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

Optimal. Leaf size=253 \[ \frac {\left (\sqrt {a} d+\sqrt {c} e\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} c^{5/4}}-\frac {\left (\sqrt {a} d+\sqrt {c} e\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} c^{5/4}}+\frac {\left (\sqrt {a} d-\sqrt {c} e\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} c^{5/4}}-\frac {\left (\sqrt {a} d-\sqrt {c} e\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} \sqrt [4]{a} c^{5/4}}+\frac {d x}{c} \]

[Out]

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

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Rubi [A]  time = 0.21, antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {1394, 1280, 1168, 1162, 617, 204, 1165, 628} \[ \frac {\left (\sqrt {a} d+\sqrt {c} e\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} c^{5/4}}-\frac {\left (\sqrt {a} d+\sqrt {c} e\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} c^{5/4}}+\frac {\left (\sqrt {a} d-\sqrt {c} e\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} c^{5/4}}-\frac {\left (\sqrt {a} d-\sqrt {c} e\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} \sqrt [4]{a} c^{5/4}}+\frac {d x}{c} \]

Antiderivative was successfully verified.

[In]

Int[(d + e/x^2)/(c + a/x^4),x]

[Out]

(d*x)/c + ((Sqrt[a]*d - Sqrt[c]*e)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(1/4)*c^(5/4)) - ((Sq
rt[a]*d - Sqrt[c]*e)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(1/4)*c^(5/4)) + ((Sqrt[a]*d + Sqrt
[c]*e)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(1/4)*c^(5/4)) - ((Sqrt[a]*d + Sqr
t[c]*e)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(1/4)*c^(5/4))

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 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 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]

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 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 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 1280

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(e*f*(f*x)^(m - 1)*
(a + c*x^4)^(p + 1))/(c*(m + 4*p + 3)), x] - Dist[f^2/(c*(m + 4*p + 3)), Int[(f*x)^(m - 2)*(a + c*x^4)^p*(a*e*
(m - 1) - c*d*(m + 4*p + 3)*x^2), x], x] /; FreeQ[{a, c, d, e, f, p}, x] && GtQ[m, 1] && NeQ[m + 4*p + 3, 0] &
& IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

Rule 1394

Int[((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[x^(n*(2*p + q))*(e + d/x
^n)^q*(c + a/x^(2*n))^p, x] /; FreeQ[{a, c, d, e, n}, x] && EqQ[n2, 2*n] && IntegersQ[p, q] && NegQ[n]

Rubi steps

\begin {align*} \int \frac {d+\frac {e}{x^2}}{c+\frac {a}{x^4}} \, dx &=\int \frac {x^2 \left (e+d x^2\right )}{a+c x^4} \, dx\\ &=\frac {d x}{c}-\frac {\int \frac {a d-c e x^2}{a+c x^4} \, dx}{c}\\ &=\frac {d x}{c}-\frac {\left (\frac {\sqrt {a} d}{\sqrt {c}}-e\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{2 c}-\frac {\left (\frac {\sqrt {a} d}{\sqrt {c}}+e\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{2 c}\\ &=\frac {d x}{c}-\frac {\left (\frac {\sqrt {a} d}{\sqrt {c}}-e\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 c}-\frac {\left (\frac {\sqrt {a} d}{\sqrt {c}}-e\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 c}+\frac {\left (\sqrt {a} d+\sqrt {c} 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}{4 \sqrt {2} \sqrt [4]{a} c^{5/4}}+\frac {\left (\sqrt {a} d+\sqrt {c} 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}{4 \sqrt {2} \sqrt [4]{a} c^{5/4}}\\ &=\frac {d x}{c}+\frac {\left (\sqrt {a} d+\sqrt {c} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} c^{5/4}}-\frac {\left (\sqrt {a} d+\sqrt {c} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} c^{5/4}}-\frac {\left (\sqrt {a} d-\sqrt {c} 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 )}{2 \sqrt {2} \sqrt [4]{a} c^{5/4}}+\frac {\left (\sqrt {a} d-\sqrt {c} 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 )}{2 \sqrt {2} \sqrt [4]{a} c^{5/4}}\\ &=\frac {d x}{c}+\frac {\left (\sqrt {a} d-\sqrt {c} e\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} c^{5/4}}-\frac {\left (\sqrt {a} d-\sqrt {c} e\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} c^{5/4}}+\frac {\left (\sqrt {a} d+\sqrt {c} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} c^{5/4}}-\frac {\left (\sqrt {a} d+\sqrt {c} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} c^{5/4}}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 293, normalized size = 1.16 \[ \frac {\left (a^{5/4} \sqrt {c} d+a^{3/4} c e\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a c^{7/4}}-\frac {\left (a^{5/4} \sqrt {c} d+a^{3/4} c e\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a c^{7/4}}+\frac {\left (a^{3/4} c e-a^{5/4} \sqrt {c} d\right ) \tan ^{-1}\left (\frac {2 \sqrt [4]{c} x-\sqrt {2} \sqrt [4]{a}}{\sqrt {2} \sqrt [4]{a}}\right )}{2 \sqrt {2} a c^{7/4}}+\frac {\left (a^{3/4} c e-a^{5/4} \sqrt {c} d\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a}+2 \sqrt [4]{c} x}{\sqrt {2} \sqrt [4]{a}}\right )}{2 \sqrt {2} a c^{7/4}}+\frac {d x}{c} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e/x^2)/(c + a/x^4),x]

[Out]

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

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fricas [B]  time = 0.88, size = 754, normalized size = 2.98 \[ \frac {c \sqrt {\frac {c^{2} \sqrt {-\frac {a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} + 2 \, d e}{c^{2}}} \log \left (-{\left (a^{2} d^{4} - c^{2} e^{4}\right )} x + {\left (a c^{4} e \sqrt {-\frac {a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} + a^{2} c d^{3} - a c^{2} d e^{2}\right )} \sqrt {\frac {c^{2} \sqrt {-\frac {a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} + 2 \, d e}{c^{2}}}\right ) - c \sqrt {\frac {c^{2} \sqrt {-\frac {a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} + 2 \, d e}{c^{2}}} \log \left (-{\left (a^{2} d^{4} - c^{2} e^{4}\right )} x - {\left (a c^{4} e \sqrt {-\frac {a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} + a^{2} c d^{3} - a c^{2} d e^{2}\right )} \sqrt {\frac {c^{2} \sqrt {-\frac {a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} + 2 \, d e}{c^{2}}}\right ) - c \sqrt {-\frac {c^{2} \sqrt {-\frac {a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} - 2 \, d e}{c^{2}}} \log \left (-{\left (a^{2} d^{4} - c^{2} e^{4}\right )} x + {\left (a c^{4} e \sqrt {-\frac {a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} - a^{2} c d^{3} + a c^{2} d e^{2}\right )} \sqrt {-\frac {c^{2} \sqrt {-\frac {a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} - 2 \, d e}{c^{2}}}\right ) + c \sqrt {-\frac {c^{2} \sqrt {-\frac {a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} - 2 \, d e}{c^{2}}} \log \left (-{\left (a^{2} d^{4} - c^{2} e^{4}\right )} x - {\left (a c^{4} e \sqrt {-\frac {a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} - a^{2} c d^{3} + a c^{2} d e^{2}\right )} \sqrt {-\frac {c^{2} \sqrt {-\frac {a^{2} d^{4} - 2 \, a c d^{2} e^{2} + c^{2} e^{4}}{a c^{5}}} - 2 \, d e}{c^{2}}}\right ) + 4 \, d x}{4 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(c*sqrt((c^2*sqrt(-(a^2*d^4 - 2*a*c*d^2*e^2 + c^2*e^4)/(a*c^5)) + 2*d*e)/c^2)*log(-(a^2*d^4 - c^2*e^4)*x +
(a*c^4*e*sqrt(-(a^2*d^4 - 2*a*c*d^2*e^2 + c^2*e^4)/(a*c^5)) + a^2*c*d^3 - a*c^2*d*e^2)*sqrt((c^2*sqrt(-(a^2*d
^4 - 2*a*c*d^2*e^2 + c^2*e^4)/(a*c^5)) + 2*d*e)/c^2)) - c*sqrt((c^2*sqrt(-(a^2*d^4 - 2*a*c*d^2*e^2 + c^2*e^4)/
(a*c^5)) + 2*d*e)/c^2)*log(-(a^2*d^4 - c^2*e^4)*x - (a*c^4*e*sqrt(-(a^2*d^4 - 2*a*c*d^2*e^2 + c^2*e^4)/(a*c^5)
) + a^2*c*d^3 - a*c^2*d*e^2)*sqrt((c^2*sqrt(-(a^2*d^4 - 2*a*c*d^2*e^2 + c^2*e^4)/(a*c^5)) + 2*d*e)/c^2)) - c*s
qrt(-(c^2*sqrt(-(a^2*d^4 - 2*a*c*d^2*e^2 + c^2*e^4)/(a*c^5)) - 2*d*e)/c^2)*log(-(a^2*d^4 - c^2*e^4)*x + (a*c^4
*e*sqrt(-(a^2*d^4 - 2*a*c*d^2*e^2 + c^2*e^4)/(a*c^5)) - a^2*c*d^3 + a*c^2*d*e^2)*sqrt(-(c^2*sqrt(-(a^2*d^4 - 2
*a*c*d^2*e^2 + c^2*e^4)/(a*c^5)) - 2*d*e)/c^2)) + c*sqrt(-(c^2*sqrt(-(a^2*d^4 - 2*a*c*d^2*e^2 + c^2*e^4)/(a*c^
5)) - 2*d*e)/c^2)*log(-(a^2*d^4 - c^2*e^4)*x - (a*c^4*e*sqrt(-(a^2*d^4 - 2*a*c*d^2*e^2 + c^2*e^4)/(a*c^5)) - a
^2*c*d^3 + a*c^2*d*e^2)*sqrt(-(c^2*sqrt(-(a^2*d^4 - 2*a*c*d^2*e^2 + c^2*e^4)/(a*c^5)) - 2*d*e)/c^2)) + 4*d*x)/
c

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giac [A]  time = 0.35, size = 247, normalized size = 0.98 \[ \frac {d x}{c} - \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} a c d - \left (a c^{3}\right )^{\frac {3}{4}} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{4 \, a c^{3}} - \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} a c d - \left (a c^{3}\right )^{\frac {3}{4}} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{4 \, a c^{3}} - \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} a c d + \left (a c^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{8 \, a c^{3}} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} a c d + \left (a c^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{8 \, a c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 266, normalized size = 1.05 \[ \frac {d x}{c}-\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, d \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{4 c}-\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, d \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{4 c}-\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, d \ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}\right )}{8 c}+\frac {\sqrt {2}\, e \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{4 \left (\frac {a}{c}\right )^{\frac {1}{4}} c}+\frac {\sqrt {2}\, e \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{4 \left (\frac {a}{c}\right )^{\frac {1}{4}} c}+\frac {\sqrt {2}\, e \ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}\right )}{8 \left (\frac {a}{c}\right )^{\frac {1}{4}} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.30, size = 240, normalized size = 0.95 \[ \frac {d x}{c} - \frac {\frac {2 \, \sqrt {2} {\left (a \sqrt {c} d - \sqrt {a} c e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} {\left (a \sqrt {c} d - \sqrt {a} c e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {\sqrt {2} {\left (a \sqrt {c} d + \sqrt {a} c e\right )} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (a \sqrt {c} d + \sqrt {a} c e\right )} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}}}{8 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.31, size = 555, normalized size = 2.19 \[ \frac {d\,x}{c}-2\,\mathrm {atanh}\left (\frac {8\,a^2\,c\,d^2\,x\,\sqrt {\frac {d^2\,\sqrt {-a\,c^5}}{16\,c^5}+\frac {d\,e}{8\,c^2}-\frac {e^2\,\sqrt {-a\,c^5}}{16\,a\,c^4}}}{2\,a^2\,d^2\,e-2\,a\,c\,e^3+\frac {2\,a^2\,d^3\,\sqrt {-a\,c^5}}{c^3}-\frac {2\,a\,d\,e^2\,\sqrt {-a\,c^5}}{c^2}}-\frac {8\,a\,c^2\,e^2\,x\,\sqrt {\frac {d^2\,\sqrt {-a\,c^5}}{16\,c^5}+\frac {d\,e}{8\,c^2}-\frac {e^2\,\sqrt {-a\,c^5}}{16\,a\,c^4}}}{2\,a^2\,d^2\,e-2\,a\,c\,e^3+\frac {2\,a^2\,d^3\,\sqrt {-a\,c^5}}{c^3}-\frac {2\,a\,d\,e^2\,\sqrt {-a\,c^5}}{c^2}}\right )\,\sqrt {\frac {a\,d^2\,\sqrt {-a\,c^5}-c\,e^2\,\sqrt {-a\,c^5}+2\,a\,c^3\,d\,e}{16\,a\,c^5}}-2\,\mathrm {atanh}\left (\frac {8\,a^2\,c\,d^2\,x\,\sqrt {\frac {d\,e}{8\,c^2}-\frac {d^2\,\sqrt {-a\,c^5}}{16\,c^5}+\frac {e^2\,\sqrt {-a\,c^5}}{16\,a\,c^4}}}{2\,a^2\,d^2\,e-2\,a\,c\,e^3-\frac {2\,a^2\,d^3\,\sqrt {-a\,c^5}}{c^3}+\frac {2\,a\,d\,e^2\,\sqrt {-a\,c^5}}{c^2}}-\frac {8\,a\,c^2\,e^2\,x\,\sqrt {\frac {d\,e}{8\,c^2}-\frac {d^2\,\sqrt {-a\,c^5}}{16\,c^5}+\frac {e^2\,\sqrt {-a\,c^5}}{16\,a\,c^4}}}{2\,a^2\,d^2\,e-2\,a\,c\,e^3-\frac {2\,a^2\,d^3\,\sqrt {-a\,c^5}}{c^3}+\frac {2\,a\,d\,e^2\,\sqrt {-a\,c^5}}{c^2}}\right )\,\sqrt {\frac {c\,e^2\,\sqrt {-a\,c^5}-a\,d^2\,\sqrt {-a\,c^5}+2\,a\,c^3\,d\,e}{16\,a\,c^5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e/x^2)/(c + a/x^4),x)

[Out]

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

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sympy [A]  time = 0.70, size = 109, normalized size = 0.43 \[ \operatorname {RootSum} {\left (256 t^{4} a c^{5} - 64 t^{2} a c^{3} d e + a^{2} d^{4} + 2 a c d^{2} e^{2} + c^{2} e^{4}, \left (t \mapsto t \log {\left (x + \frac {- 64 t^{3} a c^{4} e - 4 t a^{2} c d^{3} + 12 t a c^{2} d e^{2}}{a^{2} d^{4} - c^{2} e^{4}} \right )} \right )\right )} + \frac {d x}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(256*_t**4*a*c**5 - 64*_t**2*a*c**3*d*e + a**2*d**4 + 2*a*c*d**2*e**2 + c**2*e**4, Lambda(_t, _t*log(x
+ (-64*_t**3*a*c**4*e - 4*_t*a**2*c*d**3 + 12*_t*a*c**2*d*e**2)/(a**2*d**4 - c**2*e**4)))) + d*x/c

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