\(\int \frac {1}{(d+e x)^2 (a+b (d+e x)^2+c (d+e x)^4)} \, dx\) [618]

Optimal result

Integrand size = 30, antiderivative size = 195 \[ \int \frac {1}{(d+e x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )} \, dx=-\frac {1}{a e (d+e x)}-\frac {\sqrt {c} \left (1+\frac {b}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a \sqrt {b-\sqrt {b^2-4 a c}} e}-\frac {\sqrt {c} \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a \sqrt {b+\sqrt {b^2-4 a c}} e} \]

[Out]

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1156, 1137, 1180, 211} \[ \int \frac {1}{(d+e x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )} \, dx=-\frac {\sqrt {c} \left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a e \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} a e \sqrt {\sqrt {b^2-4 a c}+b}}-\frac {1}{a e (d+e x)} \]

[In]

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

[Out]

-(1/(a*e*(d + e*x))) - (Sqrt[c]*(1 + b/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[b^2
- 4*a*c]]])/(Sqrt[2]*a*Sqrt[b - Sqrt[b^2 - 4*a*c]]*e) - (Sqrt[c]*(1 - b/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sq
rt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*a*Sqrt[b + Sqrt[b^2 - 4*a*c]]*e)

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 1137

Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*x^2 +
c*x^4)^(p + 1)/(a*d*(m + 1))), x] - Dist[1/(a*d^2*(m + 1)), Int[(d*x)^(m + 2)*(b*(m + 2*p + 3) + c*(m + 4*p +
5)*x^2)*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[m, -1] && In
tegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

Rule 1156

Int[(u_)^(m_.)*((a_.) + (b_.)*(v_)^2 + (c_.)*(v_)^4)^(p_.), x_Symbol] :> Dist[u^m/(Coefficient[v, x, 1]*v^m),
Subst[Int[x^m*(a + b*x^2 + c*x^(2*2))^p, x], x, v], x] /; FreeQ[{a, b, c, m, p}, x] && LinearPairQ[u, v, x]

Rule 1180

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

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

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(d+e x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )} \, dx=-\frac {\frac {2}{d+e x}+\frac {\sqrt {2} \sqrt {c} \left (b+\sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \sqrt {c} \left (-b+\sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}}{2 a e} \]

[In]

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

[Out]

-1/2*(2/(d + e*x) + (Sqrt[2]*Sqrt[c]*(b + Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b - Sqrt[
b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(-b + Sqrt[b^2 - 4*a*c])*Ar
cTan[(Sqrt[2]*Sqrt[c]*(d + e*x))/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]])
)/(a*e)

Maple [C] (verified)

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

Time = 0.64 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.86

method	result	size
default	\(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,e^{4} \textit {\_Z}^{4}+4 c d \,e^{3} \textit {\_Z}^{3}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 d^{3} e c +2 b d e \right ) \textit {\_Z} +d^{4} c +b \,d^{2}+a \right )}{\sum }\frac {\left (-\textit {\_R}^{2} c \,e^{2}-2 \textit {\_R} c d e -c \,d^{2}-b \right ) \ln \left (x -\textit {\_R} \right )}{2 e^{3} c \,\textit {\_R}^{3}+6 c d \,e^{2} \textit {\_R}^{2}+6 c \,d^{2} e \textit {\_R} +2 d^{3} c +b e \textit {\_R} +b d}}{2 a e}-\frac {1}{a e \left (e x +d \right )}\)	\(168\)
risch	\(-\frac {1}{a e \left (e x +d \right )}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (16 a^{5} c^{2} e^{4}-8 b^{2} e^{4} c \,a^{4}+b^{4} e^{4} a^{3}\right ) \textit {\_Z}^{4}+\left (12 a^{2} b \,c^{2} e^{2}-7 a \,b^{3} c \,e^{2}+b^{5} e^{2}\right ) \textit {\_Z}^{2}+c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (40 a^{5} c^{2} e^{5}-22 a^{4} b^{2} c \,e^{5}+3 a^{3} b^{4} e^{5}\right ) \textit {\_R}^{4}+\left (25 a^{2} b \,c^{2} e^{3}-14 a \,b^{3} c \,e^{3}+2 b^{5} e^{3}\right ) \textit {\_R}^{2}+2 c^{3} e \right ) x +\left (40 a^{5} c^{2} d \,e^{4}-22 a^{4} b^{2} c d \,e^{4}+3 a^{3} b^{4} d \,e^{4}\right ) \textit {\_R}^{4}+\left (4 a^{4} c^{2} e^{3}-5 a^{3} b^{2} c \,e^{3}+a^{2} b^{4} e^{3}\right ) \textit {\_R}^{3}+\left (25 a^{2} b \,c^{2} d \,e^{2}-14 a \,b^{3} c d \,e^{2}+2 b^{5} d \,e^{2}\right ) \textit {\_R}^{2}+2 c^{3} d \right )\right )}{2}\)	\(310\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1339 vs. \(2 (158) = 316\).

Time = 0.28 (sec) , antiderivative size = 1339, normalized size of antiderivative = 6.87 \[ \int \frac {1}{(d+e x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/2*(sqrt(1/2)*(a*e^2*x + a*d*e)*sqrt(-((a^3*b^2 - 4*a^4*c)*e^2*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4
*a^7*c)*e^4)) + b^3 - 3*a*b*c)/((a^3*b^2 - 4*a^4*c)*e^2))*log(-2*(b^2*c^2 - a*c^3)*e*x - 2*(b^2*c^2 - a*c^3)*d
+ sqrt(1/2)*((a^3*b^4 - 6*a^4*b^2*c + 8*a^5*c^2)*e^3*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4*a^7*c)*e^
4)) - (b^5 - 5*a*b^3*c + 4*a^2*b*c^2)*e)*sqrt(-((a^3*b^2 - 4*a^4*c)*e^2*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6
*b^2 - 4*a^7*c)*e^4)) + b^3 - 3*a*b*c)/((a^3*b^2 - 4*a^4*c)*e^2))) - sqrt(1/2)*(a*e^2*x + a*d*e)*sqrt(-((a^3*b
^2 - 4*a^4*c)*e^2*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4*a^7*c)*e^4)) + b^3 - 3*a*b*c)/((a^3*b^2 - 4*a
^4*c)*e^2))*log(-2*(b^2*c^2 - a*c^3)*e*x - 2*(b^2*c^2 - a*c^3)*d - sqrt(1/2)*((a^3*b^4 - 6*a^4*b^2*c + 8*a^5*c
^2)*e^3*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4*a^7*c)*e^4)) - (b^5 - 5*a*b^3*c + 4*a^2*b*c^2)*e)*sqrt(
-((a^3*b^2 - 4*a^4*c)*e^2*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4*a^7*c)*e^4)) + b^3 - 3*a*b*c)/((a^3*b
^2 - 4*a^4*c)*e^2))) - sqrt(1/2)*(a*e^2*x + a*d*e)*sqrt(((a^3*b^2 - 4*a^4*c)*e^2*sqrt((b^4 - 2*a*b^2*c + a^2*c
^2)/((a^6*b^2 - 4*a^7*c)*e^4)) - b^3 + 3*a*b*c)/((a^3*b^2 - 4*a^4*c)*e^2))*log(-2*(b^2*c^2 - a*c^3)*e*x - 2*(b
^2*c^2 - a*c^3)*d + sqrt(1/2)*((a^3*b^4 - 6*a^4*b^2*c + 8*a^5*c^2)*e^3*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*
b^2 - 4*a^7*c)*e^4)) + (b^5 - 5*a*b^3*c + 4*a^2*b*c^2)*e)*sqrt(((a^3*b^2 - 4*a^4*c)*e^2*sqrt((b^4 - 2*a*b^2*c
+ a^2*c^2)/((a^6*b^2 - 4*a^7*c)*e^4)) - b^3 + 3*a*b*c)/((a^3*b^2 - 4*a^4*c)*e^2))) + sqrt(1/2)*(a*e^2*x + a*d*
e)*sqrt(((a^3*b^2 - 4*a^4*c)*e^2*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4*a^7*c)*e^4)) - b^3 + 3*a*b*c)/
((a^3*b^2 - 4*a^4*c)*e^2))*log(-2*(b^2*c^2 - a*c^3)*e*x - 2*(b^2*c^2 - a*c^3)*d - sqrt(1/2)*((a^3*b^4 - 6*a^4*
b^2*c + 8*a^5*c^2)*e^3*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4*a^7*c)*e^4)) + (b^5 - 5*a*b^3*c + 4*a^2*
b*c^2)*e)*sqrt(((a^3*b^2 - 4*a^4*c)*e^2*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^2 - 4*a^7*c)*e^4)) - b^3 + 3*
a*b*c)/((a^3*b^2 - 4*a^4*c)*e^2))) - 2)/(a*e^2*x + a*d*e)

Sympy [A] (verification not implemented)

Time = 2.96 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.08 \[ \int \frac {1}{(d+e x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )} \, dx=\operatorname {RootSum} {\left (t^{4} \cdot \left (256 a^{5} c^{2} e^{4} - 128 a^{4} b^{2} c e^{4} + 16 a^{3} b^{4} e^{4}\right ) + t^{2} \cdot \left (48 a^{2} b c^{2} e^{2} - 28 a b^{3} c e^{2} + 4 b^{5} e^{2}\right ) + c^{3}, \left ( t \mapsto t \log {\left (x + \frac {- 64 t^{3} a^{5} c^{2} e^{3} + 48 t^{3} a^{4} b^{2} c e^{3} - 8 t^{3} a^{3} b^{4} e^{3} - 10 t a^{2} b c^{2} e + 10 t a b^{3} c e - 2 t b^{5} e + a c^{3} d - b^{2} c^{2} d}{a c^{3} e - b^{2} c^{2} e} \right )} \right )\right )} - \frac {1}{a d e + a e^{2} x} \]

[In]

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

[Out]

RootSum(_t**4*(256*a**5*c**2*e**4 - 128*a**4*b**2*c*e**4 + 16*a**3*b**4*e**4) + _t**2*(48*a**2*b*c**2*e**2 - 2
8*a*b**3*c*e**2 + 4*b**5*e**2) + c**3, Lambda(_t, _t*log(x + (-64*_t**3*a**5*c**2*e**3 + 48*_t**3*a**4*b**2*c*
e**3 - 8*_t**3*a**3*b**4*e**3 - 10*_t*a**2*b*c**2*e + 10*_t*a*b**3*c*e - 2*_t*b**5*e + a*c**3*d - b**2*c**2*d)
/(a*c**3*e - b**2*c**2*e)))) - 1/(a*d*e + a*e**2*x)

Maxima [F]

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

[In]

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

[Out]

-integrate((c*e^2*x^2 + 2*c*d*e*x + c*d^2 + b)/(c*e^4*x^4 + 4*c*d*e^3*x^3 + c*d^4 + (6*c*d^2 + b)*e^2*x^2 + b*
d^2 + 2*(2*c*d^3 + b*d)*e*x + a), x)/a - 1/(a*e^2*x + a*d*e)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 932 vs. \(2 (158) = 316\).

Time = 0.31 (sec) , antiderivative size = 932, normalized size of antiderivative = 4.78 \[ \int \frac {1}{(d+e x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )} \, dx=-\frac {{\left ({\left (b^{8} - 9 \, a b^{6} c + 25 \, a^{2} b^{4} c^{2} - 20 \, a^{3} b^{2} c^{3} + {\left (b^{7} - 7 \, a b^{5} c + 13 \, a^{2} b^{3} c^{2} - 4 \, a^{3} b c^{3}\right )} \sqrt {b^{2} - 4 \, a c}\right )} \sqrt {2 \, a b + 2 \, \sqrt {b^{2} - 4 \, a c} a} a^{2} - 2 \, {\left (a^{2} b^{6} c - 7 \, a^{3} b^{4} c^{2} + 13 \, a^{4} b^{2} c^{3} - 4 \, a^{5} c^{4} + {\left (a^{2} b^{5} c - 5 \, a^{3} b^{3} c^{2} + 5 \, a^{4} b c^{3}\right )} \sqrt {b^{2} - 4 \, a c}\right )} \sqrt {2 \, a b + 2 \, \sqrt {b^{2} - 4 \, a c} a} {\left | a \right |} - {\left (a^{2} b^{8} - 7 \, a^{3} b^{6} c + 15 \, a^{4} b^{4} c^{2} - 10 \, a^{5} b^{2} c^{3} + {\left (a^{2} b^{7} - 5 \, a^{3} b^{5} c + 7 \, a^{4} b^{3} c^{2} - 2 \, a^{5} b c^{3}\right )} \sqrt {b^{2} - 4 \, a c}\right )} \sqrt {2 \, a b + 2 \, \sqrt {b^{2} - 4 \, a c} a}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}}}{{\left (e x + d\right )} e \sqrt {\frac {a b e^{6} + \sqrt {a^{2} b^{2} e^{12} - 4 \, a^{3} c e^{12}}}{a^{2} e^{8}}}}\right )}{8 \, {\left (a^{3} b^{6} c - 7 \, a^{4} b^{4} c^{2} + 13 \, a^{5} b^{2} c^{3} - 4 \, a^{6} c^{4} + {\left (a^{3} b^{5} c - 5 \, a^{4} b^{3} c^{2} + 5 \, a^{5} b c^{3}\right )} \sqrt {b^{2} - 4 \, a c}\right )} a^{2} e} - \frac {{\left ({\left (b^{8} - 9 \, a b^{6} c + 25 \, a^{2} b^{4} c^{2} - 20 \, a^{3} b^{2} c^{3} - {\left (b^{7} - 7 \, a b^{5} c + 13 \, a^{2} b^{3} c^{2} - 4 \, a^{3} b c^{3}\right )} \sqrt {b^{2} - 4 \, a c}\right )} \sqrt {2 \, a b - 2 \, \sqrt {b^{2} - 4 \, a c} a} a^{2} - 2 \, {\left (a^{2} b^{6} c - 7 \, a^{3} b^{4} c^{2} + 13 \, a^{4} b^{2} c^{3} - 4 \, a^{5} c^{4} - {\left (a^{2} b^{5} c - 5 \, a^{3} b^{3} c^{2} + 5 \, a^{4} b c^{3}\right )} \sqrt {b^{2} - 4 \, a c}\right )} \sqrt {2 \, a b - 2 \, \sqrt {b^{2} - 4 \, a c} a} {\left | a \right |} - {\left (a^{2} b^{8} - 7 \, a^{3} b^{6} c + 15 \, a^{4} b^{4} c^{2} - 10 \, a^{5} b^{2} c^{3} - {\left (a^{2} b^{7} - 5 \, a^{3} b^{5} c + 7 \, a^{4} b^{3} c^{2} - 2 \, a^{5} b c^{3}\right )} \sqrt {b^{2} - 4 \, a c}\right )} \sqrt {2 \, a b - 2 \, \sqrt {b^{2} - 4 \, a c} a}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}}}{{\left (e x + d\right )} e \sqrt {\frac {a b e^{6} - \sqrt {a^{2} b^{2} e^{12} - 4 \, a^{3} c e^{12}}}{a^{2} e^{8}}}}\right )}{8 \, {\left (a^{3} b^{6} c - 7 \, a^{4} b^{4} c^{2} + 13 \, a^{5} b^{2} c^{3} - 4 \, a^{6} c^{4} - {\left (a^{3} b^{5} c - 5 \, a^{4} b^{3} c^{2} + 5 \, a^{5} b c^{3}\right )} \sqrt {b^{2} - 4 \, a c}\right )} a^{2} e} - \frac {1}{{\left (e x + d\right )} a e} \]

[In]

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

[Out]

-1/8*((b^8 - 9*a*b^6*c + 25*a^2*b^4*c^2 - 20*a^3*b^2*c^3 + (b^7 - 7*a*b^5*c + 13*a^2*b^3*c^2 - 4*a^3*b*c^3)*sq
rt(b^2 - 4*a*c))*sqrt(2*a*b + 2*sqrt(b^2 - 4*a*c)*a)*a^2 - 2*(a^2*b^6*c - 7*a^3*b^4*c^2 + 13*a^4*b^2*c^3 - 4*a
^5*c^4 + (a^2*b^5*c - 5*a^3*b^3*c^2 + 5*a^4*b*c^3)*sqrt(b^2 - 4*a*c))*sqrt(2*a*b + 2*sqrt(b^2 - 4*a*c)*a)*abs(
a) - (a^2*b^8 - 7*a^3*b^6*c + 15*a^4*b^4*c^2 - 10*a^5*b^2*c^3 + (a^2*b^7 - 5*a^3*b^5*c + 7*a^4*b^3*c^2 - 2*a^5
*b*c^3)*sqrt(b^2 - 4*a*c))*sqrt(2*a*b + 2*sqrt(b^2 - 4*a*c)*a))*arctan(2*sqrt(1/2)/((e*x + d)*e*sqrt((a*b*e^6
+ sqrt(a^2*b^2*e^12 - 4*a^3*c*e^12))/(a^2*e^8))))/((a^3*b^6*c - 7*a^4*b^4*c^2 + 13*a^5*b^2*c^3 - 4*a^6*c^4 + (
a^3*b^5*c - 5*a^4*b^3*c^2 + 5*a^5*b*c^3)*sqrt(b^2 - 4*a*c))*a^2*e) - 1/8*((b^8 - 9*a*b^6*c + 25*a^2*b^4*c^2 -
20*a^3*b^2*c^3 - (b^7 - 7*a*b^5*c + 13*a^2*b^3*c^2 - 4*a^3*b*c^3)*sqrt(b^2 - 4*a*c))*sqrt(2*a*b - 2*sqrt(b^2 -
4*a*c)*a)*a^2 - 2*(a^2*b^6*c - 7*a^3*b^4*c^2 + 13*a^4*b^2*c^3 - 4*a^5*c^4 - (a^2*b^5*c - 5*a^3*b^3*c^2 + 5*a^
4*b*c^3)*sqrt(b^2 - 4*a*c))*sqrt(2*a*b - 2*sqrt(b^2 - 4*a*c)*a)*abs(a) - (a^2*b^8 - 7*a^3*b^6*c + 15*a^4*b^4*c
^2 - 10*a^5*b^2*c^3 - (a^2*b^7 - 5*a^3*b^5*c + 7*a^4*b^3*c^2 - 2*a^5*b*c^3)*sqrt(b^2 - 4*a*c))*sqrt(2*a*b - 2*
sqrt(b^2 - 4*a*c)*a))*arctan(2*sqrt(1/2)/((e*x + d)*e*sqrt((a*b*e^6 - sqrt(a^2*b^2*e^12 - 4*a^3*c*e^12))/(a^2*
e^8))))/((a^3*b^6*c - 7*a^4*b^4*c^2 + 13*a^5*b^2*c^3 - 4*a^6*c^4 - (a^3*b^5*c - 5*a^4*b^3*c^2 + 5*a^5*b*c^3)*s
qrt(b^2 - 4*a*c))*a^2*e) - 1/((e*x + d)*a*e)

Mupad [B] (verification not implemented)

Time = 9.22 (sec) , antiderivative size = 3844, normalized size of antiderivative = 19.71 \[ \int \frac {1}{(d+e x)^2 \left (a+b (d+e x)^2+c (d+e x)^4\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

- atan(((-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a
^3*b^4*e^2 + 16*a^5*c^2*e^2 - 8*a^4*b^2*c*e^2)))^(1/2)*(x*(4*a^4*c^4*e^12 - 2*a^3*b^2*c^3*e^12) + ((x*(32*a^6*
b*c^3*e^14 - 8*a^5*b^3*c^2*e^14) + 32*a^6*b*c^3*d*e^13 - 8*a^5*b^3*c^2*d*e^13)*(-(b^5 + b^2*(-(4*a*c - b^2)^3)
^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^3*b^4*e^2 + 16*a^5*c^2*e^2 - 8*a^4*b^2
*c*e^2)))^(1/2) - 16*a^5*b*c^3*e^12 + 4*a^4*b^3*c^2*e^12)*(-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2
- 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^3*b^4*e^2 + 16*a^5*c^2*e^2 - 8*a^4*b^2*c*e^2)))^(1/2) + 4*a
^4*c^4*d*e^11 - 2*a^3*b^2*c^3*d*e^11)*1i + (-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c -
a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^3*b^4*e^2 + 16*a^5*c^2*e^2 - 8*a^4*b^2*c*e^2)))^(1/2)*(x*(4*a^4*c^4*e^12 -
2*a^3*b^2*c^3*e^12) + ((x*(32*a^6*b*c^3*e^14 - 8*a^5*b^3*c^2*e^14) + 32*a^6*b*c^3*d*e^13 - 8*a^5*b^3*c^2*d*e^
13)*(-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^3*b
^4*e^2 + 16*a^5*c^2*e^2 - 8*a^4*b^2*c*e^2)))^(1/2) + 16*a^5*b*c^3*e^12 - 4*a^4*b^3*c^2*e^12)*(-(b^5 + b^2*(-(4
*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^3*b^4*e^2 + 16*a^5*c^2*e
^2 - 8*a^4*b^2*c*e^2)))^(1/2) + 4*a^4*c^4*d*e^11 - 2*a^3*b^2*c^3*d*e^11)*1i)/((-(b^5 + b^2*(-(4*a*c - b^2)^3)^
(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^3*b^4*e^2 + 16*a^5*c^2*e^2 - 8*a^4*b^2*
c*e^2)))^(1/2)*(x*(4*a^4*c^4*e^12 - 2*a^3*b^2*c^3*e^12) + ((x*(32*a^6*b*c^3*e^14 - 8*a^5*b^3*c^2*e^14) + 32*a^
6*b*c^3*d*e^13 - 8*a^5*b^3*c^2*d*e^13)*(-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c - a*c*
(-(4*a*c - b^2)^3)^(1/2))/(8*(a^3*b^4*e^2 + 16*a^5*c^2*e^2 - 8*a^4*b^2*c*e^2)))^(1/2) + 16*a^5*b*c^3*e^12 - 4*
a^4*b^3*c^2*e^12)*(-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(1
/2))/(8*(a^3*b^4*e^2 + 16*a^5*c^2*e^2 - 8*a^4*b^2*c*e^2)))^(1/2) + 4*a^4*c^4*d*e^11 - 2*a^3*b^2*c^3*d*e^11) -
(-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^3*b^4*e
^2 + 16*a^5*c^2*e^2 - 8*a^4*b^2*c*e^2)))^(1/2)*(x*(4*a^4*c^4*e^12 - 2*a^3*b^2*c^3*e^12) + ((x*(32*a^6*b*c^3*e^
14 - 8*a^5*b^3*c^2*e^14) + 32*a^6*b*c^3*d*e^13 - 8*a^5*b^3*c^2*d*e^13)*(-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) +
12*a^2*b*c^2 - 7*a*b^3*c - a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^3*b^4*e^2 + 16*a^5*c^2*e^2 - 8*a^4*b^2*c*e^2))
)^(1/2) - 16*a^5*b*c^3*e^12 + 4*a^4*b^3*c^2*e^12)*(-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b
^3*c - a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^3*b^4*e^2 + 16*a^5*c^2*e^2 - 8*a^4*b^2*c*e^2)))^(1/2) + 4*a^4*c^4*d
*e^11 - 2*a^3*b^2*c^3*d*e^11) + 2*a^3*c^4*e^10))*(-(b^5 + b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^
3*c - a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^3*b^4*e^2 + 16*a^5*c^2*e^2 - 8*a^4*b^2*c*e^2)))^(1/2)*2i - atan(((-(
b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c + a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^3*b^4*e^2
+ 16*a^5*c^2*e^2 - 8*a^4*b^2*c*e^2)))^(1/2)*(x*(4*a^4*c^4*e^12 - 2*a^3*b^2*c^3*e^12) + ((x*(32*a^6*b*c^3*e^14
- 8*a^5*b^3*c^2*e^14) + 32*a^6*b*c^3*d*e^13 - 8*a^5*b^3*c^2*d*e^13)*(-(b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 12
*a^2*b*c^2 - 7*a*b^3*c + a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^3*b^4*e^2 + 16*a^5*c^2*e^2 - 8*a^4*b^2*c*e^2)))^(
1/2) - 16*a^5*b*c^3*e^12 + 4*a^4*b^3*c^2*e^12)*(-(b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*
c + a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^3*b^4*e^2 + 16*a^5*c^2*e^2 - 8*a^4*b^2*c*e^2)))^(1/2) + 4*a^4*c^4*d*e^
11 - 2*a^3*b^2*c^3*d*e^11)*1i + (-(b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c + a*c*(-(4*a*
c - b^2)^3)^(1/2))/(8*(a^3*b^4*e^2 + 16*a^5*c^2*e^2 - 8*a^4*b^2*c*e^2)))^(1/2)*(x*(4*a^4*c^4*e^12 - 2*a^3*b^2*
c^3*e^12) + ((x*(32*a^6*b*c^3*e^14 - 8*a^5*b^3*c^2*e^14) + 32*a^6*b*c^3*d*e^13 - 8*a^5*b^3*c^2*d*e^13)*(-(b^5
- b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c + a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^3*b^4*e^2 + 16
*a^5*c^2*e^2 - 8*a^4*b^2*c*e^2)))^(1/2) + 16*a^5*b*c^3*e^12 - 4*a^4*b^3*c^2*e^12)*(-(b^5 - b^2*(-(4*a*c - b^2)
^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c + a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^3*b^4*e^2 + 16*a^5*c^2*e^2 - 8*a^4*
b^2*c*e^2)))^(1/2) + 4*a^4*c^4*d*e^11 - 2*a^3*b^2*c^3*d*e^11)*1i)/((-(b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*
a^2*b*c^2 - 7*a*b^3*c + a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^3*b^4*e^2 + 16*a^5*c^2*e^2 - 8*a^4*b^2*c*e^2)))^(1
/2)*(x*(4*a^4*c^4*e^12 - 2*a^3*b^2*c^3*e^12) + ((x*(32*a^6*b*c^3*e^14 - 8*a^5*b^3*c^2*e^14) + 32*a^6*b*c^3*d*e
^13 - 8*a^5*b^3*c^2*d*e^13)*(-(b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c + a*c*(-(4*a*c -
b^2)^3)^(1/2))/(8*(a^3*b^4*e^2 + 16*a^5*c^2*e^2 - 8*a^4*b^2*c*e^2)))^(1/2) + 16*a^5*b*c^3*e^12 - 4*a^4*b^3*c^2
*e^12)*(-(b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c + a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^
3*b^4*e^2 + 16*a^5*c^2*e^2 - 8*a^4*b^2*c*e^2)))^(1/2) + 4*a^4*c^4*d*e^11 - 2*a^3*b^2*c^3*d*e^11) - (-(b^5 - b^
2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c + a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^3*b^4*e^2 + 16*a^5
*c^2*e^2 - 8*a^4*b^2*c*e^2)))^(1/2)*(x*(4*a^4*c^4*e^12 - 2*a^3*b^2*c^3*e^12) + ((x*(32*a^6*b*c^3*e^14 - 8*a^5*
b^3*c^2*e^14) + 32*a^6*b*c^3*d*e^13 - 8*a^5*b^3*c^2*d*e^13)*(-(b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c
^2 - 7*a*b^3*c + a*c*(-(4*a*c - b^2)^3)^(1/2))/(8*(a^3*b^4*e^2 + 16*a^5*c^2*e^2 - 8*a^4*b^2*c*e^2)))^(1/2) - 1
6*a^5*b*c^3*e^12 + 4*a^4*b^3*c^2*e^12)*(-(b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c + a*c*
(-(4*a*c - b^2)^3)^(1/2))/(8*(a^3*b^4*e^2 + 16*a^5*c^2*e^2 - 8*a^4*b^2*c*e^2)))^(1/2) + 4*a^4*c^4*d*e^11 - 2*a
^3*b^2*c^3*d*e^11) + 2*a^3*c^4*e^10))*(-(b^5 - b^2*(-(4*a*c - b^2)^3)^(1/2) + 12*a^2*b*c^2 - 7*a*b^3*c + a*c*(
-(4*a*c - b^2)^3)^(1/2))/(8*(a^3*b^4*e^2 + 16*a^5*c^2*e^2 - 8*a^4*b^2*c*e^2)))^(1/2)*2i - 1/(a*e*(d + e*x))