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

Optimal. Leaf size=679 \[ \frac{x \left (c x^2 \left (20 a^2 c f+a b^2 f-12 a b (a h+2 c d)+3 b^3 d\right )+8 a^2 b c f+4 a^2 c (a h+7 c d)+a b^3 f-a b^2 (7 a h+25 c d)+3 b^4 d\right )}{8 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (\frac{-52 a^2 b c f+24 a^2 c (a h+7 c d)+a b^3 f-6 a b^2 (5 c d-3 a h)+3 b^4 d}{\sqrt{b^2-4 a c}}+20 a^2 c f+a b^2 f-12 a b (a h+2 c d)+3 b^3 d\right )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^2 \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (-\frac{-52 a^2 b c f+24 a^2 c (a h+7 c d)+a b^3 f-6 a b^2 (5 c d-3 a h)+3 b^4 d}{\sqrt{b^2-4 a c}}+20 a^2 c f+a b^2 f-12 a b (a h+2 c d)+3 b^3 d\right )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^2 \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{x \left (x^2 (a b h-2 a c f+b c d)-a b f-2 a (c d-a h)+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac{3 c (2 c e-b g) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac{3 \left (b+2 c x^2\right ) (2 c e-b g)}{4 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{-2 a g+x^2 (2 c e-b g)+b e}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]

[Out]

-(b*e - 2*a*g + (2*c*e - b*g)*x^2)/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (x*
(b^2*d - a*b*f - 2*a*(c*d - a*h) + (b*c*d - 2*a*c*f + a*b*h)*x^2))/(4*a*(b^2 - 4
*a*c)*(a + b*x^2 + c*x^4)^2) + (3*(2*c*e - b*g)*(b + 2*c*x^2))/(4*(b^2 - 4*a*c)^
2*(a + b*x^2 + c*x^4)) + (x*(3*b^4*d + a*b^3*f + 8*a^2*b*c*f + 4*a^2*c*(7*c*d +
a*h) - a*b^2*(25*c*d + 7*a*h) + c*(3*b^3*d + a*b^2*f + 20*a^2*c*f - 12*a*b*(2*c*
d + a*h))*x^2))/(8*a^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (Sqrt[c]*(3*b^3*d
+ a*b^2*f + 20*a^2*c*f - 12*a*b*(2*c*d + a*h) + (3*b^4*d + a*b^3*f - 52*a^2*b*c*
f - 6*a*b^2*(5*c*d - 3*a*h) + 24*a^2*c*(7*c*d + a*h))/Sqrt[b^2 - 4*a*c])*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]]) + (Sqrt[c]*(3*b^3*d + a*b^2*f + 20*a^2*c*f - 12*a*
b*(2*c*d + a*h) - (3*b^4*d + a*b^3*f - 52*a^2*b*c*f - 6*a*b^2*(5*c*d - 3*a*h) +
24*a^2*c*(7*c*d + a*h))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + S
qrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^2*(b^2 - 4*a*c)^2*Sqrt[b + Sqrt[b^2 - 4*a*c]])
- (3*c*(2*c*e - b*g)*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 = 10.0953, antiderivative size = 679, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{x \left (c x^2 \left (20 a^2 c f+a b^2 f-12 a b (a h+2 c d)+3 b^3 d\right )+8 a^2 b c f+4 a^2 c (a h+7 c d)+a b^3 f-a b^2 (7 a h+25 c d)+3 b^4 d\right )}{8 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right ) \left (\frac{-52 a^2 b c f+24 a^2 c (a h+7 c d)+a b^3 f-6 a b^2 (5 c d-3 a h)+3 b^4 d}{\sqrt{b^2-4 a c}}+20 a^2 c f+a b^2 f-12 a b (a h+2 c d)+3 b^3 d\right )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^2 \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right ) \left (-\frac{-52 a^2 b c f+24 a^2 c (a h+7 c d)+a b^3 f-6 a b^2 (5 c d-3 a h)+3 b^4 d}{\sqrt{b^2-4 a c}}+20 a^2 c f+a b^2 f-12 a b (a h+2 c d)+3 b^3 d\right )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^2 \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{x \left (x^2 (a b h-2 a c f+b c d)-a b f-2 a (c d-a h)+b^2 d\right )}{4 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}-\frac{3 c (2 c e-b g) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac{3 \left (b+2 c x^2\right ) (2 c e-b g)}{4 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{-2 a g+x^2 (2 c e-b g)+b e}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-(b*e - 2*a*g + (2*c*e - b*g)*x^2)/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (x*
(b^2*d - a*b*f - 2*a*(c*d - a*h) + (b*c*d - 2*a*c*f + a*b*h)*x^2))/(4*a*(b^2 - 4
*a*c)*(a + b*x^2 + c*x^4)^2) + (3*(2*c*e - b*g)*(b + 2*c*x^2))/(4*(b^2 - 4*a*c)^
2*(a + b*x^2 + c*x^4)) + (x*(3*b^4*d + a*b^3*f + 8*a^2*b*c*f + 4*a^2*c*(7*c*d +
a*h) - a*b^2*(25*c*d + 7*a*h) + c*(3*b^3*d + a*b^2*f + 20*a^2*c*f - 12*a*b*(2*c*
d + a*h))*x^2))/(8*a^2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (Sqrt[c]*(3*b^3*d
+ a*b^2*f + 20*a^2*c*f - 12*a*b*(2*c*d + a*h) + (3*b^4*d + a*b^3*f - 52*a^2*b*c*
f - 6*a*b^2*(5*c*d - 3*a*h) + 24*a^2*c*(7*c*d + a*h))/Sqrt[b^2 - 4*a*c])*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]]) + (Sqrt[c]*(3*b^3*d + a*b^2*f + 20*a^2*c*f - 12*a*
b*(2*c*d + a*h) - (3*b^4*d + a*b^3*f - 52*a^2*b*c*f - 6*a*b^2*(5*c*d - 3*a*h) +
24*a^2*c*(7*c*d + a*h))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + S
qrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*a^2*(b^2 - 4*a*c)^2*Sqrt[b + Sqrt[b^2 - 4*a*c]])
- (3*c*(2*c*e - b*g)*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((h*x**4+g*x**3+f*x**2+e*x+d)/(c*x**4+b*x**2+a)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 7.07484, size = 845, normalized size = 1.24 \[ -\frac{b c d x^3-2 a c f x^3+a b h x^3-2 a c e x^2+a b g x^2+b^2 d x-2 a c d x-a b f x+2 a^2 h x-a b e+2 a^2 g}{4 a \left (4 a c-b^2\right ) \left (c x^4+b x^2+a\right )^2}+\frac{\sqrt{c} \left (3 d b^4+3 \sqrt{b^2-4 a c} d b^3+a f b^3-30 a c d b^2+a \sqrt{b^2-4 a c} f b^2+18 a^2 h b^2-24 a c \sqrt{b^2-4 a c} d b-52 a^2 c f b-12 a^2 \sqrt{b^2-4 a c} h b+168 a^2 c^2 d+20 a^2 c \sqrt{b^2-4 a c} f+24 a^3 c h\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left (-3 d b^4+3 \sqrt{b^2-4 a c} d b^3-a f b^3+30 a c d b^2+a \sqrt{b^2-4 a c} f b^2-18 a^2 h b^2-24 a c \sqrt{b^2-4 a c} d b+52 a^2 c f b-12 a^2 \sqrt{b^2-4 a c} h b-168 a^2 c^2 d+20 a^2 c \sqrt{b^2-4 a c} f-24 a^3 c h\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{8 \sqrt{2} a^2 \left (b^2-4 a c\right )^{5/2} \sqrt{b+\sqrt{b^2-4 a c}}}+\frac{3 c (2 c e-b g) \log \left (-2 c x^2-b+\sqrt{b^2-4 a c}\right )}{2 \left (b^2-4 a c\right )^{5/2}}-\frac{3 c (2 c e-b g) \log \left (2 c x^2+b+\sqrt{b^2-4 a c}\right )}{2 \left (b^2-4 a c\right )^{5/2}}+\frac{3 d x b^4+3 c d x^3 b^3+a f x b^3+a c f x^3 b^2-6 a^2 g b^2-25 a c d x b^2-7 a^2 h x b^2-24 a c^2 d x^3 b-12 a^2 c h x^3 b-12 a^2 c g x^2 b+12 a^2 c e b+8 a^2 c f x b+20 a^2 c^2 f x^3+24 a^2 c^2 e x^2+28 a^2 c^2 d x+4 a^3 c h x}{8 a^2 \left (4 a c-b^2\right )^2 \left (c x^4+b x^2+a\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(-(a*b*e) + 2*a^2*g + b^2*d*x - 2*a*c*d*x - a*b*f*x + 2*a^2*h*x - 2*a*c*e*x^2 +
a*b*g*x^2 + b*c*d*x^3 - 2*a*c*f*x^3 + a*b*h*x^3)/(4*a*(-b^2 + 4*a*c)*(a + b*x^2
+ c*x^4)^2) + (12*a^2*b*c*e - 6*a^2*b^2*g + 3*b^4*d*x - 25*a*b^2*c*d*x + 28*a^2
*c^2*d*x + a*b^3*f*x + 8*a^2*b*c*f*x - 7*a^2*b^2*h*x + 4*a^3*c*h*x + 24*a^2*c^2*
e*x^2 - 12*a^2*b*c*g*x^2 + 3*b^3*c*d*x^3 - 24*a*b*c^2*d*x^3 + a*b^2*c*f*x^3 + 20
*a^2*c^2*f*x^3 - 12*a^2*b*c*h*x^3)/(8*a^2*(-b^2 + 4*a*c)^2*(a + b*x^2 + c*x^4))
+ (Sqrt[c]*(3*b^4*d - 30*a*b^2*c*d + 168*a^2*c^2*d + 3*b^3*Sqrt[b^2 - 4*a*c]*d -
24*a*b*c*Sqrt[b^2 - 4*a*c]*d + a*b^3*f - 52*a^2*b*c*f + a*b^2*Sqrt[b^2 - 4*a*c]
*f + 20*a^2*c*Sqrt[b^2 - 4*a*c]*f + 18*a^2*b^2*h + 24*a^3*c*h - 12*a^2*b*Sqrt[b^
2 - 4*a*c]*h)*ArcTan[(Sqrt[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]]) + (Sqrt[c]*(-3*b^4*d + 30
*a*b^2*c*d - 168*a^2*c^2*d + 3*b^3*Sqrt[b^2 - 4*a*c]*d - 24*a*b*c*Sqrt[b^2 - 4*a
*c]*d - a*b^3*f + 52*a^2*b*c*f + a*b^2*Sqrt[b^2 - 4*a*c]*f + 20*a^2*c*Sqrt[b^2 -
4*a*c]*f - 18*a^2*b^2*h - 24*a^3*c*h - 12*a^2*b*Sqrt[b^2 - 4*a*c]*h)*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*c*(2*c*e - b*g)*Log[-b + Sqrt[b^2 - 4*a*c] -
2*c*x^2])/(2*(b^2 - 4*a*c)^(5/2)) - (3*c*(2*c*e - b*g)*Log[b + Sqrt[b^2 - 4*a*c
] + 2*c*x^2])/(2*(b^2 - 4*a*c)^(5/2))

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Maple [B]  time = 0.314, size = 19742, normalized size = 29.1 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*((12*a^2*b*c^2*h - 3*(b^3*c^2 - 8*a*b*c^3)*d - (a*b^2*c^2 + 20*a^2*c^3)*f)*
x^7 - 12*(2*a^2*c^3*e - a^2*b*c^2*g)*x^6 - ((6*b^4*c - 49*a*b^2*c^2 + 28*a^2*c^3
)*d + 2*(a*b^3*c + 14*a^2*b*c^2)*f - (19*a^2*b^2*c - 4*a^3*c^2)*h)*x^5 - 18*(2*a
^2*b*c^2*e - a^2*b^2*c*g)*x^4 - ((3*b^5 - 20*a*b^3*c - 4*a^2*b*c^2)*d + (a*b^4 +
5*a^2*b^2*c + 36*a^3*c^2)*f - (5*a^2*b^3 + 16*a^3*b*c)*h)*x^3 - 4*(2*(a^2*b^2*c
+ 5*a^3*c^2)*e - (a^2*b^3 + 5*a^3*b*c)*g)*x^2 + 2*(a^2*b^3 - 10*a^3*b*c)*e + 2*
(a^3*b^2 + 8*a^4*c)*g - ((5*a*b^4 - 37*a^2*b^2*c + 44*a^3*c^2)*d - (a^2*b^3 - 16
*a^3*b*c)*f - 3*(a^3*b^2 + 4*a^4*c)*h)*x)/((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) - 1/8*integrate(((12*a^2*b*c*h - 3*(b^3*c - 8*a*b*c
^2)*d - (a*b^2*c + 20*a^2*c^2)*f)*x^2 - 3*(b^4 - 9*a*b^2*c + 28*a^2*c^2)*d - (a*
b^3 - 16*a^2*b*c)*f - 3*(a^2*b^2 + 4*a^3*c)*h - 24*(2*a^2*c^2*e - a^2*b*c*g)*x)/
(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(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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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((h*x**4+g*x**3+f*x**2+e*x+d)/(c*x**4+b*x**2+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [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((h*x^4 + g*x^3 + f*x^2 + e*x + d)/(c*x^4 + b*x^2 + a)^3,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError