3.52 \(\int \frac{d+e x}{\left (a+b x^2+c x^4\right )^3} \, dx\)

Optimal. Leaf size=474 \[ \frac{d x \left (3 b c x^2 \left (b^2-8 a c\right )+\left (b^2-7 a c\right ) \left (3 b^2-4 a c\right )\right )}{8 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{3 \sqrt{c} d \left (56 a^2 c^2-10 a b^2 c+b \left (b^2-8 a c\right ) \sqrt{b^2-4 a c}+b^4\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \sqrt{c} d \left (-\frac{56 a^2 c^2-10 a b^2 c+b^4}{\sqrt{b^2-4 a c}}-8 a b c+b^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^2 \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{6 c^2 e \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac{d x \left (-2 a c+b^2+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{3 c e \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{e \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]

[Out]

-(e*(b + 2*c*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (d*x*(b^2 - 2*a*c +
b*c*x^2))/(4*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (3*c*e*(b + 2*c*x^2))/(2*
(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (d*x*((b^2 - 7*a*c)*(3*b^2 - 4*a*c) + 3*b
*c*(b^2 - 8*a*c)*x^2))/(8*a^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (3*Sqrt[c]*
(b^4 - 10*a*b^2*c + 56*a^2*c^2 + b*(b^2 - 8*a*c)*Sqrt[b^2 - 4*a*c])*d*ArcTan[(Sq
rt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^2*(b^2 - 4*a*c)^(5/2
)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (3*Sqrt[c]*(b^3 - 8*a*b*c - (b^4 - 10*a*b^2*c +
56*a^2*c^2)/Sqrt[b^2 - 4*a*c])*d*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 -
4*a*c]]])/(8*Sqrt[2]*a^2*(b^2 - 4*a*c)^2*Sqrt[b + Sqrt[b^2 - 4*a*c]]) - (6*c^2*
e*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(5/2)

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Rubi [A]  time = 4.50264, antiderivative size = 474, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ \frac{d x \left (3 b c x^2 \left (b^2-8 a c\right )+\left (b^2-7 a c\right ) \left (3 b^2-4 a c\right )\right )}{8 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{3 \sqrt{c} d \left (56 a^2 c^2-10 a b^2 c+b \left (b^2-8 a c\right ) \sqrt{b^2-4 a c}+b^4\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \sqrt{c} d \left (-\frac{56 a^2 c^2-10 a b^2 c+b^4}{\sqrt{b^2-4 a c}}-8 a b c+b^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^2 \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{6 c^2 e \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac{d x \left (-2 a c+b^2+b c x^2\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{3 c e \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{e \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)/(a + b*x^2 + c*x^4)^3,x]

[Out]

-(e*(b + 2*c*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (d*x*(b^2 - 2*a*c +
b*c*x^2))/(4*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (3*c*e*(b + 2*c*x^2))/(2*
(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (d*x*((b^2 - 7*a*c)*(3*b^2 - 4*a*c) + 3*b
*c*(b^2 - 8*a*c)*x^2))/(8*a^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (3*Sqrt[c]*
(b^4 - 10*a*b^2*c + 56*a^2*c^2 + b*(b^2 - 8*a*c)*Sqrt[b^2 - 4*a*c])*d*ArcTan[(Sq
rt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^2*(b^2 - 4*a*c)^(5/2
)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (3*Sqrt[c]*(b^3 - 8*a*b*c - (b^4 - 10*a*b^2*c +
56*a^2*c^2)/Sqrt[b^2 - 4*a*c])*d*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 -
4*a*c]]])/(8*Sqrt[2]*a^2*(b^2 - 4*a*c)^2*Sqrt[b + Sqrt[b^2 - 4*a*c]]) - (6*c^2*
e*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(5/2)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)/(c*x**4+b*x**2+a)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 4.37103, size = 488, normalized size = 1.03 \[ \frac{1}{16} \left (\frac{8 a^2 c (3 b e+c x (7 d+6 e x))-2 a b c d x \left (25 b+24 c x^2\right )+6 b^3 d x \left (b+c x^2\right )}{a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{3 \sqrt{2} \sqrt{c} d \left (56 a^2 c^2-10 a b^2 c-8 a b c \sqrt{b^2-4 a c}+b^3 \sqrt{b^2-4 a c}+b^4\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{a^2 \left (b^2-4 a c\right )^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{3 \sqrt{2} \sqrt{c} d \left (56 a^2 c^2-10 a b^2 c+8 a b c \sqrt{b^2-4 a c}-b^3 \sqrt{b^2-4 a c}+b^4\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{a^2 \left (b^2-4 a c\right )^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{48 c^2 e \log \left (\sqrt{b^2-4 a c}-b-2 c x^2\right )}{\left (b^2-4 a c\right )^{5/2}}-\frac{48 c^2 e \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac{4 a b e+8 a c x (d+e x)-4 b d x \left (b+c x^2\right )}{a \left (4 a c-b^2\right ) \left (a+b x^2+c x^4\right )^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)/(a + b*x^2 + c*x^4)^3,x]

[Out]

((4*a*b*e + 8*a*c*x*(d + e*x) - 4*b*d*x*(b + c*x^2))/(a*(-b^2 + 4*a*c)*(a + b*x^
2 + c*x^4)^2) + (6*b^3*d*x*(b + c*x^2) - 2*a*b*c*d*x*(25*b + 24*c*x^2) + 8*a^2*c
*(3*b*e + c*x*(7*d + 6*e*x)))/(a^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (3*Sqr
t[2]*Sqrt[c]*(b^4 - 10*a*b^2*c + 56*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 8*a*b*c*Sq
rt[b^2 - 4*a*c])*d*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(a^2
*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (3*Sqrt[2]*Sqrt[c]*(b^4 - 10
*a*b^2*c + 56*a^2*c^2 - b^3*Sqrt[b^2 - 4*a*c] + 8*a*b*c*Sqrt[b^2 - 4*a*c])*d*Arc
Tan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(a^2*(b^2 - 4*a*c)^(5/2)*S
qrt[b + Sqrt[b^2 - 4*a*c]]) + (48*c^2*e*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(
b^2 - 4*a*c)^(5/2) - (48*c^2*e*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*
c)^(5/2))/16

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Maple [B]  time = 0.326, size = 3733, normalized size = 7.9 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)/(c*x^4+b*x^2+a)^3,x)

[Out]

3/16/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/
c)^2/a^2*d*x^3*(-4*a*c+b^2)^(1/2)*b^4+3*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^
2)*(-4*a*c+b^2)^(1/2)*e*ln(a^2*(2*c*x^2+(-4*a*c+b^2)^(1/2)+b))+3/16*c/(16*a^2*c^
2-8*a*b^2*c+b^4)/(4*a*c-b^2)*2^(1/2)/a^2/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan
(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*(-4*a*c+b^2)^(1/2)*b^4*d-15/8*c^2
/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)*2^(1/2)/a/((-b+(-4*a*c+b^2)^(1/2))*c)^(1
/2)*arctanh(c*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*(-4*a*c+b^2)^(1/2)*b^
2*d-15/8*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)*2^(1/2)/a/((b+(-4*a*c+b^2)^(
1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*(-4*a*c+b^2)
^(1/2)*b^2*d+3/16*c/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)*2^(1/2)/a^2/((-b+(-4*
a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*
(-4*a*c+b^2)^(1/2)*b^4*d-3/16*c/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)*2^(1/2)/a
^2/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c
)^(1/2))*b^5*d+15/8*c/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)/(x^2+1/2*b/c-1/2/c*
(-4*a*c+b^2)^(1/2))^2/a*d*x^3*(-4*a*c+b^2)^(1/2)*b^2+4*c/(16*a^2*c^2-8*a*b^2*c+b
^4)/(4*a*c-b^2)/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/c)^2*e*(-4*a*c+b^2)^(1/2)*a-
9/2*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)/(x^2+1/2*b/c-1/2/c*(-4*a*c+b^2)^(
1/2))^2*d*x^3*(-4*a*c+b^2)^(1/2)-6*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)/(x
^2+1/2*b/c-1/2/c*(-4*a*c+b^2)^(1/2))^2*d*x^3*b+6*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)/
(4*a*c-b^2)/(x^2+1/2*b/c-1/2/c*(-4*a*c+b^2)^(1/2))^2*e*x^2*a-3/2*c/(16*a^2*c^2-8
*a*b^2*c+b^4)/(4*a*c-b^2)/(x^2+1/2*b/c-1/2/c*(-4*a*c+b^2)^(1/2))^2*e*x^2*b^2-11*
c^2/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)/(x^2+1/2*b/c-1/2/c*(-4*a*c+b^2)^(1/2)
)^2*d*a*x+4*c/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)/(x^2+1/2*b/c-1/2/c*(-4*a*c+
b^2)^(1/2))^2*d*x*b^2-4*c/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)/(x^2+1/2*b/c-1/
2/c*(-4*a*c+b^2)^(1/2))^2*e*(-4*a*c+b^2)^(1/2)*a+3*c/(16*a^2*c^2-8*a*b^2*c+b^4)/
(4*a*c-b^2)/(x^2+1/2*b/c-1/2/c*(-4*a*c+b^2)^(1/2))^2*e*a*b+9/2*c^2/(16*a^2*c^2-8
*a*b^2*c+b^4)/(4*a*c-b^2)/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/c)^2*d*x^3*(-4*a*c
+b^2)^(1/2)-6*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)/(x^2+1/2/c*(-4*a*c+b^2)
^(1/2)+1/2*b/c)^2*d*x^3*b-3/16/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)/(x^2+1/2*b
/c-1/2/c*(-4*a*c+b^2)^(1/2))^2/a^2*d*x^3*b^5-5/16/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*
a*c-b^2)/(x^2+1/2*b/c-1/2/c*(-4*a*c+b^2)^(1/2))^2*d/a*x*b^4-3/16/(16*a^2*c^2-8*a
*b^2*c+b^4)/(4*a*c-b^2)/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/c)^2/a^2*d*x^3*b^5-5
/16/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/c
)^2*d/a*x*b^4+6*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)/(x^2+1/2/c*(-4*a*c+b^
2)^(1/2)+1/2*b/c)^2*e*x^2*a-3/2*c/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)/(x^2+1/
2/c*(-4*a*c+b^2)^(1/2)+1/2*b/c)^2*e*x^2*b^2+3*c/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*
c-b^2)/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/c)^2*e*a*b-11*c^2/(16*a^2*c^2-8*a*b^2
*c+b^4)/(4*a*c-b^2)/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/c)^2*d*a*x+4*c/(16*a^2*c
^2-8*a*b^2*c+b^4)/(4*a*c-b^2)/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/c)^2*d*x*b^2-9
/4*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)*2^(1/2)/a/((-b+(-4*a*c+b^2)^(1/2))
*c)^(1/2)*arctanh(c*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^3*d+3/16*c/(1
6*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)*2^(1/2)/a^2/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/
2)*arctanh(c*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*b^5*d-15/8*c/(16*a^2*c
^2-8*a*b^2*c+b^4)/(4*a*c-b^2)/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/c)^2/a*d*x^3*(
-4*a*c+b^2)^(1/2)*b^2+9/4*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)*2^(1/2)/a/(
(b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1
/2))*b^3*d-3/4/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)/(x^2+1/2/c*(-4*a*c+b^2)^(1
/2)+1/2*b/c)^2*e*b^3-1/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)/(x^2+1/2/c*(-4*a*c
+b^2)^(1/2)+1/2*b/c)^2*e*(-4*a*c+b^2)^(1/2)*b^2+1/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*
a*c-b^2)/(x^2+1/2*b/c-1/2/c*(-4*a*c+b^2)^(1/2))^2*e*(-4*a*c+b^2)^(1/2)*b^2-3/4/(
16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)/(x^2+1/2*b/c-1/2/c*(-4*a*c+b^2)^(1/2))^2*e
*b^3+5/16/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1
/2*b/c)^2*d/a*x*(-4*a*c+b^2)^(1/2)*b^3-3/16/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^
2)/(x^2+1/2*b/c-1/2/c*(-4*a*c+b^2)^(1/2))^2/a^2*d*x^3*(-4*a*c+b^2)^(1/2)*b^4-5/1
6/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)/(x^2+1/2*b/c-1/2/c*(-4*a*c+b^2)^(1/2))^
2*d/a*x*(-4*a*c+b^2)^(1/2)*b^3+6*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)*2^(1
/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2
))*c)^(1/2))*b*d+21/2*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)*2^(1/2)/((-b+(-
4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)
)*(-4*a*c+b^2)^(1/2)*d+9/4*c/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)/(x^2+1/2*b/c
-1/2/c*(-4*a*c+b^2)^(1/2))^2/a*d*x^3*b^3+5/4*c/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c
-b^2)/(x^2+1/2*b/c-1/2/c*(-4*a*c+b^2)^(1/2))^2*d*x*(-4*a*c+b^2)^(1/2)*b+9/4*c/(1
6*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)/(x^2+1/2/c*(-4*a*c+b^2)^(1/2)+1/2*b/c)^2/a*
d*x^3*b^3-5/4*c/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)/(x^2+1/2/c*(-4*a*c+b^2)^(
1/2)+1/2*b/c)^2*d*x*(-4*a*c+b^2)^(1/2)*b+21/2*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*
a*c-b^2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c
+b^2)^(1/2))*c)^(1/2))*(-4*a*c+b^2)^(1/2)*d-6*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*
a*c-b^2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c
+b^2)^(1/2))*c)^(1/2))*b*d-3*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)*(-4*a*c+
b^2)^(1/2)*e*ln(a^2*(-2*c*x^2+(-4*a*c+b^2)^(1/2)-b))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{24 \, a^{2} c^{3} e x^{6} + 36 \, a^{2} b c^{2} e x^{4} + 3 \,{\left (b^{3} c^{2} - 8 \, a b c^{3}\right )} d x^{7} +{\left (6 \, b^{4} c - 49 \, a b^{2} c^{2} + 28 \, a^{2} c^{3}\right )} d x^{5} +{\left (3 \, b^{5} - 20 \, a b^{3} c - 4 \, a^{2} b c^{2}\right )} d x^{3} + 8 \,{\left (a^{2} b^{2} c + 5 \, a^{3} c^{2}\right )} e x^{2} +{\left (5 \, a b^{4} - 37 \, a^{2} b^{2} c + 44 \, a^{3} c^{2}\right )} d x - 2 \,{\left (a^{2} b^{3} - 10 \, a^{3} b c\right )} e}{8 \,{\left ({\left (a^{2} b^{4} c^{2} - 8 \, a^{3} b^{2} c^{3} + 16 \, a^{4} c^{4}\right )} x^{8} + a^{4} b^{4} - 8 \, a^{5} b^{2} c + 16 \, a^{6} c^{2} + 2 \,{\left (a^{2} b^{5} c - 8 \, a^{3} b^{3} c^{2} + 16 \, a^{4} b c^{3}\right )} x^{6} +{\left (a^{2} b^{6} - 6 \, a^{3} b^{4} c + 32 \, a^{5} c^{3}\right )} x^{4} + 2 \,{\left (a^{3} b^{5} - 8 \, a^{4} b^{3} c + 16 \, a^{5} b c^{2}\right )} x^{2}\right )}} - \frac{-3 \, \int \frac{16 \, a^{2} c^{2} e x +{\left (b^{3} c - 8 \, a b c^{2}\right )} d x^{2} +{\left (b^{4} - 9 \, a b^{2} c + 28 \, a^{2} c^{2}\right )} d}{c x^{4} + b x^{2} + a}\,{d x}}{8 \,{\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/(c*x^4 + b*x^2 + a)^3,x, algorithm="maxima")

[Out]

1/8*(24*a^2*c^3*e*x^6 + 36*a^2*b*c^2*e*x^4 + 3*(b^3*c^2 - 8*a*b*c^3)*d*x^7 + (6*
b^4*c - 49*a*b^2*c^2 + 28*a^2*c^3)*d*x^5 + (3*b^5 - 20*a*b^3*c - 4*a^2*b*c^2)*d*
x^3 + 8*(a^2*b^2*c + 5*a^3*c^2)*e*x^2 + (5*a*b^4 - 37*a^2*b^2*c + 44*a^3*c^2)*d*
x - 2*(a^2*b^3 - 10*a^3*b*c)*e)/((a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*x^8
+ a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2 + 2*(a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c
^3)*x^6 + (a^2*b^6 - 6*a^3*b^4*c + 32*a^5*c^3)*x^4 + 2*(a^3*b^5 - 8*a^4*b^3*c +
16*a^5*b*c^2)*x^2) - 3/8*integrate(-(16*a^2*c^2*e*x + (b^3*c - 8*a*b*c^2)*d*x^2
+ (b^4 - 9*a*b^2*c + 28*a^2*c^2)*d)/(c*x^4 + b*x^2 + a), x)/(a^2*b^4 - 8*a^3*b^2
*c + 16*a^4*c^2)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/(c*x^4 + b*x^2 + a)^3,x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)/(c*x**4+b*x**2+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 34.8359, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)/(c*x^4 + b*x^2 + a)^3,x, algorithm="giac")

[Out]

Done