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

Optimal. Leaf size=496 \[ -\frac{\sqrt [4]{b} d \left (\sqrt{a} \left (a-3 b c^4\right )-\sqrt{b} c^2 \left (3 a-b c^4\right )\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} \left (a+b c^4\right )^2}+\frac{\sqrt [4]{b} d \left (\sqrt{a} \left (a-3 b c^4\right )-\sqrt{b} c^2 \left (3 a-b c^4\right )\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} \left (a+b c^4\right )^2}+\frac{\sqrt [4]{b} d \left (\sqrt{b} c^2 \left (3 a-b c^4\right )+\sqrt{a} \left (a-3 b c^4\right )\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \left (a+b c^4\right )^2}-\frac{\sqrt [4]{b} d \left (\sqrt{b} c^2 \left (3 a-b c^4\right )+\sqrt{a} \left (a-3 b c^4\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} \left (a+b c^4\right )^2}-\frac{4 b c^3 d \log (x)}{\left (a+b c^4\right )^2}+\frac{b c^3 d \log \left (a+b (c+d x)^4\right )}{\left (a+b c^4\right )^2}-\frac{\sqrt{b} c d \left (a-b c^4\right ) \tan ^{-1}\left (\frac{\sqrt{b} (c+d x)^2}{\sqrt{a}}\right )}{\sqrt{a} \left (a+b c^4\right )^2}-\frac{1}{x \left (a+b c^4\right )} \]

[Out]

-(1/((a + b*c^4)*x)) - (Sqrt[b]*c*(a - b*c^4)*d*ArcTan[(Sqrt[b]*(c + d*x)^2)/Sqrt[a]])/(Sqrt[a]*(a + b*c^4)^2)
+ (b^(1/4)*(Sqrt[a]*(a - 3*b*c^4) + Sqrt[b]*c^2*(3*a - b*c^4))*d*ArcTan[1 - (Sqrt[2]*b^(1/4)*(c + d*x))/a^(1/
4)])/(2*Sqrt[2]*a^(3/4)*(a + b*c^4)^2) - (b^(1/4)*(Sqrt[a]*(a - 3*b*c^4) + Sqrt[b]*c^2*(3*a - b*c^4))*d*ArcTan
[1 + (Sqrt[2]*b^(1/4)*(c + d*x))/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(a + b*c^4)^2) - (4*b*c^3*d*Log[x])/(a + b*c^4)^
2 - (b^(1/4)*(Sqrt[a]*(a - 3*b*c^4) - Sqrt[b]*c^2*(3*a - b*c^4))*d*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*(c +
d*x) + Sqrt[b]*(c + d*x)^2])/(4*Sqrt[2]*a^(3/4)*(a + b*c^4)^2) + (b^(1/4)*(Sqrt[a]*(a - 3*b*c^4) - Sqrt[b]*c^2
*(3*a - b*c^4))*d*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*(c + d*x) + Sqrt[b]*(c + d*x)^2])/(4*Sqrt[2]*a^(3/4)*(
a + b*c^4)^2) + (b*c^3*d*Log[a + b*(c + d*x)^4])/(a + b*c^4)^2

________________________________________________________________________________________

Rubi [A]  time = 0.894138, antiderivative size = 496, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 13, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.765, Rules used = {371, 6725, 1876, 1248, 635, 205, 260, 1168, 1162, 617, 204, 1165, 628} \[ -\frac{\sqrt [4]{b} d \left (\sqrt{a} \left (a-3 b c^4\right )-\sqrt{b} c^2 \left (3 a-b c^4\right )\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} \left (a+b c^4\right )^2}+\frac{\sqrt [4]{b} d \left (\sqrt{a} \left (a-3 b c^4\right )-\sqrt{b} c^2 \left (3 a-b c^4\right )\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt{a}+\sqrt{b} (c+d x)^2\right )}{4 \sqrt{2} a^{3/4} \left (a+b c^4\right )^2}+\frac{\sqrt [4]{b} d \left (\sqrt{b} c^2 \left (3 a-b c^4\right )+\sqrt{a} \left (a-3 b c^4\right )\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt{2} a^{3/4} \left (a+b c^4\right )^2}-\frac{\sqrt [4]{b} d \left (\sqrt{b} c^2 \left (3 a-b c^4\right )+\sqrt{a} \left (a-3 b c^4\right )\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt{2} a^{3/4} \left (a+b c^4\right )^2}-\frac{4 b c^3 d \log (x)}{\left (a+b c^4\right )^2}+\frac{b c^3 d \log \left (a+b (c+d x)^4\right )}{\left (a+b c^4\right )^2}-\frac{\sqrt{b} c d \left (a-b c^4\right ) \tan ^{-1}\left (\frac{\sqrt{b} (c+d x)^2}{\sqrt{a}}\right )}{\sqrt{a} \left (a+b c^4\right )^2}-\frac{1}{x \left (a+b c^4\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a + b*(c + d*x)^4)),x]

[Out]

-(1/((a + b*c^4)*x)) - (Sqrt[b]*c*(a - b*c^4)*d*ArcTan[(Sqrt[b]*(c + d*x)^2)/Sqrt[a]])/(Sqrt[a]*(a + b*c^4)^2)
+ (b^(1/4)*(Sqrt[a]*(a - 3*b*c^4) + Sqrt[b]*c^2*(3*a - b*c^4))*d*ArcTan[1 - (Sqrt[2]*b^(1/4)*(c + d*x))/a^(1/
4)])/(2*Sqrt[2]*a^(3/4)*(a + b*c^4)^2) - (b^(1/4)*(Sqrt[a]*(a - 3*b*c^4) + Sqrt[b]*c^2*(3*a - b*c^4))*d*ArcTan
[1 + (Sqrt[2]*b^(1/4)*(c + d*x))/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(a + b*c^4)^2) - (4*b*c^3*d*Log[x])/(a + b*c^4)^
2 - (b^(1/4)*(Sqrt[a]*(a - 3*b*c^4) - Sqrt[b]*c^2*(3*a - b*c^4))*d*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*(c +
d*x) + Sqrt[b]*(c + d*x)^2])/(4*Sqrt[2]*a^(3/4)*(a + b*c^4)^2) + (b^(1/4)*(Sqrt[a]*(a - 3*b*c^4) - Sqrt[b]*c^2
*(3*a - b*c^4))*d*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*(c + d*x) + Sqrt[b]*(c + d*x)^2])/(4*Sqrt[2]*a^(3/4)*(
a + b*c^4)^2) + (b*c^3*d*Log[a + b*(c + d*x)^4])/(a + b*c^4)^2

Rule 371

Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coefficient[v, x, 0], d = Coefficient[v,
x, 1]}, Dist[1/d^(m + 1), Subst[Int[SimplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]
] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rule 1876

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = Sum[(x^ii*(Coeff[Pq, x, ii] + Coeff[Pq, x, n/2 + ii
]*x^(n/2)))/(a + b*x^n), {ii, 0, n/2 - 1}]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ
[n/2, 0] && Expon[Pq, x] < n

Rule 1248

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(d + e*x)^q
*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x]

Rule 635

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

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 1168

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

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

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

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

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

Mathematica [C]  time = 0.124412, size = 238, normalized size = 0.48 \[ \frac{d x \text{RootSum}\left [6 \text{$\#$1}^2 b c^2 d^2+4 \text{$\#$1}^3 b c d^3+\text{$\#$1}^4 b d^4+4 \text{$\#$1} b c^3 d+a+b c^4\& ,\frac{-\text{$\#$1}^2 a d^2 \log (x-\text{$\#$1})+15 \text{$\#$1}^2 b c^4 d^2 \log (x-\text{$\#$1})+4 \text{$\#$1}^3 b c^3 d^3 \log (x-\text{$\#$1})-6 a c^2 \log (x-\text{$\#$1})-4 \text{$\#$1} a c d \log (x-\text{$\#$1})+20 \text{$\#$1} b c^5 d \log (x-\text{$\#$1})+10 b c^6 \log (x-\text{$\#$1})}{3 \text{$\#$1}^2 c d^2+\text{$\#$1}^3 d^3+3 \text{$\#$1} c^2 d+c^3}\& \right ]-4 \left (a+4 b c^3 d x \log (x)+b c^4\right )}{4 x \left (a+b c^4\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*(a + b*c^4 + 4*b*c^3*d*x*Log[x]) + d*x*RootSum[a + b*c^4 + 4*b*c^3*d*#1 + 6*b*c^2*d^2*#1^2 + 4*b*c*d^3*#1^
3 + b*d^4*#1^4 & , (-6*a*c^2*Log[x - #1] + 10*b*c^6*Log[x - #1] - 4*a*c*d*Log[x - #1]*#1 + 20*b*c^5*d*Log[x -
#1]*#1 - a*d^2*Log[x - #1]*#1^2 + 15*b*c^4*d^2*Log[x - #1]*#1^2 + 4*b*c^3*d^3*Log[x - #1]*#1^3)/(c^3 + 3*c^2*d
*#1 + 3*c*d^2*#1^2 + d^3*#1^3) & ])/(4*(a + b*c^4)^2*x)

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Maple [C]  time = 0.01, size = 188, normalized size = 0.4 \begin{align*} -{\frac{1}{ \left ( b{c}^{4}+a \right ) x}}-4\,{\frac{b{c}^{3}d\ln \left ( x \right ) }{ \left ( b{c}^{4}+a \right ) ^{2}}}+{\frac{d}{4\, \left ( b{c}^{4}+a \right ) ^{2}}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{4}b{d}^{4}+4\,{{\it \_Z}}^{3}bc{d}^{3}+6\,{{\it \_Z}}^{2}b{c}^{2}{d}^{2}+4\,{\it \_Z}\,b{c}^{3}d+b{c}^{4}+a \right ) }{\frac{ \left ( 4\,b{d}^{3}{c}^{3}{{\it \_R}}^{3}+{d}^{2} \left ( 15\,b{c}^{4}-a \right ){{\it \_R}}^{2}+4\,cd \left ( 5\,b{c}^{4}-a \right ){\it \_R}+10\,b{c}^{6}-6\,{c}^{2}a \right ) \ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{3}{d}^{3}+3\,{{\it \_R}}^{2}c{d}^{2}+3\,{\it \_R}\,{c}^{2}d+{c}^{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(b*c^4+a)/x-4*b*c^3*d*ln(x)/(b*c^4+a)^2+1/4*d/(b*c^4+a)^2*sum((4*b*d^3*c^3*_R^3+d^2*(15*b*c^4-a)*_R^2+4*c*d
*(5*b*c^4-a)*_R+10*b*c^6-6*c^2*a)/(_R^3*d^3+3*_R^2*c*d^2+3*_R*c^2*d+c^3)*ln(x-_R),_R=RootOf(_Z^4*b*d^4+4*_Z^3*
b*c*d^3+6*_Z^2*b*c^2*d^2+4*_Z*b*c^3*d+b*c^4+a))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: AttributeError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left ({\left (d x + c\right )}^{4} b + a\right )} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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