\(\int \frac {(d+e x)^2}{a+b (d+e x)^2+c (d+e x)^4} \, dx\) [615]

Optimal result

Integrand size = 30, antiderivative size = 164 \[ \int \frac {(d+e x)^2}{a+b (d+e x)^2+c (d+e x)^4} \, dx=-\frac {\sqrt {b-\sqrt {b^2-4 a c}} \arctan \left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b^2-4 a c} e}+\frac {\sqrt {b+\sqrt {b^2-4 a c}} \arctan \left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b^2-4 a c} e} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1156, 1144, 211} \[ \int \frac {(d+e x)^2}{a+b (d+e x)^2+c (d+e x)^4} \, dx=\frac {\sqrt {\sqrt {b^2-4 a c}+b} \arctan \left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} \sqrt {c} e \sqrt {b^2-4 a c}}-\frac {\sqrt {b-\sqrt {b^2-4 a c}} \arctan \left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {c} e \sqrt {b^2-4 a c}} \]

[In]

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

[Out]

-((Sqrt[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]*Sqrt[
c]*Sqrt[b^2 - 4*a*c]*e)) + (Sqrt[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]*Sqrt[c]*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 1144

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

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]

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.07 \[ \int \frac {(d+e x)^2}{a+b (d+e x)^2+c (d+e x)^4} \, dx=\frac {\left (-b+\sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )+\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {b+\sqrt {b^2-4 a c}} \arctan \left (\frac {\sqrt {2} \sqrt {c} (d+e x)}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}} e} \]

[In]

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

[Out]

((-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 - Sqrt[b^2
- 4*a*c]]*Sqrt[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
]*Sqrt[c]*Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]*e)

Maple [C] (verified)

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

Time = 0.59 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.85

method	result	size
default	\(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,e^{4} \textit {\_Z}^{4}+4 c d \,e^{3} \textit {\_Z}^{3}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 d^{3} e c +2 b d e \right ) \textit {\_Z} +d^{4} c +b \,d^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{2} e^{2}+2 \textit {\_R} d e +d^{2}\right ) \ln \left (x -\textit {\_R} \right )}{2 e^{3} c \,\textit {\_R}^{3}+6 c d \,e^{2} \textit {\_R}^{2}+6 c \,d^{2} e \textit {\_R} +2 d^{3} c +b e \textit {\_R} +b d}}{2 e}\)	\(140\)
risch	\(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,e^{4} \textit {\_Z}^{4}+4 c d \,e^{3} \textit {\_Z}^{3}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 d^{3} e c +2 b d e \right ) \textit {\_Z} +d^{4} c +b \,d^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{2} e^{2}+2 \textit {\_R} d e +d^{2}\right ) \ln \left (x -\textit {\_R} \right )}{2 e^{3} c \,\textit {\_R}^{3}+6 c d \,e^{2} \textit {\_R}^{2}+6 c \,d^{2} e \textit {\_R} +2 d^{3} c +b e \textit {\_R} +b d}}{2 e}\)	\(140\)

[In]

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

[Out]

1/2/e*sum((_R^2*e^2+2*_R*d*e+d^2)/(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=Ro
otOf(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))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 703 vs. \(2 (129) = 258\).

Time = 0.25 (sec) , antiderivative size = 703, normalized size of antiderivative = 4.29 \[ \int \frac {(d+e x)^2}{a+b (d+e x)^2+c (d+e x)^4} \, dx=\frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b^{2} c - 4 \, a c^{2}\right )} e^{2} \sqrt {\frac {1}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e^{4}}} + b}{{\left (b^{2} c - 4 \, a c^{2}\right )} e^{2}}} \log \left (\sqrt {\frac {1}{2}} {\left (b^{2} c - 4 \, a c^{2}\right )} e^{3} \sqrt {-\frac {{\left (b^{2} c - 4 \, a c^{2}\right )} e^{2} \sqrt {\frac {1}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e^{4}}} + b}{{\left (b^{2} c - 4 \, a c^{2}\right )} e^{2}}} \sqrt {\frac {1}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e^{4}}} + e x + d\right ) - \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (b^{2} c - 4 \, a c^{2}\right )} e^{2} \sqrt {\frac {1}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e^{4}}} + b}{{\left (b^{2} c - 4 \, a c^{2}\right )} e^{2}}} \log \left (-\sqrt {\frac {1}{2}} {\left (b^{2} c - 4 \, a c^{2}\right )} e^{3} \sqrt {-\frac {{\left (b^{2} c - 4 \, a c^{2}\right )} e^{2} \sqrt {\frac {1}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e^{4}}} + b}{{\left (b^{2} c - 4 \, a c^{2}\right )} e^{2}}} \sqrt {\frac {1}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e^{4}}} + e x + d\right ) - \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {\frac {{\left (b^{2} c - 4 \, a c^{2}\right )} e^{2} \sqrt {\frac {1}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e^{4}}} - b}{{\left (b^{2} c - 4 \, a c^{2}\right )} e^{2}}} \log \left (\sqrt {\frac {1}{2}} {\left (b^{2} c - 4 \, a c^{2}\right )} e^{3} \sqrt {\frac {{\left (b^{2} c - 4 \, a c^{2}\right )} e^{2} \sqrt {\frac {1}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e^{4}}} - b}{{\left (b^{2} c - 4 \, a c^{2}\right )} e^{2}}} \sqrt {\frac {1}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e^{4}}} + e x + d\right ) + \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {\frac {{\left (b^{2} c - 4 \, a c^{2}\right )} e^{2} \sqrt {\frac {1}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e^{4}}} - b}{{\left (b^{2} c - 4 \, a c^{2}\right )} e^{2}}} \log \left (-\sqrt {\frac {1}{2}} {\left (b^{2} c - 4 \, a c^{2}\right )} e^{3} \sqrt {\frac {{\left (b^{2} c - 4 \, a c^{2}\right )} e^{2} \sqrt {\frac {1}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e^{4}}} - b}{{\left (b^{2} c - 4 \, a c^{2}\right )} e^{2}}} \sqrt {\frac {1}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} e^{4}}} + e x + d\right ) \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.73 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.63 \[ \int \frac {(d+e x)^2}{a+b (d+e x)^2+c (d+e x)^4} \, dx=\operatorname {RootSum} {\left (t^{4} \cdot \left (256 a^{2} c^{3} e^{4} - 128 a b^{2} c^{2} e^{4} + 16 b^{4} c e^{4}\right ) + t^{2} \left (- 16 a b c e^{2} + 4 b^{3} e^{2}\right ) + a, \left ( t \mapsto t \log {\left (x + \frac {64 t^{3} a c^{2} e^{3} - 16 t^{3} b^{2} c e^{3} - 2 t b e + d}{e} \right )} \right )\right )} \]

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1403 vs. \(2 (129) = 258\).

Time = 0.29 (sec) , antiderivative size = 1403, normalized size of antiderivative = 8.55 \[ \int \frac {(d+e x)^2}{a+b (d+e x)^2+c (d+e x)^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 590, normalized size of antiderivative = 3.60 \[ \int \frac {(d+e x)^2}{a+b (d+e x)^2+c (d+e x)^4} \, dx=-2\,\mathrm {atanh}\left (\frac {\sqrt {-\frac {b^3+\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-4\,a\,b\,c}{8\,\left (16\,a^2\,c^3\,e^2-8\,a\,b^2\,c^2\,e^2+b^4\,c\,e^2\right )}}\,\left (x\,\left (4\,a\,c^2\,e^{12}-2\,b^2\,c\,e^{12}\right )+\frac {\left (x\,\left (8\,b^3\,c^2\,e^{14}-32\,a\,b\,c^3\,e^{14}\right )+8\,b^3\,c^2\,d\,e^{13}-32\,a\,b\,c^3\,d\,e^{13}\right )\,\left (b^3+\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-4\,a\,b\,c\right )}{8\,\left (16\,a^2\,c^3\,e^2-8\,a\,b^2\,c^2\,e^2+b^4\,c\,e^2\right )}+4\,a\,c^2\,d\,e^{11}-2\,b^2\,c\,d\,e^{11}\right )}{a\,c\,e^{10}}\right )\,\sqrt {-\frac {b^3+\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-4\,a\,b\,c}{8\,\left (16\,a^2\,c^3\,e^2-8\,a\,b^2\,c^2\,e^2+b^4\,c\,e^2\right )}}-2\,\mathrm {atanh}\left (\frac {\sqrt {\frac {\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-b^3+4\,a\,b\,c}{8\,\left (16\,a^2\,c^3\,e^2-8\,a\,b^2\,c^2\,e^2+b^4\,c\,e^2\right )}}\,\left (x\,\left (4\,a\,c^2\,e^{12}-2\,b^2\,c\,e^{12}\right )-\frac {\left (x\,\left (8\,b^3\,c^2\,e^{14}-32\,a\,b\,c^3\,e^{14}\right )+8\,b^3\,c^2\,d\,e^{13}-32\,a\,b\,c^3\,d\,e^{13}\right )\,\left (\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-b^3+4\,a\,b\,c\right )}{8\,\left (16\,a^2\,c^3\,e^2-8\,a\,b^2\,c^2\,e^2+b^4\,c\,e^2\right )}+4\,a\,c^2\,d\,e^{11}-2\,b^2\,c\,d\,e^{11}\right )}{a\,c\,e^{10}}\right )\,\sqrt {\frac {\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-b^3+4\,a\,b\,c}{8\,\left (16\,a^2\,c^3\,e^2-8\,a\,b^2\,c^2\,e^2+b^4\,c\,e^2\right )}} \]

[In]

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

[Out]

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