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

Optimal. Leaf size=746 \[ -\frac{\left (\frac{c}{d}+x\right ) \left (-4 a d^2+c^3-c d^2 \left (\frac{c}{d}+x\right )^2\right )}{16 a c \left (4 a d^2+c^3\right ) \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}-\frac{d \left (-c^{3/2} \sqrt{4 a d^2+c^3}+12 a d^2+c^3\right ) \log \left (-\sqrt{2} \sqrt [4]{c} d \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}} \left (\frac{c}{d}+x\right )+\sqrt{c} \sqrt{4 a d^2+c^3}+d^2 \left (\frac{c}{d}+x\right )^2\right )}{64 \sqrt{2} a c^{7/4} \left (4 a d^2+c^3\right )^{3/2} \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}}}+\frac{d \left (-c^{3/2} \sqrt{4 a d^2+c^3}+12 a d^2+c^3\right ) \log \left (\sqrt{2} \sqrt [4]{c} d \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}} \left (\frac{c}{d}+x\right )+\sqrt{c} \sqrt{4 a d^2+c^3}+d^2 \left (\frac{c}{d}+x\right )^2\right )}{64 \sqrt{2} a c^{7/4} \left (4 a d^2+c^3\right )^{3/2} \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}}}-\frac{d \left (c^{3/2} \sqrt{4 a d^2+c^3}+12 a d^2+c^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}}+\sqrt{2} c+\sqrt{2} d x}{\sqrt [4]{c} \sqrt{c^{3/2}-\sqrt{4 a d^2+c^3}}}\right )}{32 \sqrt{2} a c^{7/4} \left (4 a d^2+c^3\right )^{3/2} \sqrt{c^{3/2}-\sqrt{4 a d^2+c^3}}}+\frac{d \left (c^{3/2} \sqrt{4 a d^2+c^3}+12 a d^2+c^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}}-\sqrt{2} (c+d x)}{\sqrt [4]{c} \sqrt{c^{3/2}-\sqrt{4 a d^2+c^3}}}\right )}{32 \sqrt{2} a c^{7/4} \left (4 a d^2+c^3\right )^{3/2} \sqrt{c^{3/2}-\sqrt{4 a d^2+c^3}}} \]

[Out]

-((c/d + x)*(c^3 - 4*a*d^2 - c*d^2*(c/d + x)^2))/(16*a*c*(c^3 + 4*a*d^2)*(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*
x^4)) - (d*(c^3 + 12*a*d^2 + c^(3/2)*Sqrt[c^3 + 4*a*d^2])*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) - Sqrt[c^3 + 4*a*d^2]])])/(32*Sqrt[2]*a*c^(7/4)*(c^3 + 4*a*d
^2)^(3/2)*Sqrt[c^(3/2) - Sqrt[c^3 + 4*a*d^2]]) + (d*(c^3 + 12*a*d^2 + c^(3/2)*Sqrt[c^3 + 4*a*d^2])*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]])])
/(32*Sqrt[2]*a*c^(7/4)*(c^3 + 4*a*d^2)^(3/2)*Sqrt[c^(3/2) - Sqrt[c^3 + 4*a*d^2]]) - (d*(c^3 + 12*a*d^2 - c^(3/
2)*Sqrt[c^3 + 4*a*d^2])*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])/(64*Sqrt[2]*a*c^(7/4)*(c^3 + 4*a*d^2)^(3/2)*Sqrt[c^(3/2) + Sqrt[c^3 + 4*a*d^2]
]) + (d*(c^3 + 12*a*d^2 - c^(3/2)*Sqrt[c^3 + 4*a*d^2])*Log[Sqrt[c]*Sqrt[c^3 + 4*a*d^2] + Sqrt[2]*c^(1/4)*d*Sqr
t[c^(3/2) + Sqrt[c^3 + 4*a*d^2]]*(c/d + x) + d^2*(c/d + x)^2])/(64*Sqrt[2]*a*c^(7/4)*(c^3 + 4*a*d^2)^(3/2)*Sqr
t[c^(3/2) + Sqrt[c^3 + 4*a*d^2]])

________________________________________________________________________________________

Rubi [A]  time = 1.32817, antiderivative size = 746, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {1106, 1092, 1169, 634, 618, 206, 628} \[ -\frac{\left (\frac{c}{d}+x\right ) \left (-4 a d^2+c^3-c d^2 \left (\frac{c}{d}+x\right )^2\right )}{16 a c \left (4 a d^2+c^3\right ) \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}-\frac{d \left (-c^{3/2} \sqrt{4 a d^2+c^3}+12 a d^2+c^3\right ) \log \left (-\sqrt{2} \sqrt [4]{c} d \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}} \left (\frac{c}{d}+x\right )+\sqrt{c} \sqrt{4 a d^2+c^3}+d^2 \left (\frac{c}{d}+x\right )^2\right )}{64 \sqrt{2} a c^{7/4} \left (4 a d^2+c^3\right )^{3/2} \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}}}+\frac{d \left (-c^{3/2} \sqrt{4 a d^2+c^3}+12 a d^2+c^3\right ) \log \left (\sqrt{2} \sqrt [4]{c} d \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}} \left (\frac{c}{d}+x\right )+\sqrt{c} \sqrt{4 a d^2+c^3}+d^2 \left (\frac{c}{d}+x\right )^2\right )}{64 \sqrt{2} a c^{7/4} \left (4 a d^2+c^3\right )^{3/2} \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}}}-\frac{d \left (c^{3/2} \sqrt{4 a d^2+c^3}+12 a d^2+c^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}}+\sqrt{2} c+\sqrt{2} d x}{\sqrt [4]{c} \sqrt{c^{3/2}-\sqrt{4 a d^2+c^3}}}\right )}{32 \sqrt{2} a c^{7/4} \left (4 a d^2+c^3\right )^{3/2} \sqrt{c^{3/2}-\sqrt{4 a d^2+c^3}}}+\frac{d \left (c^{3/2} \sqrt{4 a d^2+c^3}+12 a d^2+c^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{\sqrt{4 a d^2+c^3}+c^{3/2}}-\sqrt{2} (c+d x)}{\sqrt [4]{c} \sqrt{c^{3/2}-\sqrt{4 a d^2+c^3}}}\right )}{32 \sqrt{2} a c^{7/4} \left (4 a d^2+c^3\right )^{3/2} \sqrt{c^{3/2}-\sqrt{4 a d^2+c^3}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((c/d + x)*(c^3 - 4*a*d^2 - c*d^2*(c/d + x)^2))/(16*a*c*(c^3 + 4*a*d^2)*(4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*
x^4)) - (d*(c^3 + 12*a*d^2 + c^(3/2)*Sqrt[c^3 + 4*a*d^2])*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) - Sqrt[c^3 + 4*a*d^2]])])/(32*Sqrt[2]*a*c^(7/4)*(c^3 + 4*a*d
^2)^(3/2)*Sqrt[c^(3/2) - Sqrt[c^3 + 4*a*d^2]]) + (d*(c^3 + 12*a*d^2 + c^(3/2)*Sqrt[c^3 + 4*a*d^2])*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]])])
/(32*Sqrt[2]*a*c^(7/4)*(c^3 + 4*a*d^2)^(3/2)*Sqrt[c^(3/2) - Sqrt[c^3 + 4*a*d^2]]) - (d*(c^3 + 12*a*d^2 - c^(3/
2)*Sqrt[c^3 + 4*a*d^2])*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])/(64*Sqrt[2]*a*c^(7/4)*(c^3 + 4*a*d^2)^(3/2)*Sqrt[c^(3/2) + Sqrt[c^3 + 4*a*d^2]
]) + (d*(c^3 + 12*a*d^2 - c^(3/2)*Sqrt[c^3 + 4*a*d^2])*Log[Sqrt[c]*Sqrt[c^3 + 4*a*d^2] + Sqrt[2]*c^(1/4)*d*Sqr
t[c^(3/2) + Sqrt[c^3 + 4*a*d^2]]*(c/d + x) + d^2*(c/d + x)^2])/(64*Sqrt[2]*a*c^(7/4)*(c^3 + 4*a*d^2)^(3/2)*Sqr
t[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 1092

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> -Simp[(x*(b^2 - 2*a*c + b*c*x^2)*(a + b*x^2 + c*x^
4)^(p + 1))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[(b^2 - 2*a*c + 2*(p + 1)
*(b^2 - 4*a*c) + b*c*(4*p + 7)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*
a*c, 0] && LtQ[p, -1] && IntegerQ[2*p]

Rule 1169

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r =
Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(
d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 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}{\left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^2} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\left (c \left (4 a+\frac{c^3}{d^2}\right )-2 c^2 x^2+d^2 x^4\right )^2} \, dx,x,\frac{c}{d}+x\right )\\ &=-\frac{\left (\frac{c}{d}+x\right ) \left (c^3-4 a d^2-c d^2 \left (\frac{c}{d}+x\right )^2\right )}{16 a c \left (c^3+4 a d^2\right ) \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}+\frac{\operatorname{Subst}\left (\int \frac{4 c^4-2 c \left (4 a+\frac{c^3}{d^2}\right ) d^2-2 \left (4 c^4-4 c \left (4 a+\frac{c^3}{d^2}\right ) d^2\right )+2 c^2 d^2 x^2}{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 )}{32 a c^2 \left (c^3+4 a d^2\right )}\\ &=-\frac{\left (\frac{c}{d}+x\right ) \left (c^3-4 a d^2-c d^2 \left (\frac{c}{d}+x\right )^2\right )}{16 a c \left (c^3+4 a d^2\right ) \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\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}} \left (4 c^4-2 c \left (4 a+\frac{c^3}{d^2}\right ) d^2-2 \left (4 c^4-4 c \left (4 a+\frac{c^3}{d^2}\right ) d^2\right )\right )}{d}-\left (4 c^4-2 c \left (4 a+\frac{c^3}{d^2}\right ) d^2-2 c^{5/2} \sqrt{c^3+4 a d^2}-2 \left (4 c^4-4 c \left (4 a+\frac{c^3}{d^2}\right ) d^2\right )\right ) 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 )}{64 \sqrt{2} a c^{11/4} \left (c^3+4 a d^2\right )^{3/2} \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}} \left (4 c^4-2 c \left (4 a+\frac{c^3}{d^2}\right ) d^2-2 \left (4 c^4-4 c \left (4 a+\frac{c^3}{d^2}\right ) d^2\right )\right )}{d}+\left (4 c^4-2 c \left (4 a+\frac{c^3}{d^2}\right ) d^2-2 c^{5/2} \sqrt{c^3+4 a d^2}-2 \left (4 c^4-4 c \left (4 a+\frac{c^3}{d^2}\right ) d^2\right )\right ) 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 )}{64 \sqrt{2} a c^{11/4} \left (c^3+4 a d^2\right )^{3/2} \sqrt{c^{3/2}+\sqrt{c^3+4 a d^2}}}\\ &=-\frac{\left (\frac{c}{d}+x\right ) \left (c^3-4 a d^2-c d^2 \left (\frac{c}{d}+x\right )^2\right )}{16 a c \left (c^3+4 a d^2\right ) \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}-\frac{\left (d \left (c^3+12 a d^2-c^{3/2} \sqrt{c^3+4 a d^2}\right )\right ) \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 )}{64 \sqrt{2} a c^{7/4} \left (c^3+4 a d^2\right )^{3/2} \sqrt{c^{3/2}+\sqrt{c^3+4 a d^2}}}+\frac{\left (d \left (c^3+12 a d^2-c^{3/2} \sqrt{c^3+4 a d^2}\right )\right ) \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 )}{64 \sqrt{2} a c^{7/4} \left (c^3+4 a d^2\right )^{3/2} \sqrt{c^{3/2}+\sqrt{c^3+4 a d^2}}}+\frac{\left (c^3+12 a d^2+c^{3/2} \sqrt{c^3+4 a d^2}\right ) \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 )}{64 a c^{3/2} \left (c^3+4 a d^2\right )^{3/2}}+\frac{\left (c^3+12 a d^2+c^{3/2} \sqrt{c^3+4 a d^2}\right ) \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 )}{64 a c^{3/2} \left (c^3+4 a d^2\right )^{3/2}}\\ &=-\frac{\left (\frac{c}{d}+x\right ) \left (c^3-4 a d^2-c d^2 \left (\frac{c}{d}+x\right )^2\right )}{16 a c \left (c^3+4 a d^2\right ) \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}-\frac{d \left (c^3+12 a d^2-c^{3/2} \sqrt{c^3+4 a d^2}\right ) \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 )}{64 \sqrt{2} a c^{7/4} \left (c^3+4 a d^2\right )^{3/2} \sqrt{c^{3/2}+\sqrt{c^3+4 a d^2}}}+\frac{d \left (c^3+12 a d^2-c^{3/2} \sqrt{c^3+4 a d^2}\right ) \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 )}{64 \sqrt{2} a c^{7/4} \left (c^3+4 a d^2\right )^{3/2} \sqrt{c^{3/2}+\sqrt{c^3+4 a d^2}}}-\frac{\left (c^3+12 a d^2+c^{3/2} \sqrt{c^3+4 a d^2}\right ) \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 )}{32 a c^{3/2} \left (c^3+4 a d^2\right )^{3/2}}-\frac{\left (c^3+12 a d^2+c^{3/2} \sqrt{c^3+4 a d^2}\right ) \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 )}{32 a c^{3/2} \left (c^3+4 a d^2\right )^{3/2}}\\ &=-\frac{\left (\frac{c}{d}+x\right ) \left (c^3-4 a d^2-c d^2 \left (\frac{c}{d}+x\right )^2\right )}{16 a c \left (c^3+4 a d^2\right ) \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}-\frac{d \left (c^3+12 a d^2+c^{3/2} \sqrt{c^3+4 a d^2}\right ) \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 )}{32 \sqrt{2} a c^{7/4} \left (c^3+4 a d^2\right )^{3/2} \sqrt{c^{3/2}-\sqrt{c^3+4 a d^2}}}-\frac{d \left (c^3+12 a d^2+c^{3/2} \sqrt{c^3+4 a d^2}\right ) \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 )}{32 \sqrt{2} a c^{7/4} \left (c^3+4 a d^2\right )^{3/2} \sqrt{c^{3/2}-\sqrt{c^3+4 a d^2}}}-\frac{d \left (c^3+12 a d^2-c^{3/2} \sqrt{c^3+4 a d^2}\right ) \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 )}{64 \sqrt{2} a c^{7/4} \left (c^3+4 a d^2\right )^{3/2} \sqrt{c^{3/2}+\sqrt{c^3+4 a d^2}}}+\frac{d \left (c^3+12 a d^2-c^{3/2} \sqrt{c^3+4 a d^2}\right ) \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 )}{64 \sqrt{2} a c^{7/4} \left (c^3+4 a d^2\right )^{3/2} \sqrt{c^{3/2}+\sqrt{c^3+4 a d^2}}}\\ \end{align*}

Mathematica [C]  time = 0.102979, size = 182, normalized size = 0.24 \[ \frac{\text{RootSum}\left [4 \text{$\#$1}^2 c^2+4 \text{$\#$1}^3 c d+\text{$\#$1}^4 d^2+4 a c\& ,\frac{\text{$\#$1}^2 c d^2 \log (x-\text{$\#$1})+12 a d^2 \log (x-\text{$\#$1})+2 \text{$\#$1} c^2 d \log (x-\text{$\#$1})+2 c^3 \log (x-\text{$\#$1})}{3 \text{$\#$1}^2 c d+\text{$\#$1}^3 d^2+2 \text{$\#$1} c^2}\& \right ]+\frac{4 (c+d x) (4 a d+c x (2 c+d x))}{4 a c+x^2 (2 c+d x)^2}}{64 a c \left (4 a d^2+c^3\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((4*(c + d*x)*(4*a*d + c*x*(2*c + d*x)))/(4*a*c + x^2*(2*c + d*x)^2) + RootSum[4*a*c + 4*c^2*#1^2 + 4*c*d*#1^3
+ d^2*#1^4 & , (2*c^3*Log[x - #1] + 12*a*d^2*Log[x - #1] + 2*c^2*d*Log[x - #1]*#1 + c*d^2*Log[x - #1]*#1^2)/(
2*c^2*#1 + 3*c*d*#1^2 + d^2*#1^3) & ])/(64*a*c*(c^3 + 4*a*d^2))

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Maple [C]  time = 0.014, size = 232, normalized size = 0.3 \begin{align*}{\frac{1}{{d}^{2}{x}^{4}+4\,cd{x}^{3}+4\,{c}^{2}{x}^{2}+4\,ac} \left ({\frac{{d}^{2}{x}^{3}}{16\,a \left ( 4\,a{d}^{2}+{c}^{3} \right ) }}+{\frac{3\,cd{x}^{2}}{16\,a \left ( 4\,a{d}^{2}+{c}^{3} \right ) }}+{\frac{ \left ( 2\,a{d}^{2}+{c}^{3} \right ) x}{8\, \left ( 4\,a{d}^{2}+{c}^{3} \right ) ac}}+{\frac{d}{16\,a{d}^{2}+4\,{c}^{3}}} \right ) }+{\frac{1}{ \left ( 256\,a{d}^{2}+64\,{c}^{3} \right ) ac}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{4}{d}^{2}+4\,{{\it \_Z}}^{3}cd+4\,{{\it \_Z}}^{2}{c}^{2}+4\,ac \right ) }{\frac{ \left ({{\it \_R}}^{2}c{d}^{2}+2\,{\it \_R}\,{c}^{2}d+12\,a{d}^{2}+2\,{c}^{3} \right ) \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)^2,x)

[Out]

(1/16*d^2/a/(4*a*d^2+c^3)*x^3+3/16/a*c*d/(4*a*d^2+c^3)*x^2+1/8/c*(2*a*d^2+c^3)/(4*a*d^2+c^3)/a*x+1/4*d/(4*a*d^
2+c^3))/(d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c)+1/64/(4*a*d^2+c^3)/a/c*sum((_R^2*c*d^2+2*_R*c^2*d+12*a*d^2+2*c^3)/
(_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*} \frac{c d^{2} x^{3} + 3 \, c^{2} d x^{2} + 4 \, a c d + 2 \,{\left (c^{3} + 2 \, a d^{2}\right )} x}{16 \,{\left (4 \, a^{2} c^{5} + 16 \, a^{3} c^{2} d^{2} +{\left (a c^{4} d^{2} + 4 \, a^{2} c d^{4}\right )} x^{4} + 4 \,{\left (a c^{5} d + 4 \, a^{2} c^{2} d^{3}\right )} x^{3} + 4 \,{\left (a c^{6} + 4 \, a^{2} c^{3} d^{2}\right )} x^{2}\right )}} + \frac{\mathit{sage}_{2}}{16 \,{\left (a c^{4} + 4 \, a^{2} c d^{2}\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)^2,x, algorithm="maxima")

[Out]

1/16*(c*d^2*x^3 + 3*c^2*d*x^2 + 4*a*c*d + 2*(c^3 + 2*a*d^2)*x)/(4*a^2*c^5 + 16*a^3*c^2*d^2 + (a*c^4*d^2 + 4*a^
2*c*d^4)*x^4 + 4*(a*c^5*d + 4*a^2*c^2*d^3)*x^3 + 4*(a*c^6 + 4*a^2*c^3*d^2)*x^2) + 1/16*integrate((c*d^2*x^2 +
2*c^2*d*x + 2*c^3 + 12*a*d^2)/(d^2*x^4 + 4*c*d*x^3 + 4*c^2*x^2 + 4*a*c), x)/(a*c^4 + 4*a^2*c*d^2)

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Fricas [B]  time = 1.82838, size = 6965, normalized size = 9.34 \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)^2,x, algorithm="fricas")

[Out]

1/64*(4*c*d^2*x^3 + 12*c^2*d*x^2 + 16*a*c*d + (4*a^2*c^5 + 16*a^3*c^2*d^2 + (a*c^4*d^2 + 4*a^2*c*d^4)*x^4 + 4*
(a*c^5*d + 4*a^2*c^2*d^3)*x^3 + 4*(a*c^6 + 4*a^2*c^3*d^2)*x^2)*sqrt(-(c^6 + 15*a*c^3*d^2 + 60*a^2*d^4 + 2*(a^3
*c^11 + 12*a^4*c^8*d^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*d^6)*sqrt(-(25*c^6*d^6 + 360*a*c^3*d^8 + 1296*a^2*d^10)/(
a^3*c^25 + 24*a^4*c^22*d^2 + 240*a^5*c^19*d^4 + 1280*a^6*c^16*d^6 + 3840*a^7*c^13*d^8 + 6144*a^8*c^10*d^10 + 4
096*a^9*c^7*d^12)))/(a^3*c^11 + 12*a^4*c^8*d^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*d^6))*log(5*c^7*d^3 + 81*a*c^4*d^
5 + 324*a^2*c*d^7 + (5*c^6*d^4 + 81*a*c^3*d^6 + 324*a^2*d^8)*x + (5*a^2*c^8*d^4 + 96*a^3*c^5*d^6 + 432*a^4*c^2
*d^8 + (a^3*c^19 + 20*a^4*c^16*d^2 + 144*a^5*c^13*d^4 + 448*a^6*c^10*d^6 + 512*a^7*c^7*d^8)*sqrt(-(25*c^6*d^6
+ 360*a*c^3*d^8 + 1296*a^2*d^10)/(a^3*c^25 + 24*a^4*c^22*d^2 + 240*a^5*c^19*d^4 + 1280*a^6*c^16*d^6 + 3840*a^7
*c^13*d^8 + 6144*a^8*c^10*d^10 + 4096*a^9*c^7*d^12)))*sqrt(-(c^6 + 15*a*c^3*d^2 + 60*a^2*d^4 + 2*(a^3*c^11 + 1
2*a^4*c^8*d^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*d^6)*sqrt(-(25*c^6*d^6 + 360*a*c^3*d^8 + 1296*a^2*d^10)/(a^3*c^25
+ 24*a^4*c^22*d^2 + 240*a^5*c^19*d^4 + 1280*a^6*c^16*d^6 + 3840*a^7*c^13*d^8 + 6144*a^8*c^10*d^10 + 4096*a^9*c
^7*d^12)))/(a^3*c^11 + 12*a^4*c^8*d^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*d^6))) - (4*a^2*c^5 + 16*a^3*c^2*d^2 + (a*
c^4*d^2 + 4*a^2*c*d^4)*x^4 + 4*(a*c^5*d + 4*a^2*c^2*d^3)*x^3 + 4*(a*c^6 + 4*a^2*c^3*d^2)*x^2)*sqrt(-(c^6 + 15*
a*c^3*d^2 + 60*a^2*d^4 + 2*(a^3*c^11 + 12*a^4*c^8*d^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*d^6)*sqrt(-(25*c^6*d^6 + 3
60*a*c^3*d^8 + 1296*a^2*d^10)/(a^3*c^25 + 24*a^4*c^22*d^2 + 240*a^5*c^19*d^4 + 1280*a^6*c^16*d^6 + 3840*a^7*c^
13*d^8 + 6144*a^8*c^10*d^10 + 4096*a^9*c^7*d^12)))/(a^3*c^11 + 12*a^4*c^8*d^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*d^
6))*log(5*c^7*d^3 + 81*a*c^4*d^5 + 324*a^2*c*d^7 + (5*c^6*d^4 + 81*a*c^3*d^6 + 324*a^2*d^8)*x - (5*a^2*c^8*d^4
+ 96*a^3*c^5*d^6 + 432*a^4*c^2*d^8 + (a^3*c^19 + 20*a^4*c^16*d^2 + 144*a^5*c^13*d^4 + 448*a^6*c^10*d^6 + 512*
a^7*c^7*d^8)*sqrt(-(25*c^6*d^6 + 360*a*c^3*d^8 + 1296*a^2*d^10)/(a^3*c^25 + 24*a^4*c^22*d^2 + 240*a^5*c^19*d^4
+ 1280*a^6*c^16*d^6 + 3840*a^7*c^13*d^8 + 6144*a^8*c^10*d^10 + 4096*a^9*c^7*d^12)))*sqrt(-(c^6 + 15*a*c^3*d^2
+ 60*a^2*d^4 + 2*(a^3*c^11 + 12*a^4*c^8*d^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*d^6)*sqrt(-(25*c^6*d^6 + 360*a*c^3*
d^8 + 1296*a^2*d^10)/(a^3*c^25 + 24*a^4*c^22*d^2 + 240*a^5*c^19*d^4 + 1280*a^6*c^16*d^6 + 3840*a^7*c^13*d^8 +
6144*a^8*c^10*d^10 + 4096*a^9*c^7*d^12)))/(a^3*c^11 + 12*a^4*c^8*d^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*d^6))) + (4
*a^2*c^5 + 16*a^3*c^2*d^2 + (a*c^4*d^2 + 4*a^2*c*d^4)*x^4 + 4*(a*c^5*d + 4*a^2*c^2*d^3)*x^3 + 4*(a*c^6 + 4*a^2
*c^3*d^2)*x^2)*sqrt(-(c^6 + 15*a*c^3*d^2 + 60*a^2*d^4 - 2*(a^3*c^11 + 12*a^4*c^8*d^2 + 48*a^5*c^5*d^4 + 64*a^6
*c^2*d^6)*sqrt(-(25*c^6*d^6 + 360*a*c^3*d^8 + 1296*a^2*d^10)/(a^3*c^25 + 24*a^4*c^22*d^2 + 240*a^5*c^19*d^4 +
1280*a^6*c^16*d^6 + 3840*a^7*c^13*d^8 + 6144*a^8*c^10*d^10 + 4096*a^9*c^7*d^12)))/(a^3*c^11 + 12*a^4*c^8*d^2 +
48*a^5*c^5*d^4 + 64*a^6*c^2*d^6))*log(5*c^7*d^3 + 81*a*c^4*d^5 + 324*a^2*c*d^7 + (5*c^6*d^4 + 81*a*c^3*d^6 +
324*a^2*d^8)*x + (5*a^2*c^8*d^4 + 96*a^3*c^5*d^6 + 432*a^4*c^2*d^8 - (a^3*c^19 + 20*a^4*c^16*d^2 + 144*a^5*c^1
3*d^4 + 448*a^6*c^10*d^6 + 512*a^7*c^7*d^8)*sqrt(-(25*c^6*d^6 + 360*a*c^3*d^8 + 1296*a^2*d^10)/(a^3*c^25 + 24*
a^4*c^22*d^2 + 240*a^5*c^19*d^4 + 1280*a^6*c^16*d^6 + 3840*a^7*c^13*d^8 + 6144*a^8*c^10*d^10 + 4096*a^9*c^7*d^
12)))*sqrt(-(c^6 + 15*a*c^3*d^2 + 60*a^2*d^4 - 2*(a^3*c^11 + 12*a^4*c^8*d^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*d^6)
*sqrt(-(25*c^6*d^6 + 360*a*c^3*d^8 + 1296*a^2*d^10)/(a^3*c^25 + 24*a^4*c^22*d^2 + 240*a^5*c^19*d^4 + 1280*a^6*
c^16*d^6 + 3840*a^7*c^13*d^8 + 6144*a^8*c^10*d^10 + 4096*a^9*c^7*d^12)))/(a^3*c^11 + 12*a^4*c^8*d^2 + 48*a^5*c
^5*d^4 + 64*a^6*c^2*d^6))) - (4*a^2*c^5 + 16*a^3*c^2*d^2 + (a*c^4*d^2 + 4*a^2*c*d^4)*x^4 + 4*(a*c^5*d + 4*a^2*
c^2*d^3)*x^3 + 4*(a*c^6 + 4*a^2*c^3*d^2)*x^2)*sqrt(-(c^6 + 15*a*c^3*d^2 + 60*a^2*d^4 - 2*(a^3*c^11 + 12*a^4*c^
8*d^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*d^6)*sqrt(-(25*c^6*d^6 + 360*a*c^3*d^8 + 1296*a^2*d^10)/(a^3*c^25 + 24*a^4
*c^22*d^2 + 240*a^5*c^19*d^4 + 1280*a^6*c^16*d^6 + 3840*a^7*c^13*d^8 + 6144*a^8*c^10*d^10 + 4096*a^9*c^7*d^12)
))/(a^3*c^11 + 12*a^4*c^8*d^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*d^6))*log(5*c^7*d^3 + 81*a*c^4*d^5 + 324*a^2*c*d^7
+ (5*c^6*d^4 + 81*a*c^3*d^6 + 324*a^2*d^8)*x - (5*a^2*c^8*d^4 + 96*a^3*c^5*d^6 + 432*a^4*c^2*d^8 - (a^3*c^19
+ 20*a^4*c^16*d^2 + 144*a^5*c^13*d^4 + 448*a^6*c^10*d^6 + 512*a^7*c^7*d^8)*sqrt(-(25*c^6*d^6 + 360*a*c^3*d^8 +
1296*a^2*d^10)/(a^3*c^25 + 24*a^4*c^22*d^2 + 240*a^5*c^19*d^4 + 1280*a^6*c^16*d^6 + 3840*a^7*c^13*d^8 + 6144*
a^8*c^10*d^10 + 4096*a^9*c^7*d^12)))*sqrt(-(c^6 + 15*a*c^3*d^2 + 60*a^2*d^4 - 2*(a^3*c^11 + 12*a^4*c^8*d^2 + 4
8*a^5*c^5*d^4 + 64*a^6*c^2*d^6)*sqrt(-(25*c^6*d^6 + 360*a*c^3*d^8 + 1296*a^2*d^10)/(a^3*c^25 + 24*a^4*c^22*d^2
+ 240*a^5*c^19*d^4 + 1280*a^6*c^16*d^6 + 3840*a^7*c^13*d^8 + 6144*a^8*c^10*d^10 + 4096*a^9*c^7*d^12)))/(a^3*c
^11 + 12*a^4*c^8*d^2 + 48*a^5*c^5*d^4 + 64*a^6*c^2*d^6))) + 8*(c^3 + 2*a*d^2)*x)/(4*a^2*c^5 + 16*a^3*c^2*d^2 +
(a*c^4*d^2 + 4*a^2*c*d^4)*x^4 + 4*(a*c^5*d + 4*a^2*c^2*d^3)*x^3 + 4*(a*c^6 + 4*a^2*c^3*d^2)*x^2)

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Sympy [A]  time = 5.8148, size = 427, normalized size = 0.57 \begin{align*} \frac{4 a c d + 3 c^{2} d x^{2} + c d^{2} x^{3} + x \left (4 a d^{2} + 2 c^{3}\right )}{256 a^{3} c^{2} d^{2} + 64 a^{2} c^{5} + x^{4} \left (64 a^{2} c d^{4} + 16 a c^{4} d^{2}\right ) + x^{3} \left (256 a^{2} c^{2} d^{3} + 64 a c^{5} d\right ) + x^{2} \left (256 a^{2} c^{3} d^{2} + 64 a c^{6}\right )} + \operatorname{RootSum}{\left (t^{4} \left (1073741824 a^{9} c^{7} d^{6} + 805306368 a^{8} c^{10} d^{4} + 201326592 a^{7} c^{13} d^{2} + 16777216 a^{6} c^{16}\right ) + t^{2} \left (491520 a^{5} c^{5} d^{4} + 122880 a^{4} c^{8} d^{2} + 8192 a^{3} c^{11}\right ) + 81 a^{2} d^{4} + 18 a c^{3} d^{2} + c^{6}, \left ( t \mapsto t \log{\left (x + \frac{- 67108864 t^{3} a^{7} c^{7} d^{8} - 58720256 t^{3} a^{6} c^{10} d^{6} - 18874368 t^{3} a^{5} c^{13} d^{4} - 2621440 t^{3} a^{4} c^{16} d^{2} - 131072 t^{3} a^{3} c^{19} + 27648 t a^{4} c^{2} d^{8} - 9216 t a^{3} c^{5} d^{6} - 5440 t a^{2} c^{8} d^{4} - 736 t a c^{11} d^{2} - 32 t c^{14} + 324 a^{2} c d^{7} + 81 a c^{4} d^{5} + 5 c^{7} d^{3}}{324 a^{2} d^{8} + 81 a c^{3} d^{6} + 5 c^{6} d^{4}} \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)**2,x)

[Out]

(4*a*c*d + 3*c**2*d*x**2 + c*d**2*x**3 + x*(4*a*d**2 + 2*c**3))/(256*a**3*c**2*d**2 + 64*a**2*c**5 + x**4*(64*
a**2*c*d**4 + 16*a*c**4*d**2) + x**3*(256*a**2*c**2*d**3 + 64*a*c**5*d) + x**2*(256*a**2*c**3*d**2 + 64*a*c**6
)) + RootSum(_t**4*(1073741824*a**9*c**7*d**6 + 805306368*a**8*c**10*d**4 + 201326592*a**7*c**13*d**2 + 167772
16*a**6*c**16) + _t**2*(491520*a**5*c**5*d**4 + 122880*a**4*c**8*d**2 + 8192*a**3*c**11) + 81*a**2*d**4 + 18*a
*c**3*d**2 + c**6, Lambda(_t, _t*log(x + (-67108864*_t**3*a**7*c**7*d**8 - 58720256*_t**3*a**6*c**10*d**6 - 18
874368*_t**3*a**5*c**13*d**4 - 2621440*_t**3*a**4*c**16*d**2 - 131072*_t**3*a**3*c**19 + 27648*_t*a**4*c**2*d*
*8 - 9216*_t*a**3*c**5*d**6 - 5440*_t*a**2*c**8*d**4 - 736*_t*a*c**11*d**2 - 32*_t*c**14 + 324*a**2*c*d**7 + 8
1*a*c**4*d**5 + 5*c**7*d**3)/(324*a**2*d**8 + 81*a*c**3*d**6 + 5*c**6*d**4))))

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \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)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError