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

Optimal. Leaf size=275 \[ -\frac{\left (3 \sqrt{c} d-\sqrt{a} e\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{7/4} c^{3/4}}+\frac{\left (3 \sqrt{c} d-\sqrt{a} e\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{7/4} c^{3/4}}-\frac{\left (\sqrt{a} e+3 \sqrt{c} d\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} c^{3/4}}+\frac{\left (\sqrt{a} e+3 \sqrt{c} d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} c^{3/4}}+\frac{x \left (d+e x^2\right )}{4 a \left (a+c x^4\right )} \]

[Out]

(x*(d + e*x^2))/(4*a*(a + c*x^4)) - ((3*Sqrt[c]*d + Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqr
t[2]*a^(7/4)*c^(3/4)) + ((3*Sqrt[c]*d + Sqrt[a]*e)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)
*c^(3/4)) - ((3*Sqrt[c]*d - Sqrt[a]*e)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(
7/4)*c^(3/4)) + ((3*Sqrt[c]*d - Sqrt[a]*e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]
*a^(7/4)*c^(3/4))

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Rubi [A]  time = 0.203391, antiderivative size = 275, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412, Rules used = {1179, 1168, 1162, 617, 204, 1165, 628} \[ -\frac{\left (3 \sqrt{c} d-\sqrt{a} e\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{7/4} c^{3/4}}+\frac{\left (3 \sqrt{c} d-\sqrt{a} e\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} a^{7/4} c^{3/4}}-\frac{\left (\sqrt{a} e+3 \sqrt{c} d\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} c^{3/4}}+\frac{\left (\sqrt{a} e+3 \sqrt{c} d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} a^{7/4} c^{3/4}}+\frac{x \left (d+e x^2\right )}{4 a \left (a+c x^4\right )} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)/(a + c*x^4)^2,x]

[Out]

(x*(d + e*x^2))/(4*a*(a + c*x^4)) - ((3*Sqrt[c]*d + Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqr
t[2]*a^(7/4)*c^(3/4)) + ((3*Sqrt[c]*d + Sqrt[a]*e)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)
*c^(3/4)) - ((3*Sqrt[c]*d - Sqrt[a]*e)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(
7/4)*c^(3/4)) + ((3*Sqrt[c]*d - Sqrt[a]*e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]
*a^(7/4)*c^(3/4))

Rule 1179

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 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 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 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 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 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 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]

Rubi steps

\begin{align*} \int \frac{d+e x^2}{\left (a+c x^4\right )^2} \, dx &=\frac{x \left (d+e x^2\right )}{4 a \left (a+c x^4\right )}-\frac{\int \frac{-3 d-e x^2}{a+c x^4} \, dx}{4 a}\\ &=\frac{x \left (d+e x^2\right )}{4 a \left (a+c x^4\right )}+\frac{\left (\frac{3 \sqrt{c} d}{\sqrt{a}}-e\right ) \int \frac{\sqrt{a} \sqrt{c}-c x^2}{a+c x^4} \, dx}{8 a c}+\frac{\left (\frac{3 \sqrt{c} d}{\sqrt{a}}+e\right ) \int \frac{\sqrt{a} \sqrt{c}+c x^2}{a+c x^4} \, dx}{8 a c}\\ &=\frac{x \left (d+e x^2\right )}{4 a \left (a+c x^4\right )}+\frac{\left (\frac{3 \sqrt{c} d}{\sqrt{a}}+e\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 a c}+\frac{\left (\frac{3 \sqrt{c} d}{\sqrt{a}}+e\right ) \int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 a c}-\frac{\left (3 \sqrt{c} d-\sqrt{a} e\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^{3/4}}-\frac{\left (3 \sqrt{c} d-\sqrt{a} e\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^{3/4}}\\ &=\frac{x \left (d+e x^2\right )}{4 a \left (a+c x^4\right )}-\frac{\left (3 \sqrt{c} d-\sqrt{a} e\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^{3/4}}+\frac{\left (3 \sqrt{c} d-\sqrt{a} e\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^{3/4}}+\frac{\left (3 \sqrt{c} d+\sqrt{a} e\right ) \operatorname{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^{3/4}}-\frac{\left (3 \sqrt{c} d+\sqrt{a} e\right ) \operatorname{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^{3/4}}\\ &=\frac{x \left (d+e x^2\right )}{4 a \left (a+c x^4\right )}-\frac{\left (3 \sqrt{c} d+\sqrt{a} e\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} c^{3/4}}+\frac{\left (3 \sqrt{c} d+\sqrt{a} e\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} a^{7/4} c^{3/4}}-\frac{\left (3 \sqrt{c} d-\sqrt{a} e\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^{3/4}}+\frac{\left (3 \sqrt{c} d-\sqrt{a} e\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^{3/4}}\\ \end{align*}

Mathematica [A]  time = 0.287757, size = 267, normalized size = 0.97 \[ \frac{\frac{\sqrt{2} \left (a^{3/4} e-3 \sqrt [4]{a} \sqrt{c} d\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{c^{3/4}}+\frac{\sqrt{2} \left (3 \sqrt [4]{a} \sqrt{c} d-a^{3/4} e\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{c^{3/4}}-\frac{2 \sqrt{2} \sqrt [4]{a} \left (\sqrt{a} e+3 \sqrt{c} d\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{c^{3/4}}+\frac{2 \sqrt{2} \sqrt [4]{a} \left (\sqrt{a} e+3 \sqrt{c} d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{c^{3/4}}+\frac{8 a x \left (d+e x^2\right )}{a+c x^4}}{32 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.05, size = 303, normalized size = 1.1 \begin{align*}{\frac{dx}{4\,a \left ( c{x}^{4}+a \right ) }}+{\frac{3\,d\sqrt{2}}{32\,{a}^{2}}\sqrt [4]{{\frac{a}{c}}}\ln \left ({ \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }+{\frac{3\,d\sqrt{2}}{16\,{a}^{2}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{3\,d\sqrt{2}}{16\,{a}^{2}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }+{\frac{e{x}^{3}}{4\,a \left ( c{x}^{4}+a \right ) }}+{\frac{e\sqrt{2}}{32\,ac}\ln \left ({ \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{e\sqrt{2}}{16\,ac}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{e\sqrt{2}}{16\,ac}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.78671, size = 1825, normalized size = 6.64 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(4*e*x^3 - (a*c*x^4 + a^2)*sqrt(-(a^3*c*sqrt(-(81*c^2*d^4 - 18*a*c*d^2*e^2 + a^2*e^4)/(a^7*c^3)) + 6*d*e)
/(a^3*c))*log(-(81*c^2*d^4 - a^2*e^4)*x + (a^6*c^2*e*sqrt(-(81*c^2*d^4 - 18*a*c*d^2*e^2 + a^2*e^4)/(a^7*c^3))
+ 27*a^2*c^2*d^3 - 3*a^3*c*d*e^2)*sqrt(-(a^3*c*sqrt(-(81*c^2*d^4 - 18*a*c*d^2*e^2 + a^2*e^4)/(a^7*c^3)) + 6*d*
e)/(a^3*c))) + (a*c*x^4 + a^2)*sqrt(-(a^3*c*sqrt(-(81*c^2*d^4 - 18*a*c*d^2*e^2 + a^2*e^4)/(a^7*c^3)) + 6*d*e)/
(a^3*c))*log(-(81*c^2*d^4 - a^2*e^4)*x - (a^6*c^2*e*sqrt(-(81*c^2*d^4 - 18*a*c*d^2*e^2 + a^2*e^4)/(a^7*c^3)) +
27*a^2*c^2*d^3 - 3*a^3*c*d*e^2)*sqrt(-(a^3*c*sqrt(-(81*c^2*d^4 - 18*a*c*d^2*e^2 + a^2*e^4)/(a^7*c^3)) + 6*d*e
)/(a^3*c))) + (a*c*x^4 + a^2)*sqrt((a^3*c*sqrt(-(81*c^2*d^4 - 18*a*c*d^2*e^2 + a^2*e^4)/(a^7*c^3)) - 6*d*e)/(a
^3*c))*log(-(81*c^2*d^4 - a^2*e^4)*x + (a^6*c^2*e*sqrt(-(81*c^2*d^4 - 18*a*c*d^2*e^2 + a^2*e^4)/(a^7*c^3)) - 2
7*a^2*c^2*d^3 + 3*a^3*c*d*e^2)*sqrt((a^3*c*sqrt(-(81*c^2*d^4 - 18*a*c*d^2*e^2 + a^2*e^4)/(a^7*c^3)) - 6*d*e)/(
a^3*c))) - (a*c*x^4 + a^2)*sqrt((a^3*c*sqrt(-(81*c^2*d^4 - 18*a*c*d^2*e^2 + a^2*e^4)/(a^7*c^3)) - 6*d*e)/(a^3*
c))*log(-(81*c^2*d^4 - a^2*e^4)*x - (a^6*c^2*e*sqrt(-(81*c^2*d^4 - 18*a*c*d^2*e^2 + a^2*e^4)/(a^7*c^3)) - 27*a
^2*c^2*d^3 + 3*a^3*c*d*e^2)*sqrt((a^3*c*sqrt(-(81*c^2*d^4 - 18*a*c*d^2*e^2 + a^2*e^4)/(a^7*c^3)) - 6*d*e)/(a^3
*c))) + 4*d*x)/(a*c*x^4 + a^2)

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Sympy [A]  time = 1.16455, size = 136, normalized size = 0.49 \begin{align*} \operatorname{RootSum}{\left (65536 t^{4} a^{7} c^{3} + 3072 t^{2} a^{4} c^{2} d e + a^{2} e^{4} + 18 a c d^{2} e^{2} + 81 c^{2} d^{4}, \left ( t \mapsto t \log{\left (x + \frac{4096 t^{3} a^{6} c^{2} e + 144 t a^{3} c d e^{2} - 432 t a^{2} c^{2} d^{3}}{a^{2} e^{4} - 81 c^{2} d^{4}} \right )} \right )\right )} + \frac{d x + e x^{3}}{4 a^{2} + 4 a c x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(65536*_t**4*a**7*c**3 + 3072*_t**2*a**4*c**2*d*e + a**2*e**4 + 18*a*c*d**2*e**2 + 81*c**2*d**4, Lambda
(_t, _t*log(x + (4096*_t**3*a**6*c**2*e + 144*_t*a**3*c*d*e**2 - 432*_t*a**2*c**2*d**3)/(a**2*e**4 - 81*c**2*d
**4)))) + (d*x + e*x**3)/(4*a**2 + 4*a*c*x**4)

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Giac [A]  time = 1.12612, size = 369, normalized size = 1.34 \begin{align*} \frac{x^{3} e + d x}{4 \,{\left (c x^{4} + a\right )} a} + \frac{\sqrt{2}{\left (3 \, \left (a c^{3}\right )^{\frac{1}{4}} c^{2} d + \left (a c^{3}\right )^{\frac{3}{4}} 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 + \left (a c^{3}\right )^{\frac{3}{4}} 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 - \left (a c^{3}\right )^{\frac{3}{4}} 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 - \left (a c^{3}\right )^{\frac{3}{4}} 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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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