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

Optimal. Leaf size=130 \[ -\frac{\sqrt{\frac{\left (\frac{6}{x}-1\right )^4-182 \left (1-\frac{6}{x}\right )^2+613}{\left (\frac{(6-x)^2}{x^2}+\sqrt{613}\right )^2}} \left (\frac{(6-x)^2}{x^2}+\sqrt{613}\right ) x^2 F\left (2 \tan ^{-1}\left (\frac{6-x}{\sqrt [4]{613} x}\right )|\frac{613+91 \sqrt{613}}{1226}\right )}{12 \sqrt [4]{613} \sqrt{3 x^4+15 x^3-44 x^2-6 x+9}} \]

[Out]

-(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*E
llipticF[2*ArcTan[(6 - x)/(613^(1/4)*x)], (613 + 91*Sqrt[613])/1226])/(12*613^(1/4)*Sqrt[9 - 6*x - 44*x^2 + 15
*x^3 + 3*x^4])

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Rubi [A]  time = 0.260063, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2069, 12, 6719, 1096} \[ -\frac{\sqrt{\frac{\left (\frac{6}{x}-1\right )^4-182 \left (1-\frac{6}{x}\right )^2+613}{\left (\frac{(6-x)^2}{x^2}+\sqrt{613}\right )^2}} \left (\frac{(6-x)^2}{x^2}+\sqrt{613}\right ) x^2 F\left (2 \tan ^{-1}\left (\frac{6-x}{\sqrt [4]{613} x}\right )|\frac{613+91 \sqrt{613}}{1226}\right )}{12 \sqrt [4]{613} \sqrt{3 x^4+15 x^3-44 x^2-6 x+9}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4],x]

[Out]

-(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*E
llipticF[2*ArcTan[(6 - x)/(613^(1/4)*x)], (613 + 91*Sqrt[613])/1226])/(12*613^(1/4)*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 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]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{9-6 x-44 x^2+15 x^3+3 x^4}} \, dx &=-\left (1296 \operatorname{Subst}\left (\int \frac{1}{3 (-6-36 x)^2 \sqrt{\frac{794448-8491392 x^2+1679616 x^4}{(-6-36 x)^4}}} \, dx,x,-\frac{1}{6}+\frac{1}{x}\right )\right )\\ &=-\left (432 \operatorname{Subst}\left (\int \frac{1}{(-6-36 x)^2 \sqrt{\frac{794448-8491392 x^2+1679616 x^4}{(-6-36 x)^4}}} \, 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{1}{\sqrt{794448-8491392 x^2+1679616 x^4}} \, dx,x,-\frac{1}{6}+\frac{1}{x}\right )}{\sqrt{9-6 x-44 x^2+15 x^3+3 x^4}}\\ &=-\frac{\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 )}{12 \sqrt [4]{613} \sqrt{9-6 x-44 x^2+15 x^3+3 x^4}}\\ \end{align*}

Mathematica [C]  time = 0.122162, size = 826, normalized size = 6.35 \[ -\frac{2 F\left (\sin ^{-1}\left (\sqrt{\frac{\left (x-\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\& ,1\right ]\right ) \left (\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\& ,2\right ]-\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\& ,4\right ]\right )}{\left (x-\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\& ,2\right ]\right ) \left (\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\& ,1\right ]-\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\& ,4\right ]\right )}}\right )|\frac{\left (\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\& ,2\right ]-\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\& ,3\right ]\right ) \left (\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\& ,1\right ]-\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\& ,4\right ]\right )}{\left (\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\& ,1\right ]-\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\& ,3\right ]\right ) \left (\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\& ,2\right ]-\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\& ,4\right ]\right )}\right ) \sqrt{\frac{x-\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\& ,1\right ]}{x-\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\& ,2\right ]}} \left (x-\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\& ,2\right ]\right )^2 \sqrt{\frac{x-\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\& ,3\right ]}{x-\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\& ,2\right ]}} \sqrt{\frac{x-\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\& ,4\right ]}{x-\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\& ,2\right ]}}}{\sqrt{\left (3 x^4+15 x^3-44 x^2-6 x+9\right ) \left (\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\& ,1\right ]-\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\& ,3\right ]\right ) \left (\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\& ,2\right ]-\text{Root}\left [3 \text{$\#$1}^4+15 \text{$\#$1}^3-44 \text{$\#$1}^2-6 \text{$\#$1}+9\& ,4\right ]\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4],x]

[Out]

(-2*EllipticF[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 - 4
4*#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 - 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] - 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 - Root[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])])/Sqrt[(9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4)*(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])]

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Maple [C]  time = 0.339, size = 1182, normalized size = 9.1 \begin{align*} \text{result 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)^(1/2),x)

[Out]

2/3*(-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=4)+RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=1))*((x-RootOf(
3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=1))/(-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=4)+RootOf(3*_Z^4+15*_Z^3-
44*_Z^2-6*_Z+9,index=1))/(x-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=2))*(-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_
Z+9,index=4)+RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=2)))^(1/2)*(x-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,ind
ex=2))^2*(-(x-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=3))/(-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=3)+R
ootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=1))/(x-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=2))*(RootOf(3*_Z^4
+15*_Z^3-44*_Z^2-6*_Z+9,index=2)-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=1)))^(1/2)*(-(x-RootOf(3*_Z^4+15*_
Z^3-44*_Z^2-6*_Z+9,index=4))/(-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=4)+RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_
Z+9,index=1))/(x-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=2))*(RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=2)
-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=1)))^(1/2)/(RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=4)-RootOf(3
*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=2))/(RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=2)-RootOf(3*_Z^4+15*_Z^3-44
*_Z^2-6*_Z+9,index=1))*3^(1/2)/((x-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=1))*(x-RootOf(3*_Z^4+15*_Z^3-44*
_Z^2-6*_Z+9,index=2))*(x-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=3))*(x-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+
9,index=4)))^(1/2)*EllipticF(((x-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=1))/(-RootOf(3*_Z^4+15*_Z^3-44*_Z^
2-6*_Z+9,index=4)+RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=1))/(x-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index
=2))*(-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=4)+RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=2)))^(1/2),((R
ootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=2)-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=3))*(-RootOf(3*_Z^4+15
*_Z^3-44*_Z^2-6*_Z+9,index=4)+RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=1))/(-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6
*_Z+9,index=3)+RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=1))/(-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=4)+
RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=2)))^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{3 \, x^{4} + 15 \, x^{3} - 44 \, x^{2} - 6 \, x + 9}}\,{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)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(3*x^4 + 15*x^3 - 44*x^2 - 6*x + 9), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{\sqrt{3 \, x^{4} + 15 \, x^{3} - 44 \, x^{2} - 6 \, x + 9}}, 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)^(1/2),x, algorithm="fricas")

[Out]

integral(1/sqrt(3*x^4 + 15*x^3 - 44*x^2 - 6*x + 9), x)

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

[Out]

Integral(1/sqrt(3*x**4 + 15*x**3 - 44*x**2 - 6*x + 9), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{3 \, x^{4} + 15 \, x^{3} - 44 \, x^{2} - 6 \, x + 9}}\,{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)^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(3*x^4 + 15*x^3 - 44*x^2 - 6*x + 9), x)