3.37 \(\int \frac{1}{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4} \, dx\)

Optimal. Leaf size=529 \[ -\frac{d \log \left (-\sqrt{2} \sqrt [4]{c} d \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}} \left (\frac{c}{d}+x\right )+\sqrt{c} \sqrt{4 a d^2+c^3}+d^2 \left (\frac{c}{d}+x\right )^2\right )}{4 \sqrt{2} c^{3/4} \sqrt{4 a d^2+c^3} \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}}}+\frac{d \log \left (\sqrt{2} \sqrt [4]{c} d \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}} \left (\frac{c}{d}+x\right )+\sqrt{c} \sqrt{4 a d^2+c^3}+d^2 \left (\frac{c}{d}+x\right )^2\right )}{4 \sqrt{2} c^{3/4} \sqrt{4 a d^2+c^3} \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}}}-\frac{d \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}}+\sqrt{2} c+\sqrt{2} d x}{\sqrt [4]{c} \sqrt{c^{3/2}-\sqrt{4 a d^2+c^3}}}\right )}{2 \sqrt{2} c^{3/4} \sqrt{4 a d^2+c^3} \sqrt{c^{3/2}-\sqrt{4 a d^2+c^3}}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}}-\sqrt{2} (c+d x)}{\sqrt [4]{c} \sqrt{c^{3/2}-\sqrt{4 a d^2+c^3}}}\right )}{2 \sqrt{2} c^{3/4} \sqrt{4 a d^2+c^3} \sqrt{c^{3/2}-\sqrt{4 a d^2+c^3}}} \]

[Out]

-(d*ArcTanh[(Sqrt[2]*c + c^(1/4)*Sqrt[c^(3/2) + Sqrt[c^3 + 4*a*d^2]] + Sqrt[2]*d*x)/(c^(1/4)*Sqrt[c^(3/2) - Sq
rt[c^3 + 4*a*d^2]])])/(2*Sqrt[2]*c^(3/4)*Sqrt[c^3 + 4*a*d^2]*Sqrt[c^(3/2) - Sqrt[c^3 + 4*a*d^2]]) + (d*ArcTanh
[(c^(1/4)*Sqrt[c^(3/2) + Sqrt[c^3 + 4*a*d^2]] - Sqrt[2]*(c + d*x))/(c^(1/4)*Sqrt[c^(3/2) - Sqrt[c^3 + 4*a*d^2]
])])/(2*Sqrt[2]*c^(3/4)*Sqrt[c^3 + 4*a*d^2]*Sqrt[c^(3/2) - Sqrt[c^3 + 4*a*d^2]]) - (d*Log[Sqrt[c]*Sqrt[c^3 + 4
*a*d^2] - Sqrt[2]*c^(1/4)*d*Sqrt[c^(3/2) + Sqrt[c^3 + 4*a*d^2]]*(c/d + x) + d^2*(c/d + x)^2])/(4*Sqrt[2]*c^(3/
4)*Sqrt[c^3 + 4*a*d^2]*Sqrt[c^(3/2) + Sqrt[c^3 + 4*a*d^2]]) + (d*Log[Sqrt[c]*Sqrt[c^3 + 4*a*d^2] + Sqrt[2]*c^(
1/4)*d*Sqrt[c^(3/2) + Sqrt[c^3 + 4*a*d^2]]*(c/d + x) + d^2*(c/d + x)^2])/(4*Sqrt[2]*c^(3/4)*Sqrt[c^3 + 4*a*d^2
]*Sqrt[c^(3/2) + Sqrt[c^3 + 4*a*d^2]])

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Rubi [A]  time = 0.895749, antiderivative size = 529, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {1106, 1094, 634, 618, 206, 628} \[ -\frac{d \log \left (-\sqrt{2} \sqrt [4]{c} d \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}} \left (\frac{c}{d}+x\right )+\sqrt{c} \sqrt{4 a d^2+c^3}+d^2 \left (\frac{c}{d}+x\right )^2\right )}{4 \sqrt{2} c^{3/4} \sqrt{4 a d^2+c^3} \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}}}+\frac{d \log \left (\sqrt{2} \sqrt [4]{c} d \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}} \left (\frac{c}{d}+x\right )+\sqrt{c} \sqrt{4 a d^2+c^3}+d^2 \left (\frac{c}{d}+x\right )^2\right )}{4 \sqrt{2} c^{3/4} \sqrt{4 a d^2+c^3} \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}}}-\frac{d \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}}+\sqrt{2} c+\sqrt{2} d x}{\sqrt [4]{c} \sqrt{c^{3/2}-\sqrt{4 a d^2+c^3}}}\right )}{2 \sqrt{2} c^{3/4} \sqrt{4 a d^2+c^3} \sqrt{c^{3/2}-\sqrt{4 a d^2+c^3}}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}}-\sqrt{2} (c+d x)}{\sqrt [4]{c} \sqrt{c^{3/2}-\sqrt{4 a d^2+c^3}}}\right )}{2 \sqrt{2} c^{3/4} \sqrt{4 a d^2+c^3} \sqrt{c^{3/2}-\sqrt{4 a d^2+c^3}}} \]

Antiderivative was successfully verified.

[In]

Int[(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4)^(-1),x]

[Out]

-(d*ArcTanh[(Sqrt[2]*c + c^(1/4)*Sqrt[c^(3/2) + Sqrt[c^3 + 4*a*d^2]] + Sqrt[2]*d*x)/(c^(1/4)*Sqrt[c^(3/2) - Sq
rt[c^3 + 4*a*d^2]])])/(2*Sqrt[2]*c^(3/4)*Sqrt[c^3 + 4*a*d^2]*Sqrt[c^(3/2) - Sqrt[c^3 + 4*a*d^2]]) + (d*ArcTanh
[(c^(1/4)*Sqrt[c^(3/2) + Sqrt[c^3 + 4*a*d^2]] - Sqrt[2]*(c + d*x))/(c^(1/4)*Sqrt[c^(3/2) - Sqrt[c^3 + 4*a*d^2]
])])/(2*Sqrt[2]*c^(3/4)*Sqrt[c^3 + 4*a*d^2]*Sqrt[c^(3/2) - Sqrt[c^3 + 4*a*d^2]]) - (d*Log[Sqrt[c]*Sqrt[c^3 + 4
*a*d^2] - Sqrt[2]*c^(1/4)*d*Sqrt[c^(3/2) + Sqrt[c^3 + 4*a*d^2]]*(c/d + x) + d^2*(c/d + x)^2])/(4*Sqrt[2]*c^(3/
4)*Sqrt[c^3 + 4*a*d^2]*Sqrt[c^(3/2) + Sqrt[c^3 + 4*a*d^2]]) + (d*Log[Sqrt[c]*Sqrt[c^3 + 4*a*d^2] + Sqrt[2]*c^(
1/4)*d*Sqrt[c^(3/2) + Sqrt[c^3 + 4*a*d^2]]*(c/d + x) + d^2*(c/d + x)^2])/(4*Sqrt[2]*c^(3/4)*Sqrt[c^3 + 4*a*d^2
]*Sqrt[c^(3/2) + Sqrt[c^3 + 4*a*d^2]])

Rule 1106

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]}, Subst[Int[SimplifyIntegrand[(a + d^4/(256*e^3) - (b*d)/(8*e) + (c - (3*d^2)/(8*
e))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2, 0] && NeQ[d, 0]] /; FreeQ[p, x] &&
PolyQ[P4, x, 4] && NeQ[p, 2] && NeQ[p, 3]

Rule 1094

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}
, Dist[1/(2*c*q*r), Int[(r - x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(r + x)/(q + r*x + x^2), x], x
]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{1}{4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4} \, dx &=\operatorname{Subst}\left (\int \frac{1}{c \left (4 a+\frac{c^3}{d^2}\right )-2 c^2 x^2+d^2 x^4} \, dx,x,\frac{c}{d}+x\right )\\ &=\frac{d \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{c} \sqrt{c^{3/2}+\sqrt{c^3+4 a d^2}}}{d}-x}{\frac{\sqrt{c} \sqrt{c^3+4 a d^2}}{d^2}-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{c^{3/2}+\sqrt{c^3+4 a d^2}} x}{d}+x^2} \, dx,x,\frac{c}{d}+x\right )}{2 \sqrt{2} c^{3/4} \sqrt{c^3+4 a d^2} \sqrt{c^{3/2}+\sqrt{c^3+4 a d^2}}}+\frac{d \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{c} \sqrt{c^{3/2}+\sqrt{c^3+4 a d^2}}}{d}+x}{\frac{\sqrt{c} \sqrt{c^3+4 a d^2}}{d^2}+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{c^{3/2}+\sqrt{c^3+4 a d^2}} x}{d}+x^2} \, dx,x,\frac{c}{d}+x\right )}{2 \sqrt{2} c^{3/4} \sqrt{c^3+4 a d^2} \sqrt{c^{3/2}+\sqrt{c^3+4 a d^2}}}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{c} \sqrt{c^3+4 a d^2}}{d^2}-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{c^{3/2}+\sqrt{c^3+4 a d^2}} x}{d}+x^2} \, dx,x,\frac{c}{d}+x\right )}{4 \sqrt{c} \sqrt{c^3+4 a d^2}}+\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{c} \sqrt{c^3+4 a d^2}}{d^2}+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{c^{3/2}+\sqrt{c^3+4 a d^2}} x}{d}+x^2} \, dx,x,\frac{c}{d}+x\right )}{4 \sqrt{c} \sqrt{c^3+4 a d^2}}-\frac{d \operatorname{Subst}\left (\int \frac{-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{c^{3/2}+\sqrt{c^3+4 a d^2}}}{d}+2 x}{\frac{\sqrt{c} \sqrt{c^3+4 a d^2}}{d^2}-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{c^{3/2}+\sqrt{c^3+4 a d^2}} x}{d}+x^2} \, dx,x,\frac{c}{d}+x\right )}{4 \sqrt{2} c^{3/4} \sqrt{c^3+4 a d^2} \sqrt{c^{3/2}+\sqrt{c^3+4 a d^2}}}+\frac{d \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{c} \sqrt{c^{3/2}+\sqrt{c^3+4 a d^2}}}{d}+2 x}{\frac{\sqrt{c} \sqrt{c^3+4 a d^2}}{d^2}+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{c^{3/2}+\sqrt{c^3+4 a d^2}} x}{d}+x^2} \, dx,x,\frac{c}{d}+x\right )}{4 \sqrt{2} c^{3/4} \sqrt{c^3+4 a d^2} \sqrt{c^{3/2}+\sqrt{c^3+4 a d^2}}}\\ &=-\frac{d \log \left (\sqrt{c} \sqrt{c^3+4 a d^2}-\sqrt{2} \sqrt [4]{c} \sqrt{c^{3/2}+\sqrt{c^3+4 a d^2}} (c+d x)+(c+d x)^2\right )}{4 \sqrt{2} c^{3/4} \sqrt{c^3+4 a d^2} \sqrt{c^{3/2}+\sqrt{c^3+4 a d^2}}}+\frac{d \log \left (\sqrt{c} \sqrt{c^3+4 a d^2}+\sqrt{2} \sqrt [4]{c} \sqrt{c^{3/2}+\sqrt{c^3+4 a d^2}} (c+d x)+(c+d x)^2\right )}{4 \sqrt{2} c^{3/4} \sqrt{c^3+4 a d^2} \sqrt{c^{3/2}+\sqrt{c^3+4 a d^2}}}-\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{2 \sqrt{c} \left (c^{3/2}-\sqrt{c^3+4 a d^2}\right )}{d^2}-x^2} \, dx,x,-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{c^{3/2}+\sqrt{c^3+4 a d^2}}}{d}+2 \left (\frac{c}{d}+x\right )\right )}{2 \sqrt{c} \sqrt{c^3+4 a d^2}}-\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{2 \sqrt{c} \left (c^{3/2}-\sqrt{c^3+4 a d^2}\right )}{d^2}-x^2} \, dx,x,\frac{\sqrt{2} \sqrt [4]{c} \sqrt{c^{3/2}+\sqrt{c^3+4 a d^2}}}{d}+2 \left (\frac{c}{d}+x\right )\right )}{2 \sqrt{c} \sqrt{c^3+4 a d^2}}\\ &=-\frac{d \tanh ^{-1}\left (\frac{\sqrt [4]{c} \left (\sqrt{2} c^{3/4}-\sqrt{c^{3/2}+\sqrt{c^3+4 a d^2}}\right )+\sqrt{2} d x}{\sqrt [4]{c} \sqrt{c^{3/2}-\sqrt{c^3+4 a d^2}}}\right )}{2 \sqrt{2} c^{3/4} \sqrt{c^3+4 a d^2} \sqrt{c^{3/2}-\sqrt{c^3+4 a d^2}}}-\frac{d \tanh ^{-1}\left (\frac{\sqrt [4]{c} \left (\sqrt{2} c^{3/4}+\sqrt{c^{3/2}+\sqrt{c^3+4 a d^2}}\right )+\sqrt{2} d x}{\sqrt [4]{c} \sqrt{c^{3/2}-\sqrt{c^3+4 a d^2}}}\right )}{2 \sqrt{2} c^{3/4} \sqrt{c^3+4 a d^2} \sqrt{c^{3/2}-\sqrt{c^3+4 a d^2}}}-\frac{d \log \left (\sqrt{c} \sqrt{c^3+4 a d^2}-\sqrt{2} \sqrt [4]{c} \sqrt{c^{3/2}+\sqrt{c^3+4 a d^2}} (c+d x)+(c+d x)^2\right )}{4 \sqrt{2} c^{3/4} \sqrt{c^3+4 a d^2} \sqrt{c^{3/2}+\sqrt{c^3+4 a d^2}}}+\frac{d \log \left (\sqrt{c} \sqrt{c^3+4 a d^2}+\sqrt{2} \sqrt [4]{c} \sqrt{c^{3/2}+\sqrt{c^3+4 a d^2}} (c+d x)+(c+d x)^2\right )}{4 \sqrt{2} c^{3/4} \sqrt{c^3+4 a d^2} \sqrt{c^{3/2}+\sqrt{c^3+4 a d^2}}}\\ \end{align*}

Mathematica [C]  time = 0.0263068, size = 71, normalized size = 0.13 \[ \frac{1}{4} \text{RootSum}\left [4 \text{$\#$1}^2 c^2+4 \text{$\#$1}^3 c d+\text{$\#$1}^4 d^2+4 a c\& ,\frac{\log (x-\text{$\#$1})}{3 \text{$\#$1}^2 c d+\text{$\#$1}^3 d^2+2 \text{$\#$1} c^2}\& \right ] \]

Antiderivative was successfully verified.

[In]

Integrate[(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4)^(-1),x]

[Out]

RootSum[4*a*c + 4*c^2*#1^2 + 4*c*d*#1^3 + d^2*#1^4 & , Log[x - #1]/(2*c^2*#1 + 3*c*d*#1^2 + d^2*#1^3) & ]/4

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Maple [C]  time = 0.059, size = 64, normalized size = 0.1 \begin{align*}{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ({d}^{2}{{\it \_Z}}^{4}+4\,cd{{\it \_Z}}^{3}+4\,{c}^{2}{{\it \_Z}}^{2}+4\,ac \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{3}{d}^{2}+3\,{{\it \_R}}^{2}cd+2\,{\it \_R}\,{c}^{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c),x)

[Out]

1/4*sum(1/(_R^3*d^2+3*_R^2*c*d+2*_R*c^2)*ln(x-_R),_R=RootOf(_Z^4*d^2+4*_Z^3*c*d+4*_Z^2*c^2+4*a*c))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{d^{2} x^{4} + 4 \, c d x^{3} + 4 \, c^{2} x^{2} + 4 \, a c}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c),x, algorithm="maxima")

[Out]

integrate(1/(d^2*x^4 + 4*c*d*x^3 + 4*c^2*x^2 + 4*a*c), x)

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Fricas [B]  time = 1.38816, size = 1770, normalized size = 3.35 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c),x, algorithm="fricas")

[Out]

1/8*sqrt(-(2*(a*c^3 + 4*a^2*d^2)*sqrt(-d^2/(a*c^9 + 8*a^2*c^6*d^2 + 16*a^3*c^3*d^4)) + 1)/(a*c^3 + 4*a^2*d^2))
*log(d^2*x + c*d + (2*a*c*d^2 + (a*c^7 + 4*a^2*c^4*d^2)*sqrt(-d^2/(a*c^9 + 8*a^2*c^6*d^2 + 16*a^3*c^3*d^4)))*s
qrt(-(2*(a*c^3 + 4*a^2*d^2)*sqrt(-d^2/(a*c^9 + 8*a^2*c^6*d^2 + 16*a^3*c^3*d^4)) + 1)/(a*c^3 + 4*a^2*d^2))) - 1
/8*sqrt(-(2*(a*c^3 + 4*a^2*d^2)*sqrt(-d^2/(a*c^9 + 8*a^2*c^6*d^2 + 16*a^3*c^3*d^4)) + 1)/(a*c^3 + 4*a^2*d^2))*
log(d^2*x + c*d - (2*a*c*d^2 + (a*c^7 + 4*a^2*c^4*d^2)*sqrt(-d^2/(a*c^9 + 8*a^2*c^6*d^2 + 16*a^3*c^3*d^4)))*sq
rt(-(2*(a*c^3 + 4*a^2*d^2)*sqrt(-d^2/(a*c^9 + 8*a^2*c^6*d^2 + 16*a^3*c^3*d^4)) + 1)/(a*c^3 + 4*a^2*d^2))) + 1/
8*sqrt((2*(a*c^3 + 4*a^2*d^2)*sqrt(-d^2/(a*c^9 + 8*a^2*c^6*d^2 + 16*a^3*c^3*d^4)) - 1)/(a*c^3 + 4*a^2*d^2))*lo
g(d^2*x + c*d + (2*a*c*d^2 - (a*c^7 + 4*a^2*c^4*d^2)*sqrt(-d^2/(a*c^9 + 8*a^2*c^6*d^2 + 16*a^3*c^3*d^4)))*sqrt
((2*(a*c^3 + 4*a^2*d^2)*sqrt(-d^2/(a*c^9 + 8*a^2*c^6*d^2 + 16*a^3*c^3*d^4)) - 1)/(a*c^3 + 4*a^2*d^2))) - 1/8*s
qrt((2*(a*c^3 + 4*a^2*d^2)*sqrt(-d^2/(a*c^9 + 8*a^2*c^6*d^2 + 16*a^3*c^3*d^4)) - 1)/(a*c^3 + 4*a^2*d^2))*log(d
^2*x + c*d - (2*a*c*d^2 - (a*c^7 + 4*a^2*c^4*d^2)*sqrt(-d^2/(a*c^9 + 8*a^2*c^6*d^2 + 16*a^3*c^3*d^4)))*sqrt((2
*(a*c^3 + 4*a^2*d^2)*sqrt(-d^2/(a*c^9 + 8*a^2*c^6*d^2 + 16*a^3*c^3*d^4)) - 1)/(a*c^3 + 4*a^2*d^2)))

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Sympy [A]  time = 0.935703, size = 88, normalized size = 0.17 \begin{align*} \operatorname{RootSum}{\left (t^{4} \left (16384 a^{3} c^{3} d^{2} + 4096 a^{2} c^{6}\right ) + 128 t^{2} a c^{3} + 1, \left ( t \mapsto t \log{\left (x + \frac{- 1024 t^{3} a^{2} c^{4} d^{2} - 256 t^{3} a c^{7} + 16 t a c d^{2} - 4 t c^{4} + c d}{d^{2}} \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d**2*x**4+4*c*d*x**3+4*c**2*x**2+4*a*c),x)

[Out]

RootSum(_t**4*(16384*a**3*c**3*d**2 + 4096*a**2*c**6) + 128*_t**2*a*c**3 + 1, Lambda(_t, _t*log(x + (-1024*_t*
*3*a**2*c**4*d**2 - 256*_t**3*a*c**7 + 16*_t*a*c*d**2 - 4*_t*c**4 + c*d)/d**2)))

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Giac [A]  time = 1.32178, size = 471, normalized size = 0.89 \begin{align*} -\frac{1}{8} \, \sqrt{-\frac{a c^{3} + 2 \, \sqrt{-a c} a c d}{a^{2} c^{6} + 4 \, a^{3} c^{3} d^{2}}} \log \left ({\left | \sqrt{-a c} d x + \sqrt{-a c} c - \sqrt{-a c^{3} + 2 \, \sqrt{-a c} a c d} \right |}\right ) + \frac{1}{8} \, \sqrt{-\frac{a c^{3} - 2 \, \sqrt{-a c} a c d}{a^{2} c^{6} + 4 \, a^{3} c^{3} d^{2}}} \log \left ({\left | \sqrt{-a c} d x + \sqrt{-a c} c - \sqrt{-a c^{3} - 2 \, \sqrt{-a c} a c d} \right |}\right ) + \frac{1}{8} \, \sqrt{-\frac{a c^{3} + 2 \, \sqrt{-a c} a c d}{a^{2} c^{6} + 4 \, a^{3} c^{3} d^{2}}} \log \left ({\left | -\sqrt{-a c} d x - \sqrt{-a c} c - \sqrt{-a c^{3} + 2 \, \sqrt{-a c} a c d} \right |}\right ) - \frac{1}{8} \, \sqrt{-\frac{a c^{3} - 2 \, \sqrt{-a c} a c d}{a^{2} c^{6} + 4 \, a^{3} c^{3} d^{2}}} \log \left ({\left | -\sqrt{-a c} d x - \sqrt{-a c} c - \sqrt{-a c^{3} - 2 \, \sqrt{-a c} a c d} \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c),x, algorithm="giac")

[Out]

-1/8*sqrt(-(a*c^3 + 2*sqrt(-a*c)*a*c*d)/(a^2*c^6 + 4*a^3*c^3*d^2))*log(abs(sqrt(-a*c)*d*x + sqrt(-a*c)*c - sqr
t(-a*c^3 + 2*sqrt(-a*c)*a*c*d))) + 1/8*sqrt(-(a*c^3 - 2*sqrt(-a*c)*a*c*d)/(a^2*c^6 + 4*a^3*c^3*d^2))*log(abs(s
qrt(-a*c)*d*x + sqrt(-a*c)*c - sqrt(-a*c^3 - 2*sqrt(-a*c)*a*c*d))) + 1/8*sqrt(-(a*c^3 + 2*sqrt(-a*c)*a*c*d)/(a
^2*c^6 + 4*a^3*c^3*d^2))*log(abs(-sqrt(-a*c)*d*x - sqrt(-a*c)*c - sqrt(-a*c^3 + 2*sqrt(-a*c)*a*c*d))) - 1/8*sq
rt(-(a*c^3 - 2*sqrt(-a*c)*a*c*d)/(a^2*c^6 + 4*a^3*c^3*d^2))*log(abs(-sqrt(-a*c)*d*x - sqrt(-a*c)*c - sqrt(-a*c
^3 - 2*sqrt(-a*c)*a*c*d)))