^{3/4} \sqrt{3 x^4+15 x^3-44 x^2-6 x+9}} \]

[Out]

-((176 - 23*(1 - 6/x)^2)*x^2)/(51759*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])

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Rubi [A]  time = 0.468024, antiderivative size = 444, normalized size of antiderivative = 1., 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 = {2069, 12, 6719, 1673, 1678, 1183, 1096, 1182, 1247, 636} \[ -\frac{\left (176-23 \left (1-\frac{6}{x}\right )^2\right ) x^2}{51759 \sqrt{3 x^4+15 x^3-44 x^2-6 x+9}}+\frac{\left (45401-3722 \left (1-\frac{6}{x}\right )^2\right ) \left (1-\frac{6}{x}\right ) x^2}{31728267 \sqrt{3 x^4+15 x^3-44 x^2-6 x+9}}+\frac{3722 \left (\left (\frac{6}{x}-1\right )^4-182 \left (1-\frac{6}{x}\right )^2+613\right ) \left (1-\frac{6}{x}\right ) x^2}{31728267 \left (\frac{(6-x)^2}{x^2}+\sqrt{613}\right ) \sqrt{3 x^4+15 x^3-44 x^2-6 x+9}}-\frac{\left (7444-145 \sqrt{613}\right ) \sqrt{\frac{\left (\frac{6}{x}-1\right )^4-182 \left (1-\frac{6}{x}\right )^2+613}{\left (\frac{(6-x)^2}{x^2}+\sqrt{613}\right )^2}} \left (\frac{(6-x)^2}{x^2}+\sqrt{613}\right ) x^2 F\left (2 \tan ^{-1}\left (\frac{6-x}{\sqrt [4]{613} x}\right )|\frac{613+91 \sqrt{613}}{1226}\right )}{207036\ 613^{3/4} \sqrt{3 x^4+15 x^3-44 x^2-6 x+9}}+\frac{3722 \sqrt{\frac{\left (\frac{6}{x}-1\right )^4-182 \left (1-\frac{6}{x}\right )^2+613}{\left (\frac{(6-x)^2}{x^2}+\sqrt{613}\right )^2}} \left (\frac{(6-x)^2}{x^2}+\sqrt{613}\right ) x^2 E\left (2 \tan ^{-1}\left (\frac{6-x}{\sqrt [4]{613} x}\right )|\frac{613+91 \sqrt{613}}{1226}\right )}{51759\ 613^{3/4} \sqrt{3 x^4+15 x^3-44 x^2-6 x+9}} \]

Antiderivative was successfully verified.

[In]

Int[(9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4)^(-3/2),x]

[Out]

-((176 - 23*(1 - 6/x)^2)*x^2)/(51759*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 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 12

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

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]

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 1183

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 1096

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] && GtQ[b^2 - 4*a*c, 0] && GtQ[c/a, 0] && LtQ[b/a, 0]

Rule 1182

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] && GtQ[b^2 - 4*a*c, 0] && GtQ[c/a, 0] && LtQ[b/a, 0]

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 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]

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 \operatorname{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 \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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]  time = 6.04799, size = 5428, normalized size = 12.23 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4)^(-3/2),x]

[Out]

Result too large to show

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Maple [C]  time = 0.016, size = 5427, normalized size = 12.2 \begin{align*} \text{output too large to display} \end{align*}

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)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (3 \, x^{4} + 15 \, x^{3} - 44 \, x^{2} - 6 \, x + 9\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

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., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\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} - 756 \, x^{2} - 108 \, x + 81}, x\right ) \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (3 x^{4} + 15 x^{3} - 44 x^{2} - 6 x + 9\right )^{\frac{3}{2}}}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (3 \, x^{4} + 15 \, x^{3} - 44 \, x^{2} - 6 \, x + 9\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

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)