\(\int \frac {(d+e x^2)^2}{(a+c x^4)^2} \, dx\) [145]

Optimal result

Integrand size = 19, antiderivative size = 349 \[ \int \frac {\left (d+e x^2\right )^2}{\left (a+c x^4\right )^2} \, dx=-\frac {e^2 x}{3 c \left (a+c x^4\right )}+\frac {x \left (3 c d^2+a e^2+6 c d e x^2\right )}{12 a c \left (a+c x^4\right )}-\frac {\left (3 c d^2+2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} c^{5/4}}+\frac {\left (3 c d^2+2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} c^{5/4}}-\frac {\left (3 c d^2-2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{5/4}}+\frac {\left (3 c d^2-2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{5/4}} \]

[Out]

-1/3*e^2*x/c/(c*x^4+a)+1/12*x*(6*c*d*e*x^2+a*e^2+3*c*d^2)/a/c/(c*x^4+a)-1/32*ln(-a^(1/4)*c^(1/4)*x*2^(1/2)+a^(
1/2)+x^2*c^(1/2))*(3*c*d^2+a*e^2-2*d*e*a^(1/2)*c^(1/2))/a^(7/4)/c^(5/4)*2^(1/2)+1/32*ln(a^(1/4)*c^(1/4)*x*2^(1
/2)+a^(1/2)+x^2*c^(1/2))*(3*c*d^2+a*e^2-2*d*e*a^(1/2)*c^(1/2))/a^(7/4)/c^(5/4)*2^(1/2)+1/16*arctan(-1+c^(1/4)*
x*2^(1/2)/a^(1/4))*(3*c*d^2+a*e^2+2*d*e*a^(1/2)*c^(1/2))/a^(7/4)/c^(5/4)*2^(1/2)+1/16*arctan(1+c^(1/4)*x*2^(1/
2)/a^(1/4))*(3*c*d^2+a*e^2+2*d*e*a^(1/2)*c^(1/2))/a^(7/4)/c^(5/4)*2^(1/2)

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {1221, 1193, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {\left (d+e x^2\right )^2}{\left (a+c x^4\right )^2} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) \left (2 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right )}{8 \sqrt {2} a^{7/4} c^{5/4}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) \left (2 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right )}{8 \sqrt {2} a^{7/4} c^{5/4}}-\frac {\left (-2 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{5/4}}+\frac {\left (-2 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{5/4}}+\frac {x \left (a e^2+3 c d^2+6 c d e x^2\right )}{12 a c \left (a+c x^4\right )}-\frac {e^2 x}{3 c \left (a+c x^4\right )} \]

[In]

Int[(d + e*x^2)^2/(a + c*x^4)^2,x]

[Out]

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

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 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 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 1193

Int[((d_) + (e_.)*(x_)^2)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-x)*(d + e*x^2)*((a + c*x^4)^(p + 1)/
(4*a*(p + 1))), x] + Dist[1/(4*a*(p + 1)), Int[Simp[d*(4*p + 5) + e*(4*p + 7)*x^2, x]*(a + c*x^4)^(p + 1), x],
x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntegerQ[2*p]

Rule 1221

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[e^q*x^(2*q - 3)*((a + c*x^4)^(p +
1)/(c*(4*p + 2*q + 1))), x] + Dist[1/(c*(4*p + 2*q + 1)), Int[(a + c*x^4)^p*ExpandToSum[c*(4*p + 2*q + 1)*(d
+ e*x^2)^q - a*(2*q - 3)*e^q*x^(2*q - 4) - c*(4*p + 2*q + 1)*e^q*x^(2*q), x], x], x] /; FreeQ[{a, c, d, e, p},
x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[q, 1]

Rubi steps \begin{align*} \text {integral}& = -\frac {e^2 x}{3 c \left (a+c x^4\right )}-\frac {\int \frac {-3 c d^2-a e^2-6 c d e x^2}{\left (a+c x^4\right )^2} \, dx}{3 c} \\ & = -\frac {e^2 x}{3 c \left (a+c x^4\right )}+\frac {x \left (3 c d^2+a e^2+6 c d e x^2\right )}{12 a c \left (a+c x^4\right )}+\frac {\int \frac {3 \left (3 c d^2+a e^2\right )+6 c d e x^2}{a+c x^4} \, dx}{12 a c} \\ & = -\frac {e^2 x}{3 c \left (a+c x^4\right )}+\frac {x \left (3 c d^2+a e^2+6 c d e x^2\right )}{12 a c \left (a+c x^4\right )}+\frac {\left (3 c d^2-2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{8 a^{3/2} c^{3/2}}+\frac {\left (3 c d^2+2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{8 a^{3/2} c^{3/2}} \\ & = -\frac {e^2 x}{3 c \left (a+c x^4\right )}+\frac {x \left (3 c d^2+a e^2+6 c d e x^2\right )}{12 a c \left (a+c x^4\right )}-\frac {\left (3 c d^2-2 \sqrt {a} \sqrt {c} d e+a e^2\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}{16 \sqrt {2} a^{7/4} c^{5/4}}-\frac {\left (3 c d^2-2 \sqrt {a} \sqrt {c} d e+a e^2\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}{16 \sqrt {2} a^{7/4} c^{5/4}}+\frac {\left (3 c d^2+2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 a^{3/2} c^{3/2}}+\frac {\left (3 c d^2+2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 a^{3/2} c^{3/2}} \\ & = -\frac {e^2 x}{3 c \left (a+c x^4\right )}+\frac {x \left (3 c d^2+a e^2+6 c d e x^2\right )}{12 a c \left (a+c x^4\right )}-\frac {\left (3 c d^2-2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{5/4}}+\frac {\left (3 c d^2-2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{5/4}}+\frac {\left (3 c d^2+2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} c^{5/4}}-\frac {\left (3 c d^2+2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} c^{5/4}} \\ & = -\frac {e^2 x}{3 c \left (a+c x^4\right )}+\frac {x \left (3 c d^2+a e^2+6 c d e x^2\right )}{12 a c \left (a+c x^4\right )}-\frac {\left (3 c d^2+2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} c^{5/4}}+\frac {\left (3 c d^2+2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} c^{5/4}}-\frac {\left (3 c d^2-2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{5/4}}+\frac {\left (3 c d^2-2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} c^{5/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.85 \[ \int \frac {\left (d+e x^2\right )^2}{\left (a+c x^4\right )^2} \, dx=\frac {-\frac {8 a^{3/4} \sqrt [4]{c} \left (a e^2 x-c d x \left (d+2 e x^2\right )\right )}{a+c x^4}-2 \sqrt {2} \left (3 c d^2+2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+2 \sqrt {2} \left (3 c d^2+2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )-\sqrt {2} \left (3 c d^2-2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )+\sqrt {2} \left (3 c d^2-2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{32 a^{7/4} c^{5/4}} \]

[In]

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

[Out]

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

Maple [C] (verified)

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

Time = 0.24 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.28

method	result	size
risch	\(\frac {\frac {e d \,x^{3}}{2 a}-\frac {\left (a \,e^{2}-c \,d^{2}\right ) x}{4 a c}}{c \,x^{4}+a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+a \right )}{\sum }\frac {\left (2 e d \,\textit {\_R}^{2}+\frac {a \,e^{2}+3 c \,d^{2}}{c}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{16 a c}\)	\(97\)
default	\(\frac {\frac {e d \,x^{3}}{2 a}-\frac {\left (a \,e^{2}-c \,d^{2}\right ) x}{4 a c}}{c \,x^{4}+a}+\frac {\frac {\left (a \,e^{2}+3 c \,d^{2}\right ) \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {d e \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{4 a c}\)	\(266\)

[In]

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

[Out]

(1/2*e*d/a*x^3-1/4*(a*e^2-c*d^2)/a/c*x)/(c*x^4+a)+1/16/a/c*sum((2*e*d*_R^2+1/c*(a*e^2+3*c*d^2))/_R^3*ln(x-_R),
_R=RootOf(_Z^4*c+a))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1596 vs. \(2 (264) = 528\).

Time = 0.72 (sec) , antiderivative size = 1596, normalized size of antiderivative = 4.57 \[ \int \frac {\left (d+e x^2\right )^2}{\left (a+c x^4\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/16*(8*c*d*e*x^3 + (a*c^2*x^4 + a^2*c)*sqrt(-(a^3*c^2*sqrt(-(81*c^4*d^8 + 36*a*c^3*d^6*e^2 + 22*a^2*c^2*d^4*e
^4 + 4*a^3*c*d^2*e^6 + a^4*e^8)/(a^7*c^5)) + 12*c*d^3*e + 4*a*d*e^3)/(a^3*c^2))*log((81*c^4*d^8 + 108*a*c^3*d^
6*e^2 + 38*a^2*c^2*d^4*e^4 + 12*a^3*c*d^2*e^6 + a^4*e^8)*x + (2*a^6*c^4*d*e*sqrt(-(81*c^4*d^8 + 36*a*c^3*d^6*e
^2 + 22*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 + a^4*e^8)/(a^7*c^5)) + 27*a^2*c^4*d^6 + 15*a^3*c^3*d^4*e^2 + 5*a^4*
c^2*d^2*e^4 + a^5*c*e^6)*sqrt(-(a^3*c^2*sqrt(-(81*c^4*d^8 + 36*a*c^3*d^6*e^2 + 22*a^2*c^2*d^4*e^4 + 4*a^3*c*d^
2*e^6 + a^4*e^8)/(a^7*c^5)) + 12*c*d^3*e + 4*a*d*e^3)/(a^3*c^2))) - (a*c^2*x^4 + a^2*c)*sqrt(-(a^3*c^2*sqrt(-(
81*c^4*d^8 + 36*a*c^3*d^6*e^2 + 22*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 + a^4*e^8)/(a^7*c^5)) + 12*c*d^3*e + 4*a*
d*e^3)/(a^3*c^2))*log((81*c^4*d^8 + 108*a*c^3*d^6*e^2 + 38*a^2*c^2*d^4*e^4 + 12*a^3*c*d^2*e^6 + a^4*e^8)*x - (
2*a^6*c^4*d*e*sqrt(-(81*c^4*d^8 + 36*a*c^3*d^6*e^2 + 22*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 + a^4*e^8)/(a^7*c^5)
) + 27*a^2*c^4*d^6 + 15*a^3*c^3*d^4*e^2 + 5*a^4*c^2*d^2*e^4 + a^5*c*e^6)*sqrt(-(a^3*c^2*sqrt(-(81*c^4*d^8 + 36
*a*c^3*d^6*e^2 + 22*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 + a^4*e^8)/(a^7*c^5)) + 12*c*d^3*e + 4*a*d*e^3)/(a^3*c^2
))) - (a*c^2*x^4 + a^2*c)*sqrt((a^3*c^2*sqrt(-(81*c^4*d^8 + 36*a*c^3*d^6*e^2 + 22*a^2*c^2*d^4*e^4 + 4*a^3*c*d^
2*e^6 + a^4*e^8)/(a^7*c^5)) - 12*c*d^3*e - 4*a*d*e^3)/(a^3*c^2))*log((81*c^4*d^8 + 108*a*c^3*d^6*e^2 + 38*a^2*
c^2*d^4*e^4 + 12*a^3*c*d^2*e^6 + a^4*e^8)*x + (2*a^6*c^4*d*e*sqrt(-(81*c^4*d^8 + 36*a*c^3*d^6*e^2 + 22*a^2*c^2
*d^4*e^4 + 4*a^3*c*d^2*e^6 + a^4*e^8)/(a^7*c^5)) - 27*a^2*c^4*d^6 - 15*a^3*c^3*d^4*e^2 - 5*a^4*c^2*d^2*e^4 - a
^5*c*e^6)*sqrt((a^3*c^2*sqrt(-(81*c^4*d^8 + 36*a*c^3*d^6*e^2 + 22*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 + a^4*e^8)
/(a^7*c^5)) - 12*c*d^3*e - 4*a*d*e^3)/(a^3*c^2))) + (a*c^2*x^4 + a^2*c)*sqrt((a^3*c^2*sqrt(-(81*c^4*d^8 + 36*a
*c^3*d^6*e^2 + 22*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 + a^4*e^8)/(a^7*c^5)) - 12*c*d^3*e - 4*a*d*e^3)/(a^3*c^2))
*log((81*c^4*d^8 + 108*a*c^3*d^6*e^2 + 38*a^2*c^2*d^4*e^4 + 12*a^3*c*d^2*e^6 + a^4*e^8)*x - (2*a^6*c^4*d*e*sqr
t(-(81*c^4*d^8 + 36*a*c^3*d^6*e^2 + 22*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 + a^4*e^8)/(a^7*c^5)) - 27*a^2*c^4*d^
6 - 15*a^3*c^3*d^4*e^2 - 5*a^4*c^2*d^2*e^4 - a^5*c*e^6)*sqrt((a^3*c^2*sqrt(-(81*c^4*d^8 + 36*a*c^3*d^6*e^2 + 2
2*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 + a^4*e^8)/(a^7*c^5)) - 12*c*d^3*e - 4*a*d*e^3)/(a^3*c^2))) + 4*(c*d^2 - a
*e^2)*x)/(a*c^2*x^4 + a^2*c)

Sympy [A] (verification not implemented)

Time = 1.01 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.79 \[ \int \frac {\left (d+e x^2\right )^2}{\left (a+c x^4\right )^2} \, dx=\operatorname {RootSum} {\left (65536 t^{4} a^{7} c^{5} + t^{2} \cdot \left (2048 a^{5} c^{3} d e^{3} + 6144 a^{4} c^{4} d^{3} e\right ) + a^{4} e^{8} + 20 a^{3} c d^{2} e^{6} + 118 a^{2} c^{2} d^{4} e^{4} + 180 a c^{3} d^{6} e^{2} + 81 c^{4} d^{8}, \left ( t \mapsto t \log {\left (x + \frac {- 8192 t^{3} a^{6} c^{4} d e + 16 t a^{5} c e^{6} - 48 t a^{4} c^{2} d^{2} e^{4} - 144 t a^{3} c^{3} d^{4} e^{2} + 432 t a^{2} c^{4} d^{6}}{a^{4} e^{8} + 12 a^{3} c d^{2} e^{6} + 38 a^{2} c^{2} d^{4} e^{4} + 108 a c^{3} d^{6} e^{2} + 81 c^{4} d^{8}} \right )} \right )\right )} + \frac {2 c d e x^{3} + x \left (- a e^{2} + c d^{2}\right )}{4 a^{2} c + 4 a c^{2} x^{4}} \]

[In]

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

[Out]

RootSum(65536*_t**4*a**7*c**5 + _t**2*(2048*a**5*c**3*d*e**3 + 6144*a**4*c**4*d**3*e) + a**4*e**8 + 20*a**3*c*
d**2*e**6 + 118*a**2*c**2*d**4*e**4 + 180*a*c**3*d**6*e**2 + 81*c**4*d**8, Lambda(_t, _t*log(x + (-8192*_t**3*
a**6*c**4*d*e + 16*_t*a**5*c*e**6 - 48*_t*a**4*c**2*d**2*e**4 - 144*_t*a**3*c**3*d**4*e**2 + 432*_t*a**2*c**4*
d**6)/(a**4*e**8 + 12*a**3*c*d**2*e**6 + 38*a**2*c**2*d**4*e**4 + 108*a*c**3*d**6*e**2 + 81*c**4*d**8)))) + (2
*c*d*e*x**3 + x*(-a*e**2 + c*d**2))/(4*a**2*c + 4*a*c**2*x**4)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 324, normalized size of antiderivative = 0.93 \[ \int \frac {\left (d+e x^2\right )^2}{\left (a+c x^4\right )^2} \, dx=\frac {2 \, c d e x^{3} + {\left (c d^{2} - a e^{2}\right )} x}{4 \, {\left (a c^{2} x^{4} + a^{2} c\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (3 \, c^{\frac {3}{2}} d^{2} + 2 \, \sqrt {a} c d e + a \sqrt {c} e^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} {\left (3 \, c^{\frac {3}{2}} d^{2} + 2 \, \sqrt {a} c d e + a \sqrt {c} e^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {\sqrt {2} {\left (3 \, c^{\frac {3}{2}} d^{2} - 2 \, \sqrt {a} c d e + a \sqrt {c} e^{2}\right )} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (3 \, c^{\frac {3}{2}} d^{2} - 2 \, \sqrt {a} c d e + a \sqrt {c} e^{2}\right )} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}}}{32 \, a c} \]

[In]

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

[Out]

1/4*(2*c*d*e*x^3 + (c*d^2 - a*e^2)*x)/(a*c^2*x^4 + a^2*c) + 1/32*(2*sqrt(2)*(3*c^(3/2)*d^2 + 2*sqrt(a)*c*d*e +
a*sqrt(c)*e^2)*arctan(1/2*sqrt(2)*(2*sqrt(c)*x + sqrt(2)*a^(1/4)*c^(1/4))/sqrt(sqrt(a)*sqrt(c)))/(sqrt(a)*sqr
t(sqrt(a)*sqrt(c))*sqrt(c)) + 2*sqrt(2)*(3*c^(3/2)*d^2 + 2*sqrt(a)*c*d*e + a*sqrt(c)*e^2)*arctan(1/2*sqrt(2)*(
2*sqrt(c)*x - sqrt(2)*a^(1/4)*c^(1/4))/sqrt(sqrt(a)*sqrt(c)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(c))*sqrt(c)) + sqrt(2
)*(3*c^(3/2)*d^2 - 2*sqrt(a)*c*d*e + a*sqrt(c)*e^2)*log(sqrt(c)*x^2 + sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(a))/(a^
(3/4)*c^(3/4)) - sqrt(2)*(3*c^(3/2)*d^2 - 2*sqrt(a)*c*d*e + a*sqrt(c)*e^2)*log(sqrt(c)*x^2 - sqrt(2)*a^(1/4)*c
^(1/4)*x + sqrt(a))/(a^(3/4)*c^(3/4)))/(a*c)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.00 \[ \int \frac {\left (d+e x^2\right )^2}{\left (a+c x^4\right )^2} \, dx=\frac {2 \, c d e x^{3} + c d^{2} x - a e^{2} x}{4 \, {\left (c x^{4} + a\right )} a c} + \frac {\sqrt {2} {\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} + \left (a c^{3}\right )^{\frac {1}{4}} a c e^{2} + 2 \, \left (a c^{3}\right )^{\frac {3}{4}} d e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{16 \, a^{2} c^{3}} + \frac {\sqrt {2} {\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} + \left (a c^{3}\right )^{\frac {1}{4}} a c e^{2} + 2 \, \left (a c^{3}\right )^{\frac {3}{4}} d e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{16 \, a^{2} c^{3}} + \frac {\sqrt {2} {\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} + \left (a c^{3}\right )^{\frac {1}{4}} a c e^{2} - 2 \, \left (a c^{3}\right )^{\frac {3}{4}} d e\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{32 \, a^{2} c^{3}} - \frac {\sqrt {2} {\left (3 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} + \left (a c^{3}\right )^{\frac {1}{4}} a c e^{2} - 2 \, \left (a c^{3}\right )^{\frac {3}{4}} d e\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{32 \, a^{2} c^{3}} \]

[In]

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

[Out]

1/4*(2*c*d*e*x^3 + c*d^2*x - a*e^2*x)/((c*x^4 + a)*a*c) + 1/16*sqrt(2)*(3*(a*c^3)^(1/4)*c^2*d^2 + (a*c^3)^(1/4
)*a*c*e^2 + 2*(a*c^3)^(3/4)*d*e)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a^2*c^3) + 1/16*
sqrt(2)*(3*(a*c^3)^(1/4)*c^2*d^2 + (a*c^3)^(1/4)*a*c*e^2 + 2*(a*c^3)^(3/4)*d*e)*arctan(1/2*sqrt(2)*(2*x - sqrt
(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a^2*c^3) + 1/32*sqrt(2)*(3*(a*c^3)^(1/4)*c^2*d^2 + (a*c^3)^(1/4)*a*c*e^2 - 2*(a
*c^3)^(3/4)*d*e)*log(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^2*c^3) - 1/32*sqrt(2)*(3*(a*c^3)^(1/4)*c^2*d^
2 + (a*c^3)^(1/4)*a*c*e^2 - 2*(a*c^3)^(3/4)*d*e)*log(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^2*c^3)

Mupad [B] (verification not implemented)

Time = 0.53 (sec) , antiderivative size = 1565, normalized size of antiderivative = 4.48 \[ \int \frac {\left (d+e x^2\right )^2}{\left (a+c x^4\right )^2} \, dx=\text {Too large to display} \]

[In]

int((d + e*x^2)^2/(a + c*x^4)^2,x)

[Out]

2*atanh((9*c^3*d^4*x*((9*d^4*(-a^7*c^5)^(1/2))/(256*a^7*c^3) - (3*d^3*e)/(64*a^3*c) - (d*e^3)/(64*a^2*c^2) + (
e^4*(-a^7*c^5)^(1/2))/(256*a^5*c^5) + (d^2*e^2*(-a^7*c^5)^(1/2))/(128*a^6*c^4))^(1/2))/(2*((27*d^6*(-a^7*c^5)^
(1/2))/(32*a^5) - (c*d^3*e^3)/8 - (a*d*e^5)/16 - (9*c^2*d^5*e)/(16*a) + (e^6*(-a^7*c^5)^(1/2))/(32*a^2*c^3) +
(5*d^2*e^4*(-a^7*c^5)^(1/2))/(32*a^3*c^2) + (15*d^4*e^2*(-a^7*c^5)^(1/2))/(32*a^4*c))) + (c*e^4*x*((9*d^4*(-a^
7*c^5)^(1/2))/(256*a^7*c^3) - (3*d^3*e)/(64*a^3*c) - (d*e^3)/(64*a^2*c^2) + (e^4*(-a^7*c^5)^(1/2))/(256*a^5*c^
5) + (d^2*e^2*(-a^7*c^5)^(1/2))/(128*a^6*c^4))^(1/2))/(2*((27*d^6*(-a^7*c^5)^(1/2))/(32*a^7) - (d*e^5)/(16*a)
- (c*d^3*e^3)/(8*a^2) - (9*c^2*d^5*e)/(16*a^3) + (e^6*(-a^7*c^5)^(1/2))/(32*a^4*c^3) + (5*d^2*e^4*(-a^7*c^5)^(
1/2))/(32*a^5*c^2) + (15*d^4*e^2*(-a^7*c^5)^(1/2))/(32*a^6*c))) + (c^2*d^2*e^2*x*((9*d^4*(-a^7*c^5)^(1/2))/(25
6*a^7*c^3) - (3*d^3*e)/(64*a^3*c) - (d*e^3)/(64*a^2*c^2) + (e^4*(-a^7*c^5)^(1/2))/(256*a^5*c^5) + (d^2*e^2*(-a
^7*c^5)^(1/2))/(128*a^6*c^4))^(1/2))/((27*d^6*(-a^7*c^5)^(1/2))/(32*a^6) - (d*e^5)/16 - (c*d^3*e^3)/(8*a) - (9
*c^2*d^5*e)/(16*a^2) + (e^6*(-a^7*c^5)^(1/2))/(32*a^3*c^3) + (5*d^2*e^4*(-a^7*c^5)^(1/2))/(32*a^4*c^2) + (15*d
^4*e^2*(-a^7*c^5)^(1/2))/(32*a^5*c)))*((a^2*e^4*(-a^7*c^5)^(1/2) + 9*c^2*d^4*(-a^7*c^5)^(1/2) - 12*a^4*c^4*d^3
*e - 4*a^5*c^3*d*e^3 + 2*a*c*d^2*e^2*(-a^7*c^5)^(1/2))/(256*a^7*c^5))^(1/2) - 2*atanh((9*c^3*d^4*x*(- (d*e^3)/
(64*a^2*c^2) - (3*d^3*e)/(64*a^3*c) - (9*d^4*(-a^7*c^5)^(1/2))/(256*a^7*c^3) - (e^4*(-a^7*c^5)^(1/2))/(256*a^5
*c^5) - (d^2*e^2*(-a^7*c^5)^(1/2))/(128*a^6*c^4))^(1/2))/(2*((27*d^6*(-a^7*c^5)^(1/2))/(32*a^5) + (c*d^3*e^3)/
8 + (a*d*e^5)/16 + (9*c^2*d^5*e)/(16*a) + (e^6*(-a^7*c^5)^(1/2))/(32*a^2*c^3) + (5*d^2*e^4*(-a^7*c^5)^(1/2))/(
32*a^3*c^2) + (15*d^4*e^2*(-a^7*c^5)^(1/2))/(32*a^4*c))) + (c*e^4*x*(- (d*e^3)/(64*a^2*c^2) - (3*d^3*e)/(64*a^
3*c) - (9*d^4*(-a^7*c^5)^(1/2))/(256*a^7*c^3) - (e^4*(-a^7*c^5)^(1/2))/(256*a^5*c^5) - (d^2*e^2*(-a^7*c^5)^(1/
2))/(128*a^6*c^4))^(1/2))/(2*((27*d^6*(-a^7*c^5)^(1/2))/(32*a^7) + (d*e^5)/(16*a) + (c*d^3*e^3)/(8*a^2) + (9*c
^2*d^5*e)/(16*a^3) + (e^6*(-a^7*c^5)^(1/2))/(32*a^4*c^3) + (5*d^2*e^4*(-a^7*c^5)^(1/2))/(32*a^5*c^2) + (15*d^4
*e^2*(-a^7*c^5)^(1/2))/(32*a^6*c))) + (c^2*d^2*e^2*x*(- (d*e^3)/(64*a^2*c^2) - (3*d^3*e)/(64*a^3*c) - (9*d^4*(
-a^7*c^5)^(1/2))/(256*a^7*c^3) - (e^4*(-a^7*c^5)^(1/2))/(256*a^5*c^5) - (d^2*e^2*(-a^7*c^5)^(1/2))/(128*a^6*c^
4))^(1/2))/((d*e^5)/16 + (27*d^6*(-a^7*c^5)^(1/2))/(32*a^6) + (c*d^3*e^3)/(8*a) + (9*c^2*d^5*e)/(16*a^2) + (e^
6*(-a^7*c^5)^(1/2))/(32*a^3*c^3) + (5*d^2*e^4*(-a^7*c^5)^(1/2))/(32*a^4*c^2) + (15*d^4*e^2*(-a^7*c^5)^(1/2))/(
32*a^5*c)))*(-(a^2*e^4*(-a^7*c^5)^(1/2) + 9*c^2*d^4*(-a^7*c^5)^(1/2) + 12*a^4*c^4*d^3*e + 4*a^5*c^3*d*e^3 + 2*
a*c*d^2*e^2*(-a^7*c^5)^(1/2))/(256*a^7*c^5))^(1/2) + ((d*e*x^3)/(2*a) - (x*(a*e^2 - c*d^2))/(4*a*c))/(a + c*x^
4)