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

Optimal. Leaf size=680 \[ -\frac {\sqrt {c} e \left (a^2 e^8-12 a c d^4 e^4+3 c^2 d^8\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \left (a e^4+c d^4\right )^3}-\frac {c^{3/4} d \left (3 a^2 e^8-12 a c d^4 e^4-2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )+c^2 d^8\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )^3}+\frac {c^{3/4} d \left (3 a^2 e^8-12 a c d^4 e^4-2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )+c^2 d^8\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )^3}-\frac {c^{3/4} d \left (3 a^2 e^8-12 a c d^4 e^4+2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )+c^2 d^8\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )^3}+\frac {c^{3/4} d \left (3 a^2 e^8-12 a c d^4 e^4+2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )+c^2 d^8\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )^3}-\frac {e^3}{2 (d+e x)^2 \left (a e^4+c d^4\right )}-\frac {4 c d^3 e^3}{(d+e x) \left (a e^4+c d^4\right )^2}-\frac {c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log \left (a+c x^4\right )}{2 \left (a e^4+c d^4\right )^3}+\frac {2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log (d+e x)}{\left (a e^4+c d^4\right )^3} \]

[Out]

-1/2*e^3/(a*e^4+c*d^4)/(e*x+d)^2-4*c*d^3*e^3/(a*e^4+c*d^4)^2/(e*x+d)+2*c*d^2*e^3*(-3*a*e^4+5*c*d^4)*ln(e*x+d)/
(a*e^4+c*d^4)^3-1/2*c*d^2*e^3*(-3*a*e^4+5*c*d^4)*ln(c*x^4+a)/(a*e^4+c*d^4)^3-1/2*e*(a^2*e^8-12*a*c*d^4*e^4+3*c
^2*d^8)*arctan(x^2*c^(1/2)/a^(1/2))*c^(1/2)/(a*e^4+c*d^4)^3/a^(1/2)-1/8*c^(3/4)*d*ln(-a^(1/4)*c^(1/4)*x*2^(1/2
)+a^(1/2)+x^2*c^(1/2))*(c^2*d^8-12*a*c*d^4*e^4+3*a^2*e^8-2*d^2*e^2*(-5*a*e^4+3*c*d^4)*a^(1/2)*c^(1/2))/a^(3/4)
/(a*e^4+c*d^4)^3*2^(1/2)+1/8*c^(3/4)*d*ln(a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(c^2*d^8-12*a*c*d^4*e
^4+3*a^2*e^8-2*d^2*e^2*(-5*a*e^4+3*c*d^4)*a^(1/2)*c^(1/2))/a^(3/4)/(a*e^4+c*d^4)^3*2^(1/2)+1/4*c^(3/4)*d*arcta
n(-1+c^(1/4)*x*2^(1/2)/a^(1/4))*(c^2*d^8-12*a*c*d^4*e^4+3*a^2*e^8+2*d^2*e^2*(-5*a*e^4+3*c*d^4)*a^(1/2)*c^(1/2)
)/a^(3/4)/(a*e^4+c*d^4)^3*2^(1/2)+1/4*c^(3/4)*d*arctan(1+c^(1/4)*x*2^(1/2)/a^(1/4))*(c^2*d^8-12*a*c*d^4*e^4+3*
a^2*e^8+2*d^2*e^2*(-5*a*e^4+3*c*d^4)*a^(1/2)*c^(1/2))/a^(3/4)/(a*e^4+c*d^4)^3*2^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.95, antiderivative size = 680, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 12, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.706, Rules used = {6725, 1876, 1168, 1162, 617, 204, 1165, 628, 1248, 635, 205, 260} \[ -\frac {c^{3/4} d \left (3 a^2 e^8-12 a c d^4 e^4-2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )+c^2 d^8\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )^3}+\frac {c^{3/4} d \left (3 a^2 e^8-12 a c d^4 e^4-2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )+c^2 d^8\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )^3}-\frac {\sqrt {c} e \left (a^2 e^8-12 a c d^4 e^4+3 c^2 d^8\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \left (a e^4+c d^4\right )^3}-\frac {c^{3/4} d \left (3 a^2 e^8-12 a c d^4 e^4+2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )+c^2 d^8\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )^3}+\frac {c^{3/4} d \left (3 a^2 e^8-12 a c d^4 e^4+2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )+c^2 d^8\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )^3}-\frac {c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log \left (a+c x^4\right )}{2 \left (a e^4+c d^4\right )^3}-\frac {4 c d^3 e^3}{(d+e x) \left (a e^4+c d^4\right )^2}-\frac {e^3}{2 (d+e x)^2 \left (a e^4+c d^4\right )}+\frac {2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log (d+e x)}{\left (a e^4+c d^4\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^3*(a + c*x^4)),x]

[Out]

-e^3/(2*(c*d^4 + a*e^4)*(d + e*x)^2) - (4*c*d^3*e^3)/((c*d^4 + a*e^4)^2*(d + e*x)) - (Sqrt[c]*e*(3*c^2*d^8 - 1
2*a*c*d^4*e^4 + a^2*e^8)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2*Sqrt[a]*(c*d^4 + a*e^4)^3) - (c^(3/4)*d*(c^2*d^8 -
12*a*c*d^4*e^4 + 3*a^2*e^8 + 2*Sqrt[a]*Sqrt[c]*d^2*e^2*(3*c*d^4 - 5*a*e^4))*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(
1/4)])/(2*Sqrt[2]*a^(3/4)*(c*d^4 + a*e^4)^3) + (c^(3/4)*d*(c^2*d^8 - 12*a*c*d^4*e^4 + 3*a^2*e^8 + 2*Sqrt[a]*Sq
rt[c]*d^2*e^2*(3*c*d^4 - 5*a*e^4))*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(c*d^4 + a*e^4)
^3) + (2*c*d^2*e^3*(5*c*d^4 - 3*a*e^4)*Log[d + e*x])/(c*d^4 + a*e^4)^3 - (c^(3/4)*d*(c^2*d^8 - 12*a*c*d^4*e^4
+ 3*a^2*e^8 - 2*Sqrt[a]*Sqrt[c]*d^2*e^2*(3*c*d^4 - 5*a*e^4))*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]
*x^2])/(4*Sqrt[2]*a^(3/4)*(c*d^4 + a*e^4)^3) + (c^(3/4)*d*(c^2*d^8 - 12*a*c*d^4*e^4 + 3*a^2*e^8 - 2*Sqrt[a]*Sq
rt[c]*d^2*e^2*(3*c*d^4 - 5*a*e^4))*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(3/4)*
(c*d^4 + a*e^4)^3) - (c*d^2*e^3*(5*c*d^4 - 3*a*e^4)*Log[a + c*x^4])/(2*(c*d^4 + a*e^4)^3)

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 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 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 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]

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 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 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 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 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 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 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]

Rubi steps

\begin {align*} \int \frac {1}{(d+e x)^3 \left (a+c x^4\right )} \, dx &=\int \left (\frac {e^4}{\left (c d^4+a e^4\right ) (d+e x)^3}+\frac {4 c d^3 e^4}{\left (c d^4+a e^4\right )^2 (d+e x)^2}+\frac {2 c d^2 e^4 \left (5 c d^4-3 a e^4\right )}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac {c \left (d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x+2 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2-2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) x^3\right )}{\left (c d^4+a e^4\right )^3 \left (a+c x^4\right )}\right ) \, dx\\ &=-\frac {e^3}{2 \left (c d^4+a e^4\right ) (d+e x)^2}-\frac {4 c d^3 e^3}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac {2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^3}+\frac {c \int \frac {d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x+2 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2-2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) x^3}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^3}\\ &=-\frac {e^3}{2 \left (c d^4+a e^4\right ) (d+e x)^2}-\frac {4 c d^3 e^3}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac {2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^3}+\frac {c \int \left (\frac {d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )+2 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2}{a+c x^4}+\frac {x \left (-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right )-2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) x^2\right )}{a+c x^4}\right ) \, dx}{\left (c d^4+a e^4\right )^3}\\ &=-\frac {e^3}{2 \left (c d^4+a e^4\right ) (d+e x)^2}-\frac {4 c d^3 e^3}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac {2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^3}+\frac {c \int \frac {d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )+2 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^3}+\frac {c \int \frac {x \left (-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right )-2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) x^2\right )}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^3}\\ &=-\frac {e^3}{2 \left (c d^4+a e^4\right ) (d+e x)^2}-\frac {4 c d^3 e^3}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac {2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^3}+\frac {c \operatorname {Subst}\left (\int \frac {-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right )-2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) x}{a+c x^2} \, dx,x,x^2\right )}{2 \left (c d^4+a e^4\right )^3}-\frac {\left (c d \left (6 c d^6 e^2-10 a d^2 e^6-\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right )\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{2 \left (c d^4+a e^4\right )^3}+\frac {\left (c d \left (6 c d^6 e^2-10 a d^2 e^6+\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right )\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{2 \left (c d^4+a e^4\right )^3}\\ &=-\frac {e^3}{2 \left (c d^4+a e^4\right ) (d+e x)^2}-\frac {4 c d^3 e^3}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac {2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^3}-\frac {\left (c^2 d^2 e^3 \left (5 c d^4-3 a e^4\right )\right ) \operatorname {Subst}\left (\int \frac {x}{a+c x^2} \, dx,x,x^2\right )}{\left (c d^4+a e^4\right )^3}-\frac {\left (c e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{2 \left (c d^4+a e^4\right )^3}+\frac {\left (c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6-\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{4 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}+\frac {\left (c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6-\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{4 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}+\frac {\left (c d \left (6 c d^6 e^2-10 a d^2 e^6+\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 \left (c d^4+a e^4\right )^3}+\frac {\left (c d \left (6 c d^6 e^2-10 a d^2 e^6+\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 \left (c d^4+a e^4\right )^3}\\ &=-\frac {e^3}{2 \left (c d^4+a e^4\right ) (d+e x)^2}-\frac {4 c d^3 e^3}{\left (c d^4+a e^4\right )^2 (d+e x)}-\frac {\sqrt {c} e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \left (c d^4+a e^4\right )^3}+\frac {2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^3}+\frac {c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6-\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}-\frac {c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6-\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}-\frac {c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log \left (a+c x^4\right )}{2 \left (c d^4+a e^4\right )^3}+\frac {\left (c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6+\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}-\frac {\left (c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6+\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}\\ &=-\frac {e^3}{2 \left (c d^4+a e^4\right ) (d+e x)^2}-\frac {4 c d^3 e^3}{\left (c d^4+a e^4\right )^2 (d+e x)}-\frac {\sqrt {c} e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \left (c d^4+a e^4\right )^3}-\frac {c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6+\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}+\frac {c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6+\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}+\frac {2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^3}+\frac {c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6-\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}-\frac {c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6-\frac {c^2 d^8-12 a c d^4 e^4+3 a^2 e^8}{\sqrt {a} \sqrt {c}}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}-\frac {c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log \left (a+c x^4\right )}{2 \left (c d^4+a e^4\right )^3}\\ \end {align*}

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Mathematica [A]  time = 0.93, size = 738, normalized size = 1.09 \[ \frac {-4 a^{3/4} e^3 \left (a e^4+c d^4\right )^2-32 a^{3/4} c d^3 e^3 (d+e x) \left (a e^4+c d^4\right )+4 a^{3/4} c d^2 e^3 (d+e x)^2 \left (3 a e^4-5 c d^4\right ) \log \left (a+c x^4\right )+16 a^{3/4} c d^2 e^3 (d+e x)^2 \left (5 c d^4-3 a e^4\right ) \log (d+e x)-\sqrt {2} c^{3/4} d (d+e x)^2 \left (10 a^{3/2} \sqrt {c} d^2 e^6+3 a^2 e^8-6 \sqrt {a} c^{3/2} d^6 e^2-12 a c d^4 e^4+c^2 d^8\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )+\sqrt {2} c^{3/4} d (d+e x)^2 \left (10 a^{3/2} \sqrt {c} d^2 e^6+3 a^2 e^8-6 \sqrt {a} c^{3/2} d^6 e^2-12 a c d^4 e^4+c^2 d^8\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )-2 \sqrt {c} (d+e x)^2 \left (-10 \sqrt {2} a^{3/2} c^{3/4} d^3 e^6+24 a^{5/4} c d^4 e^5-2 a^{9/4} e^9+3 \sqrt {2} a^2 \sqrt [4]{c} d e^8+6 \sqrt {2} \sqrt {a} c^{7/4} d^7 e^2-12 \sqrt {2} a c^{5/4} d^5 e^4-6 \sqrt [4]{a} c^2 d^8 e+\sqrt {2} c^{9/4} d^9\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+2 \sqrt {c} (d+e x)^2 \left (-10 \sqrt {2} a^{3/2} c^{3/4} d^3 e^6-24 a^{5/4} c d^4 e^5+2 a^{9/4} e^9+3 \sqrt {2} a^2 \sqrt [4]{c} d e^8+6 \sqrt {2} \sqrt {a} c^{7/4} d^7 e^2-12 \sqrt {2} a c^{5/4} d^5 e^4+6 \sqrt [4]{a} c^2 d^8 e+\sqrt {2} c^{9/4} d^9\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 a^{3/4} (d+e x)^2 \left (a e^4+c d^4\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*a^(3/4)*e^3*(c*d^4 + a*e^4)^2 - 32*a^(3/4)*c*d^3*e^3*(c*d^4 + a*e^4)*(d + e*x) - 2*Sqrt[c]*(Sqrt[2]*c^(9/4
)*d^9 - 6*a^(1/4)*c^2*d^8*e + 6*Sqrt[2]*Sqrt[a]*c^(7/4)*d^7*e^2 - 12*Sqrt[2]*a*c^(5/4)*d^5*e^4 + 24*a^(5/4)*c*
d^4*e^5 - 10*Sqrt[2]*a^(3/2)*c^(3/4)*d^3*e^6 + 3*Sqrt[2]*a^2*c^(1/4)*d*e^8 - 2*a^(9/4)*e^9)*(d + e*x)^2*ArcTan
[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + 2*Sqrt[c]*(Sqrt[2]*c^(9/4)*d^9 + 6*a^(1/4)*c^2*d^8*e + 6*Sqrt[2]*Sqrt[a]*c
^(7/4)*d^7*e^2 - 12*Sqrt[2]*a*c^(5/4)*d^5*e^4 - 24*a^(5/4)*c*d^4*e^5 - 10*Sqrt[2]*a^(3/2)*c^(3/4)*d^3*e^6 + 3*
Sqrt[2]*a^2*c^(1/4)*d*e^8 + 2*a^(9/4)*e^9)*(d + e*x)^2*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + 16*a^(3/4)*c*
d^2*e^3*(5*c*d^4 - 3*a*e^4)*(d + e*x)^2*Log[d + e*x] - Sqrt[2]*c^(3/4)*d*(c^2*d^8 - 6*Sqrt[a]*c^(3/2)*d^6*e^2
- 12*a*c*d^4*e^4 + 10*a^(3/2)*Sqrt[c]*d^2*e^6 + 3*a^2*e^8)*(d + e*x)^2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x
+ Sqrt[c]*x^2] + Sqrt[2]*c^(3/4)*d*(c^2*d^8 - 6*Sqrt[a]*c^(3/2)*d^6*e^2 - 12*a*c*d^4*e^4 + 10*a^(3/2)*Sqrt[c]
*d^2*e^6 + 3*a^2*e^8)*(d + e*x)^2*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2] + 4*a^(3/4)*c*d^2*e^3
*(-5*c*d^4 + 3*a*e^4)*(d + e*x)^2*Log[a + c*x^4])/(8*a^(3/4)*(c*d^4 + a*e^4)^3*(d + e*x)^2)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [A]  time = 0.89, size = 901, normalized size = 1.32 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(sqrt(2)*(a*c^3)^(1/4)*c^2*d^3 + 2*a*c^2*e^3 - 3*sqrt(2)*(a*c^3)^(3/4)*d*e^2)*arctan(1/2*sqrt(2)*(2*x + sq
rt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a*c^3*d^6 - 3*sqrt(2)*(a*c^3)^(1/4)*a*c^2*d^5*e + 9*sqrt(a*c)*a*c^2*d^4*e^2 +
9*a^2*c^2*d^2*e^4 - 8*sqrt(2)*(a*c^3)^(3/4)*a*d^3*e^3 - 3*sqrt(2)*(a*c^3)^(1/4)*a^2*c*d*e^5 + sqrt(a*c)*a^2*c
*e^6) + 1/4*(sqrt(2)*(a*c^3)^(1/4)*c^2*d^3 - 2*a*c^2*e^3 - 3*sqrt(2)*(a*c^3)^(3/4)*d*e^2)*arctan(1/2*sqrt(2)*(
2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a*c^3*d^6 + 3*sqrt(2)*(a*c^3)^(1/4)*a*c^2*d^5*e - 9*sqrt(a*c)*a*c^2*d
^4*e^2 + 9*a^2*c^2*d^2*e^4 + 8*sqrt(2)*(a*c^3)^(3/4)*a*d^3*e^3 + 3*sqrt(2)*(a*c^3)^(1/4)*a^2*c*d*e^5 + sqrt(a*
c)*a^2*c*e^6) + 1/8*(sqrt(2)*(a*c^3)^(1/4)*c^3*d^9 - 6*sqrt(2)*(a*c^3)^(3/4)*c*d^7*e^2 - 12*sqrt(2)*(a*c^3)^(1
/4)*a*c^2*d^5*e^4 + 10*sqrt(2)*(a*c^3)^(3/4)*a*d^3*e^6 + 3*sqrt(2)*(a*c^3)^(1/4)*a^2*c*d*e^8)*log(x^2 + sqrt(2
)*x*(a/c)^(1/4) + sqrt(a/c))/(a*c^4*d^12 + 3*a^2*c^3*d^8*e^4 + 3*a^3*c^2*d^4*e^8 + a^4*c*e^12) - 1/8*(sqrt(2)*
(a*c^3)^(1/4)*c^3*d^9 - 6*sqrt(2)*(a*c^3)^(3/4)*c*d^7*e^2 - 12*sqrt(2)*(a*c^3)^(1/4)*a*c^2*d^5*e^4 + 10*sqrt(2
)*(a*c^3)^(3/4)*a*d^3*e^6 + 3*sqrt(2)*(a*c^3)^(1/4)*a^2*c*d*e^8)*log(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/
(a*c^4*d^12 + 3*a^2*c^3*d^8*e^4 + 3*a^3*c^2*d^4*e^8 + a^4*c*e^12) - 1/2*(5*c^2*d^6*e^3 - 3*a*c*d^2*e^7)*log(ab
s(c*x^4 + a))/(c^3*d^12 + 3*a*c^2*d^8*e^4 + 3*a^2*c*d^4*e^8 + a^3*e^12) + 2*(5*c^2*d^6*e^4 - 3*a*c*d^2*e^8)*lo
g(abs(x*e + d))/(c^3*d^12*e + 3*a*c^2*d^8*e^5 + 3*a^2*c*d^4*e^9 + a^3*e^13) - 1/2*(9*c^2*d^8*e^3 + 10*a*c*d^4*
e^7 + a^2*e^11 + 8*(c^2*d^7*e^4 + a*c*d^3*e^8)*x)/((c*d^4 + a*e^4)^3*(x*e + d)^2)

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maple [B]  time = 0.01, size = 1201, normalized size = 1.77 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*e^3/(a*e^4+c*d^4)/(e*x+d)^2-4*c*d^3*e^3/(a*e^4+c*d^4)^2/(e*x+d)-6*e^7*d^2*c/(a*e^4+c*d^4)^3*ln(e*x+d)*a+1
0*e^3*d^6*c^2/(a*e^4+c*d^4)^3*ln(e*x+d)+3/4*c/(a*e^4+c*d^4)^3*(a/c)^(1/4)*a*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)
*x-1)*d*e^8-3*c^2/(a*e^4+c*d^4)^3*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x-1)*d^5*e^4+1/4*c^3/(a*e^4+c
*d^4)^3*(a/c)^(1/4)/a*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x-1)*d^9+3/8*c/(a*e^4+c*d^4)^3*(a/c)^(1/4)*a*2^(1/2)*
ln((x^2+(a/c)^(1/4)*2^(1/2)*x+(a/c)^(1/2))/(x^2-(a/c)^(1/4)*2^(1/2)*x+(a/c)^(1/2)))*d*e^8-3/2*c^2/(a*e^4+c*d^4
)^3*(a/c)^(1/4)*2^(1/2)*ln((x^2+(a/c)^(1/4)*2^(1/2)*x+(a/c)^(1/2))/(x^2-(a/c)^(1/4)*2^(1/2)*x+(a/c)^(1/2)))*d^
5*e^4+1/8*c^3/(a*e^4+c*d^4)^3*(a/c)^(1/4)/a*2^(1/2)*ln((x^2+(a/c)^(1/4)*2^(1/2)*x+(a/c)^(1/2))/(x^2-(a/c)^(1/4
)*2^(1/2)*x+(a/c)^(1/2)))*d^9+3/4*c/(a*e^4+c*d^4)^3*(a/c)^(1/4)*a*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)*d*e^
8-3*c^2/(a*e^4+c*d^4)^3*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)*d^5*e^4+1/4*c^3/(a*e^4+c*d^4)^3*(a
/c)^(1/4)/a*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)*d^9-1/2*c/(a*e^4+c*d^4)^3/(a*c)^(1/2)*arctan((1/a*c)^(1/2)
*x^2)*e^9*a^2+6*c^2/(a*e^4+c*d^4)^3/(a*c)^(1/2)*arctan((1/a*c)^(1/2)*x^2)*d^4*e^5*a-3/2*c^3/(a*e^4+c*d^4)^3/(a
*c)^(1/2)*arctan((1/a*c)^(1/2)*x^2)*e*d^8-5/4*c/(a*e^4+c*d^4)^3/(a/c)^(1/4)*2^(1/2)*ln((x^2-(a/c)^(1/4)*2^(1/2
)*x+(a/c)^(1/2))/(x^2+(a/c)^(1/4)*2^(1/2)*x+(a/c)^(1/2)))*a*d^3*e^6+3/4*c^2/(a*e^4+c*d^4)^3/(a/c)^(1/4)*2^(1/2
)*ln((x^2-(a/c)^(1/4)*2^(1/2)*x+(a/c)^(1/2))/(x^2+(a/c)^(1/4)*2^(1/2)*x+(a/c)^(1/2)))*d^7*e^2-5/2*c/(a*e^4+c*d
^4)^3/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x-1)*a*d^3*e^6+3/2*c^2/(a*e^4+c*d^4)^3/(a/c)^(1/4)*2^(1/2
)*arctan(2^(1/2)/(a/c)^(1/4)*x-1)*d^7*e^2-5/2*c/(a*e^4+c*d^4)^3/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)
*x+1)*a*d^3*e^6+3/2*c^2/(a*e^4+c*d^4)^3/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)*d^7*e^2+3/2*c/(a*e
^4+c*d^4)^3*ln(c*x^4+a)*e^7*d^2*a-5/2*c^2/(a*e^4+c*d^4)^3*ln(c*x^4+a)*e^3*d^6

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maxima [A]  time = 2.34, size = 817, normalized size = 1.20 \[ -\frac {c {\left (\frac {\sqrt {2} {\left (10 \, \sqrt {2} a^{\frac {3}{4}} c^{\frac {9}{4}} d^{6} e^{3} - 6 \, \sqrt {2} a^{\frac {7}{4}} c^{\frac {5}{4}} d^{2} e^{7} - c^{3} d^{9} + 6 \, \sqrt {a} c^{\frac {5}{2}} d^{7} e^{2} + 12 \, a c^{2} d^{5} e^{4} - 10 \, a^{\frac {3}{2}} c^{\frac {3}{2}} d^{3} e^{6} - 3 \, a^{2} c d e^{8}\right )} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {5}{4}}} + \frac {\sqrt {2} {\left (10 \, \sqrt {2} a^{\frac {3}{4}} c^{\frac {9}{4}} d^{6} e^{3} - 6 \, \sqrt {2} a^{\frac {7}{4}} c^{\frac {5}{4}} d^{2} e^{7} + c^{3} d^{9} - 6 \, \sqrt {a} c^{\frac {5}{2}} d^{7} e^{2} - 12 \, a c^{2} d^{5} e^{4} + 10 \, a^{\frac {3}{2}} c^{\frac {3}{2}} d^{3} e^{6} + 3 \, a^{2} c d e^{8}\right )} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {5}{4}}} - \frac {2 \, {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {13}{4}} d^{9} + 6 \, \sqrt {2} a^{\frac {3}{4}} c^{\frac {11}{4}} d^{7} e^{2} - 12 \, \sqrt {2} a^{\frac {5}{4}} c^{\frac {9}{4}} d^{5} e^{4} - 10 \, \sqrt {2} a^{\frac {7}{4}} c^{\frac {7}{4}} d^{3} e^{6} + 3 \, \sqrt {2} a^{\frac {9}{4}} c^{\frac {5}{4}} d e^{8} + 6 \, \sqrt {a} c^{3} d^{8} e - 24 \, a^{\frac {3}{2}} c^{2} d^{4} e^{5} + 2 \, a^{\frac {5}{2}} c e^{9}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {c}} c^{\frac {5}{4}}} - \frac {2 \, {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {13}{4}} d^{9} + 6 \, \sqrt {2} a^{\frac {3}{4}} c^{\frac {11}{4}} d^{7} e^{2} - 12 \, \sqrt {2} a^{\frac {5}{4}} c^{\frac {9}{4}} d^{5} e^{4} - 10 \, \sqrt {2} a^{\frac {7}{4}} c^{\frac {7}{4}} d^{3} e^{6} + 3 \, \sqrt {2} a^{\frac {9}{4}} c^{\frac {5}{4}} d e^{8} - 6 \, \sqrt {a} c^{3} d^{8} e + 24 \, a^{\frac {3}{2}} c^{2} d^{4} e^{5} - 2 \, a^{\frac {5}{2}} c e^{9}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {c}} c^{\frac {5}{4}}}\right )}}{8 \, {\left (c^{3} d^{12} + 3 \, a c^{2} d^{8} e^{4} + 3 \, a^{2} c d^{4} e^{8} + a^{3} e^{12}\right )}} + \frac {2 \, {\left (5 \, c^{2} d^{6} e^{3} - 3 \, a c d^{2} e^{7}\right )} \log \left (e x + d\right )}{c^{3} d^{12} + 3 \, a c^{2} d^{8} e^{4} + 3 \, a^{2} c d^{4} e^{8} + a^{3} e^{12}} - \frac {8 \, c d^{3} e^{4} x + 9 \, c d^{4} e^{3} + a e^{7}}{2 \, {\left (c^{2} d^{10} + 2 \, a c d^{6} e^{4} + a^{2} d^{2} e^{8} + {\left (c^{2} d^{8} e^{2} + 2 \, a c d^{4} e^{6} + a^{2} e^{10}\right )} x^{2} + 2 \, {\left (c^{2} d^{9} e + 2 \, a c d^{5} e^{5} + a^{2} d e^{9}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*c*(sqrt(2)*(10*sqrt(2)*a^(3/4)*c^(9/4)*d^6*e^3 - 6*sqrt(2)*a^(7/4)*c^(5/4)*d^2*e^7 - c^3*d^9 + 6*sqrt(a)*
c^(5/2)*d^7*e^2 + 12*a*c^2*d^5*e^4 - 10*a^(3/2)*c^(3/2)*d^3*e^6 - 3*a^2*c*d*e^8)*log(sqrt(c)*x^2 + sqrt(2)*a^(
1/4)*c^(1/4)*x + sqrt(a))/(a^(3/4)*c^(5/4)) + sqrt(2)*(10*sqrt(2)*a^(3/4)*c^(9/4)*d^6*e^3 - 6*sqrt(2)*a^(7/4)*
c^(5/4)*d^2*e^7 + c^3*d^9 - 6*sqrt(a)*c^(5/2)*d^7*e^2 - 12*a*c^2*d^5*e^4 + 10*a^(3/2)*c^(3/2)*d^3*e^6 + 3*a^2*
c*d*e^8)*log(sqrt(c)*x^2 - sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(a))/(a^(3/4)*c^(5/4)) - 2*(sqrt(2)*a^(1/4)*c^(13/4
)*d^9 + 6*sqrt(2)*a^(3/4)*c^(11/4)*d^7*e^2 - 12*sqrt(2)*a^(5/4)*c^(9/4)*d^5*e^4 - 10*sqrt(2)*a^(7/4)*c^(7/4)*d
^3*e^6 + 3*sqrt(2)*a^(9/4)*c^(5/4)*d*e^8 + 6*sqrt(a)*c^3*d^8*e - 24*a^(3/2)*c^2*d^4*e^5 + 2*a^(5/2)*c*e^9)*arc
tan(1/2*sqrt(2)*(2*sqrt(c)*x + sqrt(2)*a^(1/4)*c^(1/4))/sqrt(sqrt(a)*sqrt(c)))/(a^(3/4)*sqrt(sqrt(a)*sqrt(c))*
c^(5/4)) - 2*(sqrt(2)*a^(1/4)*c^(13/4)*d^9 + 6*sqrt(2)*a^(3/4)*c^(11/4)*d^7*e^2 - 12*sqrt(2)*a^(5/4)*c^(9/4)*d
^5*e^4 - 10*sqrt(2)*a^(7/4)*c^(7/4)*d^3*e^6 + 3*sqrt(2)*a^(9/4)*c^(5/4)*d*e^8 - 6*sqrt(a)*c^3*d^8*e + 24*a^(3/
2)*c^2*d^4*e^5 - 2*a^(5/2)*c*e^9)*arctan(1/2*sqrt(2)*(2*sqrt(c)*x - sqrt(2)*a^(1/4)*c^(1/4))/sqrt(sqrt(a)*sqrt
(c)))/(a^(3/4)*sqrt(sqrt(a)*sqrt(c))*c^(5/4)))/(c^3*d^12 + 3*a*c^2*d^8*e^4 + 3*a^2*c*d^4*e^8 + a^3*e^12) + 2*(
5*c^2*d^6*e^3 - 3*a*c*d^2*e^7)*log(e*x + d)/(c^3*d^12 + 3*a*c^2*d^8*e^4 + 3*a^2*c*d^4*e^8 + a^3*e^12) - 1/2*(8
*c*d^3*e^4*x + 9*c*d^4*e^3 + a*e^7)/(c^2*d^10 + 2*a*c*d^6*e^4 + a^2*d^2*e^8 + (c^2*d^8*e^2 + 2*a*c*d^4*e^6 + a
^2*e^10)*x^2 + 2*(c^2*d^9*e + 2*a*c*d^5*e^5 + a^2*d*e^9)*x)

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mupad [B]  time = 3.67, size = 1955, normalized size = 2.88 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + c*x^4)*(d + e*x)^3),x)

[Out]

symsum(log((c^7*d^5*e^6 + a*c^6*d*e^10)/(a^4*e^16 + c^4*d^16 + 4*a*c^3*d^12*e^4 + 4*a^3*c*d^4*e^12 + 6*a^2*c^2
*d^8*e^8) + root(768*a^5*c*d^4*e^8*z^4 + 768*a^4*c^2*d^8*e^4*z^4 + 256*a^3*c^3*d^12*z^4 + 256*a^6*e^12*z^4 - 1
536*a^4*c*d^2*e^7*z^3 + 2560*a^3*c^2*d^6*e^3*z^3 + 672*a^2*c^2*d^4*e^2*z^2 + 32*a^3*c*e^6*z^2 + 48*a*c^2*d^2*e
*z + c^2, z, k)*((208*a*c^7*d^7*e^7 - 48*a^2*c^6*d^3*e^11)/(a^4*e^16 + c^4*d^16 + 4*a*c^3*d^12*e^4 + 4*a^3*c*d
^4*e^12 + 6*a^2*c^2*d^8*e^8) + root(768*a^5*c*d^4*e^8*z^4 + 768*a^4*c^2*d^8*e^4*z^4 + 256*a^3*c^3*d^12*z^4 + 2
56*a^6*e^12*z^4 - 1536*a^4*c*d^2*e^7*z^3 + 2560*a^3*c^2*d^6*e^3*z^3 + 672*a^2*c^2*d^4*e^2*z^2 + 32*a^3*c*e^6*z
^2 + 48*a*c^2*d^2*e*z + c^2, z, k)*((144*a*c^8*d^13*e^4 + 16*a^4*c^5*d*e^16 + 2608*a^2*c^7*d^9*e^8 - 592*a^3*c
^6*d^5*e^12)/(a^4*e^16 + c^4*d^16 + 4*a*c^3*d^12*e^4 + 4*a^3*c*d^4*e^12 + 6*a^2*c^2*d^8*e^8) - root(768*a^5*c*
d^4*e^8*z^4 + 768*a^4*c^2*d^8*e^4*z^4 + 256*a^3*c^3*d^12*z^4 + 256*a^6*e^12*z^4 - 1536*a^4*c*d^2*e^7*z^3 + 256
0*a^3*c^2*d^6*e^3*z^3 + 672*a^2*c^2*d^4*e^2*z^2 + 32*a^3*c*e^6*z^2 + 48*a*c^2*d^2*e*z + c^2, z, k)*((896*a^4*c
^6*d^7*e^13 - 1120*a^3*c^7*d^11*e^9 - 1024*a^2*c^8*d^15*e^5 + 976*a^5*c^5*d^3*e^17 + 16*a*c^9*d^19*e)/(a^4*e^1
6 + c^4*d^16 + 4*a*c^3*d^12*e^4 + 4*a^3*c*d^4*e^12 + 6*a^2*c^2*d^8*e^8) - root(768*a^5*c*d^4*e^8*z^4 + 768*a^4
*c^2*d^8*e^4*z^4 + 256*a^3*c^3*d^12*z^4 + 256*a^6*e^12*z^4 - 1536*a^4*c*d^2*e^7*z^3 + 2560*a^3*c^2*d^6*e^3*z^3
+ 672*a^2*c^2*d^4*e^2*z^2 + 32*a^3*c*e^6*z^2 + 48*a*c^2*d^2*e*z + c^2, z, k)*((384*a^7*c^4*d*e^22 - 128*a^2*c
^9*d^21*e^2 - 128*a^3*c^8*d^17*e^6 + 768*a^4*c^7*d^13*e^10 + 1792*a^5*c^6*d^9*e^14 + 1408*a^6*c^5*d^5*e^18)/(a
^4*e^16 + c^4*d^16 + 4*a*c^3*d^12*e^4 + 4*a^3*c*d^4*e^12 + 6*a^2*c^2*d^8*e^8) + (x*(320*a^7*c^4*e^23 - 192*a^2
*c^9*d^20*e^3 - 448*a^3*c^8*d^16*e^7 + 128*a^4*c^7*d^12*e^11 + 1152*a^5*c^6*d^8*e^15 + 1088*a^6*c^5*d^4*e^19))
/(a^4*e^16 + c^4*d^16 + 4*a*c^3*d^12*e^4 + 4*a^3*c*d^4*e^12 + 6*a^2*c^2*d^8*e^8)) + (x*(80*a*c^9*d^18*e^2 - 15
36*a^2*c^8*d^14*e^6 - 2016*a^3*c^7*d^10*e^10 + 896*a^4*c^6*d^6*e^14 + 1296*a^5*c^5*d^2*e^18))/(a^4*e^16 + c^4*
d^16 + 4*a*c^3*d^12*e^4 + 4*a^3*c*d^4*e^12 + 6*a^2*c^2*d^8*e^8)) + (x*(36*a^4*c^5*e^17 - 4*c^9*d^16*e + 792*a*
c^8*d^12*e^5 + 1632*a^2*c^7*d^8*e^9 - 152*a^3*c^6*d^4*e^13))/(a^4*e^16 + c^4*d^16 + 4*a*c^3*d^12*e^4 + 4*a^3*c
*d^4*e^12 + 6*a^2*c^2*d^8*e^8)) + (x*(40*c^8*d^10*e^4 - 16*a*c^7*d^6*e^8 + 72*a^2*c^6*d^2*e^12))/(a^4*e^16 + c
^4*d^16 + 4*a*c^3*d^12*e^4 + 4*a^3*c*d^4*e^12 + 6*a^2*c^2*d^8*e^8)) + (x*(a*c^6*e^11 + c^7*d^4*e^7))/(a^4*e^16
+ c^4*d^16 + 4*a*c^3*d^12*e^4 + 4*a^3*c*d^4*e^12 + 6*a^2*c^2*d^8*e^8))*root(768*a^5*c*d^4*e^8*z^4 + 768*a^4*c
^2*d^8*e^4*z^4 + 256*a^3*c^3*d^12*z^4 + 256*a^6*e^12*z^4 - 1536*a^4*c*d^2*e^7*z^3 + 2560*a^3*c^2*d^6*e^3*z^3 +
672*a^2*c^2*d^4*e^2*z^2 + 32*a^3*c*e^6*z^2 + 48*a*c^2*d^2*e*z + c^2, z, k), k, 1, 4) - ((a*e^7 + 9*c*d^4*e^3)
/(2*(a^2*e^8 + c^2*d^8 + 2*a*c*d^4*e^4)) + (4*c*d^3*e^4*x)/(a^2*e^8 + c^2*d^8 + 2*a*c*d^4*e^4))/(d^2 + e^2*x^2
+ 2*d*e*x) + (log(d + e*x)*(10*c^2*d^6*e^3 - 6*a*c*d^2*e^7))/(a^3*e^12 + c^3*d^12 + 3*a*c^2*d^8*e^4 + 3*a^2*c
*d^4*e^8)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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