\(\int \frac {1}{x^2 (a+b (c+d x)^4)} \, dx\) [115]

Optimal result

Integrand size = 17, antiderivative size = 496 \[ \int \frac {1}{x^2 \left (a+b (c+d x)^4\right )} \, dx=-\frac {1}{\left (a+b c^4\right ) x}-\frac {\sqrt {b} c \left (a-b c^4\right ) d \arctan \left (\frac {\sqrt {b} (c+d x)^2}{\sqrt {a}}\right )}{\sqrt {a} \left (a+b c^4\right )^2}+\frac {\sqrt [4]{b} \left (\sqrt {a} \left (a-3 b c^4\right )+\sqrt {b} c^2 \left (3 a-b c^4\right )\right ) d \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (a+b c^4\right )^2}-\frac {\sqrt [4]{b} \left (\sqrt {a} \left (a-3 b c^4\right )+\sqrt {b} c^2 \left (3 a-b c^4\right )\right ) d \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (a+b c^4\right )^2}-\frac {4 b c^3 d \log (x)}{\left (a+b c^4\right )^2}-\frac {\sqrt [4]{b} \left (\sqrt {a} \left (a-3 b c^4\right )-\sqrt {b} c^2 \left (3 a-b c^4\right )\right ) d \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} \left (a+b c^4\right )^2}+\frac {\sqrt [4]{b} \left (\sqrt {a} \left (a-3 b c^4\right )-\sqrt {b} c^2 \left (3 a-b c^4\right )\right ) d \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} \left (a+b c^4\right )^2}+\frac {b c^3 d \log \left (a+b (c+d x)^4\right )}{\left (a+b c^4\right )^2} \]

[Out]

-1/(b*c^4+a)/x-4*b*c^3*d*ln(x)/(b*c^4+a)^2+b*c^3*d*ln(a+b*(d*x+c)^4)/(b*c^4+a)^2-c*(-b*c^4+a)*d*arctan((d*x+c)
^2*b^(1/2)/a^(1/2))*b^(1/2)/(b*c^4+a)^2/a^(1/2)-1/8*b^(1/4)*d*ln(-a^(1/4)*b^(1/4)*(d*x+c)*2^(1/2)+a^(1/2)+(d*x
+c)^2*b^(1/2))*((-3*b*c^4+a)*a^(1/2)-c^2*(-b*c^4+3*a)*b^(1/2))/a^(3/4)/(b*c^4+a)^2*2^(1/2)+1/8*b^(1/4)*d*ln(a^
(1/4)*b^(1/4)*(d*x+c)*2^(1/2)+a^(1/2)+(d*x+c)^2*b^(1/2))*((-3*b*c^4+a)*a^(1/2)-c^2*(-b*c^4+3*a)*b^(1/2))/a^(3/
4)/(b*c^4+a)^2*2^(1/2)-1/4*b^(1/4)*d*arctan(-1+b^(1/4)*(d*x+c)*2^(1/2)/a^(1/4))*((-3*b*c^4+a)*a^(1/2)+c^2*(-b*
c^4+3*a)*b^(1/2))/a^(3/4)/(b*c^4+a)^2*2^(1/2)-1/4*b^(1/4)*d*arctan(1+b^(1/4)*(d*x+c)*2^(1/2)/a^(1/4))*((-3*b*c
^4+a)*a^(1/2)+c^2*(-b*c^4+3*a)*b^(1/2))/a^(3/4)/(b*c^4+a)^2*2^(1/2)

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 496, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.765, Rules used = {378, 6857, 1890, 1262, 649, 211, 266, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {1}{x^2 \left (a+b (c+d x)^4\right )} \, dx=\frac {\sqrt [4]{b} d \left (\sqrt {a} \left (a-3 b c^4\right )+\sqrt {b} c^2 \left (3 a-b c^4\right )\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (a+b c^4\right )^2}-\frac {\sqrt [4]{b} d \left (\sqrt {a} \left (a-3 b c^4\right )+\sqrt {b} c^2 \left (3 a-b c^4\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} a^{3/4} \left (a+b c^4\right )^2}-\frac {\sqrt [4]{b} d \left (\sqrt {a} \left (a-3 b c^4\right )-\sqrt {b} c^2 \left (3 a-b c^4\right )\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {a}+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} \left (a+b c^4\right )^2}+\frac {\sqrt [4]{b} d \left (\sqrt {a} \left (a-3 b c^4\right )-\sqrt {b} c^2 \left (3 a-b c^4\right )\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {a}+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} \left (a+b c^4\right )^2}-\frac {\sqrt {b} c d \left (a-b c^4\right ) \arctan \left (\frac {\sqrt {b} (c+d x)^2}{\sqrt {a}}\right )}{\sqrt {a} \left (a+b c^4\right )^2}-\frac {1}{x \left (a+b c^4\right )}-\frac {4 b c^3 d \log (x)}{\left (a+b c^4\right )^2}+\frac {b c^3 d \log \left (a+b (c+d x)^4\right )}{\left (a+b c^4\right )^2} \]

[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 210

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

Rule 211

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

Rule 266

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 378

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 631

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 642

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]

Rule 649

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 1176

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 1179

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 1182

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 1262

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 1890

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 6857

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]

Rubi steps \begin{align*} \text {integral}& = d \text {Subst}\left (\int \frac {1}{(-c+x)^2 \left (a+b x^4\right )} \, dx,x,c+d x\right ) \\ & = d \text {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) \text {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) \text {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) \text {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) \text {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) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.09 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.48 \[ \int \frac {1}{x^2 \left (a+b (c+d x)^4\right )} \, dx=\frac {-4 \left (a+b c^4+4 b c^3 d x \log (x)\right )+d x \text {RootSum}\left [a+b c^4+4 b c^3 d \text {$\#$1}+6 b c^2 d^2 \text {$\#$1}^2+4 b c d^3 \text {$\#$1}^3+b d^4 \text {$\#$1}^4\&,\frac {-6 a c^2 \log (x-\text {$\#$1})+10 b c^6 \log (x-\text {$\#$1})-4 a c d \log (x-\text {$\#$1}) \text {$\#$1}+20 b c^5 d \log (x-\text {$\#$1}) \text {$\#$1}-a d^2 \log (x-\text {$\#$1}) \text {$\#$1}^2+15 b c^4 d^2 \log (x-\text {$\#$1}) \text {$\#$1}^2+4 b c^3 d^3 \log (x-\text {$\#$1}) \text {$\#$1}^3}{c^3+3 c^2 d \text {$\#$1}+3 c d^2 \text {$\#$1}^2+d^3 \text {$\#$1}^3}\&\right ]}{4 \left (a+b c^4\right )^2 x} \]

[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)

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.14 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.37

method	result	size
default	\(-\frac {1}{\left (b \,c^{4}+a \right ) x}-\frac {4 b \,c^{3} d \ln \left (x \right )}{\left (b \,c^{4}+a \right )^{2}}-\frac {d \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{4} \textit {\_Z}^{4}+4 b c \,d^{3} \textit {\_Z}^{3}+6 b \,c^{2} d^{2} \textit {\_Z}^{2}+4 b \,c^{3} d \textit {\_Z} +b \,c^{4}+a \right )}{\sum }\frac {\left (-4 b \,d^{3} c^{3} \textit {\_R}^{3}+d^{2} \left (-15 b \,c^{4}+a \right ) \textit {\_R}^{2}+4 c d \left (-5 b \,c^{4}+a \right ) \textit {\_R} -10 b \,c^{6}+6 a \,c^{2}\right ) \ln \left (x -\textit {\_R} \right )}{d^{3} \textit {\_R}^{3}+3 c \,d^{2} \textit {\_R}^{2}+3 c^{2} d \textit {\_R} +c^{3}}\right )}{4 \left (b \,c^{4}+a \right )^{2}}\)	\(184\)
risch	\(-\frac {1}{\left (b \,c^{4}+a \right ) x}-\frac {4 b \,c^{3} d \ln \left (x \right )}{b^{2} c^{8}+2 a b \,c^{4}+a^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{3} b^{2} c^{8}+2 a^{4} b \,c^{4}+a^{5}\right ) \textit {\_Z}^{4}-16 a^{3} b \,c^{3} d \,\textit {\_Z}^{3}+20 a^{2} b \,c^{2} d^{2} \textit {\_Z}^{2}-8 a b c \,d^{3} \textit {\_Z} +b \,d^{4}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-3 a^{2} b^{3} c^{12} d -a^{3} b^{2} c^{8} d +7 a^{4} b \,c^{4} d +5 d \,a^{5}\right ) \textit {\_R}^{4}+\left (4 a \,b^{3} c^{11} d^{2}-40 a^{2} b^{2} c^{7} d^{2}-44 a^{3} b \,c^{3} d^{2}\right ) \textit {\_R}^{3}+\left (-b^{3} c^{10} d^{3}+62 a \,b^{2} c^{6} d^{3}+79 a^{2} b \,c^{2} d^{3}\right ) \textit {\_R}^{2}+\left (-16 b^{2} c^{5} d^{4}-32 b c \,d^{4} a \right ) \textit {\_R} +4 b \,d^{5}\right ) x +\left (-a^{2} b^{3} c^{13}-3 a^{3} b^{2} c^{9}-3 a^{4} b \,c^{5}-c \,a^{5}\right ) \textit {\_R}^{4}+\left (3 a \,b^{3} c^{12} d -9 a^{2} b^{2} c^{8} d -11 a^{3} b \,c^{4} d +a^{4} d \right ) \textit {\_R}^{3}+\left (-b^{3} c^{11} d^{2}+46 a \,b^{2} c^{7} d^{2}+15 a^{2} b \,c^{3} d^{2}\right ) \textit {\_R}^{2}-16 b^{2} c^{6} d^{3} \textit {\_R} \right )\right )}{4}\)	\(420\)

[In]

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

[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*a*c^2)/(_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))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 78.19 (sec) , antiderivative size = 1128605, normalized size of antiderivative = 2275.41 \[ \int \frac {1}{x^2 \left (a+b (c+d x)^4\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x^2 \left (a+b (c+d x)^4\right )} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {1}{x^2 \left (a+b (c+d x)^4\right )} \, dx=\int { \frac {1}{{\left ({\left (d x + c\right )}^{4} b + a\right )} x^{2}} \,d x } \]

[In]

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

[Out]

-4*b*c^3*d*log(x)/(b^2*c^8 + 2*a*b*c^4 + a^2) + b*d^2*integrate((4*b*c^3*d^3*x^3 + 10*b*c^6 + (15*b*c^4 - a)*d
^2*x^2 - 6*a*c^2 + 4*(5*b*c^5 - a*c)*d*x)/(b*d^4*x^4 + 4*b*c*d^3*x^3 + 6*b*c^2*d^2*x^2 + 4*b*c^3*d*x + b*c^4 +
a), x)/(b^2*c^8 + 2*a*b*c^4 + a^2) - 1/((b*c^4 + a)*x)

Giac [F]

\[ \int \frac {1}{x^2 \left (a+b (c+d x)^4\right )} \, dx=\int { \frac {1}{{\left ({\left (d x + c\right )}^{4} b + a\right )} x^{2}} \,d x } \]

[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)

Mupad [B] (verification not implemented)

Time = 9.53 (sec) , antiderivative size = 2440, normalized size of antiderivative = 4.92 \[ \int \frac {1}{x^2 \left (a+b (c+d x)^4\right )} \, dx=\text {Too large to display} \]

[In]

int(1/(x^2*(a + b*(c + d*x)^4)),x)

[Out]

symsum(log(-(4*root(256*a^3*b^2*c^8*z^4 + 512*a^4*b*c^4*z^4 + 256*a^5*z^4 - 1024*a^3*b*c^3*d*z^3 + 320*a^2*b*c
^2*d^2*z^2 - 32*a*b*c*d^3*z + b*d^4, z, k)^2*b^7*c^11*d^17 - 16*root(256*a^3*b^2*c^8*z^4 + 512*a^4*b*c^4*z^4 +
256*a^5*z^4 - 1024*a^3*b*c^3*d*z^3 + 320*a^2*b*c^2*d^2*z^2 - 32*a*b*c*d^3*z + b*d^4, z, k)^3*a^4*b^4*d^16 - b
^5*d^20*x + 16*root(256*a^3*b^2*c^8*z^4 + 512*a^4*b*c^4*z^4 + 256*a^5*z^4 - 1024*a^3*b*c^3*d*z^3 + 320*a^2*b*c
^2*d^2*z^2 - 32*a*b*c*d^3*z + b*d^4, z, k)*b^6*c^6*d^18 - 60*root(256*a^3*b^2*c^8*z^4 + 512*a^4*b*c^4*z^4 + 25
6*a^5*z^4 - 1024*a^3*b*c^3*d*z^3 + 320*a^2*b*c^2*d^2*z^2 - 32*a*b*c*d^3*z + b*d^4, z, k)^2*a^2*b^5*c^3*d^17 +
176*root(256*a^3*b^2*c^8*z^4 + 512*a^4*b*c^4*z^4 + 256*a^5*z^4 - 1024*a^3*b*c^3*d*z^3 + 320*a^2*b*c^2*d^2*z^2
- 32*a*b*c*d^3*z + b*d^4, z, k)^3*a^3*b^5*c^4*d^16 + 192*root(256*a^3*b^2*c^8*z^4 + 512*a^4*b*c^4*z^4 + 256*a^
5*z^4 - 1024*a^3*b*c^3*d*z^3 + 320*a^2*b*c^2*d^2*z^2 - 32*a*b*c*d^3*z + b*d^4, z, k)^4*a^4*b^5*c^5*d^15 + 144*
root(256*a^3*b^2*c^8*z^4 + 512*a^4*b*c^4*z^4 + 256*a^5*z^4 - 1024*a^3*b*c^3*d*z^3 + 320*a^2*b*c^2*d^2*z^2 - 32
*a*b*c*d^3*z + b*d^4, z, k)^3*a^2*b^6*c^8*d^16 + 192*root(256*a^3*b^2*c^8*z^4 + 512*a^4*b*c^4*z^4 + 256*a^5*z^
4 - 1024*a^3*b*c^3*d*z^3 + 320*a^2*b*c^2*d^2*z^2 - 32*a*b*c*d^3*z + b*d^4, z, k)^4*a^3*b^6*c^9*d^15 + 64*root(
256*a^3*b^2*c^8*z^4 + 512*a^4*b*c^4*z^4 + 256*a^5*z^4 - 1024*a^3*b*c^3*d*z^3 + 320*a^2*b*c^2*d^2*z^2 - 32*a*b*
c*d^3*z + b*d^4, z, k)^4*a^2*b^7*c^13*d^15 + 16*root(256*a^3*b^2*c^8*z^4 + 512*a^4*b*c^4*z^4 + 256*a^5*z^4 - 1
024*a^3*b*c^3*d*z^3 + 320*a^2*b*c^2*d^2*z^2 - 32*a*b*c*d^3*z + b*d^4, z, k)*b^6*c^5*d^19*x + 64*root(256*a^3*b
^2*c^8*z^4 + 512*a^4*b*c^4*z^4 + 256*a^5*z^4 - 1024*a^3*b*c^3*d*z^3 + 320*a^2*b*c^2*d^2*z^2 - 32*a*b*c*d^3*z +
b*d^4, z, k)^4*a^5*b^4*c*d^15 - 184*root(256*a^3*b^2*c^8*z^4 + 512*a^4*b*c^4*z^4 + 256*a^5*z^4 - 1024*a^3*b*c
^3*d*z^3 + 320*a^2*b*c^2*d^2*z^2 - 32*a*b*c*d^3*z + b*d^4, z, k)^2*a*b^6*c^7*d^17 - 48*root(256*a^3*b^2*c^8*z^
4 + 512*a^4*b*c^4*z^4 + 256*a^5*z^4 - 1024*a^3*b*c^3*d*z^3 + 320*a^2*b*c^2*d^2*z^2 - 32*a*b*c*d^3*z + b*d^4, z
, k)^3*a*b^7*c^12*d^16 - 320*root(256*a^3*b^2*c^8*z^4 + 512*a^4*b*c^4*z^4 + 256*a^5*z^4 - 1024*a^3*b*c^3*d*z^3
+ 320*a^2*b*c^2*d^2*z^2 - 32*a*b*c*d^3*z + b*d^4, z, k)^4*a^5*b^4*d^16*x + 4*root(256*a^3*b^2*c^8*z^4 + 512*a
^4*b*c^4*z^4 + 256*a^5*z^4 - 1024*a^3*b*c^3*d*z^3 + 320*a^2*b*c^2*d^2*z^2 - 32*a*b*c*d^3*z + b*d^4, z, k)^2*b^
7*c^10*d^18*x - 248*root(256*a^3*b^2*c^8*z^4 + 512*a^4*b*c^4*z^4 + 256*a^5*z^4 - 1024*a^3*b*c^3*d*z^3 + 320*a^
2*b*c^2*d^2*z^2 - 32*a*b*c*d^3*z + b*d^4, z, k)^2*a*b^6*c^6*d^18*x - 64*root(256*a^3*b^2*c^8*z^4 + 512*a^4*b*c
^4*z^4 + 256*a^5*z^4 - 1024*a^3*b*c^3*d*z^3 + 320*a^2*b*c^2*d^2*z^2 - 32*a*b*c*d^3*z + b*d^4, z, k)^3*a*b^7*c^
11*d^17*x + 32*root(256*a^3*b^2*c^8*z^4 + 512*a^4*b*c^4*z^4 + 256*a^5*z^4 - 1024*a^3*b*c^3*d*z^3 + 320*a^2*b*c
^2*d^2*z^2 - 32*a*b*c*d^3*z + b*d^4, z, k)*a*b^5*c*d^19*x - 316*root(256*a^3*b^2*c^8*z^4 + 512*a^4*b*c^4*z^4 +
256*a^5*z^4 - 1024*a^3*b*c^3*d*z^3 + 320*a^2*b*c^2*d^2*z^2 - 32*a*b*c*d^3*z + b*d^4, z, k)^2*a^2*b^5*c^2*d^18
*x + 704*root(256*a^3*b^2*c^8*z^4 + 512*a^4*b*c^4*z^4 + 256*a^5*z^4 - 1024*a^3*b*c^3*d*z^3 + 320*a^2*b*c^2*d^2
*z^2 - 32*a*b*c*d^3*z + b*d^4, z, k)^3*a^3*b^5*c^3*d^17*x - 448*root(256*a^3*b^2*c^8*z^4 + 512*a^4*b*c^4*z^4 +
256*a^5*z^4 - 1024*a^3*b*c^3*d*z^3 + 320*a^2*b*c^2*d^2*z^2 - 32*a*b*c*d^3*z + b*d^4, z, k)^4*a^4*b^5*c^4*d^16
*x + 640*root(256*a^3*b^2*c^8*z^4 + 512*a^4*b*c^4*z^4 + 256*a^5*z^4 - 1024*a^3*b*c^3*d*z^3 + 320*a^2*b*c^2*d^2
*z^2 - 32*a*b*c*d^3*z + b*d^4, z, k)^3*a^2*b^6*c^7*d^17*x + 64*root(256*a^3*b^2*c^8*z^4 + 512*a^4*b*c^4*z^4 +
256*a^5*z^4 - 1024*a^3*b*c^3*d*z^3 + 320*a^2*b*c^2*d^2*z^2 - 32*a*b*c*d^3*z + b*d^4, z, k)^4*a^3*b^6*c^8*d^16*
x + 192*root(256*a^3*b^2*c^8*z^4 + 512*a^4*b*c^4*z^4 + 256*a^5*z^4 - 1024*a^3*b*c^3*d*z^3 + 320*a^2*b*c^2*d^2*
z^2 - 32*a*b*c*d^3*z + b*d^4, z, k)^4*a^2*b^7*c^12*d^16*x)/(a^2 + b^2*c^8 + 2*a*b*c^4))*root(256*a^3*b^2*c^8*z
^4 + 512*a^4*b*c^4*z^4 + 256*a^5*z^4 - 1024*a^3*b*c^3*d*z^3 + 320*a^2*b*c^2*d^2*z^2 - 32*a*b*c*d^3*z + b*d^4,
z, k), k, 1, 4) - 1/(a*x + b*c^4*x) - (4*b*c^3*d*log(x))/(a^2 + b^2*c^8 + 2*a*b*c^4)