\(\int \frac {(d+e x^2)^2}{a+c x^4} \, dx\) [139]

Optimal result

Integrand size = 19, antiderivative size = 297 \[ \int \frac {\left (d+e x^2\right )^2}{a+c x^4} \, dx=\frac {e^2 x}{c}-\frac {\left (c d^2+2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} c^{5/4}}+\frac {\left (c d^2+2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} c^{5/4}}-\frac {\left (c d^2-2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} c^{5/4}}+\frac {\left (c d^2-2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} c^{5/4}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {1185, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {\left (d+e x^2\right )^2}{a+c x^4} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) \left (2 \sqrt {a} \sqrt {c} d e-a e^2+c d^2\right )}{2 \sqrt {2} a^{3/4} c^{5/4}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) \left (2 \sqrt {a} \sqrt {c} d e-a e^2+c d^2\right )}{2 \sqrt {2} a^{3/4} c^{5/4}}-\frac {\left (-2 \sqrt {a} \sqrt {c} d e-a e^2+c d^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} c^{5/4}}+\frac {\left (-2 \sqrt {a} \sqrt {c} d e-a e^2+c d^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} c^{5/4}}+\frac {e^2 x}{c} \]

[In]

Int[(d + e*x^2)^2/(a + c*x^4),x]

[Out]

(e^2*x)/c - ((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]*a^(3/
4)*c^(5/4)) + ((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]*a^(
3/4)*c^(5/4)) - ((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
])/(4*Sqrt[2]*a^(3/4)*c^(5/4)) + ((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])/(4*Sqrt[2]*a^(3/4)*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 1185

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q/(a + c*x^
4), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[q]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {e^2}{c}+\frac {c d^2-a e^2+2 c d e x^2}{c \left (a+c x^4\right )}\right ) \, dx \\ & = \frac {e^2 x}{c}+\frac {\int \frac {c d^2-a e^2+2 c d e x^2}{a+c x^4} \, dx}{c} \\ & = \frac {e^2 x}{c}+\frac {\left (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}{2 \sqrt {a} c^{3/2}}+\frac {\left (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}{2 \sqrt {a} c^{3/2}} \\ & = \frac {e^2 x}{c}-\frac {\left (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}{4 \sqrt {2} a^{3/4} c^{5/4}}-\frac {\left (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}{4 \sqrt {2} a^{3/4} c^{5/4}}+\frac {\left (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}{4 \sqrt {a} c^{3/2}}+\frac {\left (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}{4 \sqrt {a} c^{3/2}} \\ & = \frac {e^2 x}{c}-\frac {\left (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 )}{4 \sqrt {2} a^{3/4} c^{5/4}}+\frac {\left (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 )}{4 \sqrt {2} a^{3/4} c^{5/4}}+\frac {\left (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 )}{2 \sqrt {2} a^{3/4} c^{5/4}}-\frac {\left (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 )}{2 \sqrt {2} a^{3/4} c^{5/4}} \\ & = \frac {e^2 x}{c}-\frac {\left (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 )}{2 \sqrt {2} a^{3/4} c^{5/4}}+\frac {\left (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 )}{2 \sqrt {2} a^{3/4} c^{5/4}}-\frac {\left (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 )}{4 \sqrt {2} a^{3/4} c^{5/4}}+\frac {\left (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 )}{4 \sqrt {2} a^{3/4} c^{5/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.91 \[ \int \frac {\left (d+e x^2\right )^2}{a+c x^4} \, dx=\frac {8 a^{3/4} \sqrt [4]{c} e^2 x-2 \sqrt {2} \left (c d^2+2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+2 \sqrt {2} \left (c d^2+2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+\sqrt {2} \left (-c d^2+2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )+\sqrt {2} \left (c d^2-2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{8 a^{3/4} c^{5/4}} \]

[In]

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

[Out]

(8*a^(3/4)*c^(1/4)*e^2*x - 2*Sqrt[2]*(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]*(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]*(
-(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]*(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])/(8*a^(3/4)*c^(5/4
))

Maple [C] (verified)

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

Time = 0.22 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.19

method	result	size
risch	\(\frac {e^{2} x}{c}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+a \right )}{\sum }\frac {\left (2 \textit {\_R}^{2} c d e -a \,e^{2}+c \,d^{2}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{4 c^{2}}\)	\(56\)
default	\(\frac {e^{2} x}{c}+\frac {\frac {\left (-a \,e^{2}+c \,d^{2}\right ) \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {d e \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{c}\)	\(228\)

[In]

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

[Out]

e^2*x/c+1/4/c^2*sum((2*_R^2*c*d*e-a*e^2+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. 1480 vs. \(2 (216) = 432\).

Time = 0.71 (sec) , antiderivative size = 1480, normalized size of antiderivative = 4.98 \[ \int \frac {\left (d+e x^2\right )^2}{a+c x^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/4*(4*e^2*x + c*sqrt(-(4*c*d^3*e - 4*a*d*e^3 + a*c^2*sqrt(-(c^4*d^8 - 12*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)/(a^3*c^5)))/(a*c^2))*log((c^4*d^8 - 4*a*c^3*d^6*e^2 - 10*a^2*c^2*d^4*e^4 - 4*a^3*
c*d^2*e^6 + a^4*e^8)*x + (a*c^4*d^6 - 7*a^2*c^3*d^4*e^2 + 7*a^3*c^2*d^2*e^4 - a^4*c*e^6 + 2*a^3*c^4*d*e*sqrt(-
(c^4*d^8 - 12*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)/(a^3*c^5)))*sqrt(-(4*c*d^3*e -
4*a*d*e^3 + a*c^2*sqrt(-(c^4*d^8 - 12*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)/(a^3*c^
5)))/(a*c^2))) - c*sqrt(-(4*c*d^3*e - 4*a*d*e^3 + a*c^2*sqrt(-(c^4*d^8 - 12*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)/(a^3*c^5)))/(a*c^2))*log((c^4*d^8 - 4*a*c^3*d^6*e^2 - 10*a^2*c^2*d^4*e^4 - 4*a^
3*c*d^2*e^6 + a^4*e^8)*x - (a*c^4*d^6 - 7*a^2*c^3*d^4*e^2 + 7*a^3*c^2*d^2*e^4 - a^4*c*e^6 + 2*a^3*c^4*d*e*sqrt
(-(c^4*d^8 - 12*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)/(a^3*c^5)))*sqrt(-(4*c*d^3*e
- 4*a*d*e^3 + a*c^2*sqrt(-(c^4*d^8 - 12*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)/(a^3*
c^5)))/(a*c^2))) + c*sqrt(-(4*c*d^3*e - 4*a*d*e^3 - a*c^2*sqrt(-(c^4*d^8 - 12*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)/(a^3*c^5)))/(a*c^2))*log((c^4*d^8 - 4*a*c^3*d^6*e^2 - 10*a^2*c^2*d^4*e^4 - 4*
a^3*c*d^2*e^6 + a^4*e^8)*x + (a*c^4*d^6 - 7*a^2*c^3*d^4*e^2 + 7*a^3*c^2*d^2*e^4 - a^4*c*e^6 - 2*a^3*c^4*d*e*sq
rt(-(c^4*d^8 - 12*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)/(a^3*c^5)))*sqrt(-(4*c*d^3*
e - 4*a*d*e^3 - a*c^2*sqrt(-(c^4*d^8 - 12*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)/(a^
3*c^5)))/(a*c^2))) - c*sqrt(-(4*c*d^3*e - 4*a*d*e^3 - a*c^2*sqrt(-(c^4*d^8 - 12*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)/(a^3*c^5)))/(a*c^2))*log((c^4*d^8 - 4*a*c^3*d^6*e^2 - 10*a^2*c^2*d^4*e^4 -
4*a^3*c*d^2*e^6 + a^4*e^8)*x - (a*c^4*d^6 - 7*a^2*c^3*d^4*e^2 + 7*a^3*c^2*d^2*e^4 - a^4*c*e^6 - 2*a^3*c^4*d*e*
sqrt(-(c^4*d^8 - 12*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)/(a^3*c^5)))*sqrt(-(4*c*d^
3*e - 4*a*d*e^3 - a*c^2*sqrt(-(c^4*d^8 - 12*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)/(
a^3*c^5)))/(a*c^2))))/c

Sympy [A] (verification not implemented)

Time = 0.73 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.80 \[ \int \frac {\left (d+e x^2\right )^2}{a+c x^4} \, dx=\operatorname {RootSum} {\left (256 t^{4} a^{3} c^{5} + t^{2} \left (- 128 a^{3} c^{3} d e^{3} + 128 a^{2} c^{4} d^{3} e\right ) + a^{4} e^{8} + 4 a^{3} c d^{2} e^{6} + 6 a^{2} c^{2} d^{4} e^{4} + 4 a c^{3} d^{6} e^{2} + c^{4} d^{8}, \left ( t \mapsto t \log {\left (x + \frac {- 128 t^{3} a^{3} c^{4} d e - 4 t a^{4} c e^{6} + 60 t a^{3} c^{2} d^{2} e^{4} - 60 t a^{2} c^{3} d^{4} e^{2} + 4 t a c^{4} d^{6}}{a^{4} e^{8} - 4 a^{3} c d^{2} e^{6} - 10 a^{2} c^{2} d^{4} e^{4} - 4 a c^{3} d^{6} e^{2} + c^{4} d^{8}} \right )} \right )\right )} + \frac {e^{2} x}{c} \]

[In]

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

[Out]

RootSum(256*_t**4*a**3*c**5 + _t**2*(-128*a**3*c**3*d*e**3 + 128*a**2*c**4*d**3*e) + a**4*e**8 + 4*a**3*c*d**2
*e**6 + 6*a**2*c**2*d**4*e**4 + 4*a*c**3*d**6*e**2 + c**4*d**8, Lambda(_t, _t*log(x + (-128*_t**3*a**3*c**4*d*
e - 4*_t*a**4*c*e**6 + 60*_t*a**3*c**2*d**2*e**4 - 60*_t*a**2*c**3*d**4*e**2 + 4*_t*a*c**4*d**6)/(a**4*e**8 -
4*a**3*c*d**2*e**6 - 10*a**2*c**2*d**4*e**4 - 4*a*c**3*d**6*e**2 + c**4*d**8)))) + e**2*x/c

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 288, normalized size of antiderivative = 0.97 \[ \int \frac {\left (d+e x^2\right )^2}{a+c x^4} \, dx=\frac {e^{2} x}{c} + \frac {\frac {2 \, \sqrt {2} {\left (c^{\frac {3}{2}} d^{2} + 2 \, \sqrt {a} c d e - a \sqrt {c} e^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} {\left (c^{\frac {3}{2}} d^{2} + 2 \, \sqrt {a} c d e - a \sqrt {c} e^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {\sqrt {2} {\left (c^{\frac {3}{2}} d^{2} - 2 \, \sqrt {a} c d e - a \sqrt {c} e^{2}\right )} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (c^{\frac {3}{2}} d^{2} - 2 \, \sqrt {a} c d e - a \sqrt {c} e^{2}\right )} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}}}{8 \, c} \]

[In]

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

[Out]

e^2*x/c + 1/8*(2*sqrt(2)*(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 + sqr
t(2)*a^(1/4)*c^(1/4))/sqrt(sqrt(a)*sqrt(c)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(c))*sqrt(c)) + 2*sqrt(2)*(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)*sqr
t(c)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(c))*sqrt(c)) + sqrt(2)*(c^(3/2)*d^2 - 2*sqrt(a)*c*d*e - a*sqrt(c)*e^2)*log(s
qrt(c)*x^2 + sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(a))/(a^(3/4)*c^(3/4)) - sqrt(2)*(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)))/c

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.07 \[ \int \frac {\left (d+e x^2\right )^2}{a+c x^4} \, dx=\frac {e^{2} x}{c} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} - \left (a c^{3}\right )^{\frac {1}{4}} a c e^{2} + 2 \, \left (a c^{3}\right )^{\frac {3}{4}} d e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{4 \, a c^{3}} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} - \left (a c^{3}\right )^{\frac {1}{4}} a c e^{2} + 2 \, \left (a c^{3}\right )^{\frac {3}{4}} d e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{4 \, a c^{3}} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} - \left (a c^{3}\right )^{\frac {1}{4}} a c e^{2} - 2 \, \left (a c^{3}\right )^{\frac {3}{4}} d e\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{8 \, a c^{3}} - \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} - \left (a c^{3}\right )^{\frac {1}{4}} a c e^{2} - 2 \, \left (a c^{3}\right )^{\frac {3}{4}} d e\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{8 \, a c^{3}} \]

[In]

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

[Out]

e^2*x/c + 1/4*sqrt(2)*((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*c^3) + 1/4*sqrt(2)*((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*c^3) + 1/8*sqrt(2)*((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*c^3) - 1/8*sqrt(2)*((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*c^3)

Mupad [B] (verification not implemented)

Time = 13.50 (sec) , antiderivative size = 1479, normalized size of antiderivative = 4.98 \[ \int \frac {\left (d+e x^2\right )^2}{a+c x^4} \, dx=\frac {e^2\,x}{c}-2\,\mathrm {atanh}\left (\frac {8\,c^3\,d^4\,x\,\sqrt {\frac {d\,e^3}{4\,c^2}-\frac {d^3\,e}{4\,a\,c}+\frac {d^4\,\sqrt {-a^3\,c^5}}{16\,a^3\,c^3}+\frac {e^4\,\sqrt {-a^3\,c^5}}{16\,a\,c^5}-\frac {3\,d^2\,e^2\,\sqrt {-a^3\,c^5}}{8\,a^2\,c^4}}}{4\,a^2\,d\,e^5-\frac {2\,d^6\,\sqrt {-a^3\,c^5}}{a^2}+4\,c^2\,d^5\,e+\frac {2\,a\,e^6\,\sqrt {-a^3\,c^5}}{c^3}-24\,a\,c\,d^3\,e^3-\frac {14\,d^2\,e^4\,\sqrt {-a^3\,c^5}}{c^2}+\frac {14\,d^4\,e^2\,\sqrt {-a^3\,c^5}}{a\,c}}+\frac {8\,a^2\,c\,e^4\,x\,\sqrt {\frac {d\,e^3}{4\,c^2}-\frac {d^3\,e}{4\,a\,c}+\frac {d^4\,\sqrt {-a^3\,c^5}}{16\,a^3\,c^3}+\frac {e^4\,\sqrt {-a^3\,c^5}}{16\,a\,c^5}-\frac {3\,d^2\,e^2\,\sqrt {-a^3\,c^5}}{8\,a^2\,c^4}}}{4\,a^2\,d\,e^5-\frac {2\,d^6\,\sqrt {-a^3\,c^5}}{a^2}+4\,c^2\,d^5\,e+\frac {2\,a\,e^6\,\sqrt {-a^3\,c^5}}{c^3}-24\,a\,c\,d^3\,e^3-\frac {14\,d^2\,e^4\,\sqrt {-a^3\,c^5}}{c^2}+\frac {14\,d^4\,e^2\,\sqrt {-a^3\,c^5}}{a\,c}}-\frac {48\,a\,c^2\,d^2\,e^2\,x\,\sqrt {\frac {d\,e^3}{4\,c^2}-\frac {d^3\,e}{4\,a\,c}+\frac {d^4\,\sqrt {-a^3\,c^5}}{16\,a^3\,c^3}+\frac {e^4\,\sqrt {-a^3\,c^5}}{16\,a\,c^5}-\frac {3\,d^2\,e^2\,\sqrt {-a^3\,c^5}}{8\,a^2\,c^4}}}{4\,a^2\,d\,e^5-\frac {2\,d^6\,\sqrt {-a^3\,c^5}}{a^2}+4\,c^2\,d^5\,e+\frac {2\,a\,e^6\,\sqrt {-a^3\,c^5}}{c^3}-24\,a\,c\,d^3\,e^3-\frac {14\,d^2\,e^4\,\sqrt {-a^3\,c^5}}{c^2}+\frac {14\,d^4\,e^2\,\sqrt {-a^3\,c^5}}{a\,c}}\right )\,\sqrt {\frac {a^2\,e^4\,\sqrt {-a^3\,c^5}+c^2\,d^4\,\sqrt {-a^3\,c^5}-4\,a^2\,c^4\,d^3\,e+4\,a^3\,c^3\,d\,e^3-6\,a\,c\,d^2\,e^2\,\sqrt {-a^3\,c^5}}{16\,a^3\,c^5}}-2\,\mathrm {atanh}\left (\frac {8\,c^3\,d^4\,x\,\sqrt {\frac {d\,e^3}{4\,c^2}-\frac {d^3\,e}{4\,a\,c}-\frac {d^4\,\sqrt {-a^3\,c^5}}{16\,a^3\,c^3}-\frac {e^4\,\sqrt {-a^3\,c^5}}{16\,a\,c^5}+\frac {3\,d^2\,e^2\,\sqrt {-a^3\,c^5}}{8\,a^2\,c^4}}}{\frac {2\,d^6\,\sqrt {-a^3\,c^5}}{a^2}+4\,a^2\,d\,e^5+4\,c^2\,d^5\,e-\frac {2\,a\,e^6\,\sqrt {-a^3\,c^5}}{c^3}-24\,a\,c\,d^3\,e^3+\frac {14\,d^2\,e^4\,\sqrt {-a^3\,c^5}}{c^2}-\frac {14\,d^4\,e^2\,\sqrt {-a^3\,c^5}}{a\,c}}+\frac {8\,a^2\,c\,e^4\,x\,\sqrt {\frac {d\,e^3}{4\,c^2}-\frac {d^3\,e}{4\,a\,c}-\frac {d^4\,\sqrt {-a^3\,c^5}}{16\,a^3\,c^3}-\frac {e^4\,\sqrt {-a^3\,c^5}}{16\,a\,c^5}+\frac {3\,d^2\,e^2\,\sqrt {-a^3\,c^5}}{8\,a^2\,c^4}}}{\frac {2\,d^6\,\sqrt {-a^3\,c^5}}{a^2}+4\,a^2\,d\,e^5+4\,c^2\,d^5\,e-\frac {2\,a\,e^6\,\sqrt {-a^3\,c^5}}{c^3}-24\,a\,c\,d^3\,e^3+\frac {14\,d^2\,e^4\,\sqrt {-a^3\,c^5}}{c^2}-\frac {14\,d^4\,e^2\,\sqrt {-a^3\,c^5}}{a\,c}}-\frac {48\,a\,c^2\,d^2\,e^2\,x\,\sqrt {\frac {d\,e^3}{4\,c^2}-\frac {d^3\,e}{4\,a\,c}-\frac {d^4\,\sqrt {-a^3\,c^5}}{16\,a^3\,c^3}-\frac {e^4\,\sqrt {-a^3\,c^5}}{16\,a\,c^5}+\frac {3\,d^2\,e^2\,\sqrt {-a^3\,c^5}}{8\,a^2\,c^4}}}{\frac {2\,d^6\,\sqrt {-a^3\,c^5}}{a^2}+4\,a^2\,d\,e^5+4\,c^2\,d^5\,e-\frac {2\,a\,e^6\,\sqrt {-a^3\,c^5}}{c^3}-24\,a\,c\,d^3\,e^3+\frac {14\,d^2\,e^4\,\sqrt {-a^3\,c^5}}{c^2}-\frac {14\,d^4\,e^2\,\sqrt {-a^3\,c^5}}{a\,c}}\right )\,\sqrt {-\frac {a^2\,e^4\,\sqrt {-a^3\,c^5}+c^2\,d^4\,\sqrt {-a^3\,c^5}+4\,a^2\,c^4\,d^3\,e-4\,a^3\,c^3\,d\,e^3-6\,a\,c\,d^2\,e^2\,\sqrt {-a^3\,c^5}}{16\,a^3\,c^5}} \]

[In]

int((d + e*x^2)^2/(a + c*x^4),x)

[Out]

(e^2*x)/c - 2*atanh((8*c^3*d^4*x*((d*e^3)/(4*c^2) - (d^3*e)/(4*a*c) + (d^4*(-a^3*c^5)^(1/2))/(16*a^3*c^3) + (e
^4*(-a^3*c^5)^(1/2))/(16*a*c^5) - (3*d^2*e^2*(-a^3*c^5)^(1/2))/(8*a^2*c^4))^(1/2))/(4*a^2*d*e^5 - (2*d^6*(-a^3
*c^5)^(1/2))/a^2 + 4*c^2*d^5*e + (2*a*e^6*(-a^3*c^5)^(1/2))/c^3 - 24*a*c*d^3*e^3 - (14*d^2*e^4*(-a^3*c^5)^(1/2
))/c^2 + (14*d^4*e^2*(-a^3*c^5)^(1/2))/(a*c)) + (8*a^2*c*e^4*x*((d*e^3)/(4*c^2) - (d^3*e)/(4*a*c) + (d^4*(-a^3
*c^5)^(1/2))/(16*a^3*c^3) + (e^4*(-a^3*c^5)^(1/2))/(16*a*c^5) - (3*d^2*e^2*(-a^3*c^5)^(1/2))/(8*a^2*c^4))^(1/2
))/(4*a^2*d*e^5 - (2*d^6*(-a^3*c^5)^(1/2))/a^2 + 4*c^2*d^5*e + (2*a*e^6*(-a^3*c^5)^(1/2))/c^3 - 24*a*c*d^3*e^3
- (14*d^2*e^4*(-a^3*c^5)^(1/2))/c^2 + (14*d^4*e^2*(-a^3*c^5)^(1/2))/(a*c)) - (48*a*c^2*d^2*e^2*x*((d*e^3)/(4*
c^2) - (d^3*e)/(4*a*c) + (d^4*(-a^3*c^5)^(1/2))/(16*a^3*c^3) + (e^4*(-a^3*c^5)^(1/2))/(16*a*c^5) - (3*d^2*e^2*
(-a^3*c^5)^(1/2))/(8*a^2*c^4))^(1/2))/(4*a^2*d*e^5 - (2*d^6*(-a^3*c^5)^(1/2))/a^2 + 4*c^2*d^5*e + (2*a*e^6*(-a
^3*c^5)^(1/2))/c^3 - 24*a*c*d^3*e^3 - (14*d^2*e^4*(-a^3*c^5)^(1/2))/c^2 + (14*d^4*e^2*(-a^3*c^5)^(1/2))/(a*c))
)*((a^2*e^4*(-a^3*c^5)^(1/2) + c^2*d^4*(-a^3*c^5)^(1/2) - 4*a^2*c^4*d^3*e + 4*a^3*c^3*d*e^3 - 6*a*c*d^2*e^2*(-
a^3*c^5)^(1/2))/(16*a^3*c^5))^(1/2) - 2*atanh((8*c^3*d^4*x*((d*e^3)/(4*c^2) - (d^3*e)/(4*a*c) - (d^4*(-a^3*c^5
)^(1/2))/(16*a^3*c^3) - (e^4*(-a^3*c^5)^(1/2))/(16*a*c^5) + (3*d^2*e^2*(-a^3*c^5)^(1/2))/(8*a^2*c^4))^(1/2))/(
(2*d^6*(-a^3*c^5)^(1/2))/a^2 + 4*a^2*d*e^5 + 4*c^2*d^5*e - (2*a*e^6*(-a^3*c^5)^(1/2))/c^3 - 24*a*c*d^3*e^3 + (
14*d^2*e^4*(-a^3*c^5)^(1/2))/c^2 - (14*d^4*e^2*(-a^3*c^5)^(1/2))/(a*c)) + (8*a^2*c*e^4*x*((d*e^3)/(4*c^2) - (d
^3*e)/(4*a*c) - (d^4*(-a^3*c^5)^(1/2))/(16*a^3*c^3) - (e^4*(-a^3*c^5)^(1/2))/(16*a*c^5) + (3*d^2*e^2*(-a^3*c^5
)^(1/2))/(8*a^2*c^4))^(1/2))/((2*d^6*(-a^3*c^5)^(1/2))/a^2 + 4*a^2*d*e^5 + 4*c^2*d^5*e - (2*a*e^6*(-a^3*c^5)^(
1/2))/c^3 - 24*a*c*d^3*e^3 + (14*d^2*e^4*(-a^3*c^5)^(1/2))/c^2 - (14*d^4*e^2*(-a^3*c^5)^(1/2))/(a*c)) - (48*a*
c^2*d^2*e^2*x*((d*e^3)/(4*c^2) - (d^3*e)/(4*a*c) - (d^4*(-a^3*c^5)^(1/2))/(16*a^3*c^3) - (e^4*(-a^3*c^5)^(1/2)
)/(16*a*c^5) + (3*d^2*e^2*(-a^3*c^5)^(1/2))/(8*a^2*c^4))^(1/2))/((2*d^6*(-a^3*c^5)^(1/2))/a^2 + 4*a^2*d*e^5 +
4*c^2*d^5*e - (2*a*e^6*(-a^3*c^5)^(1/2))/c^3 - 24*a*c*d^3*e^3 + (14*d^2*e^4*(-a^3*c^5)^(1/2))/c^2 - (14*d^4*e^
2*(-a^3*c^5)^(1/2))/(a*c)))*(-(a^2*e^4*(-a^3*c^5)^(1/2) + c^2*d^4*(-a^3*c^5)^(1/2) + 4*a^2*c^4*d^3*e - 4*a^3*c
^3*d*e^3 - 6*a*c*d^2*e^2*(-a^3*c^5)^(1/2))/(16*a^3*c^5))^(1/2)